Habilitationsschrift

zurErlangung der Venia legendi

fur das Fach Physikder

Ruprecht-Karls-UniversitatHeidelberg

vorgelegt von

Heinerich Kohler

aus Offenburg

2006

Decoherence and Fidelity

in Random Matrix Theory

and in Complex Systems

Abstract:

Quantum information devices require a high level of coherence and of fidelity to be operative. There-fore the study of decoherence and of fidelity is a topic of high current interest. Within the traditionalframework of a system–bath Hamiltonian a novel phenomenon, called quantum frustration, is stud-ied. It was first reported in 2003 by Castro Neto et al. [1]. It refers to the phenomenon thatif a system couples with two mutually conjugate variables to two independent environments theirdissipative effects partially cancel. The findings for a spin 1/2 system are compared with a har-monic oscillator which couples with position and momentum to two independent heat bath. Forthe harmonic oscillator the mutually cancellation of the two environments is weaker than for thespin system, however frustration effects, such as underdamped oscillations of the symmetrized andanti–symmetrized position correlation functions for arbitrarily strong symmetric coupling, still exist.A discussion of the results is given.

Fidelity is studied the framework of random matrix theory (RMT). This approach has proven tobe most successful in the description of the available numerical and experimental data. Withinthe formalism of the supersymmetric non–linear σ–model, exact expressions for random matrixaverages of fidelity are obtained. These are the only available analytic results for fidelity decay in thestrong coupling regime. The fidelity freeze, an anomalous slow fidelity decay for certain symmetrybreaking perturbations, predicted by Prosen and Znidaric [2] is proven to exist beyond second orderperturbation theory. This might have important implications for the design of quantum informationdevices.

The supersymmetric non–linear σ–model is the standard approach for the notoriously difficult taskof calculating RMT ensemble averages. The search for alternatives in situations where it is notapplicable leads to the study of interacting N–particle models of the Calogero–Moser–Sutherlandtype. We give an account of the different approaches for the construction of their eigenfunctions.A recursion formula for the eigenfunctions, formerly derived for a one–family particle model andan infinite system [3] will be extended to a two–family particle model and to periodic boundaryconditions. In the latter case a Bethe–Ansatz type equation arises naturally.

Contents

Published material contained in this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

1 Overview 1

2 Decoherence 5

2.1 Decoherence Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Dynamical Decoherence Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Decoherence Free Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Quantum Frustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Spin in Competing Heat Baths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Oscillator in Competing Heat Baths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Decoherence in Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Resume I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Random Matrix Theory 19

3.1 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Random Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Large N–Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4 Other observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Fidelity 27

4.1 Semiclassical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2 Random Matrix Formulation of Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Perturbative RMT Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 Exact Calculations of Fidelity 37

5.1 Supersymmetric technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Results for non symmetry breaking perturbations: Fidelity revival . . . . . . . . . . . . . . . 42

5.3 Fidelity freeze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.4 Time reversal invariance breaking: finite N results . . . . . . . . . . . . . . . . . . . . . . . . 47

5.5 Finite N results: Graded Eigenvalue method . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.6 Resume II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6 Matrix Bessel functions 53

6.1 Symmetric spaces with curvature zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.1.1 Rational CMS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.1.2 Schur polynomials, Zonal Polynomials and Jack Polynomials . . . . . . . . . . . . . . 58

6.2 Symmetric spaces with positive curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.3 Symmetric superspaces with zero curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7 Summary and Conclusion 65

A More on Quaternion elements 67

i

ii Contents

B Calculus on superalgebras 69

C Derivation of Eq. (5.4.6) 71

D Real forms and Symmetric spaces 73

E Symmetric Functions 77

Published material contained in this work

The following peer reviewed publications contain material which has been used to compile the present work.The corresponding chapters and sections are indicated in parentheses. Besides this material other authors’work is reviewed as cited in the text, and no original, unpublished work is presented unless clearly stated.The copyright remains with the respective publishers.

[4] H. Kohler and F. Sols (2006):Dissipative quantum-oscillator with two competing heat baths,New J. Phys. 8, 149 (Sec. 2.2.2).

[5] H. Kohler and F. Sols (2005):Quasiclassical Frustration,Phys. Rev. B 72, 014417(R) (Sec. 2.2.2).

[6] T. Gorin, H. Kohler, T. Prosen, T. H. Seligman and M. Znidaric (2006):Anomalous slow fidelity decay for symmetry breaking perturbations,Phys. Rev. Lett. 96, 244105 (Sec. 5.3).

[7] H. J. Stockmann and H. Kohler (2006):Fidelity freeze for a random matrix ensemble with off–diagonal perturbation,Phys. Rev. E 73, 066212 (Sec. 5.3).

[8] H. Kohler and T. Guhr (2005):Supersymmetric extensions of Calogero–Moser–Sutherland–like models:construction and some solutions,J. Phys. A 38, 9891 (Sec. 6.3).

[9] T. Guhr and H. Kohler (2005):Supersymmetry and models for two kinds of interacting particles,Phys. Rev. E 71, 045102(R) (Sec. 6.3).

[10] H. Kohler and J. Gronqvist and T. Guhr (2004):The k–point random matrix kernels obtained from one–point supermatrix models,J. Phys. A 37, 2331 (Sec. 3).

[11] H. Kohler and F. Guinea and F. Sols (2004):Quantum electrodynamic fluctuations of the macroscopic Josephson phase,Ann. Phys. (New York) 310, 127 (Sec. 2.3).

Some of the results contained in the following publications are also cited in the text.

[3] T. Guhr and H. Kohler (2002):Recursive construction of a class of radial functions. I Ordinary space,J. Math. Phys. 43, 2707.

[12] T. Guhr and H. Kohler (2002):Recursive construction of a class of radial functions. II Superspace,J. Math. Phys. 43, 2741.

[13] T. Guhr and H. Kohler (2004):Derivation of the Supersymmetric Harish–Chandra Integral for UOSp(k1/2k2),J. Math. Phys. 45, 3636.

iii

iv Published material contained in this work

1. Overview

Decoherence

Decoherence is the loss of quantum properties due to interaction with the environment. It was subject ofintensive research during the past decades. There are two main reasons for this intensive research activity.

The first reason is related to the question of how classical mechanics emerges from the underlying quantummechanical theory. The idea that classical mechanics originates from quantum mechanics in the limit ofsmall wave length or equivalently in the limit of large quantum numbers was the basis for its foundation.Classical mechanics arises in the limit of small Planck’s constant ~ in the same way as geometrical opticsarises from wave optics in the limit of small wave lengths. However, since the conceptually important workof Zeh [14] and Kubler and Zeh [15] it is by now commonly accepted that there is another mechanism oftransition from quantum to classical, which is responsible for even microscopic objects or systems with smallquantum numbers often being well described by a classical theory. This second mechanism is decoherence.By decoherence we mean the loss of quantum interference due to the interaction of a quantum system withthe environment. The key to decoherence–induced transition from quantum to classical is that a quantumobject is continuously monitored by the ubiquitous environmental degrees of freedom. The picture of thecollapse of the wave function is substituted by the process of decoherence1. This insight was most importantfor the interpretation of quantum mechanics, since it consolidates quantum mechanics as a complete theory[16].

The second reason is related to the advent of quantum information technology, a rapidly growing field withseveral subdisciplines such as quantum computation or quantum cryptology [17]. The main idea common toall quantum information devices is to take advantage of the counterintuitive consequences resulting from thesuperposition principle of quantum theory. For instance in a quantum computer the superposition principle isemployed to use two–level systems (qubits) as tiny “parallel computers”. In quantum teleportation protocolsEinstein’s “spukhafte Fernwirkung” is used to create a perfect copy of a quantum state in a region far awayfrom the original. “Gedanken” experiments, once devised to refute quantum mechanics, have become realexperiments, which use the superposition principle of quantum mechanics as a resource. The main obstacleto implementing quantum information devices is decoherence.

For macroscopic objects the process of decoherence becomes extremely fast, much faster than dissipation(energy relaxation), since it only requires the excitation or destruction of a single quantum of the thermalbath, while many quanta are necessary to change the particles energy appreciably. In this context often thefamous formula of Caldeira and Leggett for the ratio between decoherence time and relaxation time tdec/trel

= mkBT (δx)2/~2 is quoted, where δx is the distance over which interference pattern should be observable.For small but macroscopic objects m = 1 g, T = 300K and δx = 1 cm, the ratio has the dizzying value of10−40.

The loss of quantum wave coherence – also referred to as decoherence or dephasing – appears in a widerange of different physical contexts. For instance, due to its ubiquity, the quantum electrodynamic vacuumfield provides the most fundamental mechanism of decoherence that a charged particle experiences. Hereone might ask the question: Why quantum interference phenomena of charged particles are observable atall, if decoherence is as fast as indicated above. This question has indeed been investigated in some detail inRefs. [18, 19, 20]. It turns out that the QED vacuum is rather inefficient as a measuring device.

In solid state physics decoherence is responsible for the destruction of the quantum interference effects ofelectrons, which are, for instance, responsible for the weak localisation corrections to the conductance. Indisordered conductors at low temperature the dominant mechanism responsible for the loss of phase coherenceof electrons close to the Fermi surface is the Coulomb interaction with the electrons of the metal [21]. Itis widely believed that the efficiency, as measured in the inverse decoherence time, of this mechanism ofdecoherence should tend to zero for zero temperature. The experimental observation that this decoherencetime seems to saturate at a finite value for zero temperature [22, 23] has raised a fervid debate. In [24] itwas suggested that zero point fluctuations are responsible for this saturation. This has been questioned inRefs. [25, 26, 27].

These two examples show that the generic term decoherence or dephasing comprises physical problems ofwidely different conceptual and technical nature, which may yield different answers to apparently similarquestions. The message is that questions concerning decoherence do not allow for a general valid answerbut have to be investigated case by case with a careful analysis of the assumptions and premises which weremade in each case.

1The rather obscure implications arising from the question whether somebody observes the measurement processor not will be avoided in this review.

1

2 1. Overview

The literature of research articles addressing problems, which have to do with decoherence, is so vast thatany review of a reasonable size must make a selection. In the first part of this work we focus on “decoherencesuppression”. This choice is, of course, to some extent motivated by the author’s own research. In particular,a relatively novel phenomenon called quantum frustration is reviewed in more detail. It refers to the effectthat the damping of a system which couples to two independent environments with canonically conjugateobservables is smaller than it would be if it were to couple to one environment alone. It was first observedin a spin 1/2 system [1], which couples, with two spin components, to the two different magnon modes ofan antiferromagnet. A rather heuristic but very appealing explanation of the effect is the following: if twoobservers try independently to measure canonically conjugate or, in the language of quantum mechanics,non–commuting observables they fail to measure anything, due to Heisenberg’s uncertainty principle.

A similar effect albeit in a weaker form was predicted for a Josephson point contact in Ref. [11]. Herethe system’s canonically conjugate coordinates are the phase difference and the particle number differenceacross the junction. The two environments are the electromagnetic vacuum fluctuations and (bosonic)particle number fluctuations across the junction. Although the two environments are of rather differentnature, in particular they have different spectral functions, one effect of cancellation could be singled out.Both environments contribute to the uncertainty 〈φ2〉 of the macroscopic phase difference φ with oppositesigns. The particle number fluctuations tend to lock (measure) the phase and therefore reduce 〈φ2〉, whereasthe electromagnetic vector field induces additional phase fluctuations via the gauge invariant expression

φ + (2e/~c)∫ 2

1dr ·A(r), where the endpoints 1 and 2 of the integral are points deep enough in the bulk of

superconductors 1 (left) and 2 (right) where the phase is constant.

A first systematic study of quantum frustration was given in Ref. [4] for the dissipative harmonic oscillator.In that case the two independent baths couple to the position and to the momentum of the oscillator. Itturns out that the two environments cancel in some but not in all aspects. Underdamped oscillations ofthe position–position correlation function of the harmonic oscillator for arbitrary strong coupling, if theproperties of the two baths and the two couplings are exactly identical, are the most striking effect.

The first part of this work is organised as follows: in the first two sections some introductory material ondecoherence and dissipation is compiled. In particular, a precise definition of the quantities, which canbe considered as measures for coherence, is given. Moreover, the principal ideas of other mechanisms ofdecoherence suppression are sketched. In the following sections quantum frustration is considered in detailfor a spin system, Sec. 2.2.1, for the quantum oscillator Sec. 2.2.2 and for Josephson junctions Sec. 2.3.Finally, the results are discussed in a resume. The presented results are largely taken from Ref. [11, 5, 4].

Fidelity

The second part of this work deals with fidelity, a quantity which has been introduced recently and which,in many aspects, has much in common with decoherence. Fidelity is defined as the modulus squared overlapintegral |〈ψ(t)|ψ0(t)〉|2 of a wave function |ψ0(t)〉 which is propagated in time by a Hamiltonian H0 andthe same initial wave function which is propagated as |ψ(t)〉 by a slightly perturbed Hamiltonian H0 + V .It was introduced by Peres [28] who looked for a generic quantum mechanical quantity equivalent to theclassical Lyapunov exponents in the general quest for signatures of chaoticity in quantum systems. Theoverlap integral itself 〈ψ(t)|ψ0(t)〉 goes by the name fidelity amplitude.

Although fidelity started to be studied only recently, a similar quantity was first considered by Loschmidtin an attempt to refute Boltzmann’s H-theorem [29]. If one were to reverse at a given time the velocitiesof all particles of a thermodynamical system, the system would evolve from equilibrium towards the non–equilibrium initial state [30], which might have a much lower entropy. This would result in a violation ofthe second law of thermodynamics. Boltzmann’s answer [31] to Loschmidt’s argument ”Then try to do it!”made his point clear. A simultaneous reversal of all particle–velocities can only be achieved with a perfectknowledge of all positions and momenta. This requires a kind of a Maxwell’s daemon. The impossibility todevise such a Maxwell daemon is the essence of the second law of thermodynamics, the modern version ofBoltzmann’s H-theorem. In memory of the above discussion between Boltzmann and Loschmidt fidelity isdenoted Loschmidt echo by some authors [32].

Also the work of Peres remained largely unnoticed until recently. The enormous burst of activity in thefield of fidelity in recent years was triggered by the work of Jalabert and Pastawski [32], who found thatfor a coherent initial state in a chaotic system there exists a regime of perturbation strength, where fidelitydecay is only governed by the Lyapunov exponent of the system but not by the perturbation. This findingin the spirit of Peres’ original idea boosted a whole series of mostly numerical studies. Today we know thata bunch of different regimes for the behaviour of fidelity exists, depending on the nature of the system andof the perturbation as well as on different time scales and on the initial state. The different regimes will bedescribed in more detail in Chap. 4. An account of the recent developments in the subject has been given inRef. [33].

Since all of this renewed interest, fidelity has found many applications, and a series of relations to otherquantities of current interest have been discovered. For instance, fidelity is used as a benchmark for reliability

3

of quantum information devices [17]. It may be investigated as well in the description of wave propagationthrough random media [34]. High fidelity is paramount for the design of wireless communication devices [35].

Much of the similarity of the notations decoherence and fidelity has been clarified in Ref. [36]. The authorsconsidered a two–level system which is coupled to a generic environment. The system’s decoherence ismeasured by the decay rate of the off–diagonal elements of the system’s reduced density matrix. Assumingthat the interaction part of the Hamiltonian commutes with the system part (pure dephasing), the authorsfind that this decay rate is described by the overlap integral

〈Ψenv(0)|e−iHenvt+isVenvteiHenvt+isVenvt|Ψenv(0)〉 (1.0.1)

of the environment, where ±s are the two eigenvalues of the system Hamiltonian and Venv is an operatorwhich only depends on the environment’s degrees of freedom (~ = 1). This is exactly the definition of thefidelity amplitude of the environment’s state |Ψenv(0)〉, if the interaction with the system is considered asa perturbation. A representative example of a Hamiltonian with pure dephasing will be given in Sec. 2[Eq. (2.0.12)]. For a more general class of Hamiltonians Eq. (1.0.1) is a good approximation for decoherence,if one can assume that decoherence takes place on a time scale which is much shorter than relaxation time.

Recently a similar equivalence of fidelity with full counting statistics [37] was found. Full counting statistics(FCS) addresses the problem of calculating probabilities of charge transfer in a quantum wire. The crucialquantity in FCS is Pn(t), which is the probability that after a certain time n charges have passed throughthe wire. This amounts to calculate all moments 〈Q(t)n〉 of the time integrated current operator Q(t) =∫ t

0dt′I(t′). As was pointed out by Levitov and Lesovik [38, 39] it is problematic to calculate the moments

from the generating function 〈exp[iλQ(t)]〉 due to time ordering problems. They circumvented the problemby coupling a single two–level system (with eigenvalues ±s) to the wire as a measurement device. Thetransmission probabilities are then the Fourier coefficients Pn(t) =

∫ds exp(ins)χFCS(s, t) of a generating

function, which is given byχFCS = 〈e−iHwt+isVwteiHwt+isVwt〉w , (1.0.2)

where the brackets 〈. . .〉w denote a thermal average over the degrees of freedom of the wire. This expression isthe finite temperature version of Eq. (1.0.1), if the environment Hamiltonian Henv is substituted by Hw andthe interaction term Venv is substituted by Vw. Since fidelity is quite a natural quantity to be studied, notonly in quantum mechanics but also in classical wave mechanics, this list will most probably be completedin the future.

Random Matrix Theory

The main tool in the study of fidelity in this work will be random matrix theory (RMT). Since the early workof Wigner [40, 41] RMT has become a standard tool in the statistical description of complex systems. Theoriginal observation by Wigner that energy levels in complex nuclei and resonance peaks of nuclear scatteringdata have the same statistical properties as random matrices has opened up a new branch of research inspectral data analysis called spectral statistics. Today virtually everywhere where randomly fluctuatingdata are measured, spectral statistics is done as well. Scattering data in nuclear physics [42, 43, 44, 45],conductance fluctuations in mesoscopics [46, 47, 48, 49] (see Refs. [50, 51] for an overview), numerical datain QCD lattice calculations [52], stock price fluctuations in econophysics [53], neuronal activities and heartbeat rates [54] in biological science even the stopping intervals of urban busses [55, 56] have been subject ofspectral statistical analysis.

The idea that fidelity decay is a universal feature which is governed by only a few global properties of thesystem, such as chaoticity or regularity, time reversal invariance etc. is from Peres [28]. The application ofRMT to the description of fidelity seems to be due to Gorin, Prosen and Seligman [57]. Within the RMTapproach to fidelity the ensemble average of the fidelity amplitude

〈f(t)〉 =1

N

⟨tr e−2πi(H0+V )te2πiH0t

⟩(H0,V )

(1.0.3)

is studied. Here, H0 and V are two random matrices representing the system H0 and a perturbation V . Theensemble average is taken over both random matrix ensembles2. The agreement of the results obtained fromEq. (1.0.3) with experiments is striking, see Sec. 4.4, and is an a posteriori justification of the approach.

Although it is, for most applications, sufficient to calculate fidelity within perturbation theory, it turned outthat there are also important generic non–perturbative effects, such as fidelity revival, Sec. 5.2, or fidelityfreeze, Sec. 5.3. To describe these effects quantitatively exact calculations of the fidelity amplitude arenecessary.

Exact calculations of spectral quantities (and fidelity is just another spectral quantity) of RMT ensemblesare a notoriously difficult endeavour. There exist in principle four methods to obtain exact results.

2We indicate the quantities which are averaged over by an upper index. This notation we will use throughout thereview in the context with RMT.

4 1. Overview

1. The supersymmetric non–linear σ–model [58] has become the standard method to calculate ensembleaverages in RMT and in disordered systems. It is exact in the limit of infinitely large matrix dimensionN . It has been successfully employed for the calculation of fidelity in Refs. [59, 7]. The application ofthis rather complicated method to the specific problem of fidelity is explained in some detail in Sec. 5.

2. Within the supersymmetric approach there exists an alternative approach due to Guhr [60]. Thegraded eigenvalue method relies on the supersymmetric version of the Itzykson–Zuber integrationformula. The advantage of the method is that the limit of large matrix dimension must not be takenat all. The method is explained and a preliminary, unpublished result is presented in Sec. 5.5.

3. As another alternative, the method of orthogonal polynomials, pioneered by Mehta [61] has to bementioned. The method is complicated but applicable for a Gaussian orthogonal ensemble with a GUEperturbation. In Sec. 5.4 it is explained in detail and a preliminary, unpublished result is presented.

4. As a fourth method the replica trick also has to be mentioned. Although the mathematical rigor ofthe replica limit

〈logZ〉 = limn→0

〈Zn〉 − 1

n(1.0.4)

has been questioned on some occasions [62, 63], it was applied successfully in RMT and for disorderedsystems [64, 65, 66]. Recently, the application of the replica trick in RMT has been put on mathe-matically firmer grounds by the work of Kanzieper and Splittorff and Verbaarschot [67, 68, 69]. Theauthors found a connection of RMT averages to solutions of the Painleve IV differential equation andthus were able to take the replica limit n→ 0 in a clean way. In this review the replica trick will notbe used.

The second part of this work is divided into two chapters. In the first part which focusses on physical aspects,results on fidelity from semiclassical analysis, Sec. 4.1, perturbation theory, Sec. 4.3, and experiments, Sec. 4.4,are compiled. In the second, mathematical part the supersymmetric method is explained, Sec. 5.1, and resultsand effects which can only be seen in the non–perturbative regime, i. e. fidelity revival and fidelity freezeare discussed. In the last two sections, Secs. 5.4 and 5.5, two alternative approaches for exact calculation offidelity are explained.

Matrix Bessel Functions

The above mentioned Itzykson–Zuber integral formulas are crucial to obtaining exact results for randommatrix averages. For deep mathematical reasons such a compact formula only exists for the Gaussian unitaryensemble, see introductory section of Chap. 3 for the precise definition. The quest for a similar formula for theother Gaussian ensembles, namely for the Gaussian orthogonal ensemble (GOE) and the Gaussian symplecticensemble (GSE) leads naturally to the theory of matrix Bessel functions. Via the theory of Matrix Besselfunctions, RMT is connected in a beautiful way with a class of exactly solvable one–dimensional interactingmany–body systems. This relation is explained in Chap. 6.

Like vector Bessel functions matrix Bessel functions have a number of different representations: integralrepresentation, a representation as a solution of a (partial) differential equation, recursion formulas, expansionin small arguments or an asymptotic expansion are instances. The aim of this chapter is to provide acompilation of these different representations. To this end material from Ref. [8] and from Ref. [3] is used.Also an unpublished result will be presented. Related facts on symmetric spaces, which are indispensablefor the understanding of Chap. 6 are compiled in App. D.

2. Decoherence and Dissipation

The standard approach to model dissipative quantum systems which will be followed in this section isthe description with a system–bath Hamiltonian consisting of a system part HS, a bath part HB, and aninteraction part HSB

H = HS +HSB +HB . (2.0.1)

It has to be stressed that a typical many–body Hamiltonian can be split into the form (2.0.1) in many differentways. The choice of the three parts HS, HSB and HB defines the researcher’s interest rather than a specificsystem. The Hamiltonian which defines the system is usually a priori not given in the form (2.0.1). Forinstance, if one is interested in the dynamics of a charged particle in an electromagnetic field, the canonicalchoice would be to denote the Hamiltonian of the charged particle HS and the Hamiltonian of the field HB.Another example is a system of interacting electrons. Here all particles are identical and no canonical choiceof splitting is given by the form of the Hamiltonian. In order to write the Hamiltonian as it is in Eq. (2.0.1)one singles out by hand one or two electrons (depending whether one–particle or in two–particle propertiesare of interest), for instance with momenta and spins kσ and k′σ′, then the other electrons are the bath.Therefore, when a Hamiltonian is written in the form Eq. (2.0.1), it is implied that the quantities of interestare the expectation values of the system’s observables. These are obtained as averages

〈. . .〉 ≡ tr (. . .)ρS(t) , HS → R , (2.0.2)

with the reduced density matrix ρS(t). The reduced density matrix itself is defined as the partial trace overthe bath’s degrees of freedom

ρS(t) = tr Bρ(t) , HS ⊗HB → HS , (2.0.3)

where ρ(t) is the density matrix of the whole system (system + bath) and HS,B denote the Hilbert spaces ofbath and system, respectively.

There are many different approaches to calculate time evolution of the reduced density matrix. Examples arethe Feynman–Vernon influence path–integral formalism [70] or the master equation approach. The varietyof different methods is compiled in the classical textbook by Weiss [71] and with a special focus on quantumoptics in the textbook by Gardiner and Zoller [72].

The bath is by definition a large (macroscopic) collection of degrees of freedom, whereas the system itselfconsists of only a few microscopic degrees of freedom. Therefore, it is reasonable to assume that each bathdegree of freedom is only weakly perturbed by the microscopic system. Caldeira and Leggett [73, 74] gavearguments that the environment can be modelled phenomenologically by an infinite collection of harmonicoscillators. In the traditional approach [73, 75] it is also assumed that the interaction Hamiltonian is linearin the system operator as well as in the bath operators.

When an isolated system comes into contact with the environment it will relax to equilibrium. The relaxationprocess takes place on two time scales. Roughly speaking the time evolution of the diagonal elements〈x|ρS(t)|x〉 describes dissipation, whereas the decay of the off–diagonal elements describes decoherence. Thisdefinition is not very precise, since it is defined via basis dependent quantities, but it is often used in theliterature. It implicitly assumes that the interaction with the environment takes place via the positionoperator. Although this is indeed often the case it is far from being always so. For example in quantumoptics the distinguished basis is the coherent state basis and expressions as 〈x|ρS(t)|x′〉 are not so useful. Itwas pointed out first by Zurek [76] that a specific basis for the reduced density matrix is singled out by theinteraction part in Eq. (2.0.1). In a basis of eigenstates of a system operator which commutes with HSB,named pointer states by Zurek, decoherence and dissipation are measured by the decay of the off–diagonal,respectively diagonal elements of ρS.

In many respects for one continuous degree of freedom the Wigner function

WS(q, p, t) =1

π

∫d(x− x′)ei(x−x

′)p/~〈x|ρS(t)|x′〉 , q =x+ x′

2(2.0.4)

is a more natural candidate for a decoherence measure. If the Wigner function is a real function it describesthe propagation in time of a classical phase–space distribution. Therefore, the decay of ImWS(q, p, t) is oftenused as an indicator of decoherence. It is of course still basis dependent.

In any case, basis independent measures for decoherence are always preferable if their calculation is possible.The prime candidate is the von Neumann entropy S(t) = 〈ln ρS(t)〉. Another quantity which is frequentlystudied is the purity, defined as the expectation value of the density matrix itself

P(t) = 〈ρS(t)〉 = tr ρ2S(t) , (2.0.5)

5

6 2. Decoherence

where the trace operation acts only on the system’s Hibert space HS. Also the equilibrium value of thepurity at temperature T = 1/kBβ

Pβ ≡ limt→∞

P(t) , (2.0.6)

is interesting as a measure for the net efficiency of the environment in destroying coherence. On the otherhand, dissipation describes the net transfer of energy from the system to the environment during relaxation.It is measured for instance by 〈HS(t)〉.

As a prototypical example let us consider the Spin–Boson Hamiltonian, or dissipative two–level system, whichdescribes the dynamics of a localised spin coupled to a bath of harmonic oscillators (~ = 1)

H = ∆S1 + S3

∑k

(λka†k + λ∗kak) +

∑k

ωka†kak , (2.0.7)

where S1, S3 are the spin operators in the x and the z directions [Si, Sj ]− = iεijkSk and the ak are annihilation

operators [ak, a†k′ ]− = δkk′ of Bosonic modes with frequency ωk. The λk’s describe the coupling of the spin

to the Bosonic bath. For vanishing coupling the spin oscillates freely on the Bloch sphere with frequency ∆.The model (2.0.7) was studied intensively in the past with methods of all degrees of sophistication [77, 71].A review of the model is not our objective here, since we wish to illustrate the concepts introduced above.In the sequel ~ = 1 is set to one. Following Zurek’s criterion the eigenstates of S3 form the pointer basis forρS(t). In this basis we have ρS12(t) = 〈S1(t) + iS2(t)〉. Decoherence is therefore measured by the decay of|〈S1(t)〉| and of |〈S2(t)〉|, whereas dissipation is measured for instance by |〈S3(t)〉|. These expectation valuescan be calculated perturbatively [78, 79, 77, 80]. Following the exposition in [80] the decay of the elementsof the reduced density matrix are given by

〈S3(t)〉 ∝ S3(∞) + e−t/trel , 〈S±(t)〉 = 〈S±(0)〉e∓∆t−t/tdec , (2.0.8)

where S3(∞) is the equilibrium value tanh(∆/2kBT ). The decoherence time tdec and the relaxation time trel

are related as tdec = 2trel. The relaxation time itself is found to be

trel =π

2J(∆) coth

(∆

2kBT

), (2.0.9)

where the spectral function

J(ω) =∑k

|λk|2δ(ω − ωk) (2.0.10)

was introduced. It is the crucial quantity in the description of open quantum systems, where all informationabout the environment is encapsulated. It is usually assumed that the spectral function obeys a characteristicpower law for small ω,

J(ω) = 2γωαf(ω

Ω

). (2.0.11)

f(x) is a cutoff function, which has to be introduced to regularise ultraviolet divergences. Ω usually definesthe smallest time scale of the problem. Roughly one can say that the dissipative effect of an environmentdecreases with increasing exponent α. It has become popular to call environments with spectral functionswith α = 1 Ohmic, whereas environments with α > 1 (α < 1) go by the name superohmic (subohmic). Hereand throughout the review we will focus on an Ohmic environment α = 1.

The result as stated in Eqs. (2.0.8) and (2.0.9) is an approximation for small coupling strength γkBT ∆and its limitations are thoroughly discussed in the literature. In particular such features as the transitionto overdamped motion and phase transition are not captured by the above treatment. The purity is givenby P(t) = 1/2 + 2

∑3i=1〈Si〉

2. It approaches its equilibrium value exponentially with a rate 1/trel. For anOhmic bath and for zero temperature trel is given by trel = πγ∆. We notice, that in this approximationdecoherence time and relaxation time are practically identical.

The importance to distinguish between dissipation and decoherence becomes obvious in the case of puredephasing. With pure dephasing one describes the situation when the system part HS and the interactionpart HSB in Eq. (2.0.1) commute [HS, HSB]− = 0. The system Hamiltonian is a constant of motion andno net energy transfer to the environment takes place. Consequently the dissipation time trel is infinite.Nevertheless quantum coherence gets lost. To see this we consider again the Spin–Boson model in the form

H = ∆S3 + 2S3

∑k

(λka†k + λ∗kak) +

∑k

ωka†kak . (2.0.12)

We notice that her the only difference to Eq. (2.0.7) is that the spin operator in the system part has becomeS3 instead of S1. Now we have pure dephasing.

Since the diagonal elements of the reduced density matrix are constant the knowledge of the off–diagonalelement ρS12 in the S3 basis yields also the purity and the von Neumann entropy. For decoupled initial

2.1 Decoherence Suppression 7

Figure 2.1: Sketch of the pulse sequence for dynamical decoherence suppression

conditions ρ(0) = ρS ⊗ ρB the off–diagonal element can be calculated exactly. The result is, see for instanceRef. [81]

ρS12(t) = ρS12(0) exp

(− 1

π

∫ ∞0

dωJ(ω)

ω2[1− cos(ωt)] coth(βω/2)

), (2.0.13)

Time behaviour is complicated in detail but one can derive from Eq. (2.0.13) that ρS12(∞) = 0 [81].

The important message is: Although dissipation and decoherence of a system are both effects of the couplingto an environment, they are not necessarily related. In other words, strong decoherence does not necessarilyimply strong dissipation and vice versa weak (zero) dissipation might be accompanied by strong decoherence.

2.1 Decoherence Suppression

There have been considerable efforts in the past to devise mechanisms for decoherence suppression. Instancesare quantum error correction codes, projection onto decoherence–free subspaces [82, 83, 84], and onto noiselesssubsystems [85] as well as dynamical decoherence suppression. On the other hand there is an intensivesearch for suitable qubits in atomic and solid state physics, where decoherence is intrinsically small [86,87]. Optimisation of coherence properties in existing qubit systems, as for instance in qubits based on theJosephson effect [88, 89, 90], establishes a third route. A systematic, algebraic approach was recently putforth by Ritter [91]. In this section two mechanisms of decoherence suppression are described in more detail.

2.1.1 Dynamical Decoherence Suppression

The concept of dynamical decoherence suppression was introduced by Lloyd and Viola in Ref. [81] andlater by other groups [92]. The idea is to actively modify the (time–independent) system–bath HamiltonianEq. (2.0.1) with a time dependent part, for instance with a properly chosen sequence of pulses of an externalmagnetic field. Thereby, the effective interaction part of the Hamiltonian HSB is reduced or, in the mostideal case, drops out completely. By this procedure both decoherence and dissipation can, in principle, beeffectively suppressed. We sketch in the following this idea for a simple model following largely Ref. [92] andreferring the reader to the original literature for details.

The system part of the Hamiltonian Eq. (2.0.1) is acted upon by an additional periodic force at times[(n+ 1)T + nt0] with duration t0. The complete time dependent Hamiltonian reads

Htot(t) = H +Hkick

∞∑n=0

θ(t− T − 2n[T + t0])θ([2n+ 1][T + t0]− t) , (2.1.1)

where H is assumed to be of the form as given by Eq. (2.0.1). The additional part Hkick is assumed to becompletely controllable. In NMR experiments Hkick is typically given by an external magnetic field Bext,acting on a nuclear spin yielding

Hkick = −~mnucBext , (2.1.2)

where ~mnuc is the magnetic moment of the nucleus. The external field is switched on during pulses of durationt0 [93]. If the external pulse is very strong, H Hkick, time evolution of the system itself can be neglectedduring the kick

U(T, T + t0) ' exp[(−i/~)Hkickt0] , (2.1.3)

where U(t, t′) is the time evolution operator of Htot. Suppose that HS and HB are invariant under the parityoperation P , whereas HSB changes sign

PHSP = HS , PHBP = HB , PHSBP = −HSB . (2.1.4)

Dynamical decoherence suppression consists of adjusting the kick Hamiltonian Hkick in such a way thatU(T, T + t0) becomes the parity operator

U(T, T + t0) = P . (2.1.5)

Using the notation U0 = exp(−iHt/~), the time evolution of the whole system can now be written as

U(n[T + t0]) = (PU0(T ))n

=[e−(i/~)(HS−HSB+HB)T e−(i/~)(HS+HSB+HB)T

]n/2. (2.1.6)

This expression shows nicely the effect of the successive kicks in inverting the interacting part of the Hamil-tonian. One can show that in the limit T → 0, t0 → 0 the interaction part HSB cancels completely and theeffective time evolution is given by

limT+t0→0n→∞

U(n[T + t0]) = e−(i/~)(HS+HB)t , t = n(T + t0) . (2.1.7)

8 2.2 Quantum Frustration

The system–bath Hamiltonian is “actively” symmetrized with respect to a given group [83], such that allparts of the original Hamiltonian which are not invariant under the group operations drop out. In the simpleexample given in Ref. [92] and outlined in Eqs. (2.1.1) to (2.1.7) the group is just Z2, consisting of unityand the parity operator P . A generalisation to higher discrete groups is straightforward [84, 94]. As aconsequence the method is not applicable when the system part of the Hamiltonian and the interaction partshare the same symmetries. Another conclusion from Eq. (2.1.7) is that by the above described method theinteraction part of the Hamiltonian is eliminated. Thus, decoherence and dissipation are equally suppressed.

Finally, a comment on the applicability of the method is in order. It is clear that the limit of vanishing pulselength T → 0, t0 → 0 is only of academic interest. The natural question arises: how fast must the pulsefrequency be in order to suppress decoherence efficiently? The somewhat disappointing answer was given inRef. [94] for the damped harmonic oscillator and in Ref. [81] for a spin–boson model with pure dephasing, seeEq. (2.0.12) for the corresponding Hamiltonian. In both cases it turned out that decoherence is effectivelysuppressed only when the pulse frequency is of order Ω. We recall that Ω is the cutoff frequency of thespectral function of the bath. It is hard to reach this limit in realistic situations.

2.1.2 Decoherence Free Subspaces

The Hilbert space of an N qubit system has dimension 2N . It was first observed by Zanardi and Rasetti[82, 83] that under rather general assumptions on the interaction Hamiltonian not all 2N states are equallysusceptible to decoherence. Under the most favourable circumstances it is possible to single out a subspaceof states which is completely decoupled from the environment. The original idea of Rasetti and Zanardi wasrefined in later work. Here again, we sketch the main idea for a simple example [83].

Consider an N qubit Hamiltonian of the system–bath type in Eq. (2.0.1) with a fairly general form of theinteraction part (~ = 1)

HSB =

N∑i=1

∑k

(gikS

+i a†k + fikS

−i a†k + hikS

zi a†k + H.c

), (2.1.8)

where Sαi , α = z,± are the standard spin operators (~ = 1) forming a basis of the algebra sl(2,C). Thecommutation relations are [Szi , S

±j ]− = ±δijS±i , [S+

i , S−j ]− = 2δijS

zi . The coupling is described by three sets

of complex parameters gik, fik and hik.

The possibility of a decoherence free subspace arises under the assumption that the coupling to the environ-ment is the same for all qubits. If the environment is given for instance by a radiation field this assumptionis justified if the spatial separation of the qubits is smaller than the wavelength associated with the cutofffrequency of the environment, and the electric dipole approximation exp(ikx) ≈ 1 is justified for all modeswhich couple to the qubits. Then, we can set the coupling constants gik = gk, fik = fk and hik = hk for1 ≤ i ≤ N . The interaction Hamiltonian can be written in terms of the global operators Σα =

∑Ni=1 S

αi as

HSB =∑k

(gkΣ+a†k + fkΣ−a†k + hkΣza†k + H.c

). (2.1.9)

It is a rather simple observation that this interaction Hamiltonian is zero in the subspace CN which is built bythe singlet representations of the complete system’s Hilbert space (C2)⊗N . This holds independently of thebath state. The dimension of CN is the number of inequivalent singlet representations in the Clebsch–Gordandecomposition of sl(2,C)⊗N . Using standard results of representation theory [95] one finds

CN =N !

[N/2]![N/2 + 1]!= exp

[N ln 2− 3

2lnN + . . .

], (2.1.10)

for even N (for odd N no singlet representations exist). Thus there exists, at least theoretically, the possibilityto construct a Hilbert space which is completely decoupled from the environment. The ratio of the dimensionof the decoherence free subspace to the dimension of the complete Hilbert space scales in leading order asN−3/2. The question, if the implementation of quantum computation algorithms exclusively on decoherencefree subspaces is feasible, is a topic of current research [96, 97].

2.2 Quantum Frustration

Another possible mechanism of suppressing decoherence is Quantum Frustration. This term was coined byCastro Neto, Novais et al. [1, 98]. If a system couples to two independent environments with two differentobservables which do not commute, both environments are competing in “measuring” their correspondingobservable. Due to the non–commutativity of the two observables this measurement cannot be successful. Asa result the two environments partially cancel each other and the effective strength of the coupling decreases.This situation was studied in Refs. [1, 98] for a spin 1/2 system and for a system with a continuous degreeof freedom in Refs. [99, 100, 101, 5, 4].

2.2.1 Spin in Competing Heat Baths 9

2.2.1 Spin in Competing Heat Baths

In Ref. [1, 98] the dynamics of an impurity spin in an antiferromagnetic environment was studied. Theimpurity spin interacts with the two Goldstone modes of the Heisenberg antiferromagnet in the orderedphase. After an s wave expansion of the Goldstone modes (magnons) the problem reduces to a generalisedSpin–Boson model

H = ∆S1 + i∑k>0

[S2λk(a1k − a†1k) + S3µk(a2k − a†2k)

]+∑k>0

vk(a†1ka1k + a†2ka2k) . (2.2.1)

Here Si are the spin operators of the localised impurity and ∆ is the Bloch oscillation frequency of the freespin. The a1k, a2k are bosonic annihilation operators of the magnon modes with linear dispersion relationω(k) = vk. The coupling of the spin to the two independent magnon modes is described by the two sets ofcoupling constants

λk =√

4πγ1kv/√L , µk =

√4πγ2kv/

√L , (2.2.2)

where L is the linear dimension of the system. The velocity of v the magnon modes as well as ~ are set toone in the following. The dimensionless coupling constants γ1 and γ2 measure the strength of the bath. Thespectral functions J1(ω) =

∑k λ

2kδ(ω − k) and J2(ω) =

∑k µ

2kδ(ω − k) of both baths obey an Ohmic power

law for small frequencies J1(ω) = 2γ1ω, J2(ω) = 2γ2ω. They are regularised by a cutoff D ∆.

In Ref. [1, 98] the Hamiltonian was analysed by a renormalisation group analysis. The renormalisation groupequations (RGE) for the coupling constants and the dimensionless transition frequency h = ∆/D are [1]

dγ1

dl= −γ1γ

22 − γ1h

2 ,dγ2

dl= −γ2

1γ2 − γ2h2

dh

dl= (1− γ2

1 − γ22)h , (2.2.3)

where dl = d lnD is the differential of the flow parameter. These equations reduce to the renormalisationgroup equations of the usual Spin–Boson Hamiltonian [77], when one of the two coupling constants is set tozero. The non–linear flow which is symmetric for the two couplings γ1 and γ2 has a trivial fixed point ath = γ1 = γ2 = 0 and two non–trivial fixed points at h = γ1 = 0, γ2 = 1 and h = γ2 = 0, γ1 = 1, indicatinga phase transition. At the phase transition the impurity spin becomes localised in one of its eigenstates or,loosely speaking, it is “measured” [102, 103]. This corresponds to the Kosterlitz–Thouless phase transitionof the Kondo model from a triplet ground state to a singlet ground state [104]. At the point where the twobare couplings become exactly the same γ1 = γ2 ≡ γ, the RGE’s become

dγ

dl= −γ3 − γh2 ,

dh

dl= (1− 2γ2)h , (2.2.4)

and the dimensionless transition frequency h always, even for very large γ, scales towards infinity (for h ≈ 1the perturbative RG breaks down). As a result no Kosterlitz–Thouless phase transition occurs. In Ref. [1, 98]this was attributed to the fact that the two spin components S1 and S2 do not commute. In a numericalanalysis the authors also calculated the response function

χ′′⊥(ω) = Im

∫ ∞0

dteiωt〈[S3(t), S3(0)]−〉 . (2.2.5)

The behaviour of this function is frequently used as an indicator of underdamped or overdamped dynamics ofthe localised spin. For the uncoupled spin χ′′ has poles at the frequencies ±∆. When the coupling is switchedon the two peaks broaden and eventually merge to a single peak at zero. This is commonly referred to asthe transition from underdamped to overdamped oscillations in the context with the Spin–Boson problem[71]. In other words, if the response function χ′′, or more precisely χ′′/ω, has still a maximum away fromzero this is interpreted as a signature of underdamped dynamics or, equivalently, of coherent transitions. Ifno maximum occurs, the dynamics are overdamped.

Applying this criterion, they found that in the isotropic case (γ1 = γ2 = γ) the oscillations between thetwo spin states are always underdamped (coherent) even in the regime where the usual dissipative two–levelsystem (γ1 = γ, γ2 = 0) yields overdamped oscillations, see Fig. 2.2.

In conclusion, the competition of the two independent environments not only prevents a localisation of thespin (the most extreme case of decoherence) but reduces the effective coupling so much that the oscillationsof the impurity spin remain underdamped. The authors interpreted the underdamped oscillations of thespin–spin correlation functions as a signature of annihilated decoherence.

10 2.2 Quantum Frustration

Figure 2.2: Above: Response function χ′′⊥(ω), Eq. (2.2.5) for the model Eq. (2.2.1) for symmetric couplingγ1 = γ2 = 0.59 and for ultra asymmetric coupling γ1 = 0.59, γ2 = 0. Taken from Ref. [1]. Below: Positionresponse function Dq(ω), Eq. (2.2.19) for model Eq. (2.2.6). The marginal case of the q–oscillator, (γq, γp) =

(√

2, 0), (dashed line) develops a maximum when the parameter γp is tuned to a symmetric coupling (γq, γp) =

(√

2,√

2) (full line). Taken from Ref. [4]. The maximum at symmetric coupling is much more pronouncedfor the spin model.

2.2.2 Oscillator in Competing Heat Baths

Motivated by the work [1, 98] of Castro Neto et al. the influence of two independent baths on a genericsystem with a continuous degree of freedom has been investigated in Refs. [5, 4]. To this end a harmonicoscillator is a prime candidate for two reasons. On the one hand, in contrast to the model (2.2.1) for thequantum oscillator, many time dependent quantities can be evaluated exactly even for non–equilibrium initialconditions. The other and more fundamental reason is that the symmetrical roles played by position andmomentum make the harmonic oscillator the natural ground to study the difference and the interplay (whenboth are present) between position and momentum coupling.

Starting with the work of Magalinskii [105] and Ullersma [106, 107, 108, 109] the study of the dissipativeharmonic oscillator has a long history [110, 111, 112, 113, 114, 115, 116, 117, 118, 119], see Ref. [120] or thetextbook by Weiss [71] for a review and a comprehensive list of references. However, relatively little attentionhas been paid to effects arising from the coupling to different system variables. The system variable whichcouples to the heat bath is most often assumed to be the position. This model will be called q–oscillatorin the remainder of this section. Its choice was favoured in Ref. [73]. There it was argued that a complexdissipative environment can be modelled phenomenologically by a bath of harmonic oscillators, with thecoupling parameters chosen to yield a macroscopic Langevin equation for the q variable. However, whenthe dissipative model is derived from a microscopic Hamiltonian one has no choice and should take theinteraction Hamiltonian as derived from first principles. This happens, for instance, for the dynamics ofthe phase difference in Josephson contacts to be discussed in Sec. 2.3. The coupling of a system to a singlebath with its momentum variable is referred to as “anomalous coupling” [121] and has been considered inRefs. [121, 122].

Starting point is the following general form of a Hamiltonian describing a particle in an oscillator potentialwhich is coupled with momentum and position to two independent baths

H =ωq2q2 +

∑k

ωk

∣∣∣∣aqk +λkωkq

∣∣∣∣2 +ωp2p2 +

∑k

ωk

∣∣∣∣apk +µkωkp

∣∣∣∣2 , (2.2.6)

where the short-hand notation |a|2 = a†a has been used. The two baths are described by Bosonic operators

[aqk, a†qk′ ]− = δkk′ and [apk, a

†pk′ ]− = δkk′ . All operators are dimensionless [q, p] = i and ~ = 1. The form of

Eq. (2.2.6) avoids the notion of mass in order to highlight the symmetry between q and p. It is related tothe usual form of a harmonic oscillator with mass m and frequency ω0 by ωp = 1/m and

√ωqωp = ω0. The

model (2.2.6) displays the highest degree of symmetry between the canonically conjugate observables q andp. The two environments are characterised by the parameters λk and µk through the spectral functions

Jq(ω) = 2∑k

|λk|2δ(ω − ωk) , Jp(ω) = 2∑k

|µk|2δ(ω − ωk) . (2.2.7)

The Hamiltonian (2.2.6) is equivalent to the Hamiltonian (2.2.1) for a large spin S with the correspondence

q ∼ S1/√

2S and p ∼ S2/√

2S.

2.2.2 Oscillator in Competing Heat Baths 11

Two more comments are in order. First, if we consider for instance the first (q dependent) part of Eq. (2.2.6)and expand the modulus squared

Hq =

(ωq2

+∑k

λ2k

ωk

)q2 +

∑k

λk(aqk + a†qk)

=1

2

(ωq +

∫ ∞0

Jq(ω)

ωdω

)q2 +

∑k

λk(aqk + a†qk) , (2.2.8)

it is seen that a counterterm appears which renormalises the frequency ωq to ωq+δωq with δωq =∫∞

0Jq(ω)d lnω.

It has been pointed out in various occasions [73, 71] that such counterterms are necessary to yield the Hamil-tonian (2.2.6) bound from below for all values of ωq and ωp.

Second we notice that in Eq. (2.2.6) the system–bath interaction is represented as the coupling of the variablesq and p to a system of otherwise independent oscillators. The importance of this statement becomes clearerif we consider for instance a charged particle in an electromagnetic field, where a “velocity–coupling model”[123] seems to apply. Minimal coupling (p→ p − eA/c) generates not only an interaction term p ·A butalso a diamagnetic term ∝ A2 which can be interpreted as an interaction between the effective oscillators.A unitary transformation acting on p − (e/c)A, removes the coupling p ·A. This happens at the expenseof generating a coupling ∝ q ·E between the position and the electric field E. In this new representation noquadratic field term is left, i.e. the charge couples to a set of independent photons. Thus, in the languagewe adopt here, a charged particle couples to the electromagnetic field through its position q.

The Hamiltonian (2.2.6) is studied in Ref. [4] for spectral functions Jn(ω) = 2γnωαn (Ωαn−1π)−1, n = q, p

with arbitrary exponents αn and a cutoff frequency Ω which is assumed to be much larger than all otherfrequency scales. Here, we only report on the case of two Ohmic spectral functions Jn(ω) = 2γnω/π, n = q, p.

Equilibrium quantities

Elimination of the bath variables yields the Heisenberg equations of motion for q and p

q(t) = ωpp(t) +

∫ t

−∞dsKp(t− s)p(s) + Fp(t) ,

−p(t) = ωqq(t) +

∫ t

−∞dsKq(t− s)q(s) + Fq(t). (2.2.9)

The response kernel is defined as

Kn(t) =

∫ ∞0

Jn(ω)

ωcos(ωt)dω , n = q, p, (2.2.10)

and the force operators are given by

Fq(t) =∑

λkaqk exp(−iωkt) + H.c , and Fp(t) =∑

µkapk exp(−iωkt) + H.c . (2.2.11)

The equilibrium autocorrelation functions of the central oscillator C(±)nn (t) = 1

2〈[n(t), n(0)]±〉 with n = q, p

can be calculated exactly by standard methods. For instance

C(+)qq (t) =

1

π

∫ ∞0

|χ(ω)|2 cos(ωt) coth(βω/2)Im

(Jq(ω)

∣∣∣ωp + Jp(ω)∣∣∣2 + ω2Jp(ω)

)dω , (2.2.12)

C(−)qq (t) =

1

π

∫ ∞0

|χ(ω)|2 sin(ωt)Im

(Jq(ω)

∣∣∣ωp + Jp(ω)∣∣∣2 + ω2Jp(ω)

)dω . (2.2.13)

In these expressions a generalised form of the dynamical susceptibility was introduced

χ−1(ω) = ω20 − ω2 − ωqJp(ω)− ωpJq(ω) + Jq(ω)Jp(ω) , (2.2.14)

where Jn(ω) is the Riemann transform of the spectral function Jn(ω)

J(ω) = ω2P∫ ∞

0

J(ω′)

ω′(ω′2 − ω2

)dω′ − isgn (ω)π

2J(|ω|) . (2.2.15)

The time evolution of the symmetrized autocorrelation function (2.2.12) is governed by the poles of thegeneralised susceptibility. For Ohmic environments the real part of the Riemann transform vanishes and χ−1

is a quadratic polynomialχ−1 = ω2

0 − i(ωqγp + ωpγq)ω − (1 + γqγp)ω2 , (2.2.16)

12 2.2 Quantum Frustration

Figure 2.3: Stripes of underdamped oscillations of the symmetrized auto correlation function Eq. (2.2.12)

in the (γq, γp) plane for three different values of the parameter η = (ωq/ωp)1/2.

with zeros at

ω± =ω0

(1 + γqγp)1/2

(−iκ±

√1− κ2

)= −iτ−1 ± ζ , κ =

ωpγq + ωqγp

2ω0 (1 + γqγp)1/2

. (2.2.17)

The loci ω± of the zeros are either purely imaginary or a pair of complex conjugates depending on whether

κ is greater or smaller than 1. If the solutions have a real part, the time evolution of C(±)qq (t) is damped

but oscillatory, purely imaginary solutions are associated with overdamped oscillations. Thus, κ < 1 isthe commonly accepted criterion to distinguish between underdamped and overdamped oscillations. The

underdamped region lies in a stripe of width ∆ = 4η(1 + η4

)−1/2, with η ≡ (ωq/ωp)

1/2, limited by the

graphs of the functions f(γq) = γq/η2 ± 2/η. In Fig. 2.3 the stripes of underdamped oscillations of the

symmetrized autocorrelation function, marked in the (γq, γp) plane are plotted for three different values of η.It is seen that for symmetric coupling γq = γp = γ the central oscillator is always underdamped. This is theanalogue of the underdamped oscillations of an impurity spin observed in Ref. [1] and discussed in Sec. 2.2.1.

Denoting by C(±)qq (ω) the Fourier transform of C

(±)qq (t), a calculation of Dq(ω) ≡ C(−)

qq (ω)/ω allows for a directcomparison with the results for the spin susceptibility, Eq. (2.2.5). Of special interest is the slope of Dq(ω)near ω = 0. Since limω→∞Dq(ω) = 0, the condition for the existence for Dq(ω) displaying a maximum canbe written as

D′q(0) > 0, (2.2.18)

which may be viewed as an indicator of underdamped oscillations or, equivalently, coherent transitions1, seealso the remark after Eq. (2.2.5). For Ohmic damping, one finds

Dq(ω) =γqω

2p + γp(1 + γqγp)ω

2

[(1 + γqγp)ω2 − ω20 ]2 + (γqωp + γpωq)2ω2

, (2.2.19)

and the critical curve for γp is thus given by the relation γcritp = γq (γ2

q/η2 − 2)/η2. In Fig. 2.2 Dq(ω) is

plotted for symmetric coupling and for the q–oscillator. The marginal case of the q–oscillator γq =√

2,γp = 0 develops a maximum for symmetric coupling, although the peak is much less pronounced than forthe spin 1/2 system.

For Ohmic damping and as well as for high temperatures and for zero temperature, the integral in Eq. (2.2.12)can be performed and the position–position autocorrelation function can be evaluated exactly. It decaysexponentially at a rate ∝ τ−1 for high temperatures and algebraically ∝ t−2 for zero temperature like theq–oscillator [71]. The exact expressions can be found in Ref. [4].

As outlined in the introductory section of Chap. 2 equilibrium purity can serve as a measure for the efficiencyof the environment in destroying quantum coherence. For a harmonic oscillator in thermal equilibrium, thereduced Wigner function is [71]

Wβ(q, p) =1

2π [〈q2〉β〈p2〉β ]1/2exp

(− q2

2〈q2〉β− p2

2〈p2〉β

), (2.2.20)

which leads to an equilibrium purity P−2β = 4〈q2〉β〈p2〉β (~ = 1). At zero temperature the mean squares

〈q2〉∞, 〈p2〉∞ can be calculated exactly

〈q2〉∞ =r(κ)

4κ(1 + γqγp)

[γqη2

+ γp(1− 2κ2)]+

γpπ

(1 + γpγq) lnΩ√

1 + γqγpω0

+O(Ω−1) , (2.2.21)

with the function

r(κ) =1

π√κ2 − 1

lnκ+√κ2 − 1

κ−√κ2 − 1

. (2.2.22)

For small γq, γp, r(κ) = 1− 2κ/π +O(κ2) and the position mean square becomes

〈q2〉∞ =1

2η− γq

2η2+ γp

(ln

Ωpω0− 1

2

)+O(γq, γp) . (2.2.23)

1Remarkably, this criterion is at variance with the commonly accepted criterion κ < 1 even for the classical dampedq–oscillator. The fact that a distinction between overdamped and underdamped time evolution is not unique seems tohave been missed in the literature. For a discussion of this issue see Ref. [5].

2.2.2 Oscillator in Competing Heat Baths 13

Owing to the symmetry in p and q, 〈p2〉∞ is obtained by exchanging q and p everywhere in Eqs. (2.2.21),(2.2.23) and in Eq. (2.2.17). In contrast to the result for the q–oscillator both 〈q2〉∞ and 〈p2〉∞ exhibit alogarithmic dependence on the cutoff frequency. Consequently, the zero temperature equilibrium purity isdominated by the square of the logarithmic cutoff, P∞ ∝ ln−2 Ω. For the q–oscillator P∞ ∝ ln−1 Ω. Thisindicates that coherence is reduced as compared with the q–oscillator.

Time evolution

Although equilibrium decoherence (as measured by the product 〈q2〉β〈p2〉β) is enhanced by the additionalnoise term, one may wonder whether for low temperatures the decoherence time becomes larger than forthe damped q–oscillator. To answer this question exhaustively one has to calculate the time evolution ofthe purity for an arbitrary initial state. In the following we consider a decoupled initial density matrixρ(0) = ρS(0) ⊗ ρB(0). The bath is assumed to be in thermal equilibrium at t = 0. For the choice of thesystem part ρS(0) two cases are particularly interesting: a coherent (Gaussian) state and the superpositionof two Gaussian wave packets as initial state. For the q–oscillator the decoherence of a decoupled initial statewas calculated in Refs. [114, 118, 124].

We first focus on a coherent state

WS(q, p, 0) =1

πexp

[−η (q − q0)2 − 1

η(p− p0)2

], (2.2.24)

as initial state. This case should present the greatest robustness against decoherence [118]. For the two bathHamiltonian (2.2.6) the Ohmic case with general coupling constants γq and γp is cumbersome and can befound in Ref. [4]. Here, we focus on the symmetric situation

γq = γp = γ , η = 1 . (2.2.25)

Although purity can be calculated exactly the resulting expressions yield little insight. We just state thelimiting expressions for zero temperature

PG(t) '

e−Ωt , for 0 ≤ t . Ω−1

P∞[1 +

1

ω20t

2

2γ

1 + γ2cos(Λt)e−t/τ

], for Ω−1 t→∞ ,

(2.2.26)

with P−1∞ = 2〈q2〉∞ and Λ = ω0/(1 + γ2) is related to the oscillator frequency. From Eq. (2.2.26) it can be

seen that coherence is reduced immediately after the start of the coupling. Although afterwards it decreasesmore slowly, on a time scale τ , for larger couplings the curves of PG(t) for different values of γ never cross,i.e., PG(t) is a monotonously decreasing function of γ for all t.

An initial slip in a time of order of the inverse cutoff frequency 1/Ω as described in Eq. (2.2.26) also occursfor the q–oscillator [114]. However, in that case its effect on purity is much less severe. A detailed analysisreveals that the purity evolution of the q–oscillator is insensitive to the initial slip stemming from the useof decoupled initial conditions: At t ∼ Ω−1 the purity is still approximate unity, decreasing afterwards at arate ∝ γq. Purity for the q–oscillator has been extensively studied in Ref. [124].

The second choice is a superposition of two Gaussian wave packets as initial condition. This case has beenstudied for a single bath by Caldeira and Leggett [74], see also [100, 101, 99]. It displays two different aspectsof decoherence. On the one hand, there is the decoherence which either wave packet would experience alone.This part is essentially described by PG(t) [see Eq. (2.2.26)]. It was called Gaussian purity in Ref. [4]. Onthe other hand, there is the decoherence due to the spatial separation with distance a of the two packets.This second contribution is expected to become increasingly important when the distance a between the twopackets becomes large. The initial wave function is the sum of two Gaussian wave packets

ψ(x) =1

c4√

2πσ2

(e− (x+a/2)2

4σ2 + e− (x−a/2)2

4σ2

), (2.2.27)

where c is a normalisation constant. In the most symmetric case σ = 1/√

2, the purity can be expressedcompactly in terms of the Gaussian purity as [5, 4]

Pa(t) =PG(t)

2

1 +cosh2

[a2

4(φ(t)PG(t)− 1

2)]

cosh2 (a2/8)

. (2.2.28)

The function φ(t) is a complicated function [4] which evolves from φ(0) = 1 to limt→∞ φ(t) = 0 at an average∼ exp(−γt) and the single wave packet purity PG(t) is given in Eq. (2.2.26). It is useful to introduce withPrel(t) = Pa(t)/PG(t) the relative purity as a new quantity. As expected, the Prel(t) → 1 as a → 0, andPrel(t)→ 1/2 as a→∞.

14 2.3 Decoherence in Josephson Junctions

Interestingly, the structure of (2.2.28) is such that as time passes and φ(t)PG(t) evolves from 1 to 0, Prel(t)starts at unity, which corresponds to a pure state, then decreases and finally, at large times, goes back tounity. When a is large Prel(t) decays rapidly to 1/2 on a timescale ∼ 1/4a2γ. There it stays for a timewhich increases with distance as ∼ γ−1 ln a. Afterwards it returns to one. The ratio 1/2 can be rightlyinterpreted as resulting from the incoherent mixture of the two wave packets. Thus, somewhat surprisinglyPrel(t) becomes unity again at large times, as if coherence among the two wave packets were eventuallyrecovered. The physical explanation lies in the ergodic character of the long time evolution, with both wavepackets evolving towards the equilibrium configuration [see Eq. (2.2.21)].

2.3 Decoherence in Josephson Junctions

Josephson junctions are the principle ingredient for SQUIDS which are a prime candidate for qubits. Onedrawback of Josephson junction based qubits is their strong susceptibility to decoherence due to their rel-atively large spatial dimension and their embedding in a solid state environment. Several sources of de-coherence in Josephson link based qubits have been explored in the past, among them localised two–levelsystems as a possible source of 1/f noise [125, 126, 127] and the coupling to phonon modes in the conduc-tor [128, 129]. However the experimentally observed coherence decay [130] has still not found a completelysatisfactory theoretical explanation.

Fluctuations of the electrodynamical vacuum are an intrinsic source of decoherence in Josephson junctions,which has been considered in Ref. [131, 11]. The macroscopic phase is locked in the bulk of the superconduc-tor. In a Josephson junction the screening of the electromagnetic field is weaker and the macroscopic phasedifference φ accumulates uncertainty due to the vacuum fluctuations of the EM field. The system is repre-sented by the canonically conjugate variables, particle number difference N and phase difference φ acrossthe junction [N,φ] = i. As we will see, the EM field can be interpreted as an additional environment of thesystem. This additional environment is in competition with the standard environment due to quasiparticletunnelling across the junction. The situation of quantum frustration described in the previous section arisesnaturally, albeit in a weaker form.

The standard way to describe the dynamics of the macroscopic phase in Josephson links is the resistively andcapacitively shunted junction (RCSJ) model [132, 133, 134], which contemplates an ideal Josephson junctionshunted by a resistor and a capacitor. The resistor models the dissipative effect of incoherent quasiparticletunnelling through the junction, while the capacitor accounts for the charging energy, which plays the roleof the kinetic energy of the phase. In the absence of driving currents, the RSCJ model reads

N(t) = −EJ~

sinφ(t)− ~4e2R

φ(t) +1

2eI(t) , (2.3.1)

φ(t) =EC~N(t) . (2.3.2)

We have introduced the notation EJ ≡ ~Ic/2e and EC ≡ 4e2/C for the Josephson coupling energy and thecharging energy, respectively (Ic and C are the critical current and the capacitance of the junction). I(t)is a stochastic process with zero mean. At high temperatures it is related to the resistance by Einstein’srelation 〈I(t)I(0)〉 = 2kBTδ(t)/R, where R is the resistance of the junction in the normal state. Eq. (2.3.2) isrecognised as the ac–Josephson relation written in terms of the relative Cooper pair number, which generatesa chemical potential difference µ through the interaction energy EC = ∂µ/∂N .

The effect of the EM vacuum fluctuations was introduced in Ref. [11] through the following argumentation:written in the language of the Coulomb gauge (as is standard in these contexts [132, 133]), Eqs. (2.3.1)and (2.3.2) relate the phase to gauge invariant quantities. More specifically, they include the effect of the

longitudinal electric field, which in its simplest form yields a circulation∫ 2

1E‖ · dr = 2eN/C. Therefore, one

is entitled to replace the phase in (2.3.2) by its gauge invariant expression φ = φ1−φ2 + (2e/~c)∫ 2

1dr ·A(r),

where 1 and 2 are points deep enough in the bulk of superconductors 1 (left) and 2 (right). The resultingequation is again interpreted in the particular language of the transverse gauge, in which the vector potentialis characteristically related to the transverse electric field via cE⊥ = −A. Then one can write Eq. (2.3.2) ina form which treats the transverse and longitudinal electronic field on the same footing:

φ =2e

~

(∫ 2

1

E‖ · dr +

∫ 2

1

E⊥ · dr). (2.3.3)

Thus, one may view the transverse EM modes as the cause of additional voltage fluctuations V (t) =∫ 2

1E⊥ ·dr

with zero mean 〈V (t)〉 = 0. Writing the dissipative term in Eq. (2.3.1) as a retarded expression

(~/4e2R)φ(t) =

∫ t

−∞Γqp(t− t′)φ(t′)dt′ (2.3.4)

2.3 Decoherence in Josephson Junctions 15

and introducing a dissipative term which is related to the fluctuating voltage V (t) by a fluctuation–dissipationtheorem, a set of equations of motion is obtained

N(t) = −EJ~

sinφ(t)−∫ t

−∞Γqp(t− t′)φ(t′)dt′ +

1

2eI(t) (2.3.5)

φ(t) =EC~N(t) +

∫ t

−∞ΓEM(t− t′)N(t′)dt′ +

2e

~V (t) . (2.3.6)

This is exactly the form of Eq. (2.2.9) with a cosine potential instead of the harmonic potential. They arecompleted by the relations

ΓEM(t) =

∫ ∞0

JEM(ω)

ωcos(ωt)dω (2.3.7)

Γqp(t) =

∫ ∞0

Jqp(ω)

ωcos(ωt)dω , (2.3.8)

and the fluctuation-dissipation theorem

〈V (t)V (0)〉 =~2

8e2

∫ ∞0

JEM(ω) cos(ωt) coth(~βω/2)dω (2.3.9)

〈I(t)I(0)〉 = 2e2

∫ ∞0

Jqp(ω) cos(ωt) coth(~βω/2)dω . (2.3.10)

The derivation of Eqs. (2.3.5) and (2.3.6) presented here was based on intuitive and rather heuristic ar-guments. However, it has to be stressed that the Hamiltonian, which yields Eqs. (2.3.5) and (2.3.6) asHeisenberg equations, was derived from a microscopic tunnelling Hamiltonian in Ref. [11]. Ambegaokar,Eckern and Schon derived [135, 136] the dynamics of the macroscopic phase difference as described byEq. (2.3.1) from a microscopic Hamiltonian. Their work was extended in Ref. [11] to the case of additionalvoltage fluctuations. The resulting effective Hamiltonian of Caldeira–Leggett type reads

H = EJ (1− cosφ) +EC2N2 + φ

∑i

λi(bi + b†i ) +N∑kε

µkε(akε + a†kε)

+∑i

~ωib†i bi +∑kε

~ωka†kεakε +N2∑kε

µ2k

~ωk+ φ2

∑i

λ2i

~ωi. (2.3.11)

It has exactly the general structure of Eq. (2.2.6). The bosonic operators bi describe fluctuations of par-ticle number difference due to quasiparticle tunnelling across the junction. The spectral function of theseexcitations is a rather complicated function which can be found for instance in Ref. [133]. For 0 < T . Tcit is fairly well approximated by an Ohmic spectral density Jqp(ω) = ~ω

2πe2R, which is the basis for the

RCSJ model Eq. (2.3.1). The parameters λi are adjusted to yield Jqp(ω) = 2~−2∑i λ

2i δ(ω − ωi). The other

set of bosonic operators akε describes the vacuum modes of the EM field. In general, their spectral densitydepends on the geometry of the junction. For point contacts it can be approximated by the superohmic

spectral function of the free EM vacuum. JEM(ω) = 8αd2

3πc2ω3, where α ≈ 1/137 is the fine structure constant

and d is the thickness of the junction. The natural cutoff frequency is given by the inverse square root of thesurface area A of the junction. The two last terms in Eq. (2.3.11) are counterterms. They arose naturally inthe derivation of Eq. (2.3.11), see Refs. [136, 11] for details, and were not introduced ad hoc as in Eq. (2.2.6).

The equilibrium phase uncertainty was calculated in Ref. [11] within a harmonic approximation for the cosinepotential. For large resistance R 2πe2/~ it was found that

〈φ2〉 =1

2

√ECEJ− EC~

8EJRe2+

2αf2

3π+

8α2f4~9π2e2R

(1 +

π2

4

), (2.3.12)

where f is the aspect ratio of the junction, i. e. the ratio between thickness of the tunnelling region andsquare root of the surface area F of the contacts. This expression features the competing character of thetwo environments nicely. The first term is the phase uncertainty due to the free oscillations of the phasedifference, the second term is due to quasiparticle tunnelling. It reduces phase uncertainty. For small normalresistance it becomes increasingly important and will, for very small values of R, eventually lead to a phaselocking or, equivalently, to a vanishing of the Josephson effect. The third term in Eq. (2.3.12) is due to thevacuum fluctuations of the EM-field. As expected, it is small, being of order α, which is characteristic ofQED effects. The structure of the junction enters via the aspect ratio. Again this was expected, since theaspect ratio is proportional to the dipole moment of the junction. As is seen from Eq. (2.3.11) the modes ofthe EM–vacuum couple to the dipole via the charge number difference. The fourth term in Eq. (2.3.12) is ajoint contribution of both baths.

16 2.4 Resume I

Since the two leading order contributions of the two environments have opposite sign, the result Eq. (2.3.12)can be considered as a signature of quantum frustration.

Finally, a comment on observability of the EM–vacuum fluctuations in Josephson contacts is in order. Tothis end we consider Eq. (2.3.12) and neglect the contribution of quasiparticle tunnelling. Then, for thethird term in Eq. (2.3.12) to be the leading contribution, the surface area must be small. However, for smallsurfaces the capacitative energy EC becomes large since the capacitance scales ∼ F . The Josephson couplingenergy scales as well as ∼ F , since the critical current Ic is related to the resistance of the normal junctionR by the formula of Ambegaokar and Baratoff [137]

Ic(T ) =π∆(T )

2eRtanh

∆(T )

2kBT(2.3.13)

where ∆(T ) is the temperature dependent modulus of the order parameter. Thus, the first term scales as∼ F−1 with the surface area in exactly the same way as the third one. Reduction of the surface area leadsto no enhancement of the third term with respect to the first. Therefore, due to the smallness of α the firstterm will be dominant in all realistic situations.

2.4 Resume I

In this section a mechanism of decoherence suppression called “Quantum frustration” [1] was consideredin some detail. It was first noted in Ref. [1] for the (cylindrically) symmetric spin–boson model with S =1/2, i.e. in a model which has no classical analogue. “Quantum frustration” of the spin can pictorially bedescribed as the result of having two observers attempting to measure simultaneously, with equal efficiency,two non-commuting components of the spin. Because of the uncertainty principle, both of them fail tomeasure anything. The most striking effects observed in Ref. [1] are: 1) the absence of a Kosterlitz–Thoulessphase transition for exactly symmetric coupling and 2) underdamped oscillations of the spin susceptibilityfor arbitrarily strong coupling.

These findings are contrasted with the model of an oscillator whose position and momentum variable couplelinearly to independent heat baths. Here the effect is due to the mutually conjugate character of position andmomentum. In the symmetric case, where both baths are Ohmic and their coupling strengths are the same,the motion of the central oscillator is underdamped for all coupling strengths. This indicates that also forthe harmonic oscillator the two baths partially cancel each other. This cancellation was called “quasiclassicalfrustration” in Ref. [5] because the dissipative oscillator may describe a large spin impurity coupled to themagnon bath in a ferromagnetic medium [138]. The observed cancellations are more moderate than for aspin 1/2. For example, the antisymmetrized equilibrium position position correlator, which is the analogueto the spin susceptibility for an oscillator develops only a tiny maximum for symmetric coupling even in themost favourable choice of parameters, see Fig. 2.2.

As for decoherence we have seen in the specific example of a decoupled initial state that, in destroyingquantum coherence, two baths are always more efficient than one bath. This is true for both the overdampedand the underdamped regime. This demonstrates that, at least for the harmonic oscillator, underdampeddynamics is by no means a reliable signature of high global coherence, which we identify with purity here.Equilibrium purity is also reduced in presence of a second bath.

The results provide evidence that decoherence and dissipation are not necessarily correlated. We have seenthat, depending on the situation, an increase of dissipation can be accompanied by either a reduction or anincrease of decoherence. Vice versa, a source of decoherence may or may not lead to dissipation. In viewof these results the conclusions of Castro Neto et al. for the spin system have at least to be questioned.They interpreted underdamped dynamics of the spin susceptibility as a signature of reduced decoherence.But at least for the harmonic oscillator antisymmetrized correlation functions are not a reliable indicator ofcoherence. In Ref. [5] it was argued that the dimensionality of Hilbert space is the key difference in bothproblems. Evolving in the continuum, the quantum oscillator can be considerably degraded by the effect ofthe environment, as shown in (2.2.26). It is only when the two wave packets recombine because of ergodicitythat relative purity is recovered. By contrast, the spin-1/2 magnetic impurity lives in a two-dimensionalspace. The only possible effect of the environment is to flip the spin.

Exact dependence on dimensionality as well as the question whether the occurrence of frustration in theclassical regime is a general property or an artifact of the harmonic oscillator deserve further investigation.Quantum frustration is an interesting and intriguing phenomenon and it is worthwhile to study other systemscoupled to two environments from the viewpoint of quantum frustration (one instance might be the two–channel Kondo model [139]).

3. Random Matrix Theory

Random matrix theory (RMT) was developed by Wigner and Dyson [41, 40, 140, 141, 142, 143] as a statisticalapproach to the description of spectra of complex many–body systems, originally nuclei. By now, it hasbecome a standard tool in the description of the spectra of 1) complex systems like nuclei or atoms 2)classically chaotic systems and 3) disordered systems. Its importance for chaotic quantum systems is basedon the conjecture [144, 145] by Giannoni, Bohigas and Schmit which has been probed in many numericalstudies and by now is widely accepted, although it is, despite of important recent advances [146, 147, 148,149, 150], still not completely proved. Literally it states that the Spectra of time reversal invariant systemswhose classical analogues are K systems show the same fluctuation properties as predicted by the Gaussianorthogonal ensembles [145]. Often the rather restrictive requirement of a K system, a maximally mixingsystem [151, 152], is substituted by the much weaker condition of ergodicity. In more recent developmentsRMT has also found applications in as different fields as QCD [52] and econophysics [53]. The immenseamount of publications on the field of RMT is reviewed in many good reviews [45, 153] and books [143, 61].Therefore, here we limit ourselves to two aspects: on the one hand, we compile the basic concepts of RMTwhich are indispensable for the understanding of its application to fidelity decay. On the other hand, wereview some very recent developments in general RMT to which the author contributed and which were notcovered by the last review on RMT of Guhr et al. in 1998 [153].

In RMT the local statistical behaviour of the energy levels of a system is emulated by the energy fluctuations ofGaussian N×N random matrices. The spectra of the systems are thereby classified into a very small amountof universality classes, which are only distinguished by the overall symmetry of the underlying Hamiltonian.In the original classification of Wigner and Dyson there exist only three different universality classes. Dysonwas also the first to realize the connection of RMT ensembles with Riemannian symmetric spaces [154].Later, Dyson’s ”threefold way” has been completed by Oppermann [155] and by Altland and Zirnbauer [156].The authors showed that a random matrix ensemble corresponds to any Riemannian symmetric space asclassified by Cartan [157, 158]. For instance, the classical RMT ensembles (GOE, GUE, GSE) together withthe chiral RMT ensembles are associated with coset spaces of the special linear group SL(N,C) and therebytreated in a unified way. Zirnbauer and Altland also identified a series of RMT ensembles associated withthe coset spaces of the orthogonal group SO(N,C) and for the symplectic group Sp(N,C). Oppermann [155]as well as Zirnbauer and Altland [156] found an application in disordered superconductors. More details onsymmetric spaces can be found in App. D. Here, we only review the original three universality classes ofWigner and Dyson.

• For a spin–independent system which is invariant under time–reversal operation all entries of theHamiltonian can be chosen real in an appropriate basis. Accordingly, the random matrix average hasto be taken over all real symmetric matrices. The ensemble is called Gaussian orthogonal ensemble(GOE) for its invariance property under the action of the orthogonal group.

• For systems, where time–reversal invariance is broken, the corresponding random matrix should bechosen from an ensemble of Hermitean complex matrices, which is called Gaussian unitary ensemble(GUE). It is invariant under unitary transformations.

• Finally, for time–reversal invariant systems with half–integer spin and no rotational symmetry a basiscan be found, where the Hamiltonian can be written as a N ×N matrix with quaternionic entries. Forthe entries qik the condition of selfduality holds, which is defined as

qik = qki , q = q(0)e(0) +3∑

n=1

q(n)e(n) , q = q(0)e(0) −3∑

n=1

q(n)e(n) , (3.0.1)

where the components q(0) and q(n) are real and the basis vectors can be chosen as e(0) = 1 and e(n) asiσn with the standard Pauli matrices σn. We distinguish between the scalar product of two quaternionsqiqj = tr qiqj/2 and the quaternion valued product qi · qj denoting the usual matrix multiplication.This ensemble is called Gaussian symplectic ensemble (GSE) and is invariant under the action of theso–called unitary symplectic group USp.

• Here, we also introduce the so-called Poisson ensemble. It consists of matrices, where no correlationsbetween eigenvalues exist. The average is over an ensemble of diagonal matrices. In contrast to theother ensembles mentioned above, it bears no rotational invariance.

In Gaussian RMT, for all three ensembles, the invariant probability measure can be written as

PNβ(H)d[H] = ANβ exp

(− β

4a2trH2

)d[H] , (3.0.2)

17

18 3.1 Correlation functions

where d[H] denotes a product of differentials over all independent matrix entries of the corresponding en-semble. The normalisation constant is given by

ANβ = 2βN(N−1)/4

(β

4a2π

)N/2+βN(N−1)/4

. (3.0.3)

The integer β is the Dyson index and has the values β = 1 for the GOE, β = 2 for the GUE and β = 4 forthe GSE. It was introduced by Dyson [140, 141, 142] who interpreted the eigenvalue statistics of Gaussianrandom matrix ensembles within the framework of classical statistical mechanics.

The equivalence to a classical one–dimensional many–body problem is most easily seen, when the probabilitymeasure is written in angle–eigenvalue coordinates H → U−1

β XUβ :

PNβ(H)d[H] ∝ PNβ(X)|∆N (X)|βd[X]dµ(Uβ),

∝ exp

(−β

N∑i=1

x2i

4a2− β

∑i<j

ln |xi − xj |

)d[X]dµ(Uβ), (3.0.4)

where ∆N (X) =∏n<m(xn − xm) is the Vandermonde determinant. Uβ is an orthogonal (β = 1), unitary

(β = 2) or unitary symplectic (β = 4) matrix. The differential dµ(Uβ) denotes the invariant measure over the

corresponding group. d[X] =∏Ni=1 dxi is the measure of the radial coordinates. With a suitable redefinition

of (3.0.4) one can also associate the Poisson ensemble with the Dyson index β = 0.

We define the ensemble average

〈. . .〉N,β =

∫(. . .)PNβ(H)d[H] =

1

C(β)N N !

∫(. . .)

N∏i=1

pβ(xi)|∆N (X)|βd[X], (3.0.5)

where pβ(x) = exp(−βx2/4a2)/√

4πa2/β is the Gaussian measure. The second equality in (3.0.5) holdsonly for rotational invariant observables, for which the integral over the group is trivial. The normalisationconstant can be evaluated using Selberg’s integral [159, 61]

C(β)N =

1

N !

∫ N∏i=1

pβ(xi)|∆N (X)|βd[X]

=1

N !2N/2

(β

4a2

)−N/2−βN(N−1)/4

[Γ(1 + β/2)]−NN∏j=1

Γ(1 + βj/2) . (3.0.6)

The left hand side of Eq. (3.0.4) is the partition function of a one–dimensional gas confined in a harmonicoscillator potential with a logarithmic repulsion which can be nicely interpreted as two–dimensional Coulombinteraction [140, 141, 142]. In this picture the Dyson index β has the usual meaning from thermodynamicsas an inverse temperature and can be continued canonically to arbitrary positive values. The implicationsof a generalisation to β ∈ R+ will be discussed in Chap. 6. The variance of the measure (3.0.2) will be fixed

by setting a = 1/√

2 in the remainder of the chapter.

3.1 Correlation functions

The most important observables in RMT are the energy k–point correlators

R(β)k (x1, . . . , xk) =

N !

C(β)N (N − k)!

∫ ∞−∞

dxk+1 . . .

∫ ∞−∞

dxN

N∏i=1

pβ(xi)|∆N (X)|β

=1

πk

⟨k∏p=1

Im tr1

x−p −H

⟩N,β

. (3.1.1)

Here and in the following we define the symbol x− by

f(x−) = f(x− iη) , (3.1.2)

where η is infinitesimal. Both definitions of the k–point correlators in Eq. (3.1.1) are basically equivalent1

and both have been used as starting point for the derivation of the solutions. For all three classical ensemblesthe solution for the k–point function can be written as a quaternion determinant (Pfaffian)

R(β)k (x1, . . . , xk) =

√det R (3.1.3)

1Strictly speaking the two definitions are equivalent only for the one–point function k = 1. For a general k–correlator the second definition yields additional terms containing at least one δ–function, for instance δ(xq−xp)(. . .),where the dots in the bracket denote a correlation function of order k − 1. Only after omitting all contributionscontaining δ–functions in the second definition both are equivalent.

3.1 Correlation functions 19

of the Hermitean selfdual matrix R. Since the k–point function depends only on the determinant of R, itsentries are not uniquely defined. Indeed, they have been given in several forms in the past. Here, we statethree different forms of the entries of R and denote them Kij , Lij and L′ij , respectively.

Sums of oscillator wave functions

The original form of the quaternion matrix R was derived by Mehta, Gaudin and Dyson [160, 161, 162, 163,164]. The entries of R are given by

Kij =

[K

(β)N (xi, xj) DK

(β)N (xi, xj)

IK(β)N (xi, xj) K

(β)N (xj , xi)

]. (3.1.4)

For β = 1, 4 the symbols D and I denote the antisymmetrized differential and integral operators

Df(x, y) =1

2(∂yf(x, y)− ∂xf(y, x)) , (3.1.5)

If(x, y) =1

2

∫ ∞−∞

dtε(x− t)f(t, y)− 1

2

∫ ∞−∞

dtε(y − t)f(t, x)− δβ1ε(x− y),

where ε(x) = sgn(x)/2. For β = 2: Df = If = 0. The antisymmetry of the operators I and D guaranteesthe selfduality of the matrix R. From Eq. (3.1.4) it is apparent that all information of the energy k–point

correlators is encoded in one single function K(β)N (xi, xj) which is called matrix kernel. For the GUE the

matrix kernel is given by an incoherent sum of oscillator wave functions

K(2)N (xi, xj) =

N−1∑n=0

φn(xi)φn(xj) ,

φn(x) =1√

2nn!√πHn(x)e−

x2

2 , (3.1.6)

where Hn(x) is the standard Hermite polynomial [165]. The GOE and GSE kernels contain the GUE kernelplus an additional term. They are given by

K(1)N (xi, xj) = K

(2)N (xi, xj) +

√N

2φN−1(xi)

∫ ∞−∞

ε(xj − t)φN (t)dt

+φN−1(xi)

(∫∞−∞ φN−1(t)dt

)−1

for N odd,

0 for N even,(3.1.7)

for the GOE and by

K(4)N (xi, xj) =

1√2K

(2)2N+1(

√2xi,√

2xj)

+

√2N + 1

2φN−1(

√2xi)

∫ ∞−∞

ε(xj − t)φN (√

2t)dt (3.1.8)

for the GSE. It is worthwhile to notice that formulae (3.1.6), (3.1.7), and (3.1.8) remain basically unchangedif instead of Gaussian probability density (3.0.2) a different probability measure is considered [166]. In par-ticular the determinantal structure (GUE) and the structure of a quaternion determinant remain unaffectedby a different measure function. A recipe for the construction of the extra terms in the GOE and the GSEkernels for an arbitrary measure function was given Ref. [167].

Integral representations:

There exists an alternative representation of the matrix kernel in terms of a supersymmetric matrix integral[10]

L(β)N (xq, xp) =

1

γπ

1

xp − xq

Imβ2

8γ4

∫exp

(− β

2|γ|Strσ2

)Sdet−βN/2|γ|(σ − x−)d[σ] , (3.1.9)

where γ = 1 for β = 1, 2 and γ = −2 for β = 4. Here, Sdet and Str denote the supertrace and superdetermi-nant, for a precise definition see App. B. The supermatrix σ is a Hermitean 2× 2 matrix for the GUE and

20 3.1 Correlation functions

a 4× 4 supermatrix with an orthosymplectic symmetry for the GOE and the GSE. Explicitly, σ reads

σ =

[z1 ρ∗

ρ iz2

], σ =

√ca

√cd λ∗ −λ√

cd√cb µ∗ −µ

λ µ√−cw 0

λ∗ µ∗ 0√−cw

, (3.1.10)

where the first form applies for the GUE and the second form applies for the GOE and the GSE with c = 1and c = −1, respectively. The variables a, b, d and w are real commuting, while λ, λ∗ and µ, µ∗ are complexanticommuting. In the GUE case the anticommuting entries are complex, whereas z1 and z2 are real. Theintegration measure d[σ] is over all independent entries of the supermatrix σ ( see Appendix B for furtherdetails on supermatrices). The diagonal supermatrix

x = diag (xp, xp, xq, xq) for β = 1, 4 , and x = diag (xp, xq) for β = 2 (3.1.11)

contains the energy arguments. The map of the integral representation Eq. (3.1.9) on the form as derivedby Mehta can be found in Ref. [60] for the GUE and in Ref. [10] for the GOE and the GSE.

The representation (3.1.9) is a fourfold integral for the GUE and an eightfold integral for the GOE and theGSE. It can be simplified by a transformation into angle–eigenvalue coordinates akin to the step performedbefore Eq. (3.0.4). The angular coordinates are now given by the supergroup U(1|1) for β = 2 and bya supermanifold called UOSp(2|2) for β = 1, 4 (see App. B for the precise definitions). The non–trivialintegration over the angular coordinates was done in Ref. [60] for U(1|1) and in Ref. [12] for UOSp(2|2).

After integration over the unitary supergroup U(1|1) the unitary case reduces to the double integral

L(2)N (xp, xq) = − 1

π2

∫ +∞

−∞

∫ +∞

−∞

ds1ds2

s1 − is2exp

(−(s1 + xp)

2 + (is2 + xq)2)

(is2)N Im1

(s−1 )N. (3.1.12)

The integration over UOSp(2|2) yields triple integrals, given by

L(1)N (xp, xq) = − 1

8π2

∫ +∞

−∞

∫ +∞

−∞

∫ +∞

−∞

|s11 − s21|ds11ds21ds2

(s11 − is2)2(s21 − is2)2

exp

(−1

2(s11 + xp)

2 − 1

2(s21 + xp)

2 + (is2 + xq)2

)(2(xp − xq)(s11 − is2)(s21 − is2) + (s11 + s21 − 2is2))

(is2)N Im1

(s−11)N/2(s−21)N/2, (3.1.13)

for the GOE and by

L(4)N (xp, xq) = − 1

8π2

∫ +∞

−∞

∫ +∞

−∞

∫ +∞

−∞

|s11 − s21|ds11ds21ds2

(is11 − s2)2(is21 − s2)2

exp

(1

2

(is11 +

√2xp)2

+1

2

(is21 +

√2xp)2

−(s2 +

√2xq)2)

(2(√

2xq −√

2xp)

(is11 − s2)(is21 − s2) + (is11 + is21 − 2s2))

(is11is21)N Im1

(s−2 )2N(3.1.14)

for the GSE. The double integral Eq. (3.1.12) and the triple integrals Eq. (3.1.13) and Eq. (3.1.14) can beconsidered as the minimal integral representations of the matrix kernel, similar to the integral representationof the oscillator wave function [165]

φn(x) =

√n!

2n√π

1

πex

2/2

∫ ∞−∞

dse−(s−x)2Im1

(s−)n+1√2n

n!√π

1√πe−x

2/2

∫ ∞−∞

dse(is+x)2(−is)n .

The relation between the integral representation L(β)N and K

(β)N is for all three ensembles given by [60, 10]

K(β)N (xq, xp) = exp[γ(x2

p − x2q)/2]L

(β)N (xp, xq) ≡ W (xp, xq)L

(β)N (xp, xq) . (3.1.15)

3.1 Correlation functions 21

Notice that the arguments of L(β)N and K

(β)N are interchanged in Eq. (3.1.15).

A major virtue of the representation of the kernels as a supersymmetric matrix integral Eq. (3.1.9) is that

it establishes a relation between the matrix kernels L(β)N and the ensemble average over the ratio of two

characteristic polynomials. It is well known that the supersymmetric matrix integral on the right hand sideof Eq. (3.1.9) can be viewed as an integral representation of the ensemble average of characteristic polynomials

γ(xp − xq)L(β)N (xp, xq) =

⟨(det(x−p −H)

det(x−q −H)

)γ⟩N,β

. (3.1.16)

The ratio of determinants on the left hand side is the ensemble average of the generating function of the leveldensity. The level density is obtained from Eq. (3.1.16) by differentiating the left hand side by xq − xp andsetting xq = xp afterwards. However, it comes as a surprise that away from the point xq = xp the generatingfunction of the level density is identical with the full two–point matrix kernel.

Using Eq. (3.1.4) together with Eqs. (3.1.5) and (3.1.15) the quaternion entry Lij of R is given by

Lij =

[L

(β)N (xi, xj) W−1(xi, xj)DW (xi, xj)L

(β)N (xi, xj)

W−1(xi, xj)IW (xi, xj)L(β)N (xi, xj) L

(β)N (xj , xi)

]. (3.1.17)

SinceW−1(xi, xj) =W (xj , xi) the quaternion determinant det R remains unchanged as compared to Eq. (3.1.4).

For practical purposes the integral representations Eqs. (3.1.12), (3.1.13), (3.1.14) are important since theyallow in a simple way for a generalisation to ensembles with broken rotational invariance. For β = 2 theintegral representation (3.1.12) was used in Ref. [168] to evaluate the energy–energy correlator for a randommatrix ensemble, where rotational invariance is broken by an admixture of a Poissonian ensemble. Morespecifically, the author considered an RMT Hamiltonian of the form H = H(0) +λH(1), where H(1) is chosenfrom a GUE ensemble and H(0) is chosen from a Poissonian ensemble. As a function of the transitionparameter λ, the spectral statistics change from Poissonian (λ = 0) to GUE behaviour(λ =∞). This modelwas also considered by Pandey in Ref. [169]2.

The GUE integral representation (3.1.12) was also used in Refs. [171, 172] to investigate the spectral statistics

of a random matrix H(1) coupled to an external, deterministic matrix H(0). In Ref. [173] Brezin and Hikami

used again Eq. (3.1.12) to couple a GUE random matrix to an external, deterministic matrix H(0) with onlytwo different eigenvalues, each of them N/2 times degenerate. Tuning the transition parameter λ such thatthe gap between the two eigenvalues exactly vanishes, they investigated the density of states at the criticalpoint. A similar analysis ought to be possible also for the GOE and the GSE with the integral representationsas given in Eq. (3.1.13) and (3.1.14).

Limit of averages over characteristic polynomials

Borodin and Strahov [174] found a general connection of the matrix R to the averages over ratios of charac-teristic polynomials than Eq.(3.1.16). For the GUE they reproduce Eq. (3.1.16). For the GOE and the GSEcase they find, introducing the definition

Dα(x) =

Im(detα(x− −H)) , α ∈ R−

detα(x−H)) , α ∈ R+(3.1.18)

the following representations for the quaternionic entries Eq. (3.1.4) of R

L′ij =1

γ

1

xi−xj〈Dγ(xi)D−γ(xj)〉N,β

C(β)N−2/γ

C(β)N

(xi − xj)〈Dγ(xi)Dγ(xj)〉N−2/γ,β

C(β)N+2/γ

C(β)N

(xi − xj)〈D(xi)−γD(xj)

−γ〉N+2/γ,β1

xj−xi〈Dγ(xj)D−γ(xi)〉N,β

,

(3.1.19)

where again γ = 1 for β = 1, 2 and γ = −2 for β = 4.3 A comparison of L′ij with Lij in Eq. (3.1.17) and withEq. (3.1.16) reveals that the diagonal entries of Lij and L′ij are identical. It seems natural to conclude thatalso the off–diagonal entries should be identical, but this is not the case. In App. A the relation between theoff–diagonal elements of L′ij and Lij and Kij is derived.

In conclusion there exist three equivalent forms for the quaternion determinant det R, connected by a simi-larity transformation. 1) The original representation derived by Mehta [61], 2) a (minimal) integral represen-tation [Eqs. (3.1.12), (3.1.13) and (3.1.14), together with Eq. (3.1.17)], derived in Refs. [60, 10] and 3) as anaverage over characteristic polynomials Eq. (3.1.19). The three forms were derived by completely differentmethods.

2The same problem for the GOE and the GSE is actually much harder, but some recent advances were made inRefs. [12, 170].

3For the GOE Eq. (3.1.19) has been proved only for even matrix dimension.

22 3.2 Random Polynomials

3.2 Random Polynomials

Random polynomials are closely related to the energy–energy correlation functions. Theory of randompolynomials deals with the calculation of the moments and correlation functions of characteristic polynomialsof an N ×N random matrix H

Z(x, y;α, β) =

∏ni=1 detαi(xi −H)∏mj=1 detβj (yj −H)

, αi, βi ∈ R+ , (3.2.1)

where x and α denote the sets of n arguments xi, αi, 1 ≤ i ≤ n and y and β denote the sets of m arguments yi,βi, 1 ≤ i ≤ m. The xi and yi are in general complex numbers and the yi have a non–vanishing imaginary part.They form a class of observables, which is of considerable current interest [175, 176, 177, 178, 179, 180, 181,182, 183, 184]. They are frequently studied for various types of random matrices H and in various differentcontexts. One, maybe the most prominent, reason is the intimate relation between the statistics of the zerosof the Riemann zeta function on the critical line and the statistics of the zeros of random polynomials, seeRef. [175, 178] for a review and a complete list of references. Another reason is their relation to the theoryof orthogonal polynomials, see Deift’s [176] book for a review and a comprehensive list of references.

Although an exhaustive review on this topic is beyond our scope, the work of Borodin and Strahov [174]has to be mentioned, since in some sense it completes and summarises our current knowledge about averages(over the classical Gaussian ensembles) of characteristic polynomials. They were able to calculate the averageof Z for all three classical invariant ensembles and for an arbitrary weight function (not only for a Gaussianweight). The main result of Ref. [174] is that for an arbitrary number n of determinants in the numeratorand of m determinants in the denominator the expectation value 〈Z〉N,β always decomposes into a Pfaffianform. For the GOE and the GSE the result reads

〈Z(x, y;α, β)〉N,β =CN+n−m

CN

∏i,j(xi − yj)

∆n(x)∆m(y)Pf Q

(β)N , (3.2.2)

where α = (1, . . . 1) and β = (1, . . . 1). Q(β)N is the skewsymmetric (n+m)× (n+m) matrix with entries

Q(β)N (xi, xj) =

CN+(n−m−2)/|γ|

CN+(n−m)/|γ|(xi − xj)〈Z(xi, xj ; 1, 1)〉N+(n−m−2)/|γ|,β ,

Q(β)N (yi, yj) =

CN+(n−m+2)/|γ|

CN+(n−m)/|γ|(yi − yj)〈Z(yi, yj ; 1, 1)〉N+(n−m+2)/|γ|,β ,

Q(β)N (xi, yj) =

1

xi − yj〈Z(xi, yj ; 1, 1)〉N,β , (3.2.3)

where, as before, γ = 1 for β = 1 and γ = −2 for β = 4. The ensemble average is defined as in Eq. (3.0.5)but with an arbitrary measure function p(z). The case of arbitrary integer powers is easily obtained byintroducing degeneracies. In the GOE case this result has only been proved for even matrix dimension N .

A similar result also holds for the GUE, which has also been addressed by other authors [185, 186, 187, 188].

Following Ref. [174], we define four sets of arguments x(1) with n1 elements, x(2) with n2 elements as well

as y(1) with m1 and y(2) with m2 elements, such that n1 −m1 = n2 −m2 = S. Then the ensemble averageover the GUE can be written as

〈Z(x(1), x(2), y(1)y(2);α, β)〉N2 =C

(2)N+S

C(2)N

∏i,j(x

(1)i − y

(1)j )

∏i,j(x

(2)i − y

(2)j )

∆n1(x(1))∆n2(x(2))∆m1(y(1))∆m2(y(2))det W

(2)N , (3.2.4)

where W(2)N is a (n1 +m2)× (n2 +m1) matrix

W(2)N (x

(1)i , x

(2)j ) =

CN+S−1

CN+S〈Z(x

(1)i , x

(2)j ; 1, 1)〉N+S−1,2 ,

W(2)N (x

(1)i , y

(1)j ) =

1

x(1)i − y

(1)j

〈Z(x(1)i , y

(1)j ; 1, 1)〉N+S,2 ,

W(2)N (y

(2)i , x

(2)j ) =

1

y(1)i − x

(2)j

〈Z(y(2)i , x

(2)j ; 1, 1)〉N+S,2 ,

W(2)N (y

(2)i , y

(1)j ) =

CN+S+1

CN+S〈Z(y

(2)i , y

(1)j ; 1, 1)〉N+S+1,2 . (3.2.5)

Eqs. (3.2.2) to (3.2.5) mean the following in words: To calculate the average of Z for an arbitrary ratio ofcharacteristic polynomials with an arbitrary integer power αi, βj ∈ N, 1 ≤ i ≤ n, 1 ≤ j ≤ m it is enoughto calculate the averages of products and ratios of two determinants as given in Eqs. (3.2.3) and (3.2.5).These results are restricted to integer powers of determinants αi, βj ∈ N. Up to now, nobody derived similarformulas for rational or even real powers αi, βj . It is an open question if the beautiful structure of a Pfaffiansurvives in these cases.

3.3 Large N–Limit 23

3.3 Large N–Limit

Most important for comparison with experimental and numerical data are the k–point correlators on theunfolded scale. The unfolding procedure consists of two steps. In a first step the density of states is calculatedin the limit of infinite matrix dimension N → ∞. For the pure RMT ensembles considered here it is givenby Wigner’s semicircle

ρ(x) ≡ limN→∞

R1(x) =

1π

√2N − x2 if |x|2 ≤ 2N ,

0 otherwise.(3.3.1)

Only in the region x = 0 it is approximately constant. There, the mean level spacing D = ρ−1(0) is given by

D =√π2/2N . (3.3.2)

Dyson himself considered it as a drawback of Gaussian random matrix ensembles that their spectral densityis a semicircle rather than a constant. It led him ultimately to introduce the circular ensembles [61], whichwill not be considered here.

In the next step of unfolding an energy interval in the spectrum is chosen, which is small enough such thatthe level density itself can be assumed approximately a constant. In principle for Wigner’s semicircle anypoint but the edges can be chosen. For convenience, however, usually the center of the spectrum is taken.For a general k–point correlation function the limit N → ∞ is taken such that (xp − xq)/D remains finite.By this procedure the non–universal dependence of the spectral correlator on the energy–density is removed.

For the matrix kernels K(β)N as given by Eqs. (3.1.6), (3.1.7), and (3.1.8) the large N limit is taken by using

the Christoffel–Darboux formula for oscillator wavefunctions [189, 61]

N−1∑n=0

φn(x)φn(y) =

√N

2

φN (x)φN−1(y)− φN (y)φN−1(x)

x− y . (3.3.3)

Formula (3.3.3) allows to take the limit N → ∞ using the asymptotic formula for oscillator wave functions[190]

limN→∞

(−1)N/2(Nπ2/2)1/4φN (x) = cos(πξ) , limN→∞

(−1)N/2(Nπ2/2)1/4φN+1(x) = sin(πξ) , (3.3.4)

where ξ = x/D and N even. Plugging (3.3.4) into Eq. (3.3.3) yields directly

limN→∞

K(2)N (xq, xp) =

sin(r)

Dr≡ s(r)

D, r = π(xq − xp)/D , (3.3.5)

which only depends on the microscopic energy difference. One can show (see App. A.10 of Ref. [61]) thatthe additional terms appearing in the GOE kernel (3.1.7) and the GSE kernel (3.1.8) do not contribute inthe large N–limit. Thus, on the unfolded scale the kernels of the three ensembles become identical yieldingthe universal sine kernel (3.3.5).

After unfolding the microscopic energy correlation functions depend on the spectral density only in a trivialway. For instance, the microscopic two–point function

X(β)2 (r) = lim

N→∞R

(β)2 (x1, x2) (3.3.6)

is given by

D2X(1)2 (r) =

(1− s2(r)− ds(r)

dr

∫ ∞r

s(r′)dr′), r = π(x1 − x2)/D

D2X(2)2 (r) =

(1− s2(r)

), r = π(x1 − x2)/D

D2X(4)2 (r) =

(1− s2(2r) +

ds(2r)

dr

∫ r

0

s(2r′)dr′), r = π(x1 − x2)/D . (3.3.7)

3.4 Other observables

In this section we compile definitions and results of some other spectral observables which are relevant in thedescription of fidelity decay.

24 3.4 Other observables

Spectral form–factor

The spectral form factor bβ2(t) will be important in the next chapters in the context with fidelity. It isdefined as the Fourier transform of the energy–energy correlator (3.3.7),

bβ2(t) =

∫ ∞−∞

exp(i2πrt)[1−X(β)2 (r)]dr (3.4.1)

and it is particularly useful for the RMT formulation of fidelity,

b12(t) = t ln(2t+ 1)−

2t− 1, 0 < t < 1 ,1 + t ln(2t− 1), 1 < t ,

b22(t) =

t− 1, 0 < t < 1 ,

0, 1 < t ,

b42(t) =

1− t

2+ t

4ln |t− 1|, 0 < t < 2 ,

0, 2 < t .(3.4.2)

In the symplectic case, b42(t) has an integrable singularity at t = 1, which translates into an oscillatory

behaviour of the energy–energy correlator X(4)2 (r).

For an RMT Hamiltonian H(λ) which depends parametrically on a dimensionless parameter λ other spectralquantities can be considered, and were investigated in the past. In the context with fidelity decay the levelcurvature is particularly important.

Level curvature

For definiteness consider the parametric Hamiltonian H(λ) = cosλH0 + sinλH1, where both H0 and H1 aretaken from the same Gaussian ensemble. Then the level curvature distribution was defined as [191]

Sβ(x) = D

N∑n=1

⟨δ

(x− ∂2xn(λ)

∂λ2

)δ(xn)

⟩(H0,H1)

N,β

, (3.4.3)

where the xn are the eigenvalues of H(λ). The level curvature ∂2xn(λ)/∂λ2|λ=0, respectively the levelcurvature distribution (3.4.3) were studied in Refs. [192, 193, 194] and in Refs. [195, 196] to investigate thetransition from regular to chaotic motion. Exact analytic results were obtained in the large N limit for allthree ensembles [191, 197, 198]. The remarkably simple result is

Sβ(r) ∝ (1 + r2)−(2+β)/2 , (3.4.4)

in terms of the dimensionless curvature

r =xD

πβ〈(dxn(λ)/dλ)2〉(H0,H1). (3.4.5)

The mean square level velocity 〈(dxn(λ)/dλ)2〉(H0,H1) is independent of the index n. The form (3.4.4) hasbeen conjectured correctly by Zakrzewski and Delande in Ref. [199].

4. Fidelity

The notation of fidelity has been introduced by Peres [28] in an attempt to distinguish chaoticity andregularity in quantum systems. He looked for a generic quantum mechanical equivalent to the classicalLyapunov exponents, addressing one central question of quantum chaos, namely the quest for signaturesof chaoticity in quantum systems. The fidelity amplitude f(t) is defined as the overlap integral of a wavefunction, which is propagated in time by a Hamiltonian H0, with the same initial wave function which ispropagated in time by a slightly perturbed Hamiltonian H0 + V

f(t) = 〈ψ|e−i(H0+V )teiH0t|ψ〉 . (4.0.1)

Fidelity itself is the modulus squareF (t) = |f(t)|2 . (4.0.2)

Usually only F (t) is amenable to experiments, whereas often, in particular in the RMT framework, f(t) ismore convenient for calculations. Equivalently the fidelity amplitude can be expressed as the expectationvalue of a unitary operator called echo operator

M(t) = e−i(H0+V )teiH0t , (4.0.3)

which refers to the terminology Loschmidt echo for the quantity described by Eq. (4.0.2). It is used as oftenas fidelity in the literature. This notation alludes to the commonplace picture of an echo. A signal emittedat time t = −t0 evolves freely until time t ≈ 0. Then it is reflected at some boundaries. At a time t ≈ t0one expects an almost perfect image of the initial signal. Instead of boundary reflections one can think of aswitch of the system Hamiltonian H0 → −H0 at time t = 0. The imperfections of the switch are subsumedin the perturbation V . Such switches can indeed be obtained by standard techniques in NMR experiments[200, 201].

Fidelity has been subject of intensive research in the recent years. Apart from the above mentioned signi-ficance for the understanding of the transition from quantum mechanics to classical chaoticity, it is studiedin quantum information theory [202, 203]1. There it has become a standard benchmark for reliability ofquantum information devices [17].

In the above contexts fidelity is a measure of stability of a Hamiltonian H0 against an uncontrolled per-turbation V . Mutatis mutandis fidelity can be used to probe the behaviour of an unknown system by acontrolled perturbation [205, 202]. Recently Kaplan found another interesting application: He used fidelityof eigenstates to test uncertainty of quantisation [206].

Fidelity has been studied for a large variety of different systems, perturbations and initial conditions. Itturned out that it shows a surprisingly rich behaviour depending on the various parameters, which enter inits definition. These are i) the system, ii) the initial state, iii) the structure and strength of the perturbationand iv) different time scales.

i) In most studies the unperturbed system, defined by H0, is chosen as a strongly chaotic quantumsystem, either generic (RMT) [207, 59] or in numerical studies [208, 209, 210, 211] a representativeexample, as a stadium billiard [209] or the quantised kicked top [211, 212]. The opposite regime ofregularity is also addressed but to a lesser extent [212, 213, 207]. Intermediate situations of weakchaoticity or transition ensembles have scarcely been addressed.

ii) When studying fidelity, mainly two types of initial states are considered. A coherent state as the“most classical” quantum state is especially suited as initial wave function (CIS) for semiclassicalapproximations of the echo operator. The opposite to a coherent state is a random initial state,defined as an incoherent mixture of all eigenstates. It is obtained from the definition Eq. (4.0.1) bythe substitution 〈ψ| . . . |ψ〉 → tr (. . .)/N . Random initial states (RIS) are usually considered in thecontext of random matrix approaches, on the one hand because of the nice invariance properties ofthe trace which is indispensable for analytical treatment. On the other hand, arguments were putforward [212, 2, 207, 6] that a random matrix average is always also an average over the initial wavefunction or, in other words, random matrix averages of the fidelity are independent from the initialwave function.

1Definition (4.0.1) is a special case of a more general definition of fidelity used in quantum information theory [204].

There, F =(tr√ρ1ρ2

)2, where ρi are arbitrary density matrices. For pure states ρi = |ψi〉〈ψi|, this reduces to F =

|〈ψ1|ψ2〉|2.

25

26 4.1 Semiclassical Results

iii) Fidelity decay is best studied for chaotic systems and coherent initial conditions. In this case depend-ing on the perturbation strength at least four qualitatively different regimes exist. As perturbationincreases there is a crossover from Gaussian decay to the Fermi golden–rule regime of linear decay tothe Lyapunov regime with perturbation independent decay and finally to the strong coupling regime.For an RIS and in RMT the Lyapunov regime cannot be achieved since in random matrix ensemblesthe Lyapunov exponent is infinity by construction. It is worthwhile noticing that the concept of fidelityis sensible even in the regime V H0. In the limit 〈|V |2〉 /〈|H0|2〉 → ∞ time evolution of the systemH0 is frozen and fidelity becomes equal to the survival probability |〈ψ(t)|ψ(0)〉|2 with respect to thetime evolution |ψ(t)〉 = exp(iV t) |ψ(0)〉.As for the structure of the perturbations it has been pointed out by Prosen and Znidaric [2, 214] thatthe diagonal and the off-diagonal matrix elements of the perturbation in the system’s eigenbasis playrather different roles. They postulated a fidelity freeze for purely off–diagonal perturbations. Thefidelity freeze and related subjects are reviewed in detail in Sec. 5.3.

iv) For chaotic quantum systems two time scales are particularly important: the Ehrenfest time tE andHeisenberg time tH . Ehrenfest time is given by tE ≈ ln(1/~)/λL, where λL is the maximal Lyapunovexponent. It is the time, needed for a minimal wave packet, to spread effectively over the accessiblephase space. For t < tE quantum motion is exponentially unstable like the classical one [213]. AftertE the system starts to “feel” its quantum nature. Heisenberg time is defined as tH = ~/D, where Dis the mean level spacing of the system. The quantum relaxation process takes place during the timet < tH . Typically tE tH . In RMT Ehrenfest time is zero and Heisenberg time is the only time scaleof the system.

4.1 Semiclassical Results

The first estimates of fidelity were given by Peres himself who argued on rather general arguments that thefidelity of a regular system should be larger than that of a classically chaotic system. He calculated fidelityfor a general non–stochastic system Hamiltonian H0 and perturbation V in stationary perturbation theoryand found that for large times fidelity approximately equals the inverse participation ratio (IPR) S =

∑|cn|4

of the initial state |ψ0〉 =∑cn|n〉. In a classically regular system a generic initial state has a large IPR

S . 1. In contrast a generic initial state of a chaotic Hamiltonian has a small IPR. For small times he founda quadratic decay

F ' 1− (∆V )2t2/~ , (∆V )2 = 〈V 2〉 − 〈V 〉2 (4.1.1)

independently whether the system is regular or chaotic. The brackets denote an expectation value of theinitial state |ψ0〉. He concluded that the fidelity of a chaotic quantum system always decays faster than thatof a regular system. It turned out that the real situation is much more complicated. It exhibits a largevariety of phenomena, which are not captured by Peres’ estimate, depending on the parameters specified inthe introductory section of Chap. 4.

Many studies on fidelity have been triggered by the work of Jalabert and Pastawski, who investigated fidelityof a Gaussian wave packet with initial spread σ, momentum p0 and velocity vF in a disordered potentialV (x) [32]. The authors choose a Gaussian disorder potential in d dimensions

V (x) =

N∑α=1

uα(2πξ)d/2

exp

[− 1

2ξ2(x−Xα)2

], 〈uαuβ〉 = u2δαβ (4.1.2)

mainly for mathematical convenience. The positions of the impurities Xα are uniformly distributed in thevolume with density n = N/V . The assumption of finite correlation length of the disorder potential wascrucial for their approach1. Using the semiclassical approximation for the propagator, the fidelity amplitudeis written as

f(t) =

(σ2

π~2

)d/2 ∫dx∑s,s′

√CsCs′ exp

[i

~(Ss − Ss)−

iπ

2(µs − µs)

]

exp

[− σ2

2~2

[(ps − p0)2 + (ps − p0)2]] , (4.1.3)

where S =∫ t

0L(q , q )dt′ is the action along the classical trajectory of the unperturbed (Ss) and the perturbed

(Ss) system. µs is the Maslov index which counts the number of conjugate points along the trajectory s. Cs is

the Jacobian Cs = | detBs| with (Bs)kl = (∂2Ss/∂x(i)k ∂x

(f)l ), where x(i) is the initial point and x(f) is the end

point of the trajectory. The approximate form Eq. (4.1.3) is valid in the regime ξ σ λF , i. e. the initialspread of the wave packet is small compared to the potential fluctuations but large compared to the initialde Broglie wave length of the particle. This condition is essential for the applicability of the semiclassical

1The case of δ–correlated disorder has been addressed in Ref. [215].

4.1 Semiclassical Results 27

perturbation theory used by the authors. For the calculation of fidelity itself a second integration over x′

and paths s′ and s′ is required. Keeping only trajectories s ≈ s and s′ ≈ s′ they obtain

|f(t)|2 =

(σ2

π~2

)d ∫dxdx′

∑s,s′

CsCs′ exp

[i

~(∆Ss −∆Ss′)

]

exp

[−σ

2

~2

[(ps − p0)2 + (ps′ − p0)2]] , (4.1.4)

where ∆Ss = −∫ t

0dt′V [q s(t

′)] is the disorder dependent part of the action. The authors split the ex-pression (4.1.4) into a diagonal and a non–diagonal contribution. The non–diagonal contribution containstrajectories s and s′ exploring different regions in phase space, therefore the disorder average 〈. . .〉dis of |f(t)|2can be taken for each path separately⟨

exp

[i

~∆ (Ss −∆Ss′)

]⟩dis

≈⟨

exp

[i

~∆Ss

]⟩dis

⟨exp

[− i~

∆Ss′]⟩

dis

. (4.1.5)

This leads to an exponential decay with a rate determined by the disorder potential through the elastic meanfree path l = 2

√π~2v2

F ξ/u2n. For the diagonal terms the two trajectories xs(t) and xs′(t) are approximately

the same. The difference of the two actions can be approximated as

∆Ss −∆Ss′ =

∫ t

0

dt′V [xs(t′)](xs(t

′)− xs′(t′))

=

∫ t

0

dt′V [xs(t′)]B−1(t′)

(p s(t

′)− p s′(t′))

=

∫ t

0

dt′V [xs(t′)]B−1(t′)B(t)

(x− x′

)(4.1.6)

using the Jacobian B, for details see [32]. For chaotic systems the time behaviour of B is governed bythe Lyapunov exponents and the authors made the approximation B ' exp(−λLt) where λL is some meanLyapunov exponent of the unperturbed system. This approximation should be valid for times λLt 1. Asa result the diagonal contribution to Eq. (4.1.4) is again an exponential with a decay rate given by the meanLyapunov exponent.

The main result of Ref. [32] can be formulated as follows. The disorder averaged fidelity can be written as a

sum of a diagonal contribution F (d)(t) and of a non–diagonal contribution F (nd)(t)

〈F (t)〉dis = F (nd)(t) + F (d)(t) = e−tvF /l +A

te−λLt , λ−1

L t tE . (4.1.7)

The prefactor of the second contribution is given by A = m/tad/2, with a = (d−1)u2n/ 4λL vF (4πξ2)(d−1)/2.tE is Ehrenfest time. For small perturbations the non–diagonal contribution is dominant, in particular forvanishing perturbation it guarantees normalisation |f(t)|2 = 1. The main conclusion of Eq. (4.1.7) is the exis-tence of an intermediate regime of the perturbation l < vF /λL, and of a time window specified in Eq. (4.1.7),where fidelity is independent of the perturbation strength. The intermediate regime is characterised by twoconditions. On the one hand perturbation is classically weak in the sense that the classical paths are notaffected by V on a length given by the mean free path l = 4(kF ξ)

2l. On the other hand perturbation isquantum mechanically strong vF /λL l. We notice again that in order to reach this limit the correlationlength ξ must be large.

The findings of Jalabert and Pastawski were confirmed by numerical simulations [209, 210, 208]. In [209]it was shown that the non–diagonal contribution of Eq. (4.1.7) is a much more universal result, which canbe related to perturbative random matrix theory. Random matrix theory predicts an exponential decay, seeSec. 4.3,

F (nd) = exp

(−Γt

~

), Γ =

2π〈|V |2〉D

, (4.1.8)

where 〈|V |2〉 is an average over the matrix elements of the perturbation and D is the mean level spacing ofthe unperturbed system. Γ goes by the name Breit–Wigner spreading width in nuclear physics and Thoulessenergy in mesoscopics [153]. In this regime fidelity is the Fourier transform of the local density of statesLDOS [211, 208] which has Lorentzian form in perturbative RMT (see Sec. 4.3). For the specific exampleof a stadium billiard and with the same disorder potential as in Ref. [32], given by Eq. (4.1.2), Cuchiettiet al. [209] could work out 〈|V |2〉 and relate it to the elastic mean free path as Γ = ~vF /l. The regime ofexponential decay of fidelity with the decay rate as given by Eq. (4.1.8) is referred to as “Fermi golden–rule”regime. It has also been addressed in Refs. [211, 216].

28 4.1 Semiclassical Results

Cerruti and Tomsovic [217, 218] investigated fidelity for a generic chaotic quantum system H0 and a pertur-bation which depends parametrically on one parameter λ. In the regime of Gaussian decay

F (t) ≈ exp(−λ2σ2

Et2/~2) (4.1.9)

the decay rate depends on σ2E , which is the variance of the level velocity ∂Ei/∂λ|λ=0 of the eigenvalues Ei

of H(λ). This quantity has been related to the perturbation V within a semiclassical approach [219] by σ2E

≈ 2gK(E)D/(π~β), where 2g/β is the number of classical orbits with the same action. β is the Dyson indexof the universality class, D is the mean level spacing and K(E) is the classical action diffusion constant onthe energy surface E, defined as [220]

K(E) =

∫ ∞0

〈V [p (0), r(0), λ]V [p (t), r(t), λ]〉dt . (4.1.10)

The average in Eq. (4.1.10) is over the primitive periodic orbits of a very large period. The physical picture isthat the action difference of two long orbits which are continuously deformable into each other as a functionof λ is the result of a diffusion process. As the long orbit explores the available phase space randomly,sometimes its action decreases, sometimes it increases as a function of λ. The diffusion constant is the slopeof the energy level.

In the Fermi golden–rule regime F (t) ≈ exp(−λ2σ2

W t/~2)

they find a decay rate which also depends on the

classical action diffusion constant σ2W =2K(E). Thus they could estimate the time scale of the crossover from

Gaussian (quadratic) decay to Fermi golden–rule regime as t = πβtH/g, which is essentially the Heisenbergtime ~/D.

Prosen and Znidaric [221, 222, 212] studied the fidelity amplitude in the semiclassical as well as in theperturbative limit by a different technique. They used a discrete time step decomposition of the echooperator Eq. (4.0.3) into time steps of unit length

M(t) =

t−1∏t′=0

exp

(iδ

~Vt′

)= T exp

(iδ

~

t−1∑t′=0

Vt′

)(4.1.11)

where Vt is the perturbation in the interaction picture and T denotes time ordering. Expanding the exponentin powers of the perturbation strength δ a relation between fidelity and the two–point time correlator of theperturbation was found

F (t) = 1− δ2

2~2

t−1∑t′,t′′=0

T C(t′, t′′) +O(δ3) , C(t, t′) = 〈VtVt′〉 . (4.1.12)

The brackets denote an expectation value with respect to the initial state. The authors analysed thisformula in various regimes for the correlator C(t′, t′). In the ergodic regime the initial state can be takenas an incoherent mixture (RIS) and the correlator C(t, t′) becomes translation invariant. The double sum

in Eq. (4.1.12) simplifies to tC(0) +∑t−1t′=1(t − t′)C(t′). If moreover the correlation function C(t) decays

sufficiently fast on a time scale tmix, one can neglect the second term∑t−1t′=1 t

′C(t′) in the above sum andobtains a linear time dependence for the fidelity

F (t) = 1− δ2

~2σ0t+O(δ3) , σ0 =

1

2C(0) +

∞∑t=0

C(t) . (4.1.13)

The quantity σ0 can be interpreted as a type of action transport coefficient, which has a well defined classicallimit σcl. It plays the role of the action diffusion constant (4.1.10) in the approach of Cerutti and Tomsovic.Higher orders in δ can be be taken into account by exponentiating the l. h. s. of Eq. (4.1.13). The authorsargued that the assumption of ergodicity is usually justified for globally chaotic classical motion.

The other regime considered by the authors corresponds to integrable or quasi–integrable quantum dynamics,characterised by a non–vanishing time average of the correlation function Eq. (4.1.12). Due to the non–ergodicdynamics the non–vanishing time average

C = limt→∞

1

t2

t−1∑t′,t′′=0

T C(t′, t′′) (4.1.14)

also depends on the choice of the initial state. Assuming that there exists a time scale tav after which the limiton the l. h. s. of Eq. (4.1.14) is reached, Eq. (4.1.12) yields for non–ergodic dynamics the simple quadraticdecay for the fidelity amplitude

F (t) = 1− δ2

~2Ct2 +O(δ3) . (4.1.15)

4.2 Random Matrix Formulation of Fidelity 29

A general formula for was also given [212]

f(t) = 〈exp(iδV /~)〉 , V = limt→∞

1

t

∑Vt′ (4.1.16)

in terms of the time averaged operator V .

For random initial states the authors argued that the time averaged operator C reaches a classical (~–independent) limit Ccl. Comparing Eq. (4.1.15) with Eq. (4.1.13) they could give an estimate for theperturbation strength δ < ~Ccl/σcl for which fidelity decays faster for a non–ergodic system than for anergodic system. Notice that this is the opposite of Peres’ prediction, which however is only valid for a highIPR of the initial state. However, a similar estimate was also given for a coherent initial state. In that casethe regime of perturbation strength, where fidelity of a non–ergodic system decays faster than fidelity of aergodic system, is much harder to reach.

A major virtue of the work of Prosen and Znidaric was that it expressed the purely quantum mechanicalquantity “fidelity” in terms of averaged correlation functions σ0 and C which are classical quantities or, moreexactly, which have well defined classical limits.

However, the authors pointed out that quantum dynamics of a finite and bound system has always a discretespectrum, whereas quantum ergodicity requires an infinite Hilbert space dimension [223]. Semiclassicallythe dimension of the relevant Hilbert space is given by N = V/(2π~)d, where V is the classical phase–spacevolume and d is the number of degrees of freedom. Therefore true quantum ergodicity is obtained either inthe limit ~→ 0 or in the thermodynamical limit d→∞. The authors gave estimates of 1/N corrections ofthe above findings.

There were also efforts to construct a classical analogue to fidelity [213, 221] as an overlap integral in classicalphase space of two phase–space densities propagated by two slightly different Hamiltonian flows from thesame initial phase–space density, for details see Ref. [221].

4.2 Random Matrix Formulation of Fidelity

For the understanding of the role of RMT in the quantitative and qualitative description of fidelity decayin chaotic quantum systems the equivalence between chaotic quantum systems and random matrices, asformulated in the conjecture by Bohigas, Gianonni and Schmit [144, 145] is particularly important. Despiteits remarkable success in the description of chaotic quantum systems it is a priori not clear that a randommatrix formulation for fidelity decay is useful. It was pointed out in Sec. 3 that RMT emulates the spectralproperties of real systems on the unfolded scale. However, in real systems in general fidelity depends also onthe spectral density of the unperturbed and of the perturbed system. This dependence is non–universal inthe sense that it cannot be captured by an RMT model. Any comparison of experimental or numerical datawith RMT results should therefore take place after unfolding of the RMT model as well as of the data. Onthe theoretical side a correct unfolding is not always easy to achieve.

It was already pointed out in the introduction that a random matrix formulation of fidelity involves twoensemble averages, an average over the unperturbed matrix H0 as well as over the perturbation V . They arechosen from two independent random matrix ensembles [59]. Moreover, in the RMT approach it is naturalto study fidelity and fidelity amplitude for random initial conditions, i. e. for an appropriate average overthe system eigenstates. Choosing the initial state from an ensemble of equally distributed eigenstates witha random phase, |ψ〉 =

∑cn|n〉 with cnc∗m = δnm/N

2 the expressions become

〈F (t)〉 =1

N2

⟨tr e−2πi(H0+V )te2πiH0ttr e−2πiH0te2πi(H0+V )t

⟩(H0,V )

(4.2.1)

for the fidelity and

〈f(t)〉 =1

N

⟨tr e−2πi(H0+V )te2πiH0t

⟩(H0,V )

(4.2.2)

for the fidelity amplitude. The factor 2π is introduced in the definition for mathematical convenience (h = 1instead of ~ = 1). The brackets denote an ensemble average as defined in Eq. (3.0.5)2 over the ensembles H0

and V . Technically it is often easier to consider the Fourier transforms of 〈f(t)〉 and 〈F (t)〉, respectively

〈r(E1, E2)〉 =1

N

⟨tr Im

1

E−1 −H0 − VIm

1

E+2 −H0

⟩(H0,V )

, (4.2.3)

and

〈R(E11, E21, E12, E22)〉 =

1

N2

⟨tr Im

1

E−11 −H0 − VIm

1

E+21 −H0

tr Im1

E−12 −H0 − VIm

1

E+22 −H0

⟩(H0,V )

.(4.2.4)

2To simplify notation here and in the following the subindices N, β are suppressed.

30 4.3 Perturbative RMT Results

In practice fidelity itself is of more interest than the fidelity amplitude. However, since the Fourier transformof fidelity is a 4–point average over two RMT ensembles its exact calculation is technically highly demandingand has not yet been achieved, not even in the simplest case of H0 and V being two GUE ensembles.However, there is strong numerical evidence [6, 224] that 〈F (t)〉 ≈ |〈f(t)〉|2 (in words: taking the modulussquare ”commutes” with the ensemble average) in many cases. In these cases it will be sufficient to calculatethe ensemble average of the fidelity amplitude 〈f(t)〉. This assumption is also confirmed by the resultsobtained from perturbation theory [207], see Section 4.3.

Recently, Schafer et al. introduced yet another type of fidelity [225]. Guided by analogies with compoundnuclei [226] the authors considered a scattering system. The idea is to consider instead of the resolvent thescattering matrix [226, 227] in Eq. (4.2.3)

S(0)(E) = 1− 2iπW †(E)G0(E)W (E) , G−10 = E −H0 −W (E)W †(E) (4.2.5)

where for n open channels W (E) is an n × N matrix containing the couplings of the system to the openchannels. H0 is chosen from a random matrix ensemble. A second scattering matrix with a slightly differentsystem part H0 + λV is introduced correspondingly

S(λ)(E) = 1− 2iπW †(E)Gλ(E)W (E) , G−1λ = E −H0 − λV −W (E)W †(E) . (4.2.6)

The authors defined the scattering fidelity amplitude as

f(λ)ab (t) =

〈S(0)∗ab (t)S

(λ)ab (t)〉(H0,V )√

〈|S(0)ab (t)|2〉(H0,V )〈|S(λ)

ab (t)|2〉(H0,V )

, (4.2.7)

where the hat symbol denotes the Fourier transform. The authors showed that for small perturbations and

for random initial states the scattering fidelity amplitude for any scattering matrix element S(0)ab , S

(λ)ab reduces

to the fidelity amplitude Eq. (4.0.1) of the closed system.

4.3 Perturbative RMT Results

The simple, general arguments of Peres predict a quadratic (Gaussian) decay of fidelity. This can easilybe understood within RMT. The Gaussian decay for small perturbations is caused by the diagonal part ofthe perturbation. For small perturbations V can be truncated to Vdiag, its diagonal part in the basis ofeigenfunctions of H0. In this regime the fidelity amplitude reduces to [217, 218]

〈f(t)〉 =1

N

⟨tre2πi(H0+Vdiag)te−2πiH0t

⟩(H0,V )

=1

N

⟨∑n

e2πiVnnt

⟩(V )

= e−t2

2 〈V 2diag〉(V )

, (4.3.1)

where the last equation holds due to the Gaussian nature of the average. This calculation is equivalent tofirst–order perturbation theory. The results can be extended to the Fermi golden–rule regime. This hasbeen done in various occasions [209, 211]. Expanding the advanced Green’s function (E−H0−V + iη)−1 inpowers of the perturbation yields Dyson’s equation

〈G(E+)〉(V ) = G0

⟨1

1− V G0V G0

⟩(V )

= G01

1− 〈V G0V G0〉(V ). (4.3.2)

In the eigenbasis of H0 the averaged Green’s function 〈G(E(+))〉(V ) is diagonal

〈Gnn(E+)〉(V ) =1

E + iη − En − Σn, Σn =

∑m

⟨|Vnm|2

⟩(V )(G0)m . (4.3.3)

The decay rate is defined as the imaginary part of the self energy Σn. It is given by Γn = 2π∑⟨|Vnm|2

⟩δ(E −Em) or after an appropriate average as Γ = 2π

⟨|V |2

⟩/D, where D is the mean level spacing. For the

average echo operator Eq. (4.0.3) one obtains in the eigenbasis of H0

〈Mnn′(t)〉(V ) = δnn′θ(t) exp

(−Γt

~

). (4.3.4)

This is the so called ”Fermi golden–rule” regime of fidelity.

Gorin, Prosen and Seligman derived expressions for the fidelity amplitude as well as for the fidelity ofGaussian RMT ensembles within time dependent second order perturbation theory [207]. They found that

4.3 Perturbative RMT Results 31

RMT is capable of reproducing both the “Fermi golden–rule” regime as well as the Gaussian decay regimein a unified way. They considered the Hamiltonian

H = H0 + λV , (4.3.5)

which depends parametrically on the mean perturbation strength λ. The system H0 is taken from either ofthe three ensembles. The mean level spacing is set to one in the band center and the perturbation is chosenfrom the same ensemble as the system with the variance given by

〈VijVkl〉 =

δikδjl + δilδjk (GOE),δilδjk (GUE),δilδjk − 1

2δikδlj (GSE),

(4.3.6)

where in the GSE case the matrix elements are quaternions. The echo operator Eq. (4.0.3) is expanded inpowers of VI(t), which denotes the perturbation in the interaction picture

MI(t) = 1− 2πiλ

∫ t

0

dt′VI(t′)− 4π2λ2

∫ t

0

∫ t′

0

VI(t′)VI(t

′′)dt′dt′′ . (4.3.7)

The ensemble average is taken first over the perturbation and then over the system. The final result reads

〈f(t)〉 = 1− ε Cβ(t) , with ε = 4π2λ2, (4.3.8)

where Cβ(τ) is given by

Cβ(t) =t2

β+t

2−∫ t

0

∫ τ

0

bβ2(τ ′)dτ ′dτ . (4.3.9)

Time is measured in units of the Heisenberg time and bβ2 is the spectral form–factor of the system HamiltonianH0 given in Eq. (3.4.2). The appearance of bβ2 in Eq. (4.3.9) is not surprising owing to the formal similarityof 〈r(E1, E2)〉 in Eq. (4.2.3) to the energy–energy correlator as defined in Eq. (3.1.1). The double integral inEq. (4.3.9) can be performed in all three cases [207]. For the GOE we state only the leading order terms forlarge and small times compared with Heisenberg–time tH :

C1(t) =

t2 + 1

12ln(2t) + 1

6+O(t−1) , t→∞ ,

12

(t+ t2

)+O(t3) , t→ 0 ,

C2(t) =t

2+

t3

6, t < 1 ,

t2

2+ 1

6− t

2, t > 1 ,

C4(t) =t2

4+

60t−60t2+17t3

144− 2−3t+t3

24ln |1− t|, t < 2 ,

19, t > 2 .

(4.3.10)

The expression for the symplectic case was not given in Ref. [207] but was implicitly used in Ref. [59]. Sincethe result is invariant under an additional averaging over the initial conditions, the authors argued thatthe same result (4.3.8) would have been obtained by averaging over the initial states instead of the systemHamiltonian H0.

The range of validity of the linear response approximation can be somewhat extended by exponentiating Eq.(4.3.8),

〈f(t)〉 = e−ε Cβ(t) , β = 1, 2, 4 . (4.3.11)

The authors argued that the errors of the approximation should be fairly small for λ ∼ 0.1 and negligible forλ ∼ 0.01 (corresponding to ε = 0.4 and 0.004, respectively), which was backed later by the exact analyticresults [59]. From Eq. (4.3.10) the transition from “Fermi golden–rule” regime before Heisenberg time toGaussian for times larger than Heisenberg time can be seen nicely. For t < 1 the linear term dominates,whereas for t > 1 the quadratic term becomes increasingly important. The same analysis yields for aPoissonian ensemble, where no correlations between eigenvalues exist

〈f(t)〉 = e−ε(t2+ t

2 ) . (4.3.12)

Due to the absence of correlations in the spectrum, fidelity decays faster than for chaotic systems. SincePoissonian ensembles describe classically regular systems, this seems to be a contradiction to Peres’ result.However, one has to keep in mind that Eq. (4.3.12) was derived for ergodic (RIS) initial conditions, whereasPeres’ result was derived for an initial state with high IPR. For regular systems with only a small chaoticperturbation (as is assumed in perturbation theory) Peres’ assumption on the initial state seems to be morerealistic.

A similar analysis was also performed for fidelity itself [207]. The result is

〈F (t)〉 = |〈f(t)〉|2 +2

βεt2S +O(t4, ε2) , (4.3.13)

32 4.4 Experiments

where S denotes the inverse participation ratio of the initial state. In the perturbative regime and for aninitial state with a small IPR the second term in Eq. (4.3.13) vanishes and fidelity is just the modulus squareof the fidelity amplitude. In the opposite case, where the initial state is an eigenstate of the unperturbedHamiltonian the IPR is one and the second term in Eq. (4.3.13) exactly cancels the quadratic term of Cβ(t),Eq. (4.3.10). Up to now nothing is known about the range of validity of the perturbative result Eq. (4.3.13).

4.4 Experiments

Experiments exclusively devised to measure fidelity were not performed until recently [228, 229]. However,there exists a series of experiments that were originally devised for other objectives but that were laterrevisited as measurements of fidelity. In particular, two types of experiments have to be mentioned.

i) In spin echo experiments magnetisation is locally injected into a spin Sn at position n of an open or closedspin chain. Afterwards it diffuses into the chain. After a certain time t and a series of radio–frequency(rf) pulses the magnetisation of Sn eventually reaches again a maximum, which can be interpreted as anLoschmidt echo. Such effects have been observed for more than fifty years in the context of refocusing freeinduction decays [230, 231].

For instance, in the experiment considered in Refs. [231, 232] the spin of a rare 13C in a crystal of ferrocene(C5H5)2Fe is injected into the proton by a series of rf pulses. Thereafter the spin diffuses freely among thefive protons in one ferrocene ring. The rest of the crystal is a weakly interacting environment. After timeτ1 the sign of the Hamiltonian is reversed by an appropriate radio–frequency pulse. Afterwards the probeevolves freely again during a time period τ2. At time t = τ1 + τ2 the local magnetisation of the bond protonis transferred back to the 13C by yet another series of (rf) pulses. Then it can be measured. The localpolarisation exhibits a clear maximum at a time τ2 . τ1. The ratio of the initial value to the value of themaximum can be interpreted as fidelity [232].

ii) Recently Lobkis and Weaver reported an elastomechanic experiment [233, 234] on an aluminium block. Thegeometry of the probe was chosen irregular with nonparallel faces and a defocusing hole. The wave equationof elastomechanics allows for three types of solutions i) longitudinal solutions, ii) shear wave solutions andiii) surface Raleigh waves. At the boundary mode conversion occurs such that due to multiple reflection atthe irregular boundaries the wave dynamics becomes chaotic. Such a chaotic sound wave is called ”coda” inacoustics [235]. The authors measured the cross correlation of a signal induced at time t = 0 into the probefor varying temperature T1 and T2

X(ε, t) =

∫∆tST1(τ)ST2(τ [1 + ε])dτ√∫

∆tS2T1

(τ)dτ∫

∆tS2T2

(τ [1 + ε])dτ, (4.4.1)

where the integration is over a time window ∆t centered on t. The change in temperature has two effects onthe system. First, the density of states and therefore the Heisenberg time tH changes with temperature. Itcan be approximately determined by a Weyl expansion [236] of the density of states,

tH ≈ 4πV

(2

c3S+

1

c3L

)ω2 + (surface term)ω + . . . . (4.4.2)

V is the volume of the probe and the dots denote terms of smaller powers in frequency ω, containing other,more complex information on the geometry of the probe. The ”trivial” effect, an overall change of Heisenbergtime, was eliminated by varying ε and considering the maximum D(t) = − lnXmax(t). The function D(t)called distortion in the original work was later [237] identified with the single channel scattering fidelity asdefined in Eq. (4.2.7). The ”non–trivial” perturbation is due to the different change with temperature ofthe shear wave velocity cS and the longitudinal wave velocity cL, which cannot be taken into account bya proper rescaling of TH and ultimately prevents D(t) from being unity. The Raleigh wave contributionswere neglected. The results of the experiment were revisited in Ref. [237] as a fidelity measurement. Theexperimental data were compared with the predictions of perturbative RMT as stated in Eqs. (4.3.8) (4.3.9)and (4.3.10) with an appropriately estimated dimensionless perturbation strength. Figure 4.1 shows thatexperimental data is in excellent agreement with RMT prediction.

The scattering fidelity has also been measured in a quasi two dimensional microwave cavity (billiard) bySchafer et al.[225, 238]. Also in these experiments good agreement was found with the predictions of RMT.

Finally we mention that as early as in 1997 Gardiner et al. proposed an experimental configuration for thedirect measurement of fidelity of the ion motion [228] within an ion trap.

4.4 Experiments 33

Figure 4.1: Above: Signals taken in the same time window are compared for different sample temperatures.An offset has been added to aid visibility, taken from Ref. [233]. Middle: Normalised cross correlationfunction X(ε) for the “big block” taken from 2 msec window on wide–band signals at age 40 msec andtemperatures of 33 and 37 C. Mean dilation (shift of Heisenberg time ∆tH/tH) is 1.116 × 10−3, takenfrom Ref. [233]. Below: The distortion of the aluminium block as measured in Ref. [233] compared with therandom matrix prediction Eqs. (4.3.8) and (4.3.9). The thin jagged lines correspond to measurements inthe frequency ranges 200 kHz, 600 kHz, 700 kHz and 800kHz (from bottom to top). The thick lines showεC1(t) according to Eq. (4.3.9). The values of the dimensionless perturbation strength were estimated fromthe material constants and Eq. (4.4.2), such that the fit is parameter free. For the thin lines only the linearterm in Eq. (4.3.9) has been taken into account. Figure is taken from Ref. [237].

34 4.4 Experiments

5. Exact Calculations of Fidelity

In certain situations the ensemble averages in the calculation of the fidelity amplitude can be performedexactly. Thereby, the perturbative results of Sec. 4.3 can be extended to the regime of arbitrary strongperturbation. In [59] the case, where the perturbation and the system Hamiltonian belong to the samesymmetry class, have been calculated for the GUE and the GOE case. In [7] the fidelity freeze has beenanalysed within an exact supersymmetric calculation. Whereas the supersymmetric method as used inRefs. [59, 239] is restricted to the limit of infinite matrix dimension, for some ensembles fidelity can also beinvestigated to a certain extent for finite N . This is the case for a GUE system with a GUE perturbationand for a GUE system with a GOE perturbation [240]. Finite N results for fidelity are important for thegeneral theory of random matrices. They often yield insight into deep mathematical structures of the RMTensembles which get lost in the limit N →∞ [61].

5.1 Supersymmetric technique

The concept of supersymmetry was developed by Wess and Zumino [241] in relativistic field theory. Sincethen the theory of supergroups and superalgebra has become an important discipline in mathematics. Todayone can safely say that theory of superalgebra is developed almost to the same level as the theory of classicalLie algebras. A classification of superalgebras akin to Cartan’s classification of Lie algebras was given byKac [242, 243], see also the work of Parker and Serganova [244, 245]. Also superanalysis has been developedto a high degree of sophistication1. Some basic facts on superalgebras and useful identities are compiled inAppendix B.

In condensed matter theory supersymmetry has been introduced by Efetov [58] as a mathematical tool. Theuse of the method is explained in several good books and review articles [247, 227]. As in field theory, super-symmetry involves a symmetry between bosonic and Fermionic degrees of freedom. However, it is importantto notice that the Fermionic degrees of freedom have no physical meaning. Integrals over anticommuting(Grassmann) variables are only introduced as an extremely useful bookkeeping device to express positivepowers of determinants as Gaussian integrals via the identity

detA = (2π)N∫d[ξ]e−

12ξ†Aξ , d[ξ] =

N∏dξ∗ndξn . (5.1.1)

This equation holds for any matrix A. Here and in the following we apply the standard convention anduse Latin letters for commuting variables and Greek letters for anticommuting variables. Together with theusual expression of the inverse of a determinant as a Gaussian integral over commuting variables we obtainthe important identity ∫

d[z ]e−12z †Az = 1 , d[z ] =

N∏dξ∗ndξndzndz

∗n , (5.1.2)

with the definitions

A =

[A 00 A

], z T = (z1, . . . , zN , ξ1, . . . , ξN ) , (5.1.3)

where now A is a supermatrix and z T is a supervector, see Appendix B. Eq. (5.1.3) is the basis for allapplications of supersymmetry in problems of disorder and chaos. It allows to express the ensemble averageof many observables in terms of a supersymmetric generating function Z(J), defined as an average of ratiosof determinants

Z(J) =

∏Nn=1 det(Hn + Jn)∏Nn=1 det(Hn − Jn)

. (5.1.4)

The form and number of matrices Hn and Jn depends on the specific problem under consideration. Thenumber N defines the order of the generating function. By definition the generating function is normalisedto one at the origin

Z(0) = 1 . (5.1.5)

This automatic normalisation of the generating function is a big advantage to the replica trick and yieldsthe supersymmetric method, if applicable, often preferable to the latter [62, 227]. It can be considered asthe combination of Bosonic (commuting) and Fermionic (anticommuting) replica trick, in such a way that

1An early review on superanalysis and superalgebra was given in Berezins book [246], which unfortunately remainedunfinished.

35

36 5.1 Supersymmetric technique

the ambiguous limit of zero replica Eq.(1.0.4) must not be taken at all2. In contrast, the replica trick hasa wider range of applicability. Until recently, the application of the supersymmetric method seemed to berestricted to Gaussian probability distributions (3.0.2), but in Refs. [248, 249] it was shown that with theappropriate modifications the supersymmetric method can also be applied to a much larger and very generalclass of probability distributions.

In the following, the supersymmetric calculation is explained for the decay of the fidelity amplitude. As arepresentative example the case of symmetry conserving perturbations is taken, i. e. H0 and V belong tothe same RMT ensemble, GOE, GUE or GSE, respectively. To treat the three cases in a unified way weintroduce the parameter

d = 1 GUEd = 2 GOE and GSE.

(5.1.6)

Starting point is the fidelity amplitude for random initial states in its Fourier transformed form 〈r(E1, E2)〉as given in Eq. (4.2.3). As in Sec. 4.3 a perturbation parameter λ is introduced by writing the completeHamiltonian as H = H0 + λV . The calculation of 〈r(E1, E2)〉 is mapped on a supersymmetric problem byusing the generating function

Z(J) =det(E−1 −H0 + J1)

det(E−1 −H0 − J1)

det(E+2 −H0 − λV + J2)

det(E+2 −H0 − λV − J2)

, (5.1.7)

where J1 and J2 are fully occupied source matrices. This expression is identical with the generating functionof the parametric energy–energy correlator [250]. The latter is defined as

K(E1, E2, λ) = 〈tr δ(E1 −H(λ))tr δ(E2 −H(0))〉 , (5.1.8)

where the random matrix H(λ) depends on a parameter λ. The difference between fidelity and the parametricenergy–energy correlator is hidden in the structure of the source terms. The parametric energy–energycorrelator is derived as ∂J1∂J2Z|J=0 and the source terms are scalars.

The connection between the generating function and the Fourier transform of the fidelity amplitude 〈r(E1, E2)〉as defined in Eq. (4.2.3) is established by the formula

tr1

AB=

1

2

∑n,m

∂

∂J1nm

∂

∂J2mn

det(A+ J1)

det(A− J1)

det(B + J2)

det(B − J2)

∣∣∣∣∣J1=J2=0

, (5.1.9)

which holds for any (non–singular) matrix A and B. With formula (5.1.9) the averaged fidelity amplitude isgiven by

〈r(E1, E2)〉 =∑n,m

∂

∂J1nm

∂

∂J2mn〈Z(J)〉(H0,V ) . (5.1.10)

In the supersymmetric approach the energy arguments are lumped into a 4 × 4 diagonal supermatrix forβ = 2 and into a 8× 8 matrix for β = 1, 4

E = 1d ⊗ diag (E+1 , E

+1 , E

−2 , E

−2 ) . (5.1.11)

The generating function written as a Gaussian integral is given by

Z(J) =

∫d[Ψ] exp

(iΨ†L1/2 (E− 14d ⊗H0 + J) L1/2Ψ

)exp

(iλΨ†A(12d ⊗ V )ΨA

), (5.1.12)

where the wavevectors ΨT = (ΨA,ΨR) are 4dN dimensional supervectors. Depending on the sign of theimaginary increments ±iη of the energy arguments one distinguishes between an advanced part ΨA and aretarded part ΨR of the wavefunction. For the GUE the advanced part is given explicitly by ΨT

A = (sA, ξA),where each of the entries sA, ξA itself is an N vector with commuting (sA) or anticommuting (ξA) complexentries. For the GOE the form of the vectors is ΨT

A = (sA, rA, ξA, ξ∗A), where each entry sA, rA, is a vector

with real entries. The correct form of the wavevector for the symplectic ensemble was given by Wegner [251]:ΨTA = (sA, ξA), where sA is an N vector with quaternionic entries and ξA is an N vector with anticommuting

quaternionic entries, for instance

sAk =

[zAk −vAkv∗Ak z∗Ak

], ξAk =

[ζAk ζ∗AkηAk η∗Ak

], (5.1.13)

2Here again the work of Kanzieper and of Splittdorff and Verbaarschot [67, 68, 69] must be mentioned whichdescribes a method how the replica limit can be taken in an unambiguous way in situations where a connection toPainleve equations exists.

5.1 Supersymmetric technique 37

with complex zAk, vAk ζAk, ηAk. An element of the wavevector Ψ carries two indices Ψnµ, where the Latinindex n labels the components in N dimensional ordinary vector space and µ labels the components in 4ddimensional superspace. The notation Ψn denotes the 4d dimensional projection of Ψ onto a fixed n. Viceversa Ψµ denotes the N dimensional projection of Ψ onto a fixed µ. Likewise, the supermatrix J, for instance,carries four indices Jklµν and Jkl denotes the 4d× 4d block for fixed k, l and Jµν denotes the N ×N blockfor fixed µ, ν.

One more general remark on the notation is in order. A general supermatrix is defined as a 2 × 2 blockmatrix with commuting entries in the diagonal blocks and anticommuting entries in the off–diagonal blocks[246], see App. B. This ordering of the blocks is referred to as Boson–Fermion block (BF) notation, whichin many respects is the most natural and most useful way of writing a supermatrix. However, in the presentcontext it is convenient to reorder the blocks such that all entries, associated with the advanced part ofthe Green’s function, appear in the upper left block and the entries associated with the retarded Green’sfunction, appear in the lower right block. This notation is commonly called advanced–retarded block (AR)notation. The notations are equivalent, related by the linear transformation

σ(BF) =

[σBB σBF

σFB σFF

]→ σ(AR) =

[σAA σAR

σRA σRR

]= P−1σ(BF)P , (5.1.14)

where

P =

1d

1d

1d

1d

. (5.1.15)

Both notations will be used in the following. An important quantity entering in Eq. (5.1.12) is the matrixL = L ⊗ 1N , where L = diag (1d,−1d, 12d) in BF notation, which plays the role of a metric in superspace.Its form is dictated by the locus of the imaginary increments of the energy arguments. In order to yield theintegrals over the commuting components of the field Ψ convergent, the metric has to be chosen indefinite inthe Boson–Boson block. Since an integral over Grassmann variables is always convergent, in the Fermion–Fermion block the metric can be chosen as unit matrix [227].

The principal difference of Eq. (5.1.12) to the corresponding generating functions of the energy–energycorrelator or the parametric energy–energy correlator Eq. (5.1.8) is the source matrix J. It is a 4dN ⊗ 4dNmatrix which is fully occupied in ordinary space, i. e. in the variables k, l. In superspace it is given inadvanced–retarded block notation by

Jkl = 12 ⊗ diag (J1kl, J2kl) (GUE)Jkl = 14 ⊗ diag (J1kl, J2kl) (GOE)

Jkl = 12 ⊗ diag

([J1kl11 J1k112

J1kl21 J1kl22

],

[J2kl11 J2k112

J2kl21 J2kl22

])(GSE) .

Jkl has off–diagonal elements in superspace for a GSE ensemble.

In Eq. (5.1.12) the ensemble average over H0 and V can be taken easily with the Gaussian probabilitymeasure Eq. (3.0.2). As in Sec. 4.3, the variance is chosen such that the density of states is one in the centerof the band and the variance of the perturbation is given by Eq. (4.3.6). One finds for the averages in allthree cases

〈. . . 〉(H0) = exp

(− N

βπ2StrB2

), 〈. . . 〉(V ) = exp

(−λ

2

βStr(BRR)2

). (5.1.16)

Most conveniently, the wave function only appears in form of a supermatrix

B = L1/2N∑n

ΨnΨ†nL1/2 . (5.1.17)

This allows to eliminate the quartic terms of the wavefunctions by a Hubbard–Stratonovich transformation

exp

(−1

2StrB2

)∝∫d[σ] exp

(−1

2Strσ2 − iStrσB

). (5.1.18)

The symmetry of the supermatrix B should be reflected in the supermatrix σ. For one thing, the supermatrixσ should be Hermitean in all three cases. For the GUE this is the only constraint. For the GOE and theGSE the matrix has an additional symmetry called orthosymplectic. The explicit form of the supermatricesis given in Appendix B. It is noteworthy that, due to the non–compact metric L, the identity (5.1.18) ismuch less trivial than it appears, because the metric yields the quadratic form in the exponent of Eq. (5.1.18)non–positive definite. The obvious way of proving Eq. (5.1.18) by shifting σ → σ + B under the integral ishindered by the boundary conditions which have to be imposed on the integration domain in order to yieldall integral convergent. A careful analysis of the validity of the Hubbard–Stratonovich transformation forvarious parametrisations [58, 227, 252, 253, 254] of σ has been given recently [255].

38 5.1 Supersymmetric technique

In an intermediate step a second supermatrix is introduced via a Hubbard–Stratonovich transformation forBRR. The integral over this second supermatrix can be done by Gaussian integration. Then 〈r(E1, E2)〉 isgiven by the supersymmetric matrix model

〈r(E1, E2)〉 ∝ 1

N3D∫d[σ] exp

[iΨ†(σ ⊗ 1N + J)Ψ

]exp

(−βπ

2

4NStr(σ + E)2 +

βπ4λ2

4N2κN (λ)Strσ2

RR

)∣∣∣∣J=0

, (5.1.19)

where for all three ensembles

D =∑kl

Str∂

∂J1kl

∂

∂J2lk− Str

∂

∂J1klM

∂

∂J2lkMT . (5.1.20)

In Eqs. (5.1.20) the supermatrix ∂/∂J has been introduced in the self–explaining notation (∂/∂J)µν = ∂/∂Jµνif Jµν 6= 0 and (∂/∂J)µν = 0 if Jµν = 0. Moreover, the matrix M was introduced which swaps the Bosonand the Fermion blocks

M

[σBB σBF

σFB σFF

]MT =

[σFF σFB

σBF σBB

]. (5.1.21)

In Eq. (5.1.19), κN (λ) = 1 + λ2π2/N . In the limit N →∞, κN = 1 since λ is of order O(1). In contrast tothe original work [59], here the source term is still present in the supersymmetric matrix model. While forthe GOE and the GUE this seems to be an unnecessary complication, it is crucial to treat also the symplecticcase [256]. In all three cases the derivatives after the source variables can now be performed using [256]

∑kl

∂2

∂J1klµνJ2lkµ′ν′Sdet(σ ⊗ 1N + J)

∣∣∣∣∣J=0

=∂2

∂σRAµν′∂σARµ′νSdetNσ . (5.1.22)

Using partial integration in Eq. (5.1.19) one finds that the derivatives after the source terms simply yielda preexponential term. The Fourier transform of the Fidelity amplitude can therefore be written as anexpectation value of a preexponential function G(σ) weighted with the exponential of an effective action

〈r(E1, E2)〉 =

∫d[σ]G(σ) exp (−L(σ)) . (5.1.23)

The specific form of G(σ) varies for the three ensembles and can for the GOE and the GUE be found in theoriginal literature [59].

G(σ) =

Str σARPσRAP GOEStr σARPσRAP GUE

Str σARPσRAP − Str σARPMσRAMTP GSE ,

with the matrix P = diag (1d,−1d). The Lagrangian which governs the action is given by

L(σ) = − βπ2

4N |γ|Strσ2 − βN

2|γ|Str ln(σ −E) +βπ4

4N2|γ|Strσ2RR , (5.1.24)

where terms which vanish for N →∞ have been omitted. We recall the definition of γ = 1 for the GOE andfor the GUE and γ = −2 for the GSE. Here we mention that a similar expression to Eq. (5.1.23) and (5.1.24)can also be derived for the fidelity itself. The only change is in the dimensionality of the supermatrix σ. Forthe fidelity as defined in Eq. (4.2.1) the corresponding σ–model would imply an 8× 8 σ–matrix for the GUEand a 16× 16 σ–matrix for the GOE and the GSE.

Eq. (5.1.23) and Eq. (5.1.24) can be evaluated in the limit N → ∞ by a saddle–point approximation. Acomparison with literature [62] shows that the effective Lagrangian Eq. (5.1.24) differs from the effectiveLagrangian of the two–point energy correlator only in the last term in the effective action (5.1.24). Thisterm breaks the symmetry between the advanced and the retarded blocks explicitly. It vanishes for large Nand thus does not contribute to the saddle–point equation, which is therefore the same as for the two–pointenergy correlator [58]

δL(σ)

δσ= 0 ,

π2

Nσ +

N

σ − E= 0 , (5.1.25)

where E = (E1 + E2)/2. The diagonal solutions of Eq. (5.1.25)

σ(D)n =

E

2± iN

π

√1− Eπ

2N, n = 1, 2 (5.1.26)

are purely imaginary for E in the band center. Only those saddle–points can be reached by a contourdeformation in the complex plane which have the imaginary increment on the same side as the imaginary

5.1 Supersymmetric technique 39

increment of the energy. Therefore for the retarded block the negative sign in Eq. (5.1.26) has to be chosenand for advanced block the positive sign.

The unfolding procedure as described in Sec. 3 is not unique, since there exist two different densities ofstate. The unperturbed system H0 has the density of states ρ0(E) =

√1− πE/2N . The perturbed system

H0 + λV has a λ dependent density of states. With the variance chosen as in Eq. (4.3.6) it reads ρλ(E) =√κN (λ)− πE/2N . We recall the definition of κN (λ) = 1 +λ2π2/N as given below Eq. (5.1.19). In principle

the mathematically cleanest way to remove this ambiguity would be to choose the variances of H0 and Vsuch that the mean level spacing is constant for arbitrary λ. In principle this could be accomplished by anappropriate choice of the variance of the perturbation. In the limit N →∞ this procedure is not necessarysince κN (λ) = 1, if λ is of order of the mean level spacing.

Unfolding in the center of the spectrum amounts to setting E1 +E2 = 0 and E1−E2 = πE/2N yielding thediagonal saddle–point solution

σ(D) =iN

πL . (5.1.27)

The σ–matrix is diagonalised by elements of the supergroup U(1, 1|2) for the GUE and by elements ofUOSp(2, 2|4) for the GOE and the GSE, see App. B. Written in the coset decomposition [253, 58, 247, 227]σ reads

σ = T0Rσ(D)R−1T−1

0 , R ∈

[U(1|1)]2 for β = 2[UOSp(2|2)]2 for β = 1, 4.

(5.1.28)

The matrix T0 parametrises the coset U(1, 1|2) /[U(1|1)]2 for the GUE and UOSp(2, 2|4) / [UOSp(2|2)]2 forthe GOE and for the GSE. The diagonal solution of the saddle point equation are invariant under the actionof the subgroup parametrised by R. Hence the integration over R becomes trivial.

The original matrix integral (5.1.23) is an integral over 16 variables for the GUE and over 32 variables forthe GOE and the GSE. By virtue of the saddle–point approximation the number of integration variables isreduced to 8 for the GUE and to 16 for the GOE and the GSE. Eq. (5.1.23) now becomes an integral overthe T0 matrix

〈r(E)〉 ∝∫dµ(T0)G(T0T

†0 ) exp

(−βE

4StrT0T

†0 +

βπ2

4Str(T0T

†0 )RR

), (5.1.29)

where the subscript (RR) denotes the retarded block in the AR notation. Matrix integrals of the form(5.1.29) are called non–linear σ–models due to the non–linear constraint

LT †0 L = T−10 (5.1.30)

for the matrix variable T0. Together with Eq. (5.1.28) it allowed for the replacement

σ → iN

πT0T

†0 (5.1.31)

everywhere in Eq. (5.1.29). The remaining task is to integrate over the coset manifold parametrised by T0.To this end an explicit parametrisation has to be chosen. Here, we limit ourselves to mention the two mostfrequently used parametrisations of Efetov [58, 247, 250] and of Verbaarschot et al. [227, 59, 7] and refer thereader to Efetovs book and to Ref. [227] for the explicit formulas. Using the parametrisation of Ref. [227]the results as stated in Eq. (5.2.1), (5.2.2), and (5.3.12) can be derived.

We finish this section with a general comment on the saddle–point approximation. The choice of the imag-inary increments on opposite sides of the real axis has been most important for the application of thesaddle–point approximation. It enforces the parametrisation of the σ–matrix in terms of non–compact su-permanifolds U(1, 1|2) and UOSp(2, 2|4), which is crucial to obtain correct results [252, 58, 62, 227]. Thechoice of the imaginary increments on the same side of the real axis allows for a parametrisation of the σ–matrix in terms of compact supermanifolds U(2|2) and UOSp(4|4). Within the saddle–point approximationthis parametrisation would have given unavoidably a trivial result, i. e. a result which is independent of theenergy difference E = E1 − E2.

It has to be mentioned that it is possible to obtain correct results also with a compact parametrisation ofthe σ–matrix, if the saddle–point approximation is avoided and the level number N remains finite. Guhr [60]was able to derive the finite N energy–energy correlation function within the supersymmetric approach usinga supersymmetric variant of the Itzykson–Zuber formula [257, 258], and without employing the saddle–pointapproximation [see Eq. (5.4.3) and Theorem 5 in Sec. 5.4 for the Itzykson–Zuber formula in ordinary and insuperspace]. Thereby he proved the above statement for the GUE ensemble. Recently, Fyodorov and Strahov[259] showed that the Itzykson–Zuber group integrals over the compact unitary group U(p+ q) and over thenon–compact unitary group U(p, q) yield equivalent results, for the exact statement see Sec. 5.4. Althoughit was not stated explicitly in Ref. [259] it seems a straightforward observation that the same result holdsalso for the corresponding supergroups U(q, p|m). This can be considered an a posteriori justification of acompact parametrisation for finite N . Guhr himself gave another justification of a compact parametrisation

40 5.2 Results for non symmetry breaking perturbations: Fidelity revival

with a construction involving an Ingham–Siegel integral [248, 249]. For the GOE and the GSE up to nownobody was able to give a direct proof. In the present context it is also possible to apply the graded eigenvaluemethod used by Guhr in Ref. [60]. The corresponding calculation [260] will be given in some detail in Sec. 5.4.

5.2 Results for non symmetry breaking perturbations: Fidelity revival

With the supersymmetric technique outlined in Sec. 5.1 the fidelity amplitude was calculated exactly for theGUE and for the GOE. The final result is an integral for the GUE [59, 239]

〈f(t)〉 =1

t

∫ min(t,1)

0

dy(1 + t− 2y)e−(ε/2)t(1+t−2y) (5.2.1)

and a double integral for the GOE [59, 239]

〈f(t)〉 =

∫ t

max(0,t−1)

du

∫ u

0

vdv√[u2 − v2][(u+ 1)2 − v2]

(t− u)(1− t+ u)

(v2 − t2)2[t(2u+ 1− t) + v2] e− ε2 [t(2u+1−t)+v2] . (5.2.2)

For the GUE the integral can be performed. The result can be written in terms of the universal sine kernels(x) Eq. (3.3.5)

〈f(t)〉 = −1

t

d

dx

e−xt s(ixt) , t ≤ 1e−xts(ix) , t > 1

, x =εt

2. (5.2.3)

For the GSE the result will be given elsewhere [256]. Here t is given in units of Heisenberg time. In Fig. 5.1the graphs for the GOE GUE and GSE are plotted. The most remarkable feature is a revival of the fidelityamplitude at Heisenberg time. This revival is a purely non perturbative effect. It was first reported byStockmann and Schafer [239]. It was also found numerically in a spin–chain model by Pineda et al. [261].The explanation in terms of a Debye–Waller factor is due to Stockmann [239]. The following arguments arein the spirit of Ref. [239].

The neutron or x-ray diffraction patterns in solid state physics are not sharp as one would expect for a rigidlattice but are smeared out by lattice vibrations. In the simplest approximation the decrease of the diffractionmaxima is described by a temperature dependent Debye–Waller factor [262]. The Bragg scattering amplitudeof a lattice

F (k,k′) =1

N

∫dV ρ(r) exp

[i(k− k′)r

](5.2.4)

has a maximum, whenever k− k′ is a multiple of a reciprocal lattice vector. The maximum would be sharpif the scatterers were at fixed positions ρ(r) =

∑δ(r − nR), ni ∈ N. At finite temperatures the scatterers

are moving thermally around their equilibrium position. Within the simplest approximation this uncertaintyis taken into account by a Gaussian distribution of the scatterers

ρ(r) =∑n

(βω2M

2π

)d/2exp

[−ω

2Mβ

2(r− nR)2

], ni ∈ N . (5.2.5)

In Eq. (5.2.5) it was assumed that the scatterers have mass M and are moving in a d dimensional harmonicoscillator potential. The scattering amplitude is thereby reduced by a temperature dependent Debye–Wallerfactor fDW = exp

(−|k− k′|2/2ω2Mβ

).

The survival probability amplitude of a quantum state 〈ψ| exp(2πiHt)|ψ〉 has much in common with thescattering amplitude of a one dimensional crystal. Eigenvalues are identified with the position of the scatterersand time plays the role of the wavevector difference. For an ergodic initial state (RIS) as considered in RMTthe analogy becomes almost perfect

〈ψ| exp(2πiHt)|ψ〉 → 1

Ntr exp(2πiHt) =

1

N

N∑n=1

exp (2πiEnt)

=

∫dEρ(E) exp (2πiEt) . (5.2.6)

If the spectrum were completely rigid En = nD, n ∈ N (harmonic oscillator) the revival at Heisenberg time1/D would be perfect. Spectral rigidity due to the level repulsion is characteristic for general invariant matrixensembles and in particular for Gaussian RMT ensembles. A typical feature is the repulsion of neighboringenergy levels with a characteristic power |Ei − Ei+1|β , due to the Vandermonde determinant in the jointprobability density (3.0.4). Dyson’s index, β = 1, 2, 4 is here again interpreted as an inverse temperature inthe spirit of Dyson’s Brownian motion [140, 141, 142]. The Gaussian symplectic ensemble is the most rigidone, therefore the Debye–Waller factor should be smallest. Vice versa the orthogonal ensemble represents

5.3 Fidelity freeze 41

Figure 5.1: Fidelity amplitude for the GSE (full line) is more pronounced than for the GUE (dashed dottedline). For the GOE, corresponding to the highest temperature the revival peak is least pronounced. GSEcurve are numerical simulations. Taken from Ref. [239].

the lattice with the highest temperature and the revival should be weakest. This is exactly the behaviourseen in Fig. 5.1.

The above explanation suggest that this effect is only seen for very strong perturbations. As was pointed outin the introduction to Chap. 4, only in this limit fidelity is approximately similar to the survival probability.On the other hand fidelity is always a monotonously decreasing function with perturbation strength (actually,the limit of infinite perturbation strength λ→∞ yields an infinitely rapid decay of fidelity on the unfoldedscale). The revival is therefore best seen for large perturbation (when the approximation by the survivalprobability is valid) but for small values of fidelity itself.

Another consequence of the heuristic explanation of fidelity revival by the rigidity of a one–dimensionaleigenvalue lattice is that for the magnitude of the fidelity revival the spectral statistics of the perturbationis important. This is backed by a comparison of the exact RMT results of a GOE with a GOE perturbation,Eq. (5.2.2), with the corresponding results for a GOE with a GUE perturbation [240].

5.3 Fidelity freeze

Of particular importance in the general context of quantum information processing are situations wherethe fidelity stays close to one for a long time. These situations might also be interesting in the context ofclassical multichannel wireless communication [263, 264]. Prosen and Znidaric [2, 214] proposed a class ofperturbations where this scenario can be achieved. They coined the notation fidelity freeze for this situation.The authors made the crucial observation that the roles of the diagonal and the non–diagonal elements ofthe perturbation in the eigenbasis of the system are rather different. This can be most easily seen by thesimple perturbative treatment presented in Eq. (4.3.1), see also [217]. For small perturbations only thediagonal elements of the perturbation in the system eigenbasis contribute to the fidelity decay. However, ageneric perturbation V can always be decomposed into a diagonal part and a residual part which is purely offdiagonal, V = V + Vres with Vresjj = 0. Using the methods developed in Refs. [222, 2] the authors exploredthe consequences of a purely residual perturbation. In Ref. [2] a classically regular system was considered. InRef. [214] the results of Ref. [2] were generalised to general, in particular to chaotic, quantum dynamics. InRef. [214] the perturbation is assumed to be a commutator of some observable with the system HamiltonianV = i

~ [W,H0]−. A commutator of two Hermitean operators being antihermitean, this automatically yields Vresidual. Moreover, it yields matrix elements of V between levels of H0 close to each other small. Assuminga mixing time t1 after which time correlation functions vanish semiclassically, see Sec.4.1 the authors foundthat in the time window t1 ≤ t ≤ t2 fidelity freezes to a plateau value given by

Fplat = 1− δ2

~2

(∆W (0)2 + ∆clW

2) , ∆W (t)2 = 〈W (t)2〉 − 〈W (t)〉2 , (5.3.1)

and ∆clW2 = 〈W 2〉cl − 〈W 〉2cl, where 〈. . .〉cl denotes an appropriate average over a classical set. Here δ is

the perturbation strength as defined in Eq. (4.1.11). For a coherent state as initial state ∆W (0)2 is of order~. Therefore ∆W (0)2 ∆clW

2 can be neglected whereas for a random initial state we have ∆W (0)2 =∆clW

2. Therefore 1−Fplat is systematically twice as high for coherent initial conditions as for random initialconditions. Also an estimate for t2, the time when the plateau ends was given

t2 ≈ min

√tH∆clW 2

σclδ2,

∆clW2

σclδ2

, (5.3.2)

where σcl is the classical limit of the action transport coefficient σ0 defined in Eq. (4.1.13).

Besides the case, when the perturbation can be written as a time derivative (or commutator with the unper-turbed Hamiltonian), one can identify three other different physical situations in which quantum freeze oranomalous slow fidelity decay occurs. These indeed physically most important cases are: (i) a perturbationwhich breaks an antiunitary symmetry (e.g. time-reversal) in an optimal way, meaning that the perturbationanticommutes with the antiunitary symmetry, e.g. switching on the magnetic field. The other two cases cor-respond to a mean field approach in which the diagonal part of the perturbation is moved to the unperturbedHamiltonian. There one can consider unperturbed Hamiltonians, with (ii) and without (iii) an antiunitarysymmetry.

In the RMT approach a random Hamiltonian

H = H0 + λV , Vjj = 0 (5.3.3)

42 5.3 Fidelity freeze

with a residual perturbation V corresponding to a perturbation with vanishing diagonal elements, is con-sidered. All other matrix elements are independent Gaussian variables with variance 〈|Vij |2〉 = 1 − δij .Variance of H0 is taken such that mean level spacing D = 1 in the center of the band. Corresponding to thethree cases discussed above the perturbation is modelled by

(i) an ensemble of purely imaginary antisymmetric matrices. This corresponds to a Zirnbauer–Altlandensemble, related to the symmetric space DB, see App. D. This is the symmetry class of the firstquantised Bogoliubov–de Gennes Hamiltonian [156].

(ii) an ensemble of real symmetric matrices with deleted diagonal and

(iii) an ensemble of Hermitean matrices with deleted diagonal.

A perturbative treatment as outlined in Sec. 4.3 gives the result

〈f (freeze)β (t)〉 = 1− εC(freeze)

β (t)

=t

2−∫ t

0

dt′∫ t′

0

bβ2(t′′)dt′′ , ε = 4π2λ2 (5.3.4)

where bβ2(t) is the spectral form factor of the system H0 as defined in Eq. (3.4.2). The result (5.3.4) dependsonly on the symmetry class of H0 and holds for any of the three perturbations i), ii), iii). In comparisonwith the results from Section 4.3 in Eq. (4.3.10), where the perturbation was in the same universality classas the system, the quadratic time dependence is missing. This is in accordance to the heuristic argument atthe end of Section 4.3 that the quadratic decay of fidelity in time is exclusively due to the diagonal elementsof the perturbation in the eigenbasis of H0. For H0 chosen from the GOE one finds from Eq. (5.3.4) andEqs. (4.3.10)

C(freeze)1 (t) =

ln(2t) + 2

12+O(t−1 ln t) , t > 1 (5.3.5)

This yields a logarithmically slow decay of fidelity

〈f (freeze)1 (t)〉 ≈ 1− ε

12[2 + ln(2t)] +O(λ4t2) t > 1 : GOE. (5.3.6)

For H0 chosen from the GUE, Eq. (5.3.4) and the corresponding formula in Eqs. (4.3.10) yield

C(freeze)2 (t) =

t2− t2

2+ t3

6, t ≤ 1

16, t > 1 .

(5.3.7)

As a result for times longer than Heisenberg time Eq. (5.3.4) predicts the fidelity to freeze on a value

fplateau2 = 1− ε

6: GUE . (5.3.8)

Both results Eq. (5.3.5) and Eq. (5.3.7), follow directly from Eq. (5.3.4). Thus, they are also valid for anyof the three ensembles used for the perturbation, (i), (ii), and (iii). A similar result can also be obtained forthe symplectic ensemble with

C(freeze)4 (t) =

60t−60t2+17t3

144− 2−3t+t3

24ln |1− t| , t ≤ 2

19, t > 2 .

(5.3.9)

and the plateau value

fplateau4 = 1− ε

9: GSE . (5.3.10)

For large times and small perturbations, the fidelity amplitude can be expressed as the Fourier transform ofthe level curvature distribution [33] which was defined in Sec. 3.4. Using the analytic result (3.4.4) one finds[6, 33]

〈f(t)〉 =

τ K1(τ) : GOE(1 + τ) e−τ : GUE

(τ2 + 3τ + 3) e−τ : GSE, τ =

εt

2, (5.3.11)

Where K1(τ) is the modified Bessel function of the second type [165].

One should stress that diagonal elements of the perturbation vanish also in the presence of a discrete orcontinuous unitary symmetry R, of H0, which anti-commutes with V , RV = −V R. However, it turns outthat its effect on fidelity enhancement is less drastic than the predictions of Eqs. (5.3.6) and (5.3.8), becauseof the lack of correlations between different subspectra of H0. As a result, the asymptotic growth of thecorrelation integral is linear C(t) ∝ t, for times before and after the Heisenberg time.

For the case i) of H0 taken from the GOE and a purely imaginary antisymmetric perturbation the averagefidelity can be obtained exactly by supersymmetry techniques in the limit of large dimension N . The

5.3 Fidelity freeze 43

Figure 5.2: Ensemble average of the fidelity amplitude 〈f(t)〉 with H0 taken from the GOE and a purelyimaginary antisymmetric perturbation (solid line, calculated from Eq. (5.3.12)) for different perturbationstrengths ε = 4π2λ2. For comparison the result from the linear response approximation Eq. (4.3.11) (dashedline), and for a GOE perturbation Eq. (5.2.2) (dashed-dotted line) are shown as well. Taken from Ref. [7].

Figure 5.3: Comparison of different approaches for small perturbation on a logarithmic scale. The ana-lytic result (full line) Eq. (5.3.12) is well approximated by the linear response approximation (dashed line)Eq. (5.2.2). Immediately beyond the time, when the linear response formula fails the asymptotic result(long dashed) Eq. (5.3.11) describes fidelity decay well. Results of RMT simulations (points) and for a GOEperturbation (dashed-dotted line) are also plotted. Taken from Ref. [6].

calculation is as outlined in 5.1. However the details, in particular the integration over the saddle–pointmanifold is much more involved than for the case of a pure GOE perturbation [7]. The result is a doubleintegral

〈f(t)〉 = 2

t∫max(0,t−1)

du

u∫0

v dv√[u2 − v2][(u+ 1)2 − v2]

(t− u)(1− t+ u)

(v2 − t2)2

[1 + ε(t2 − v2)][t(2u+ 1− t) + v2]e−ε2

[t(2u+1−t)−v2] . (5.3.12)

Formally there is only a tiny difference to the non–symmetry breaking case Eq. (5.2.2) [59, 239]: an additionalfactor [1+ ε(t2−v2)] in the integrand, and a minus sign with the v2 term in the exponent, where in the GOEcase there is a plus sign. However the physical consequences of the formally tiny difference are dramatic. InFigure 5.2 the fidelity amplitude is plotted for different theoretical approaches. Fidelity is always by ordersof magnitude higher as compared to a perturbation taken from the GOE. As a conclusion fidelity freeze isnot an artefact of perturbation theory but persists in the strong coupling regime ε & 1.

In [6] the exact RMT result Eq. (5.3.12) was compared with numerical simulations on a quantised kickedtop, which has a chaotic classical limit [265]. The one step Floquet propagator was chosen as

U = U0Uλ , U0 = P12 e−iγS2P

12 , Uλ = e−iλS1 , γ = π/2.4 , (5.3.13)

with P = e−iαS23/2S−iS3 . S1,2,3 are standard spin operators. U0 is time-reversal invariant, and the per-

turbation S1 is antisymmetric in the eigenbasis of U0. The “symmetrisation” of U0 is essential for V toanticommute with the time-reversal symmetry. The spin is S = 200. A fixed initial state is chosen randomlyand the fidelity is averaged over 400 realizations of the propagator U where for each realization a parameterα from a Gaussian distribution of width 1 centered around 30 is drawn. In this regime the classical kickedtop is fully chaotic [266].

The results of fidelity decay for different strengths of perturbation are shown in Fig. 5.4. The perfectagreement with the square of the theoretical result (5.3.12) for the fidelity amplitude is striking for tworeasons. First no fit parameters have been adjusted in Fig. 5.4. The dimensionless perturbation strength εin Eq. (5.3.12) is obtained as ε = 2Nσcl(Sλ)2 = 4λ2S3σcl, where σcl = 0.153 is the integral of the classicalcorrelation function, see Eq. (4.1.13) and comment thereafter, calculated using the corresponding classicalmap, see Ref. [222] for more details. Heisenberg time is tH = 2S.

Second in Fig. 5.4 the square root of the theoretical result for the fidelity amplitude is plotted against thefidelity itself of the quantum kicked top. It was pointed out at the end of Sec. 4.3 that for small perturbationfidelity is well approximated by the square root of the fidelity amplitude. However there is no obvious reasonwhy this approximation should hold for large perturbation strength. Therefore it comes as a surprise thatthere are no significant differences in the curves of the numerical result and the analytical result even forstrong perturbations ε = 4.9.

In [6] also the cases ii) and iii) were treated numerically and compared to the fidelity of a quantised kickedtop. In these cases rotational symmetry of H0 is broken by the perturbation. The exact calculation of fidelityfor these cases poses a theoretical challenge for the supersymmetric method.

Figure 5.4: Fidelity freeze for a quantised kicked top. The perturbation is chosen imaginary antisymmetric,whereas the system has time–reversal invariance (GOE) symmetry. Dashed lines give the numerical simu-lations, solid lines give the square of the exact result Eq. (5.3.12) for the fidelity amplitude. Taken fromRef. [6].

44 5.4 Time reversal invariance breaking: finite N results

5.4 Time reversal invariance breaking: finite N results

The GUE has a higher symmetry than the GOE and the GSE. It is invariant under the full unitary group,whereas the GOE and the GSE are invariant “only” under the action of subgroups of the unitary group.This fact allows in many cases for a calculation of spectral quantities for the GUE, whose calculation seemshopeless if the GOE or GSE would be considered instead. Therefore it should be possible to derive exactresults for the fidelity amplitude in cases where a GUE ensemble either as system H0 or as perturbation V isinvolved. Up to now only one case, namely the case where the system Hamiltonian is chosen from a GOE andthe perturbation is taken from the GUE was considered [240]. This ensemble describes the situation wheretime reversal invariance of the system is broken for instance by an external magnetic field. This transitionhas been studied first by Mehta and Pandey [267] who skillfully derived all energy correlation functions ofthe ensemble for finite N .

Starting point is the expression for the fidelity amplitude Eq. (4.2.2) for a parametric Hamiltonian H =H0 + λV . After a shift and transforming H0 → U−1

1 xU1 and V → U−12 aU2 in angle eigenvalue coordinates

the smaller group U1 ∈ O(N) is absorbed into U2 ∈ U(N) by invariance of the Haar measure. The averagedfidelity amplitude Eq. (4.2.2) becomes

f(t) ∝∫d[a]d[x]dµ(U2)|∆N (a)|∆2

N (x)P1(a/λ)

P2([a− U†2xU2]/λ)tr

(eixtU2e

−iat/√

1−λ2U†2

), (5.4.1)

where the probability densities P2 and P1 of Eq. (3.0.2) have been used. This expression involves a non–trivialintegration over the unitary group U(N). Now the peculiarity of the GUE and the unitary group comes intoplay, which allows for a closed solution of the integral. As early as in 1956 Harish–Chandra derived thefollowing theorem [257, 268]

Theorem 1 (Harish–Chandra) Let G be a compact semi-simple group and g, g′ elements of its maximallycommuting (Cartan) subalgebra g0, [269], then∫

U∈Gexp

(trU−1gUg′

)dµ(U) =

1

|W |∑s∈W

1

π(g)π(s(g′))exp

(tr s(g)g′

), (5.4.2)

where π(g) is defined as the product of all positive roots of H0 and W is the Weyl reflection group of G with|W | elements.

For G = U(N) the Cartan subalgebra consists of diagonal matrices and the product over the positive rootsyields the Vandermonde determinant. As a result the celebrated Itzykson–Zuber formula [257, 258, 270] isobtained:

Theorem 2 (Harish–Chandra–Itzykson–Zuber formula) For two diagonal matrices x and k∫U∈U(N)

exp(itrU−1xUk

)dµ(U)∫

U∈U(N)dµ(U)

=

N−1∏j=0

j!detNnm

[eiknxm

]∆N (x)∆N (k)

. (5.4.3)

Remarkably, a similar result holds also for the group integral over the pseudo–unitary group U(p, q) [268, 181].

Theorem 3 (Harish–Chandra, Fyodorov, Strahov) Define x = diag (x11, . . . , xp1, x12 . . . , xq2) and k =diag (k11, . . . , kp1, k12 . . . , kq2), then∫

U∈U(p,q)

exp(trU−1xUk

)dµ(U) = const.

detpnm[eikn1xm1

]detqnm

[eikn2xm2

]∆N (x)∆N (k)

, (5.4.4)

where ∆N (x) is the usual Vandermonde determinant

∆N (x) =

p∏n<m

(xn1 − xm1)

q∏n<m

(xn2 − xm2)

p,q∏n,m

(xn1 − xm2) . (5.4.5)

The integral on the left hand side of Eq. (5.4.3) over an arbitrary group or symmetric space with diagonal xand k is a symmetric function in two sets of arguments x1, . . . , xN and k1, . . . , kN . It is called matrix Bessel

function Φ(β)N (x, k), see Chap. 6.

For O(N) or USp(2N) diagonal matrices do not belong to the algebra of the group. Therefore the orthogonalor the unitary–symplectic group are not included in the Harish–Chandra theorem. For this reason GOE andGSE averages are in almost all applications much harder to obtain than GUE averages.

5.4 Time reversal invariance breaking: finite N results 45

The group integral in Eq. (5.4.1) is also over the unitary group but it is not of the Itzykson–Zuber type. It hasfour diagonal matrices as argument: two diagonal matrices appear in the exponent as in the Itzykson–Zuberintegral, the other two diagonal matrixes appear in a preexponential term. The integration is non–trivial,but in consideration of the Itzykson–Zuber formula it comes not as a surprise that the result can be expressedin a compact form.

Theorem 4 Define the four diagonal N ×N matrices x, k, s, r. Then the following integral formula holds∫U∈U(N)

dµ(U)tr(sUrU†

)eitr(xUkU

†) =

BN∆N (x)∆N (k)

N∑i,j

risjei(N−2)xjkidetNm 6=j,n 6=i [gji(xm, kn)] , (5.4.6)

where the function gij is given by

gji(xn, km) = eikmxn+ixjki − eikmxn+ixjki − eikmxj+ikixn(xn − xj)(km − ki)

= −ieikmxj+ikixn∫ xn−xj

0

t(km − ki)ei(km−ki)tdt , m 6= i , n 6= j . (5.4.7)

BN is a normalisation constant.

The theorem is proved most conveniently by using the recursion formula for matrix Bessel functions [3]. Theproof is given in App.C. Using this integration theorem the group integral in Eq. (5.4.1) can be performedin one step. The fidelity is written as

fλ(t) ∝∫dzdbGN (z, b) (5.4.8)

exp

[i√

2v2λ2t

(z − b√

1− λ2

)− b2

4λ2v2(1− λ2)− z2

4λ2v2+

zb

2λ2v2

], (5.4.9)

where GN (z, b) is still a 2(N − 1) fold integral over the two sets of eigenvalues an and xn. It reads

GN (z, b) =

∫da1 . . . daN−1dx1 . . . dxN−1∆N−1(x)e−

12

∑x2m

sgn(∆N−1(a)) detN−1nm [h(an, xm, z, b)] (5.4.10)

h(an, xm, z, b) = e−xm(z−b)∫ xm

0

dtt|an|e− a2n

2(1−λ2)−an

(b

1−λ2−z−t

). (5.4.11)

The integration over the set of eigenvalues an can be performed using the method of integration over alternate

variables [61]. The result is a Pfaffian form of a skewsymmetric matrix H with entries

H(xn, xm, z, b) =

∫ds [h(s, xn, z, b)H(s, xm, z, b)− h(s, xm, z, b)H(s, xn, z, b)]

H(s, xn, z, b) =

∫ s

h(s′, xn, z, b)ds′ . (5.4.12)

Expanding the Vandermonde determinant in Eq. (5.4.11) it is easily seen that GN (z, b) = det1/2[η] is alsothe Pfaffian of the skewsymmetric (N − 1) × (N − 1) matrix η. The entries of η are given as a threefoldintegral

ηnm(z, b) =

∫dxdydsφn(x)φm(y) [h(s, x, z, b)H(s, y, z, b)− h(s, y, z, b)H(s, x, z, b)] . (5.4.13)

The integrals can be performed. But we do not state here the lengthy result, which contains a vast amount ofHermite polynomials of the first and of the second kind [271]. Up to now it was impossible to take the limitN → ∞ in the above expression. The reason for this lies in the complicated function h in Eq. (5.4.11). Itobviates the use of the Mehta–Mahoux theorem, which allows for the recursive integration of determinantalfunctions under certain conditions, [61]. Put another way, the matrix η is not separable, i. e. it can not bewritten as a dyadic η = qq† of a vector q with quaternionic entries.

46 5.5 Finite N results: Graded Eigenvalue method

5.5 Finite N results: Graded Eigenvalue method

It was mentioned at the end of Sec. 5.1 that for the GUE it is also possible to do a finite N calculation withinthe supersymmetric approach, i. e. avoiding the saddle–point approximation described in Sec. 5.1. The reasonwhy this calculation is feasible for the GUE but up to now not for the GOE and the GSE is very similar tothe arguments given in Sec. 5.4. There it was argued that the group integral Eq. (5.4.3) has a closed solutionfor an integration over the unitary group due to Harish–Chandra’s theorem (5.4.2) but not for an integrationover the orthogonal or the unitary symplectic group. It was proved in Refs. [60, 272, 273] for the unitarysupergroup U(n|m) and in Ref. [13] for the manifold UOSp(k1|k2) that a result similar to Harish–Chandra’sresult holds also for supergroups. For the unitary supergroup it states the following [60, 272].

Theorem 5 (Supersymmetric Itzykson–Zuber formula) Let r = diag (r11, . . . , r1n , ir21, . . . ir2m) ands = diag (s11, . . . , s1n, is21, . . . is2m) be diagonal supermatrices, then∫

U∈U(n|m)

exp(iStrU−1rUs

)=

1

B(s)B(r)detnjk[exp(ir1js1k)]detmjk[exp(ir2js2k)] , (5.5.1)

where B(s) is the Berezinian, which in the case that m = n is a Cauchy determinant [274]

B(s) =

∏i<k(s1i − s1k)

∏i<k(is2i − is2k)∏

i,j(s1i − is2j)= detnij

[1

s1i − is2j

]. (5.5.2)

However, in complete analogy to the situation in ordinary space, the structure of the integrals involved in thesupersymmetric approach is such that only for the calculation of the fidelity for the GUE the supersymmetricversion of Harish–Chandra’s result can be employed. Thereby a finite N expression in terms of a twofoldintegral is found. How this is done is presented now in some detail [260].

Starting point is the Fourier transform of the fidelity amplitude Eq.(4.2.3) 〈r(E1, E2)〉. After averaging overtwo GUE ensembles and a twofold Hubbard Stratonovich transformation, see [59] for details, it can be writtenas

〈r(E1, E2)〉 ∝ 1

N

∫d[σ]d[ρ] Str(σRAPσARP )

Sdet−N (σ + κ−E±) exp

(−Strσ2 − 1

λ2Str κ2

). (5.5.3)

In contrast to Sec. 5.1 the variances of the system H0 and of the perturbation V are both set to one. InEq. (5.5.3) the matrix σ is a Hermitean 4×4 supermatrix in AR notation Eq. (5.1.14). The second matrix κ isnon zero only in the AA block σ1 = diag (κ, 02), where κ is a Hermitean 2× 2 supermatrix, P = diag (1,−1)and E± = diag (E+

1 , E+1 , E

−2 , E

−2 ). The first term in the integrand is conveniently expressed in terms of

derivatives of exp(StrσJ) after a source matrix J with entries only in the off–diagonal blocks JAR and JRA.After a series of shifts σ → σ + E− κ, and κ→ κ+ E1 one arrives at

〈r(E1, E2)〉 ∝ 1

NStr

(∂

∂JARP

∂

∂JRAP

)∫d[σ]d[κ] Sdet−Nσ±

exp

(−Str (σ − E)2 − 1

λ2Str (κ+ E1)2 +

1

4StrJ2

)∣∣∣∣J=0

, (5.5.4)

with the 4× 4 matrix

E =

[κ JAR/2

JRA/2 −E2

], (5.5.5)

in the AR notation. After these preparatory steps 〈r(E1, E2)〉 is now in the suited form for the main stepof the graded eigenvalue method which is to use the integral theorem Eq. (5.5.1). A transformation inangle eigenvalue coordinates σ → U−1sU , with U ∈ U(2|2) and κ → U−1

A kUA, UA ∈ U(1|1), and diagonalsupermatrices k = diag (k1, ik2) and s = diag (s11, s12, is21, is22) yields

〈r(E1, E2)〉 ∝ 1

NStr

(∂

∂JARP

∂

∂JRAP

)∫d[s]d[k]

B2(s)B2(k)dµ(U)dµ(UA) Sdet−Ns±

exp

(−Str

(s2 + E2)− 2StrU−1sUE − 1

λ2Str (k + E1)2 +

1

4StrJ2

)∣∣∣∣J=0

. (5.5.6)

5.6 Resume II 47

Now the integral over dµ(U) can be performed in one step by using the integral theorem Eq. (5.5.1)

〈r(E1, E2)〉 ∝ 1

NStr

(∂

∂JARP

∂

∂JRAP

)∫dµ(UA)d[k]B2(k)

exp

(− 1

λ2Str (k + E1)2 +

1

4StrJ2

)1

B(ε)deti,j=1,2

[LN (ε1i, ε2j)

]∣∣∣∣J=0

LN (ε1i, ε2j) =

∫ds1ds2

s1 − is2

(is2)N

(s1 ± iη)Nexp

(− (s1 + ε1i)

2 + (is2 + ε2j)2) . (5.5.7)

Here εij , i, j = 1, 2 are the eigenvalues of E , which still depend on the source terms JAR, JRA and on UA. Thederivatives after the source term would be a nuisance, if there were not the inverse Berezinian B−1(ε). Tozeroth order in J , the diagonal matrix ε, which contains the eigenvalues of E reads ε = diag (k1,−E2, ik2,−E2)in BF notation. The degeneracy of a Fermionic and Bosonic eigenvalue is a highly welcome feature in thepresent situation, since it yields immediately B−1(ε)|J=0 = 0. The degeneracy is not lifted but in secondorder in J . Therefore one is allowed to act with the operator Str∂JARP∂JRAP onto B−1(ε) only and to setJ = 0 everywhere else

Str

(∂

∂JARP

∂

∂JRAP

)B−1(ε)

∣∣∣∣J=0

=(k1 − ik2)2

2(E2 + k1)(E2 + ik2). (5.5.8)

Since this result is independent of UA, the integration over UA is trivial. One main virtue of the gradedeigenvalue method is that the determinant structure inherent in the RMT ensembles is not destroyed. The

fourfold integral over the eigenvalues s decomposes into a product of twofold integrals L(ε1i, ε2j). Comparison

with Eq. (3.1.12) reveals that LN is, up to a prefactor, identical with the GUE matrix kernel L(2)N . Collecting

the results one finds

〈r(E1, E2)〉 ∝ 1

N

∫dk1dk2

ik1k2exp

(− 1

λ2Str (k + E1)2

)(L

(2)N (k1, ik2)L

(2)N (E2, E2)− L(2)

N (k1, E2)L(2)N (E2, ik2)

). (5.5.9)

This is a representation of the Fourier transformed fidelity amplitude as a twofold integral. This result isexact for finite N . However, there seems to be no way for further simplification in the general case.

In the scaling limit the GUE kernel becomes the universal sine kernel Eq. (3.3.5). Unfolding by E1 − E2 →Dx, λ → Dλ and rescaling k1 → Dk1, k2 → Dk2, with D = π/2

√N one obtains

〈r(x)〉 ∝∫dk1dk2

ik1k2exp

(− 1

λ2Str (k + x)2

)(sin(k1 − ik2)

k1 − ik2− sin(k1) sin(ik2)

k1ik2

). (5.5.10)

The fidelity amplitude is obtained by Fourier transformation

〈f(t)〉 =

∫dxeitx〈r(x)〉 . (5.5.11)

5.6 Resume II

Fidelity shows a rich behaviour depending on systems, perturbation, initial state, and time. Different ap-proaches for calculation and the regimes of qualitatively different behaviour have been reviewed with a specialemphasis on the random matrix approach to fidelity. The random matrix approach is especially well suitedto reveal the universal features of fidelity of a generic chaotic or regular quantum system. The crossover fromquadratic (Gaussian) decay to linear decay (Fermi golden–rule regime) is captured by the RMT approachin a unifying way. The perturbation independent Lyapunov decay in the semiclassical regime is the onlyphenomenon that is not and cannot be described by the very definition of RMT. Excellent almost astonishingagreement with numerical simulations but also with the few accessible experimental data justify the RMTapproach to fidelity.

RMT is by now the only method to make reliable predictions about fidelity in the strong coupling limit,i. e. in the limit where perturbation strength is of the order or larger than the mean level spacing of thesystem. It predicts new generically non–perturbative phenomena such as a fidelity revival at Heisenberg timeor the fidelity freeze. Fidelity can be used as a benchmark of reliability in quantum information technology.Therefore, in particular the discovery of the fidelity freeze might have impact on the design of quantuminformation devices. It gives a clear criterion of how to design a quantum information device in order toavoid fidelity decay. The diagonal elements of the perturbation in the system’s eigenbasis have to be avoidedat any expense even at the expense of very large off–diagonal elements.

From the mathematical point of view fidelity is a novel observable in random matrix theory. Its relationto other observables in RMT is an interesting question which has to be addressed in more detail. In the

48 5.6 Resume II

landscape of RMT observables it is situated somehow between the spectral form factor bβ2(t), the Fouriertransform of the parametric density–density correlator, Eq. (5.1.8), the Fourier transform of the parametriclevel curvature distribution Eq. (3.4.3) and the variance of the parametric level velocity . Which observableit approaches in which limit is only partially understood.

By applying the supersymmetric technique exact closed expressions could be derived in the limit of infinitematrix dimension. Results were derived for the case that the perturbation belongs to the same symmetryclass but also for one case, where time reversal invariance of the system is broken by the perturbation (albeitin a very special way, such that the orthogonal invariance of the system survives). This case exhibits thefidelity freeze. Other scenarios of fidelity freeze can be treated within perturbation theory but defied exactanalytical treatment. These other cases contemplate a perturbation with deleted diagonal elements in theeigenbasis of the system. Such a perturbation breaks the rotational invariance of the ensemble. Up to nownobody was able to deal with RMT ensembles with time reversal invariance but with broken rotationalsymmetry. Indeed, the problem has much in common with a classical problem of RMT [275, 143]: thetransition from regularity to chaos can be addressed within RMT by considering an ensemble H = H0 + λVwith a regular (Poissonian) H0 and a chaotic part V taken from a GOE. In this case rotational invariance isbroken by adding a diagonal matrix, in the former case by subtracting a diagonal matrix.

Although the treatment of ensembles with broken rotational symmetry within a non–linear σ–model is anextremely interesting challenge, there exist other methods which can be employed alternatively. First, themethod of Mehta and Pandey and second the graded eigenvalue method of Guhr. With both methods, finiteN results could be derived for special cases. At first glance, both methods seem to be restricted to applicationsin connection with the GUE, since both rely on an Itzykson–Zuber integration formula. However, there havebeen recent advances towards an Itzykson–Zuber type formula, which is applicable also for the GOE andthe GSE [12]. These advances are closely related with the theory of matrix Bessel functions, which will bereviewed in the next section.

6. Matrix Bessel functions and exactly solvable systems

In the last two sections of the previous chapter the importance of the Itzykson–Zuber formula for the exactcalculation of RMT averages has been pointed out in various occasions. The fact that a compact formulalike the Itzykson–Zuber formula is not available for the GOE and the GSE makes the treatment of theseensembles more difficult. The investigation of the group integrals over the orthogonal and over the symplecticgroup leads to the theory of harmonic analysis on symmetric spaces.

Symmetric spaces have been classified long ago by Cartan. A coarse classification is given by the root systemsof the classical Lie algebras. Each classical root system itself comprises a whole series of different symmetricspaces, each of them can be further classified by its curvature in a symmetric space of compact, non–compactand flat type. See App. D for introductory material on symmetric spaces.

A symmetric space allows for a set of independent invariant differential operators, which are constructed fromthe Casimir operators of the Lie algebra. The simplest one is the Laplace–Beltrami operator, which is thegeneralisation of the commonplace Laplace operator in Euclidean space. Consider for instance the Laplaceoperator in Euclidean R3. It can canonically be decomposed into a radial part and an angular part (angularmomentum operator). The eigenfunctions of the radial part are (vector) Bessel functions. It turns out thata similar decomposition is possible for all Riemannian symmetric spaces. The eigenfunctions of the radialpart of the Laplace–Beltrami operator are the so called zonal spherical functions [157, 158]. We will also callthem matrix Bessel functions, since they can be considered the matrix generalisation of the vector Besselfunction. They reduce to vector Bessel functions, if matrices (symmetric spaces) of rank one are considered.

As we will see, these functions are also the eigenfunctions of a class of one–dimensional interacting many–body systems, known as Calogero–Moser–Sutherland (CMS) Hamiltonians. In this context the GUE ensemblecorresponds to a system where the interaction vanishes, whereas the GOE and the GSE ensembles correspondto an interacting particle system. This is an appealing and very intuitive explanation why the Gaussianunitary ensembles is so much handier than the other ensembles. In the context with CMS systems Dyson’sindex β defines the the coupling strength of the particle interaction. Therefore it has lost its geometricalmeaning and can consequently take any real positive value.

In this section we review the theory of Matrix Bessel functions. Their connection to exactly solvable manybody Hamiltonians of the (CMS) type is pointed out and different approaches for the construction of theeigenfunctions are reviewed. We limit ourselves to the matrix Bessel function associated with the root spaceAN−1, In principle it should be possible to extend all results, which are presented here, also to the symmetricspaces of the root spaces BN , CN , DN and to the exceptional root spaces. However in many cases this hasnot yet been done. A complete compilation of results for all root systems will be presented elsewhere [276].For convenience of the reader some results of the author’s PhD thesis [277] are included.

6.1 Symmetric spaces with curvature zero

We consider the eigenvalue equation of the Laplace–Beltrami operator in a symmetric space of curvaturezero associated with the root space AN−1. It is nothing, but the plane wave equation in matrix space

∆HeitrKH = −trK2eitrKH , (6.1.1)

where

∆H = tr

(∂

∂H

)2

,

(∂

∂H

)ij

=∂

∂Hij. (6.1.2)

H and K are matrix realizations of a flat Riemannian symmetric space ip as listed in Table D.1, for A, AI,AII, AIII. The metric g is defined by the invariant length tr (dH)2 =

∑dHαg

αβdHβ , where the sum of αand β goes over all independent matrix entries of H. Consider for instance ip of the symmetric space of typeAIII. It is represented by matrices of the form1.

H =

[0p BB† 0q

], B : complex p× q , p+ q = N . (6.1.3)

Any symmetric space allows naturally for a radial decomposition of its coordinates by H → U−1xU , whereU is the compact subgroup as given in the sixth column of Table D.1 in App. D, and x contains the radial

1The usual d dimensional vector plane wave equation is given by the symmetric spaces BDI in Table D.1 of the Liealgebra so(d+ 1).

49

50 6.1 Symmetric spaces with curvature zero

variables. The number of radial coordinates equals the rank of the symmetric space given in the last columnof Table D.1. The eigenvalue equation (6.1.1) reads in radial coordinates

(∆x + LU ) exp(trUxU−1k

)= −tr k2 exp

(trUxU−1k

). (6.1.4)

The diagonal matrix k contains the radial coordinates of K. For the symmetric spaces A, AI, AII and AIIIthe radial operator ∆x is given by

∆x =

N∑n=1

1

|∆N (x)|2∂

∂xn|∆N (x)|2 ∂

∂xn, for A,

N∑n=1

1

|∆N (x)|∂

∂xn|∆N (x)| ∂

∂xn, for AI,

N∑n=1

1

|∆N (x)|4∂

∂xn|∆N (x)|4 ∂

∂xn, for AII,

p∑n=1

1

∆2p(x2)

∏pi xi

∂

∂xn∆2p(x

2)

p∏i=1

xi∂

∂xn, p ≤ q , for AIII.

(6.1.5)

For the symmetric space AIII a derivation of the radial operator can be found for instance in the textbookby Hua [278]. The Vandermonde determinant with a certain power is a common feature to all four radialLaplaceans. This power is closely related to Dyson’s index, which itself is defined as the multiplicity of theunderlying root system [279], see App. D. Here it is important to notice that the underlying root system isnot the root system of the Lie algebra (which would be by definition AN−1 in all four cases), but the so calledrestricted root system of the symmetric space. The restricted root system is in some cases equal to the rootsystem of the algebra, in some cases it is different. In the present case for the symmetric spaces A, AI, AIIit is indeed identical with AN−1, but for the symmetric space AIII it is given by Bp if p is odd and by Cp ifp is even. Since here we wish to focus mainly on the root system AN−1, in the sequel we mainly concentrateon the symmetric spaces A, AI, AII, the space AIII being more closely related to the root systems B and C.

The angular operator LU depends on the parameters of the compact subgroup U as given in the sixth columnof Table D.1 of App. D. In general, since the group is compact the metric will be coordinate dependent andLU is a complicated operator. From now on we will focus exclusively on the radial part.

An eigenfunction to the radial part of the Laplace–Beltrami operator is obtained from the matrix plane waveby integrating over the angular part of the plane wave, i. e. over one of the groups O(N), U(N), USp(2N),or SU(q)⊗ SU(p)⊗U(1). The plane wave equation becomes

∆xΦN (x, k) = −tr k2 ΦN (x, k), (6.1.6)

ΦN (x, k) =

1

Vol

∫U∈U(N)

dµ(U) exp(itrUxU−1k

), for A,

1

Vol

∫U∈O(N)

dµ(U) exp(itrUxU−1k

), for AI,

1

Vol

∫U∈USp(2N)

dµ(U) exp(itrUxU−1k

), for AII,

1

Vol

∫U∈SU(p)⊗SU(q)⊗U(1)

dµ(U) exp(itrUxU−1k

), for AIII.

(6.1.7)

The factor Vol−1 guarantees normalisation. These integrals are in general referred to as zonal sphericalfunctions [157, 158]. We observe, see Table D.1, that the symmetric spaces A and AIII have Dyson index 2.In these cases the solutions to Eq. (6.1.6) can be given in a closed form. For A the Itzykson–Zuber integralformula Eq. (5.4.3) is recovered. For AIII the solution is given by the Berezin–Karpelevich integral formula[280]

ΦN (x, k) ∝detpij=1[J0(xikj)]

∆p(x2)∆p(k2), for AIII . (6.1.8)

For the other symmetric spaces AI and AII associated with Dyson index 1, 4 no such compact expressionsfor the zonal, spherical functions are available.

6.1.1 Rational CMS model

The first three radial Laplaceans of Eq. (6.1.5) differ only in the power β of the Vandermonde determinant.A natural generalisation consists in considering the eigenvalue equation(

N∑n=1

1

|∆N (x)|β∂

∂xn|∆N (x)|β ∂

∂xn

)Φ

(β)N (x, k) = −

(N∑n=1

k2n

)Φ

(β)N (x, k) , (6.1.9)

6.1.1 Rational CMS model 51

for arbitrary β ∈ R+. For other values than β = 1, 2, 4 the function Φ(β)N (x, k) has no geometrical inter-

pretation, yet. In Eq. (6.1.9), Φ(β)N (x, k) is a function of E ≡

∑Nn=1 k

2n only. Nevertheless, we expect the

eigenfunctions Φ(β)N (x, k) to be a generalisation of the group integrals (6.1.7). Therefore, we are looking

for solutions which depend on a set of N parameters ki, such that the following symmetry and boundaryconditions, which are obvious from the group integrals, are fulfilled:

i) Symmetry under permutations of all xi and all ki:

Φ(β)N (. . . , xi, . . . , xj , . . . , k) = Φ

(β)N (. . . , xj , . . . , xi, . . . , k) ,

Φ(β)N (x, . . . , ki, . . . , kj , . . .) = Φ

(β)N (x, . . . , kj , . . . , ki, . . .) . (6.1.10)

ii) Symmetry under exchange of x and k:

Φ(β)N (x, k) = Φ

(β)N (k, x) . (6.1.11)

iii) Normalization:

Φ(β)N (0, k) = 1 Φ

(β)N (x, 0) = 1 . (6.1.12)

We call these solutions to Eq. (6.1.9) for arbitrary β ∈ R+ also matrix Bessel functions. The connection toone–dimensional interacting particle models of the Calogero–Moser–Sutherland (CMS) type for β > 0 is seenby using the ansatz

Φ(β)N (x, k) =

Ψ(β)N (x, k)

∆β/2N (x)∆

β/2N (k)

. (6.1.13)

The eigenvalue equation (6.1.9) is reduced to a Schrodinger equation(N∑n=1

∂2

∂x2n

− 1

2

∑n<m

β(β − 2)

(xn − xm)2

)Ψ

(β)N (x, k) = −

(N∑n=1

k2n

)Ψ

(β)N (x, k) . (6.1.14)

The N variables xn, n = 1, . . . , N are interpreted as the positions of the particles. The set of N variableskn, n = 1, . . . , N plays the role of a set of quantum numbers. More precisely the kn are the momenta ofthe asymptotically free particles [281]. The specific model Eq. (6.1.14) is called rational CMS model [282] orfree CMS model. The parameter β ≥ 0 measures the strength of the inverse quadratic interaction2, whichcan be attractive β < 2 or repulsive β > 2. The minimal value of the coupling constant −1/4 is obtainedfor β = 1. For β = 2 and β = 0, the model is interaction free. In this case the Fermionic solutions toEq. (6.1.14) are Slater determinants of plane waves. Together with Eq. (6.1.13) the Itzykson–Zuber formulais readily derived. In this case the ansatz (6.1.13) can be considered a separation ansatz for the much morecomplicated Laplace equation (6.1.9). In all known cases such a separation ansatz seems to work only for aDyson index β = 2. The specific model has been first investigated by Sutherland [283, 284].

From condition i) and from Eq. (6.1.13) it can be seen that the wave function Ψ(β)N (x, k) obtains under

particle exchange a complex phase

TnΨ(β)N (x, k) = exp(−iπβ/2)Ψ

(β)N (x, k) , (6.1.15)

where Tn is the transposition operator

TnΨ(β)N (x1, . . . , xn, xn+1, . . . , k) = Ψ

(β)N (x1, . . . , xn+1, xn, . . . , k) . (6.1.16)

Due to the symmetry condition i) and (6.1.13) the behaviour under particle exchange of the wave function is

governed exclusively by the Vandermonde determinant ∆β/2N (x). Due to relation (6.1.15) the model (6.1.14)

is frequently used as paradigm for systems with anionic statistics [285, 286, 287].

As for exact solvability one more remark is in order. By construction it is evident that the interacting many–body Hamiltonian (6.1.14) is exactly solvable for the couplings β = 1, 2, 4. Via Eq. (6.1.13) and Eq. (6.1.9)

a set of solutions Ψ(β)N (x, k) is constructed which is labelled by exactly N independent conserved quantities

(good quantum numbers) ki. This is one definition of exact solvability. Conservation of the asymptoticallyfree momenta is a characteristic feature of exactly solvable interacting many–body problem [281, 288]. Ina multi–particle scattering process only the momenta of the particles are reshuffled but their values remainunchanged, the scattering matrix factorises in two body scattering matrices. Although it is natural to deducefrom exact solvability of the model for β = 1, 2, 4 the exact solvability for arbitrary β such a conclusion isnot allowed. Indeed exact solvability of Eq. (6.1.14) for arbitrary positive β was proved by different methods[289, 281].

The symmetric spaces derived form the root series BN , CN and DN are also related to Schrodinger equations,but with a different interaction [279].

2From the form of the coupling it is clear that we can restrict ourselves to β ≥ 1. Since the solutions, we will obtaincan be extended analytically to β ∈ R+ we will do so, although the physical content of these solutions is questionable[283].

52 6.1 Symmetric spaces with curvature zero

Solution for two particles

For two particles the Schrodinger equation can be solved by a separation ansatz for all β ≥ 0. The resultcan be written in terms of vector Bessel functions Jν(z) as [283, 279]

Φ(β)2 (x, k) = Γ([β + 1]/2) exp

(i(x1 + x2)(k1 + k2)

2

)J(β−1)/2 (z/2)

z(β−1)/2, z = (x1 − x2)(k1 − k2) . (6.1.17)

An even index β corresponds to a Bessel function of half–integer order or, equivalently, to a spherical Besselfunction [165]. The wave function is given by

Ψ(β)2 (x, k) = Γ([β + 1]/2) exp

(i(x1 + x2)(k1 + k2)

2

)√zJ(β−1)/2 (z/2) , z = (x1 − x2)(k1 − k2) ,

∼ Γ([β + 1]/2) exp

(i(x1 + x2)(k1 + k2)

2

)4(z/2)β/2 , z → 0

∼ Γ([β + 1]/2) exp

(i(x1 + x2)(k1 + k2)

2

)cos(z/2− πβ/4) , z →∞ . (6.1.18)

From the asymptotic form the two–body scattering phase shift πβ/2 can be read off.

Recursion formula

A solutions Φ(β)N (x, k) of the eigenvalue equation (6.1.9) can be expressed in terms of a recursion formula [3]

in the particle number. This formula holds for arbitrary positive β

Φ(β)N (x, k) =

∫dµ(x′, x) exp

[i

(N∑n=1

xn −N−1∑n=1

x′n

)kN

]Φ

(β)N−1(x′, k) . (6.1.19)

where Φ(β)N−1(x′, k) is the solution of (6.1.9) for N −1 and k denotes the set kn, n = 1, . . . , (N −1). Moreover

x′ denotes the set of integration variables x′n, n = 1, . . . , (N − 1). The measure is given by

dµ(x′, x) = G(β)N

∆N−1(x′)∏n,m |xn − x

′m|β/2−1

∆N (x)β−1d[x′] . (6.1.20)

We also state the equivalent recursion formula for the eigenfunctions Ψ(β)N (x, k) of the Schrodinger Eq. (6.1.14)

Ψ(β)N (x, k) =

∫dµ(x′, x) exp

[i

(N∑n=1

xn −N−1∑n=1

x′n

)kN

]Ψ

(β)N−1(x′, k) , (6.1.21)

where now Ψ(β)N−1(x′, k) is the solution of the Schrodinger equation for N−1 particles and k denotes the set of

quantum numbers kn, n = 1, . . . , (N − 1). The x′ are again the integration variables x′n, n = 1, . . . , (N − 1).The integration measure is now given by

dµ(x′, x) = G(β)N

N−1∏i=1

(kN − ki)β/2( ∏

n,m |xn − x′m|

∆N (x)∆N−1(x′)

)β/2−1

d[x′] . (6.1.22)

The value of the normalisation constant G(β)N can be found in Ref. [3]. The inequalities

xn ≤ x′n ≤ xn+1 , n = 1, . . . , (N − 1) (6.1.23)

define the domain of integration in both cases. Moreover we have d[x′] =∏Nn=1 dx

′n. For β = 1, 2, 4 the

recursion formula (6.1.19) is equivalent to the group integrals Eq. (6.1.9). The measure, Eq. (6.1.20), togetherwith the integration domain is nothing but a very special parametrisation, a modification [3] of the Gelfand–Tzetlin coordinates [290, 291]. They parametrise the unit sphere, i. e. the coset Uβ(N)/Uβ(N − 1), whereUβ(N) denotes one of the groups U(N), O(N) or USp(2N). For arbitrary β the integral (6.1.21) has notyet found a geometrical interpretation. However the corresponding measure function could be considered as

a β deformation of the unitary unit sphere. It was shown in Ref. [3] that Φ(β)N (x, k) meets with all three

conditions i), ii) , iii) defined in the beginning of the section.

The second form of the recursion formula contains a k dependent factor. This factor was introduced to main-tain the exchange symmetry ii) between x and k. From the viewpoint of a quantum mechanical Schrodingerequation there is no need to keep this symmetry.

6.1.1 Rational CMS model 53

Asymptotic expansion

The commonplace vector Bessel functions Jν(z) can be expressed as an asymptotic series, which proves to bemost convenient in many physical applications in particular in scattering problems. In the celebrated Hankelexpansion of the Bessel function [165] the following ansatz is made

H(ν)± (z) =

exp(±iz)√z/π

w(ν)(z−1) , (6.1.24)

where H± is the Hankel function and w(ν) is a polynomial function with w(ν)(0) = 1. The coefficients aµ inthe expansion

w(ν)(z−1) =

∞∑µ=0

aµz−µ (6.1.25)

can be determined recursively

aµ+1 =1

2i(µ+ 1)

(µ(µ+ 1)− ν2 − 1

4

)aµ . (6.1.26)

From Eq. (6.1.26) it is readily derived that the condition for the asymptotic series to terminate is µ = ν+1/2.This can only be fulfilled for Bessel functions of half integer order ν = 1/2, 3/2 . . .. In that case the asymptoticexpansion terminates after the ν’th step, otherwise it is infinite.

Matrix Bessel functions Φ(β)N (x, k) can in principle be constructed by an asymptotic expansion akin to the

Hankel ansatz for vector Bessel functions for arbitrary positive β. To this end the limit (xi − xj) → ∞,∀xi, xj and (ki − kj) → ∞, ∀ki, kj is considered. For β = 1, 2, 4 this is equivalent to an evaluation of thegroup integral Eq. (6.1.7) in a saddle–point approximation. Following Ref. [3] we make the ansatz

Φ(β)N,ω(x, k) =

exp(i∑Nn=1 xnkω(n)

)|∆N (x)∆N (k)|β/2

W(β)N,ω(x, k) , (6.1.27)

where ω is an element of the permutation group SN . The full solution Φ(β)N (x, k) is obtained as the linear

combination of the functions (6.1.27)

Φ(β)N (x, k) =

1

N !

∑ω∈SN

(−1)π(ω)Φ(β)N,ω(x, k) . (6.1.28)

Here, π(ω) is the parity of the permutation. W(β)N,ω(x, k) solves the differential equation

Lx,ω(k) W(β)N,ω(x, k) = 0 , (6.1.29)

where the operator is given by

Lx,ω(k) =

N∑n=1

∂2

∂x2n

+ 2i

N∑n=1

kω(n)∂

∂xn− β

(β

2− 1

) ∑n<m

1

(xn − xm)2. (6.1.30)

In [3] it was shown that Eq. (6.1.30) also hold if x and ω(k) are interchanged. Introducing the scaling variable

zω(nm) = (kω(n) − kω(m))(xn − xm) , (6.1.31)

an asymptotic expansion of W(β)N,ω(zω(nm)) can be put forward by the ansatz

W(β)N,ω(x, k) =

∑µ

aµ12µ13···µ(N−1)N∏n<m z

µnmω(nm)

(6.1.32)

with coefficients aµ12µ13···µ(N−1)N. However, the recursive determination of the coefficients is most cumber-

some and feasible only for small matrix dimension [3]. The leading contribution in the asymptotic expansion

of Φ(β)N (x, k) is given by

Φ(β)N (x, k) ∼ detNnm[eixnkm ]

(∆N (x)∆N (k))β/2

∼ ei∑ki∑xi/N

∑ω∈SN

(−1)π(ω)∏n<m

eizω(nm)/N

(znm)β/2, (6.1.33)

54 6.1 Symmetric spaces with curvature zero

For β = 2 the leading contribution of the asymptotic expansion is already the exact result Eq. (5.4.3). Thisis a special case of a general localisation theorem due to Duistermaat and Heckman [292, 293]. The result forN = 2 particles suggests to identify Dyson’s index with the order of the vector Bessel function as β = 2ν+ 1.Roughly, one can say that matrix Bessel function with even index β correspond to vector Bessel functionswith odd index. Therefore it has been conjectured [3] that for all even β and for all N the asymptotic

expansion of Φ(β)N (x, k) should be finite. This conjecture is confirmed by the exact results obtained for small

matrix dimension, for β = 4 and N = 3, and N = 4 in Ref. [3] and for N = 3 and for arbitrary even β inRef. [294].

Bethe–Ansatz

Sutherland studied the Hamiltonian (6.1.14) with the asymptotic Bethe–ansatz [283, 284]. The essence ofhis idea was that even if Bethe’s ansatz in its original form [295, 296, 297] cannot be applied for particleswith long–range interaction, one can find a Bethe–ansatz solution in the asymptotic regime (xi − xj)→∞,∀xi, xj and ki 6= kj , ∀i 6= j. Writing Eq. (6.1.33) as

ψ(β)N (x, k) =

∑ω∈SN

A(ω) exp(i∑n

kω(n)xn) , (6.1.34)

the coefficients A(ω) are related to the two–body scattering phase–shift θ(k) as

A(. . . , n, n+ 1, . . .)

A(. . . , n+ 1, n, . . .)= − exp(iθ(kn − kn+1)) , (6.1.35)

where θ(k) = πβsgn(k)/2 is obtained from the two–particle problem. Bethe’s ansatz yields for Bosons withδ– interaction a set of equations for ki [298, 299, 296, 297]

kiL = 2πIi −∑j 6=i

θ(ki − kj) , i = 1, . . . , N , Ii ∈ Z . (6.1.36)

In the ground state the Ii are different, adjacent integers centered around zero. Eq. (6.1.36) becomes in thefield–theoretical limit i→ x/L = y a Fredholm integral equation for the density of states ρ(k) = dy/dk.

ρ(k) =1

2π+

∫ k0

−k0dk′ρ(k′)

d

dkθ(k, k′) . (6.1.37)

The integration domain k0 is defined by

n ≡ N

L=

∫ k0

−k0dkρ(k) , ε ≡ E

L=

∫ k0

−k0dkk2ρ(k) . (6.1.38)

The remarkable observation of Sutherland was that this ansatz is exact in the thermodynamical limit [284].With θ′(k) = πβδ(k) one finds for the density of states and for the energy–density

ρ(k) =1

2πβ, ε(n) =

1

3π2β2n3 , (6.1.39)

which for β = 1, 2, 4 agrees with the results obtained from random matrix theory [283].

6.1.2 Schur polynomials, Zonal Polynomials and Jack Polynomials

Vector Bessel functions have a well defined convergent expansion in small arguments. Such an expansion

can also be put forward for Φ(β)N (x, k). This yields an expansion in an infinite sum of symmetric functions

in x and k. How this expansion works is most easily seen for the symmetric space A, where the compactItzykson–Zuber result Eq. (5.4.3) is available. There, the expansion in small arguments consists in expandingall exponentials in the determinant

detNnm[eixnkm ] =∑

n1...nN

(N∏j=1

(ikj)nj

nj !

)|xn1 . . .xnN | , (6.1.40)

where the vector (xk)T = (xk1 , . . . , xkN ) was defined. The determinant on the right hand side vanishes

when two ni’s are equal. The first non–vanishing terms are therefore given by one of the combinationsni = ω(i) − 1, 1 ≤ i ≤ N , where ω is an element of the permutation group SN . Each permutation yieldsjust the Vandermonde determinant ∆N (x) as a common factor. The sum over all permutations is exactlythe Vandermonde determinant ∆N (ik)

∑ω∈SN

(−1)π(ω)N∏i=1

(iki)nω(i) = ∆N (ik) . (6.1.41)

6.1.2 Schur polynomials, Zonal Polynomials and Jack Polynomials 55

Zonal polynomials Schur functions Jack polynomials Young tableau∑i xi

∑i xi

∑i xi [1]∑

i x2i + 2

3

∑i<j xixj

∑i x

2i +

∑i<j xixj

∑i x

2i + 2β

2+β

∑i<j xixj [2]

∑i<j xixj

∑i<j xixj

∑i<j xixj [12]∑

i x3i + 3

4

∑i6=j x

2ixj

+ 12

∑i<j<k xixjxk

∑i x

3i +

∑i 6=j x

2ixj

+∑i<j<k xixjxk

∑i x

3i + 3β

2β+2

∑i6=j x

2ixj +

3β2

(β+2)(β+1)

∑i<j<k xixjxk

[3]

32

∑i 6=j x

2ixj

+ 278

∑i<j<k xixjxk

2∑i 6=j x

2ixj

+ 4∑i<j<k xixjxk

3(β+2)2(β+1)

∑i 6=j x

2ixj +

9β(β+2)

2(β+1)2

∑i<j<k xixjxk

[2, 1]

Table 6.1: Simplest Schur symmetric functions, zonal polynomials, and Jack polynomials and their Youngtableau in ”C” normalisation.

The two Vandermonde determinants cancel exactly the singular denominator in Eq. (5.4.3) and the first term

in a small x expansion of Φ(2)N (x, k) is a constant. The next order term is obtained by choosing the set of

integers as ni = ω(i) − 1, where i ∈ 1, . . . , N − 1, N + 1, which is the only possibility of choice to yield a

linear term. If we carry on this procedure we get an expansion of Φ(2)N (x, k) in terms of Schur polynomials

[300] defined as

C(2)λ (x) =

const.

∆N (x)

∣∣∣∣∣∣∣xN−1+l1

1 xN−2+l21 . . . xlN1

.... . .

xN−1+l1N xN−2+l2

N . . . xlNN

∣∣∣∣∣∣∣ . (6.1.42)

The normalisation constant varies in literature, see below. Schur functions form a basis in the space ofsymmetric functions. They are labelled by a partition λ = (l1, . . . , lm) of length m(λ) and weight l(λ) =∑mi=1 li. If the partition is ordered lexicographically l1 ≥ l2 ≥ . . . ≥ lm it is called a Young tableau. Young

tableaux are represented by m(λ) rows of boxes of length li. Another convenient notation for a Young tableau

is λ = [ln11 ln2

2 . . . lnm(λ)

m(λ) ], where ni denotes the multiplicity of rows with the same length. It is often useful to

introduce the following ordering relation between two Young tableaux λ = (l1, . . . , lm) and κ = (k1, . . . , kn). We say λ > κ if li > ki and lj = kj for 1 ≤ j < i. Examples of the smallest Schur functions and theircorresponding Young tableaux are given in Tab. 6.1. The expansion of the determinant detNnm[eixnkm ] inSchur functions becomes for higher order Young tableau increasingly cumbersome. The result is the following

expansion of Φ(2)N (x, k), respectively of the Itzykson–Zuber formula (5.4.3)

Φ(2)N (x, k) = 1 +

∞∑l=1

∑l(λ)=l,λ

C(2)λ (x)C

(2)λ (ik)

l!C(2)λ (1N )

, (6.1.43)

which is normalized such that

(trx)j =

(N∑i=1

xi

)j=

∑l(λ)=j,λ

Cλ(x) . (6.1.44)

The non–trivial normalisation constant C(2)λ (1N ) can be found for instance in Ref. [301]. The left hand

side of Eq. (6.1.43) is a complicated and rather unhandy infinite sum of symmetric functions, which hasbeen derived from the Itzykson–Zuber formula. It would be very difficult to do the derivation in the otherdirection.

The group integral Eq. (6.1.9) over the orthogonal group, β = 1, is also discussed in mathematical literaturein the context with multivariate statistics and special functions with matrix arguments [302, 303]. Thesolution of Eq. (6.1.9) over the orthogonal group, which corresponds to the expansion (6.1.43) can be foundin Muirhead’s book. The equivalent to Schur polynomials for the orthogonal group goes by the name zonalpolynomials. They are uniquely defined by the properties

1. C(1)λ (x) is symmetric in all variables and homogeneous of degree l

2. The term of highest weight is xl11 . . . xlmm or

C(1)λ (x) = cλx

l11 . . . xlmm + . . . . (6.1.45)

56 6.1 Symmetric spaces with curvature zero

3. C(1)λ (x) are eigenfunctions to the operator

∆∗(1) =

N∑n=1

x2n∂2

∂x2n

+∑n6=m

x2n

xn − xm∂

∂xn

∆∗(1)C

(1)λ (x) = [ρλ + l(N − 1)]C

(1)λ (x) , (6.1.46)

with ρλ =∑Nn=1 ln(ln − n).

4. Normalisation: following Muirhead [303] we normalise zonal polynomials such that they have unitcoefficients in the expansion of (trx)k according to Eq. (6.1.44). This is the so called ”C” normalisation.Other normalisations which are used are the ”P” normalisation, which sets the coefficient of the lowestorder polynomial [1N ] to N ! and the ”J” normalisation which is monic and defined by orthogonalitywith respect to the following scalar product of two power sums 3

〈pλ, pκ〉 = δλκ

(2

β

)−l(λ) m(λ)∏j=1

jnjnj ! , (6.1.47)

where pλ and pκ are two power sums (App. E) and ni is the number of rows in λ equal to i.

The explicit construction of zonal polynomials is extremely cumbersome. For instance the first non–trivial

zonal polynomials C(1)

[2] (x) and C(1)

[1,1](x) can be obtained as follows: From condition 2 and from normalisationwe get

C(1)

[2] (x) =

N∑i=1

x2i + g

∑i<j

xixj , C(1)

[12](x) = (2− g)

∑i<j

xixj . (6.1.48)

To find the constant g one must plug in the explicit form (6.1.48) into the differential operator Eq. (6.1.46).It is this step makes the construction so messy for higher partitions. In the present example it is sufficient

to evaluate C(1)

[12](x). It is found

∆∗(1)∑i<j

xixj = (N − 2)∑i<j

xixj , (6.1.49)

and therefore g = 1. A recipe how to construct zonal polynomials recursively in terms of monomial symmetricfunctions (App. E) is given in Muirhead’s book [303]. The virtue of zonal polynomials is that they have thefollowing invariance property∫

U∈O(N)

C(1)λ (U−1xUk)dµ(U) =

C(1)λ (x)C

(1)λ (k)

C(1)λ (1N )

. (6.1.50)

With this invariance it follows immediately that the expansion of the matrix Bessel function in zonal poly-nomials is given by

Φ(β)N (x, k) =

∞∑l=0

∑l(λ)=l,λ

C(1)λ (x)C

(1)λ (k)

l!C(1)λ (1N )

. (6.1.51)

The generalisation of Schur polynomials and zonal polynomials to arbitrary positive β is called Jack polyno-

mial [304, 274]. They4J(β)λ (x) form yet another basis in the space of symmetric functions in the N variables

xi. To be consistent with literature we use the ”J” normalisation defined above. Jack polynomials are definedas the eigenfunctions of the operator N∑

n=1

x2n∂2

∂x2n

+ β∑n 6=m

x2n

xn − xm∂

∂xn

J(β)λ (x) =

[ρ

(β)λ + l(N − 1)

]J

(β)λ (x) , (6.1.52)

where ρ(β)λ =

∑Nn=1 ln[ln−1−β(n−1)]. It was shown during the last decades that many of the properties of

zonal polynomials and of Schur polynomials carry also over to Jack polynomials [274]. However the invarianceproperty (6.1.50) has no straightforward extension to arbitrary β. The virtue of Jack polynomials is that

3The definition of power sums as well as a compilation of the standard bases of the algebra of symmetric functionsis given in App. E.

4Here we keep the parameter β to label the Jack polynomials. In most of the literature on Jack polynomials [274]the label α = 2/β is used instead of β.

6.2 Symmetric spaces with positive curvature 57

there have been proven a series of powerful integral theorems for them. MacDonald conjectured and laterproved [274] the following orthogonality relation, which was also proved by Kadell [305]

〈κ|λ〉β ≡∫ L

0

dx1 . . .

∫ L

0

dxNJ(β)κ (z∗)J

(β)λ (z)|∆N (z)|β = A(κ)δκλ , (6.1.53)

where zj = exp(2πixj/L). The complicated normalisation constant A(κ) can be found for instance inMacDonald’s book [274].

In a series of papers Okounkov and Olshanski [306, 307, 308, 309] derived combinatorial formulas for Jackpolynomials. An interesting relation to the recursion formulas (6.1.21) and (6.2.7) provides the followingrecursion formula [306]

J(β)λ (x) =

1

B(λ,N)

∫ x1

x2

dx′1 . . .

∫ xN−1

xN

dxN−1µ(x, x′)J(β)λ (x′)

µ(x, x′) =∆N−1(x′)

∏Ni=1

∏N−1j=1 |xi − x

′j |β/2−1

∆β−1N (x)

, (6.1.54)

where J(β)λ (x) is a Jack polynomial. The partition λ is the same on both sides. B(λ,N) is again a complicated

normalisation constant which can be found in Ref. [306]. The function µ(x, x′) is identical with the measurefunction (6.1.1). In Sec. 6.1.1 it was argued that µ(x, x′) is the β deformation of the unitary, orthogonal orsymplectic unit sphere. Eq. (6.1.54) states an invariance property of Jack polynomials under an integrationwith the integration kernel µ(x, x′). Loosely speaking, it might be considered the β deformed version of theinvariance property of zonal polynomials Eq. (6.1.50), for one matrix x or k being of rank one.

6.2 Symmetric spaces with positive curvature

Now we consider a space with positive curvature. For instance the symmetric space A with positive curvatureis given by the space of unitary matrices. One can define a metric on this space by the invariant length element−(L/2π)2tr (ln dU)2, where L introduces a length scale. The radial part of the Laplace–Beltrami operator isgiven by

∆∗(β) =

N∑n=1

1

∆βN (e2πix/L)

∂

∂xn∆β(e2πix/L)

∂

∂xn

= −(

2π

L

)2∑n=1

z2i∂2

∂z2i

+ β∑i 6=j

z2i

zi − zj∂

∂zi

, zi = e2πixi/L , (6.2.1)

where ∆N (e2πix/L) =∏i<j [exp(2πixi/L) − exp(2πixj/L)]. The index β takes the the values β = 1 for AI

β = 2 for A and β = 4 for AII. We do not consider here the compact symmetric space AIII. From Eq. (6.2.1)

a Hamiltonian is obtained by adjunction with ∆β/2N (z).

∆∗(β) → ∆β/2N (z)∆∗(β)∆

−β/2N (z) ≡ HCS . (6.2.2)

The corresponding Hamiltonian is defined again for any positive β

HCS =

N∑n=1

∂2

∂x2n

− β(β

2− 1

) ∑n<m

(π/L)2

sin2[π(xn − xm)/L]. (6.2.3)

This is the Calogero–Sutherland Hamiltonian [310], which has been studied extensively in the past [310,311, 312, 313]. It is the periodic version of Eq. (6.1.14) describing N particles moving on a ring with the1/x2 interaction between two particles at distances x + nL/π, n ∈ Z. The particles are confined and theHamiltonian has a well defined ground state, which can be given explicitly

Ψ∗(β)0 (x) =

1

AβN∆β/2N (e2πx/L) . (6.2.4)

Since by construction Jack polynomials are eigenfunctions of the Laplace–Beltrami operator (6.2.1), theeigenfunctions of Eq. (6.2.3) can be expressed in terms of Jack symmetric functions

Ψ∗(β)N (x) = Ψ

∗(β)0 (x)J

(β)λ (z)

N∏n=1

exp

(2πiK

N∑i=1

xi/L

), (6.2.5)

where K is the center of mass momentum which can be an arbitrary integer.

58 6.3 Symmetric superspaces with zero curvature

In the context with the Calogero–Sutherland Hamiltonian Jack polynomials were introduced by Forrester[312, 313, 314] for rational β. The representation (6.2.5) of the eigenfunctions was extraordinarily successful.Ha [315, 285] derived exact expressions for the one–particle Green’s function and for the dynamical density–density correlator

⟨Ψ∗(β)0 |ρ(x, t)ρ(x′, 0)|Ψ∗(β)

0

⟩, with ρ(x, t) = eiHCSt

N∑n=1

δ(x− xn)e−iHCSt , (6.2.6)

in the thermodynamical limit and at zero temperature. For the values β = 1, 2, 4 his results were derivedearlier by Simons, Lee, and Altshuler using a non–linear σ–model approach [316, 317]. For these values eventhermodynamically exact expressions for the two–particle Green’s function have been derived [318].

Recursion formula

A solution to Eq. (6.2.1) can also be given in terms of a recursion formula

Ψ(β)N (x, k) =

∫dµ(x′, x) exp

[i

(N∑n=1

xn −N−1∑n=1

x′n

)kN

]Ψ

(β)N−1(x′, k) , (6.2.7)

where Ψ(β)N−1(x′, k) is the solution of the Schrodinger equation (6.2.1) for N−1, k denotes the set of quantum

numbers kn, n = 1, . . . , (N − 1) and x′ the set of integration variables x′n, n = 1, . . . , (N − 1). Up to anormalisation constant the integration measure is

dµ(x′, x) =

( ∏n,m | sin[π(xn − x′m)/L]|∏N

n<m sin[π(xn − xm)/L]∏N−1n<m sin[π(x′n − x′m)/L]

)β/2−1

d[x′] . (6.2.8)

Here, d[x′] =∏Nn=1 dx

′n. The integration domain is defined by the inequalities

0 ≤ xn ≤ x′n ≤ xn+1 ≤ L , n = 1, . . . , (N − 1) (6.2.9)

As in the zero curvature case the above recursion formula is equivalent to group integrals over O(N), U(N)

and USp(2N) for β = 1, 2, 4. The recursion formula Eq. (6.2.7) is proved by plugging explicitly Ψ(β)N (x, k)

into the eigenvalue equation, Eq. (6.2.3) [319]. It is seen from Eq. (6.2.7) that the wave function obtains anadditional phase πβ/2 by interchanging to adjacent particles. Since the system is now constrained to a circle,the wave numbers kn have to be chosen such that the wave function fulfils periodic boundary conditions

Ψ(β)N (x1, . . . , xN ) = Ψ

(β)N (x2, . . . , xN , x1 + L) . (6.2.10)

This yields for kN the constraint

LkN = 2πI + π

(β

2− 1

), I ∈ Z . (6.2.11)

Eq. (6.2.11) is very similar to the Bethe–ansatz type equation for the momenta found by Sutherland [283,289, 311]. Up to now the curvature one recursion formula Eq. (6.2.7) has not been analysed further.

6.3 Symmetric superspaces with zero curvature

The theory of matrix Bessel functions can naturally be extended to symmetric superspaces, which have beenclassified by Zirnbauer [320]. There are only two root systems in superalgebra which allow for an arbitrarynumber of dimensions, namely the root system of the general linear superalgebra and the root system of theorthosymplectic superalgebra. Here only the symmetric superspaces connected with the root system of thegeneral linear superalgebra gl(n|m) are considered. They are labelled A|A, AI|AII and AII|AI in Ref. [320].

The corresponding formula for symmetric superspaces derived from the orthosymplectic algebra can be foundin Ref. [8]. As before a plane wave equation in supermatrix space serves as starting point

Str

(∂

∂σ

)2

eiStrσρ = −Strρ2eiStrσρ , (6.3.1)

where ρ and σ are either Hermitean (A|A) or Hermitean selfdual matrices (AI|AII or AII|AI) and thesupermatrix ∂/∂σ is defined by (

∂

∂σ

)ij

=∂

∂σij(6.3.2)

6.3 Symmetric superspaces with zero curvature 59

Zirnbauer’snotation

A B (β1, β2) angular space

A|A Hermitean Hermitean (2, 2) U(n|m)

AI|AII symmetric Hermiteanselfdual

(1, 4) UOSp(n|2m)

AII|AI Hermiteanselfdual

symmetric (4, 1) UOSp(2n|m)

Table 6.2: Classification of curvature zero Riemannian symmetric superspaces derived from GL(n,m),adapted from Zirnbauer [320]. A denotes the Boson–Boson block and B the Fermion–Fermion block accordingto Eq. (B.1)

0

1

2

3

4

5

1 2 3 4 5

β1

β2

β1 = 4/β2

β1 = β2

dAII|AI

dAI|AII

d A|A

Figure 6.1: Parameter space in the (β1, β2) plane of known solutions. Taken from Ref. [8].

The radial part of the Laplace–Beltrami operator now depends on n Bosonic and m Fermionic radial variables.It has in all three cases the form

∆(c,β1,β2)s =

1√β1

n∑j=1

1

B(c,β1,β2)nm (s)

∂

∂sj1B(c,β1,β2)nm (s)

∂

∂sj1

+1√β2

m∑j=1

1

B(β1,β2)nm (s)

∂

∂sj2B(c,β1,β2)nm (s)

∂

∂sj2. (6.3.3)

The parameters β1 and β2 are as specified in Tab. 6.2. The role of the Vandermonde determinant is takenby the Berezinian, see App. B. Here it is stated in a more general form

B(c,β1,β2)nm (s) =

∏nj<k(sj1 − sk1)β1

∏mj<k(sp2 − sq2)β2∏

j,k(sp1 − csq2)√β1β2

, (6.3.4)

where the parameters β1 and β2 can have arbitrary positive values. The parameter c can take the values

c = ±i. The functions B(c,β1,β2)nm (s) induce a differential operator which yields the Hamiltonian

H =

n∑j=1

p2j1

2M1+

m∑j=1

p2j2

2M2+

∑j<k

g11

(sj1 − sk1)2 −∑j<k

g22

(sj2 − sk2)2 −∑j,k

g12

(sj1 − sk2)2, (6.3.5)

with canonically conjugate variables, [ppj , sql] = δpqδjl. The coupling constants are functions of β1 and β2

g11 =√β1

(β1

2− 1

), g22 =

√β2

(β2

2− 1

),

g12 =1

2

(√β1 −

√β2

)(1

2

√β1β2 + 1

)(6.3.6)

and one positive mass M1 =√β1/4 and one negative mass M2 = −

√β2/4. The Hamiltonian Eq. (6.3.5)

is a two–family generalisation of the rational CMS Hamiltonian Eq. (6.1.14). An interpretation of their

60 6.3 Symmetric superspaces with zero curvature

physical content has been given in Ref. [9]. Here only their mathematical properties are discussed. For theparameter values specified in Tab. 6.2 a solution to Eqs. (6.3.3) and (6.3.6) can be obtained by integrationover a compact supermanifold

Ψ(s1, s2, r1, r2) =

√B

(c,β1,β2)nm (s)B

(c,β1,β2)nm (r)

∫dµ(U) exp

(UsU−1r

)(6.3.7)

where the integral goes over a supergroup U is as specified in Tab. 6.2. The radial coordinates r1j , j ≤ nand r2k, k ≤ m can again be interpreted as the asymptotically free momenta of particles of type 1 and type2 related to the energy (eigenvalue of H) as

E =

(n∑j=1

r2j1

2M1+

m∑j=1

r2j2

2M2

). (6.3.8)

For general values of (β1, β2) the solutions to (6.3.6) are not known. In particular exact solvability is notproved for general (β1, β2). However, there exist two one–parameter subspaces in the β1, β2 plane, wheresolutions can be constructed [8]. One subspace is defined by the condition β1 = β2. In this case the interactionbetween particles of different types vanishes and the Hamiltonian (6.3.5) is a sum of two rational CMSHamiltonians Eq. (6.1.14). The other one–parameter subspace is the hyperbola, defined by the condition β1

= 4/β2, see Fig. 6.1. On this parabola a recursion formula can be constructed akin to Eqs. (6.1.21) and (6.2.7).It involves an integration over commuting variables as well as over anticommuting variables. Its completeform and its proof was given in Ref. [8]. We do not state it here, since the expression for the integrationmeasure, corresponding to Eqs. (6.1.20), (6.1.22) and (6.2.8) is rather lengthy, involving products of alltype of differences between Bosonic and Fermionic and between primed and unprimed variables, alltogether10 products. It is quite remarkable that exactly on the hypebola and only on the hyperbola Sergeev andVeselov were able to construct a supersymmetric extension of Jack symmetric polynomials [321, 282, 322].The reason for this might lie in a strong–weak coupling duality symmetry of the Hamiltonian Eq. (6.3.5) onthe hyperbola. This means the following. Denote H(β, n,m) the Hamiltonian on the hyperbola with n typeone particles and m type two particles then it holds

H(β, n,m) = H(4/β,m, n) . (6.3.9)

This strong weak coupling duality is a generalisation of the duality of the orthogonal and the symplecticgroup. It is not clear if this remarkable symmetry is responsible for exact solvability of the many–bodyHamiltonian (6.3.5) or if the parameter space of exact solutions can be extended to the whole (β1, β2) plane.

7. Summary and Conclusion

Quantum information devices require a high level of coherence and of fidelity to be operative. Therefore thestudy of decoherence and of fidelity is a topic of high current interest. The approaches, which were chosenfor the calculation of decoherence and of fidelity were quite different. For the analysis of decoherence thetraditional approach of a system–bath Hamiltonian, pioneered by Caldeira and Leggett was chosen. In thisapproach the environment is dynamical and the quantities of interest are obtained from a partial trace overthe dynamical bath variables.

Within this framework a novel phenomenon, called quantum frustration, was studied. It was first reportedin 2003 by Castro Neto et al. [1]. It refers to the phenomenon that if a system couples with two mutuallyconjugate variables to two independent environments their dissipative effects partially cancel. The originalfindings of Ref. [1] for a spin 1/2 system are compared with a harmonic oscillator, which couples with positionand momentum to two independent heat baths. This scenario is by no means esoteric, it can be found forinstance in Josephson junctions. For the harmonic oscillator the mutual cancellation of the two environmentsis weaker than for the spin system, however frustration effects, such as underdamped oscillations of thesymmetrized and anti–symmetrized position correlation functions for arbitrarily strong symmetric coupling,still exist. A discussion of the findings was given in the resume of Chap. 2.

Fidelity was studied within the framework of random matrix theory (RMT). Fidelity is obtained as an RMTensemble average over typical realizations of a chaotic system and a chaotic perturbation. This approach hasproved to be very successful in the description of the available numerical and experimental data.

Supersymmetric calculations were performed to obtain exact expressions for RMT averages of fidelity in thestrong coupling regime. Up to now these are the only available results (apart from numerical simulations)on fidelity decay in the strong coupling regime. Two important effects are found: first, the fidelity revivaldiscovered by Stockmann and Schafer [239]. Second, it was shown that fidelity can freeze on a plateau closeto one for large time intervals, if the diagonal elements of the perturbation in the system’s eigenbasis vanish.An exact analytical calculation shows that this is a genuinely non–perturbative effect, i. e. fidelity decaysmore slowly for a strong perturbation with off–diagonal elements than for a weak perturbation with diagonalelements. This gives an important guideline for the construction of quantum information devices.

Random matrix theory is more than a tool in the description of chaotic and disordered systems. It is a self–contained discipline at the interface between mathematics and physics. In this work both aspects of RMT wereexplored. In the physical part it proved to be an extremely useful tool in the description of chaotic quantumsystems. The exact calculation of RMT averages is a big challenge mathematically. The supersymmetric non–linear σ–model, the standard approach, is by now restricted to rotationally invariant ensembles. The searchfor alternative approaches leads to the study of matrix Bessel functions. They correspond to eigenfunctionsof an exactly solvable system of the Calogero–Moser–Sutherland type. The interaction constant is relatedto Dyson’s index β. For β = 2 the system becomes interaction free, corresponding to the Itzykson–Zuberformula. Different approaches for the construction of eigenfunctions were presented. A recursion formulafound earlier [3] was extended to two–type particle models and to periodic boundary conditions. In the lattercase a Bethe–Ansatz type equation arises naturally.

In this work results mainly of previously published papers are compiled. However some calculations andresults of this work are unpublished. Although it was stated in situ when an unpublished result was presented,we compile them here again. First, the integral theorem Eq. (5.4.6) and the finite N calculation with theresult written as a double integral over a Pfaffian form as stated in Eq. (5.4.8) and Eq. (5.4.13) is unpublished.Second, the finite N calculation with the graded eigenvalue method as described in Sec. 5.5 is unpublished.Third, the recursion formula for matrix Bessel function, which was first found for symmetric spaces ofcurvature zero in Ref. [3] and later extended to symmetric superspaces in Ref. [8], was extended here tosymmetric spaces of curvature one, or, equivalently, for the CMS–Hamiltonian of particles moving on a ring.Also the Bethe–Ansatz type equation Eq. (6.2.11) for the particle momenta is original.

61

62 7. Summary and Conclusion

Appendix A. More on the quaternion elements Kij, Lij

and L′ij

In this appendix a relation between the off-diagonal elements of the quaternionic entries Lij and Kij re-spectively L′ij as given in Eqs. (3.1.4), (3.1.17), and (3.1.19) for the GOE and the GSE is derived. Therelation between Lij and Kij was already determined in the main text in Eqs. (3.1.17) and (3.1.15). It waspointed out already in the main text that the diagonal entries of Lij and L′ij are identical. This is just thestatement of Eq. (3.1.16). Since the quaternionic determinant is the same in all three representations alsothe product of the two off–diagonal elements, which enters in the Pfaffian, are the same. This means, forinstance comparing Eq. (3.1.4) and Eq. (3.1.19) it must hold[

DK(β)N (xi, xj)

] [IK

(β)N (xj , xi)

]=

C(β)

N−2/γC(β)

N+2/γ

[C(β)N ]2

(xi − xj)2〈Dγ(xi)Dγ(xj)〉N−2/γ,β〈D−γ(xi)D−γ(xj)〉N+2/γ,β . (A.1)

We recall the definition Eq. (3.1.18) of D. Focussing on the GOE (β = 1, γ = 1) we are now going to prove

〈D(xi)D(xj)〉N−2,1 =exp(x2

i /2 + x2j/2)

c(A)N1 (xi − xj)

DK(1)N (xi, xj) . (A.2)

Due to Eq. (A.1) this automatically yields the second identity

〈D−1(xi)D−1(xj)〉N+2,1 =exp(−x2

i /2− x2j/2)

c(B)N1 (xi − xj)

IK(1)N (xi, xj) (A.3)

where the two normalisation constants c(A)N1 and c

(B)N1 are related by

c(A)N1 c

(B)N1 =

C(β)

N−2/γC(β)

N+2/γ

[C(β)N ]2

. (A.4)

As will become clear below, the second identity, Eq. (A.3), is much harder to prove directly. To proveEq. (A.2), the determinants on the right hand side are written as integrals over Grassmann variables. Afterperforming the Gaussian integrals and removing the quartic terms by a Hubbard–Stratonovich transformationwe can write

〈D(xi)D(xj)〉N−2,1 = c0

∫d[σ] exp

(1

2tr (iσ + x)2

)detN/2−1(iσ) , (A.5)

where σ is now an ordinary Hermitean selfdual 4×4 matrix and x = diag (xi, xi, xj , xj). c0 is a normalisationconstant, which is independent of N . The matrix integral (A.5) can be evaluated by changing to angle–eigenvalue coordinates, σ → U−1isU , where U ∈ USp(4) and is = diag (is1, is1, is2, is2). The integrationmeasure transforms as d[σ]→ (is1− is2)4ds1ds2dµ(U). The integral over the group USp(4) yields the matrix

Bessel function Φ(4)2 (2s, x), which was calculated in Ref. [3], see Eq. (6.1.17) in Chap. 6,

Φ(4)2 (2s, x) =

1

(xi − xj)2(is1 − is2)2

(1− 1

(xi − xj)(is1 − is2)

)e2is1xi+2is2xj + (xi ↔ xj) . (A.6)

The right hand side of Eq. (A.5) can now be written as a double integral

〈D(xi)D(xj)〉N−2,1 = c0

∫(is1 − is2)4(is1)N−2(is2)N−2Φ

(4)2 (2s, x) exp

(1

2tr (is)2 +

1

2trx2

)ds1ds2 . (A.7)

Plugging the expression (A.6) for Φ(4)2 (2s, x) into Eq. (A.7), the double integral can be written as a sum

of products of two single integrals, which are just integral representations of Hermite polynomials, seeEq. (3.1.15). Using the Christoffel–Darboux formula, the result can be expressed in terms of the GUE

matrix kernel K(2)N−1, see Eq. (3.1.6),

〈D(xi)D(xj)〉N−2,1 =π2c02N−2

(N − 2)!ex

2j/2+x2i /2

xi − xj

[D − xi − xj

2

]K

(2)N−1(xj , xi) , (A.8)

63

64 Appendix A. More on Quaternion elements

where D is the anti–symmetrized differential operator as defined in Eq. (3.1.5). This is almost the finalresult. It remains to show that [

D − xi − xj2

]K

(2)N−1(xj , xi) = DK

(1)N (xj , xi) (A.9)

Using the explicit forms (3.1.7) and (3.1.7) together with the Christoffel–Darboux formula and the followingidentity for oscillator wave functions√

N − 1

2φN−2(x) = φ′N−1(x) +

√N

2φN (x) , (A.10)

one can show that Eq. (A.9) holds. Since the normalisation constant c0 is independent of N , it is mostconveniently obtained from Eq. (A.7) at the point xi = xj = 0 and for N = 4. The result is c0 = 2−4.

Therefore Eq. (A.3) is proven with the constant c(A)N1 = 2N+2/π2(N − 2)!.

A similar reasoning for the second identity (A.3) is much harder. Performing for (A.3) the steps, which leadto Eq. (A.7) before, one finds now

〈D−1(xi)D−1(xj)〉N+2,1 ∝∫d[s]|∆4(s)| exp

(−1

2tr s2 − 1

2trx2

)(A.11)

Φ(1)4 (is, x)Im

1

(s−1 s−2 )N/2+1

Im1

(s−3 s−4 )N/2+1

. (A.12)

Here, s = diag (s1, . . . , s4) and x is defined as before. The matrix Bessel function Φ(1)4 (is, x) is defined as an

integral over the orthogonal group SO(4) [see Eq.(6.1.6)]. The complicated structure of Φ(1)4 (is, x) makes it

so hard to evaluate the right hand side of Eq. (A.11) further.

Finally we mention that for the GSE the relations, which correspond to Eqs. (A.2) and (A.3) read as follows

〈D−2(xi)D−2(xj)〉N+1,4 =exp(−x2

i − x2j )

c(A)N4 (xi − xj)

DK(4)N (xi, xj) ,

〈D2(xi)D2(xj)〉N−1,4 =exp(+x2

i + x2j )

c(B)N4 (xi − xj)

IK(4)N (xi, xj) . (A.13)

We observe that the role of the differential operator D and the integral operator I are swapped as comparedwith the GOE. Combining this result with the GOE result, we can write compactly

〈Dγ(xi)Dγ(xj)〉N−2/γ,β =exp[γ(x2

i + x2j )/2]

c(A)Nβ (xi − xj)

DK(β)N (xi, xj) ,

〈D−γ(xi)D−γ(xj)〉N+2/γ,β =exp[−γ(x2

i + x2j )]/2

c(B)Nβ (xi − xj)

IK(β)N (xi, xj) , (A.14)

which holds for the GOE (β = 1, γ = 1) and for the GSE (β = 4, γ = −2).

Appendix B. Calculus on superalgebras

A supermatrix is a block matrix whose diagonal blocks have commuting and whose off–diagonal blocks haveanti–commuting entries

σ =

[A κρ iB

], (B.1)

where A is quadratic m×m and B is a n×n matrix. The block A is called Boson–Boson block and the block Bis called Fermion–Fermion block. In the physical literature, the Fermion–Fermion block is frequently labelledwith an imaginary unit. In mathematical literature the imaginary unit is usually omitted [246, 242, 243].

Due to the anti–commuting blocks κ and ρ different definitions of the invariants are required. The supertraceis defined as

Strσ = trA− itrB (B.2)

which guarantees the cyclic invariance. The superdeterminant is defined as

Sdetσ = det−1(iB) det(A+ iκB−1ρ) = det(A)det−1(iB − ρA−1κ) . (B.3)

This definition is consistent withSdetσ = exp(Str lnσ) (B.4)

Most of the properties of ordinary determinants carry over to superdeterminants if the usual definition ofthe algebraic minor is replaced by the more general form σ(ik) = ∂

∂σikSdetσ . Kramer’s rule for the inverse

matrix is particularly useful

(σ−1)ik = Sdet−1(σ)σ(ki) . (B.5)

A classification of supermatrices in superalgebras was given by Kac [242]. Altogether, he distinguished9 different classes of classical Lie–superalgebras, labelled by capital Latin letters A(m,n), B(m,n), C(n),D(m,n) , F(4), G(3), in the spirit of Cartans classification of semisimple Lie–algebras in ordinary space. Inthe context of RMT the general supersymmetric algebra gl(n|m) = A(m,n) and the orthosymplectic algebraosp(n|m), corresponding to B(m,n) and D(m,n) are most important. The gl(n|m) comprises all matrices ofthe form Eq. (B.1). The orthosymplectic algebra comprises all matrices in gl(n|m) which fulfil the additionalcondition

σL + LσT = 0 , L = diag (1n, e(2) ⊗ 1m), σ ∈ osp(n|m) . (B.6)

The concept of a transposed and Hermitean conjugate matrix in ordinary space have analogues in superspace.The transposed matrix to σ is defined as

σT =

[AT ρT

−κT iBT

]. (B.7)

The Hermitean adjoint to σ is defined as

σ† =

[A† ρ†

−κ† iB†

]. (B.8)

Thus a Hermitean (m+ n) supermatrix σ ∈ herm(n|m) has the form

σ =

[A ρ†

ρ iB

], with

A = A† ,B = B† .

(B.9)

The entries in the off–diagonal block ρ are independent anti–commuting variables ρij , i ≤ n, j ≤ m. Thereforeσ has m2 +n2 independent commuting entries and 2mn independent anticommuting entries. For m = n thenumber of commuting and anticommuting entries is the same. The complex conjugate of an anticommutingvariable is defined by

(α∗)∗ = −α , (αβ)∗ = α∗β∗ , (B.10)

such that(α∗α)∗ = α∗α ≡ |α|2 (B.11)

is a real commuting number. The concepts of a symmetric matrix and of Hermitean selfduality in ordinaryspace have no exact equivalent in superspace. Both symmetries are included in one supersymmetry whichis called orthosymplectic. Hermitean Supermatrices with an additional orthosymplectic symmetry σ ∈SymSd(n|m) have the form

σ =

[A ρ†

ρ iB

], with I :

A = AT

Bij = Bjior II :

B = BT

Aij = Aji. (B.12)

65

66 Appendix B. Calculus on superalgebras

where the entries of B (case I) and A (case II) are quaternionic and we recall the definition of a dual quaternionEq. (3.0.1). Now the entries of the off–diagonal block ρ are not all independent. The orthosymplecticsymmetry requires a joint concept of reality and selfduality for the Grassmannian entries. The entriesof ρ are chosen as ρ(i+1)j = ρ∗ij , such that Eq. (B.6) is fulfilled. A Hermitean orthosymplectic matrixσ ∈ SymSd(n|m) has n(n+ 1)/2 +m(2m− 1) commuting and 2mn anticommuting variables. The numberof commuting and anticommuting variables is equal for n = 2m.

In the same way as Hermitean matrices in ordinary space are diagonalised by unitary matrices Hermiteansupermatrices are diagonalised by elements of the unitary supergroup. The unitary supergroup consists ofall invertible supermatrices which fulfil the additional condition u†u = 1. The coordinate transformation ofσ ∈ Herm(n,m) to angle–eigenvalue coordinates σ → u−1su induces a non–trivial Jacobian, which is calledBerezinian for supermatrices

σ → u−1su : d[σ]→ B(s)dµ(u)

n∏i=1

ds1i

n∏j=1

ds2j , (B.13)

where dµ(u) is the Haar measure of the unitary supergroup U(n|m) and with 1

B(s) =

∏i<k(s1i − s1k)

∏i<k(is2i − is2k)∏

i,j(s1i − is2j). (B.14)

The supergroup U(n|m) is compact. Similarly the supermatrices which diagonalise σ ∈ SymSd(n|m) forma compact supermanifold. It consists of all unitary supermatrices which fulfil the additional condition uTLu= L. The coordinate transformation of σ ∈ SymSd(n|m) to angle–eigenvalue coordinates σ → u−1su isgiven by

σ → u−1su : d[σ]→ B(s)dµ(u)

n∏i=1

ds1i

n∏j=1

ds2j , (B.15)

where dµ(u) is the Haar measure of the unitary orthosymplectic supermanifold UOSp(n|2m) and

B(s) =

∏i<k |s1i − s1k|

∏i<k(is2i − is2k)4∏

i,j(s1i − is2j)2(B.16)

is the Berezinian.

Integration over an anticommuting variable is defined by the two integration rules:∫dξξ = 1/

√2π , and

∫dξ = 0 . (B.17)

The normalisation varies in literature.

1Strictly speaking this Berezinian is only one part of the full integration measure, which includes additional termsdue to boundary contributions from the non–compact integration manifold, usually called Efetov–Wegner terms. Theyoccur when in a variable transformation a real or complex commuting variable is shifted by a nilpotent part and theintegration domain is non–compact. A prescription for their construction was given by Rothstein [323].

Appendix C. Derivation of Eq. (5.4.6)

In this appendix the integral theorem [see Eq. (5.4.6)]

I ≡∫U∈U(N)

dµ(U)tr(sUrU†

)eitr(xUkU

†)

=BN

∆N (x)∆N (k)

∑i,j

risjei(N−2)xjkidetmn [gji(xm, kn)] , (C.1)

where the function gij is given by

gji(xn, km) = eikmxn+ixjki − eikmxn+ixjki − eikmxj+ikixn(xn − xj)(km − ki)

(C.2)

= −ieikmxj+ikixn∫ xn−xj

0

t(km − ki)ei(km−ki)tdt , m 6= i , n 6= j ,

is proved.

For the proof the recursion formulae for matrix Bessel functions (6.1.21) is used. We denote eij the N ×Nmatrix with entries (eij)kl = δikδjl. Then we can write the preexponential term in the integrand of the lefthand side of Eq. (C.1) as a sum

tr(sUrU†) =∑i

ritr(sUeiiU†) . (C.3)

In each term of the sum we rotate eii = V (i)e11V(i)†. This rotation also swaps the entries k1 and ki of k.

After absorbing in each term V (i) in U using the invariance of the measure we arrive at

I =

∫U∈U(N)

dµ(U)tr(sUrU†

)etr(xUkU†)

=∑i

ri

∫U∈U(N)

dµ(U)tr(sUe11U

†)etr(xUk(i)U†) . (C.4)

Here k(i) is the matrix k with the entries ki and k1 interchanged. We observe that the preexponential onlydepends on the first column U1 of U , U1 ∈ U(N)/U(N − 1). Therefore we can directly apply the recursionformula Eq. (6.1.21). After plugging in the known Itzykson–Zuber result Eq. (5.4.3) for the unitary groupU(N − 1) we arrive at

I =∑i,j

ri

∫ x1

x2

. . .

∫ xN−1

xN

dx′1 . . . dx′N−1

∆N (x)∆N−1(k 6=i)

sj∏i(xj − x

′n)∏

i 6=j(xj − xi)exp

(∑n

xnki −∑n

x′nki

)detN−1,N

n=1,m 6=i

[ex′nkm

](C.5)

Since the integrand is a determinant, the remaining integrals can be performed separately

I =∑i,j

risj∆N (x)∆N (k)

∏m 6=i(ki − km)∏n6=j(xj − xn)

e∑n xnkidetN−1,N

n=1,m 6=i [Fji(xn, xn+1, km)] .

The integrals in the functions Fij are elementary

Fji(xn, xn−1, km) =

∫ xn

xn−1

dx′n(xj − x′n)ex′n(km−ki)

= Gij(xn, km)−Gij(xn−1, km)

Gij(xn, km) =xj − xnkm − ki

(1 +

1

(xj − xn)(km − ki)

)exn(km−ki) . (C.6)

Using properties of determinants we finally arrive at Eq. 5.4.6.

67

68 Appendix C. Derivation of Eq. (5.4.6)

Appendix D. Real forms and Symmetric spaces

In this appendix the classification of real forms and of symmetric spaces, as developed by Cartan, is brieflyreviewed. Its main purpose is to give an elementary explanation of the entries of Tab. D.1. For a more exten-sive introduction the reader is referred to the textbooks of Helgason [157, 158] or Gilmore [269]. Harmonicanalysis on symmetric spaces is also treated in the textbook by Barut and Raczka [324]. A very well writtenintroduction to the topic amenable to physicists was given in Ref. [325].

There exist four classical complex semisimple Lie algebras: sl(N,C), so(2N+1,C), sp(2N,C) and so(2N,C).They are denoted by AN−1, BN , CN , and DN . According to Cartan’s classification each of the complexsemisimple Lie algebra g has as many real forms as involutive automorphisms T exist on the algebra. Thealgebra is split by any T naturally into two orthogonal subspaces which transform under T as

g = k+ p , T (g) = T (k) + T (p) = k− p . (D.1)

This is the Cartan decomposition of the complex algebra g. The subspace k is closed under multiplication[k, k] ∈ k, whereas [p, p] ∈ k and [p, k] ∈ p. On the Lie algebra g there is naturally defined an inner product by〈g, g′〉 = tr Ad(g)Ad(g′) where g, g′ ∈ g and Ad(g) is the adjoint representation of the algebra, defined bythe action of the Lie algebra onto itself

Ad(g)h = [g, h]− , g, h ∈ g . (D.2)

Instead of the adjoint representation the trace of any faithful representation D′ can be used to define the innerproduct by 〈g, g′〉 = i(D′)tr D′(g)D′(g′). The rational number i(D) is called the index of the representationD. For applications in the context with RMT and exactly solvable many–body systems it is often preferableto use the inner product in the defining representation of the Lie algebra. A semisimple Lie algebra is calledcompact iff 〈g, g〉 < 0 , ∀g ∈ g. A real form of a complex Lie algebra g is stable under a given involutiveautomorphism

T (g) = g, ∀g ∈ g . (D.3)

Suppose a compact real form g+ is given, then another real form g− is obtained by acting with an involutiveautomorphism T on g and applying Weyl’s unitary trick p→ ip such that Eq. (D.3) is fulfilled. If g is compact,g− is by construction non compact. Cartan observed that only three types of involutive automorphisms exist:

I) complex conjugation, denoted by K

II) JN = σx ⊗ 1N ,

III) Ip,q = diag (1p,−1q).

A classification of all real forms is obtained by acting successively with these automorphisms and combina-tions of them onto the maximal compact real form. It was shown by Cartan that this procedure is indeedexhaustive.

As an example let us choose the complex special linear algebra sl(N,C). Its maximal compact real form issu(N). It consists of all antihermitean matrices of rank N

su(N) 3 H =

[A B−B† C

],

A† = −A q × qC† = −C p× pB complex q × p .

(D.4)

We now apply the automorphisms I), II), and III) on H ∈ su(N).

I) Acting with K on su(N) yields the decomposition

su(N) = isym(N) + so(N) , (D.5)

where sym(N) is the vector space of real symmetric N ×N matrices. Applying Weyl’s unitary trickyields sl(N,R) as another real form. It is not compact and has a maximal compact subalgebra, namelyso(N), which is the generating algebra of the orthogonal group. The symmetric space was named AIby Cartan.

69

70 Appendix D. Real forms and Symmetric spaces

II) Now we suppose the dimension of H is even H ∈ su(2N). Acting with JNK on H yields

JNH∗J−1N =

[C∗ BT

−B∗ A∗

]=

[(A− CT )/2 (B +BT )/2−(B +BT )∗/2 (C +A∗)/2

]∈ k−

[(A+ CT )/2 (B −BT )/2(B −BT )∗/2 (C −A∗)/2

]∈ p . (D.6)

Setting (A− CT )/2 ≡ A and (B +BT )/2 ≡ B we can write the compact subspace k as[A B−B† −AT

]∈ k = usp(2N) ,

A = −A† N ×NB complex N ×N . (D.7)

These are the generators of the compact group USp(2N). We apply Weyl’s unitary trick on the

subspace p and define i(A+ CT )/2 ≡ A and i(B −BT )/2 ≡ B:[A B

B† −AT]∈ ip = sd(2N) ,

A = A† N ×N

B complex N ×N . (D.8)

This is the vector space of Hermitean selfdual 2N × 2N matrices. The non–compact real algebrag = k+ ip goes by the name su∗(2N).

III) Finally we apply the automorphism Ip,q onto su(N)

Ip,qHI−1p,q =

[A −BB† C

]=

[A 00 C

]∈ k+

[0 B−B† 0

]∈ p . (D.9)

Weyl’s unitary trick yields the non–compact real form AIII consisting of matrices

H =

[A −iBiB† C

],

A† = −A q × qC† = −C p× pB complex q × p ,

(D.10)

with the maximal compact subalgebra su(q) +su(p)+ u(1).

Although the subspace p does not form a subalgebra, it shares many properties of an algebra. In particularon p also an inner product is defined. The exponential exp(p) is called a Riemannian globally symmetricspace. It can also be written as a coset G/K of a (real) group, generated by the (real) algebra g and acompact subgroup K generated by a maximal compact subalgebra k. A Riemannian symmetric space iscalled compact (positive curvature) if 〈p, p〉 < 0 ∀p ∈ p and non–compact (negative curvature) if 〈p, p〉 >0 ∀p ∈ p. A non–compact symmetric space G−/K is obtained from a compact symmetric space G+/K byWeyl’s unitary trick p→ ip.

The subspace ip itself can also be considered a symmetric space with zero curvature by a construction due toGindikin [326]. In this construction the real group G0 is the semi–direct product of the compact subgroupK with the subspace ip. It is denoted by G0 = K p. It is defined by its action on x ∈ ip

G0(x) = k−1xk + ip , k ∈ K, and x, ip ∈ ip . (D.11)

So we can associate with each compact subgroup K a triplet of symmetric spaces X±, X 0:

X+ = exp(p) = G+/K, G+ = exp(g+) , g+ = k+ pX− = exp(ip) = G−/K, G− = exp(g−) , g− = k+ ipX 0 = ip = G0/K, G0 = K p .

(D.12)

In Table D.1 the classes of symmetric spaces are listed in their version with positive curvature (3rd column),with zero curvature (4rd column) and with negative curvature (5th column).

To understand the relation to Dyson’s index, the concept of root multiplicity must be introduced. A semisim-ple Lie algebra g splits into a maximal commuting (Cartan) subalgebra g0 with [g, g′]− = 0 for g, g′ ∈ g0 andinto a set of root vectors, denoted by e such that

[g, eα]− ≡ Ad(g)eα = α(g)eα , (D.13)

with g ∈ g0 and eα ∈ e. e forms a vector space which is of the same dimension as the Cartan subalgebra,denoted the rank of the algebra. The second equality in Eq. (D.13) has the form of an eigenvalue equationwith eigenvalues α(g), which are called roots. Equivalently they are the zeros of the characteristic equation

(Ad(g)− α) = 0 . (D.14)

71

The number of root vectors with the same modulus of the root |α(g)| is the root multiplicity. If eα is a rootvector to the root α(g), for general reasons [269] also e−α is a root vector with root −α(g). Therefore anysemisimple Lie algebra has a root multiplicity of at least 2.

Now consider a symmetric spaces realized as exp(p). Since by the Cartan decomposition (D.1) the algebra issplit into two disjoint subsets k and p, the root vectors eα ∈ e belong either to the invariant subalgebra k or top. The set of roots lying in p is called restricted root system. In the restricted root system the multiplicity ofthe roots has in general changed as compared to the root system of the algebra. But not only the multiplicitymight change, also the structure of the root space, as described by a Dynkin diagram might be different.The multiplicity of the roots in the restricted root system is identified with the Dyson index.

As an example consider again the complex Lie algebra sl(N,C). In a basis where the Cartan subalgebra isdiagonal, the root vectors are given by the matrices eij with entries (eij)kl = δikδjl. eij is a root vector to

the root α(g) = gi − gj for g0 3 g =∑Ni=1 gieii. The negative root vector is obviously given by eji with root

gj − gi.

If we now consider the Cartan decomposition I) we see that neither eij nor eji are invariant under theinvolutive automorphism K. However we can easily obtain two invariant root vectors as linear combinations(eij ± eji)/2. The combination with the minus sign belongs to the invariant subspace k = so(N), whereasthe plus sign belongs to the subspace ip = sym(N). The Cartan subalgebra itself belongs to sym(N). Thusa basis of sym(N) is given by (eij + eji)/2, g0, ∀i < j. Since g0 ∈ sym(N) the root system for sym(N) isthe same as for sl(N,C) but with root multiplicity 1 (Dyson’s index).

For the complementary subspace k = so(N) we can write a basis as (eij − eji)/2, ∀i < j. The originalCartan subalgebra being in p we must construct a new “restricted” Cartan subalgebra which commutes withall elements of k. This “restricted” Cartan subalgebra defines a new root system, which is denoted restrictedroot system. In the present case, since k = so(N) generates the orthogonal group the restricted root systemis trivially nothing but the root system B(N−1)/2 of so(N,C) if N is odd and DN/2 of so(N,C) if N is even.In both cases the root multiplicity (Dyson index) is 2.

For the Cartan decomposition III) the roles of k and p are reversed. Here the original Cartan subalgebralies completely in k = su(q) +su(p) + u(1), whereas for p a restricted Cartan subalgebra of rank min(p, q)must be constructed. It can be shown that in this case the restricted root system is of the type Bq or Cqdepending whether q is odd or even. The root multiplicity (Dyson index) is 2.

72 Appendix D. Real forms and Symmetric spaces

root

spac

e

Car

tan

’scl

assi

fi-

cati

onco

mp

act

non

com

-p

act

ipco

mp

act

sub

-gro

up

Dyso

nin

dex

β

ran

kof

cose

tA

N−1

AS

U(N

)S

L(N,C

)H

erm

itea

nS

U(N

)2

N−

1A

IS

U(N

)S

L(N,R

)H

erm

itea

nsy

mm

etri

cS

O(N

)1

N−

1A

IIS

U(2N

)S

U∗ (

2N

)H

erm

itea

nse

lfd

ual

US

p(2N

)4

N−

1

AII

IS

U(q

+p)

SU

(q,p

)H

erm

itea

nb

lock

off

–d

iagon

al

SU

(q)⊗

SU

(p)

⊗U

(1)

2m

in(p,q

)

BN,D

NB

DS

O(2N

+1)

SO

(2N

+1,C

)re

al

sym

met

ric

SO

(2N

+1)

2N−

1B

DS

O(2N

)S

O(2N,C

)re

al

sym

met

ric

SO

(2N

)2

N−

1

BD

IS

O(p′+q′

)S

O(p′ ,q′

)sy

mm

etri

cb

lock

off

–d

iagon

al

SO

(q′ )⊗

SO

(p′ )

2m

in(p′ ,q′

)

CN

CU

Sp

(2N

)S

p(2N,C

)H

erm

itea

nse

lfd

ual

US

p(2N

)2

NC

IU

Sp

(2N

)S

p(2N,R

)co

mp

lex

sym

met

ric

SU

(N)

4N

CII

US

p(2p

+2q)

US

p(2p,2q)

com

ple

xsy

mp

lect

icb

lock

off

–d

iagon

al

US

p(2q)⊗

US

p(2

p)

2m

in(p,q

)

DN

DII

IS

O(2

N)

SO∗ (

2N

)co

mp

lex

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sym

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(N)

1N/2

Tab

leD

.1:

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csp

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s,ex

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onal

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=p

+q.p′+q′

=2N

or

2N

+1.

Table

isadapte

dfr

om

Gilm

ore

[269].

Appendix E. Symmetric Functions

The algebra of symmetric functions ΛN has several important bases. All of them can be labelled by apartition, respectively a Young tableau λ. In the main text Schur polynomials, zonal polynomials and Jacksymmetric functions were discussed. The standard bases are

1. Monomial symmetric polynomials mλ are defined by

mλ(x1, . . . , xN ) =∑ω∈SN

xλω(1)

1 xλω(2)

2 . . . xλω(N)

N , (E.1)

summed over all distinct permutations ω of λ = (λ1, λ2, . . . , λN ).

2. Elementary symmetric polynomials are defined by

∞∑k=0

ektk =

∏i≥1

(1 + xit). (E.2)

For each partition λ we define eλ = eλ1eλ2 . . . .

3. Similarly complete symmetric polynomials are defined by

∞∑k=0

hktk =

∏i≥1

(1− xit)−1 (E.3)

and hλ = hλ1hλ2 . . . .

4. Finally the power sums are defined as

pk = xk1 + xk2 + . . . , (E.4)

where again for any partition λ, pλ = pλ1pλ2 . . . .

It is well-known [274] that each of these sets of functions with l(λ) ≤ N form a basis in ΛN .

73

74 Appendix E. Symmetric Functions

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