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Doctoral Thesis

Symmetry-breaking ground states of a spin-boson model

Author(s): Funck, Pierre

Publication Date: 1995

Permanent Link: https://doi.org/10.3929/ethz-a-001534402

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Diss. eth No

Symmetry-Breaking Ground States

of a Spin–Boson Model

A dissertation submitted to theSwiss Federal Institute of Technology Zurich

for the degree ofDoctor of Natural Sciences

presented byPierre FunckDipl. Natw. eth

born September , citizen of Luxembourg

accepted on the recommendation ofProf. H. Primas, examiner

pd Dr A. Amann, co-examiner

Abstract

The spin–boson model consists of a two-level system coupledto infinitely many harmonic oscillators and is regarded as aprototype for a ‘small’ system, e.g., a molecule, interactingwith its environment, e.g., the electromagnetic field. Main in-terest is focused to the question whether a ground state of thespin–boson system is breaking some symmetry with respectto which the Hamiltonian is invariant. For certain values ofthe system parameters (level splitting of the spin, frequenciesof the field modes, and coupling strength) only one groundstate exists, which is invariant with respect to the symme-try mentioned above, while for other values of the parameterstwo symmetry-breaking ground states arise. In the latter casethese two ground states are separated by a superselection rule,i.e., superpositions between the two ground states are forbid-den. Superselection rules only arise for systems with infinitelymany degrees of freedom. The Ritz–Rayleigh variation princi-ple allows, with a rigorous theoretical background, to investi-gate such systems within the framework of traditional Hilbert-space quantum mechanics. The variation principle is carriedout with an ansatz where the state of the bosonic system istaken to be in a superposition of coherent states. The pa-rameter ranges where symmetry-breaking ground states ariseare investigated, and the possibilities and shortcomings of theansatz are discussed.

Summarium

Simulacrum spin-bosonicum constat ex duarum planitierumenergeticarum systemate copulato cum oscillatoribus harmo-nicis in numero infinito atque exemplar ‘parvi’ videtur syste-matis, sicut moleculæ, nexi vicino ejus, sicut campo electro-magnetico. Præcipue examinatur quæstio num violet statusinfimus systematis spin-bosonici symmetriam quamdam quæHamiltonianum immutatum relinquit. Simulacri parametris(videlicet fisso ambos inter planities energeticas, frequentiismodorum campi nexusque magnitudine) definitos habentibusvalores, unus modo exstat status infimus qui est immutabilis asymmetria supra dicta, cum autem oriantur status infimi duosymmetriam violantes parametris alios valores habentibus. Inhac rerum condicio ambo status infimi regula superselectionisseparati sunt, id est, superpositiones inter ambos status infi-mos non fieri licet. Regulæ superselectionis modo in systema-tibus infinito cum numero graduum libertatis oriuntur. Perprincipium variationis secundum Ritz–Rayleigh talia syste-mata examinari possunt prisco formalismo mechanicæ quan-ticæ spatio Hilbertiano utente. Principium variationis perfici-tur ex conjectura systema bosonicum in superpositione sta-tuum cohærentium esse. Inquiruntur valores parametrorumubi status infimi symmetriam violantes oriuntur, atque emo-lumenta exilitatesque hujus rationis aguntur.

Zusammenfassung

Das Spin–Bosonen-Modell besteht aus einem 2-Niveau-Sy-stem, das an unendlich viele harmonische Oszillatoren ge-koppelt ist. Es wird als Prototyp betrachtet fur ein ”klei-nes“ System, z. B. eine Molekel, das mit seiner Umgebung,z. B. dem elektromagnetischen Feld, in Wechselwirkung steht.Diese Arbeit behandelt hauptsachlich die Frage, wann einGrundzustand des Spin–Bosonen-Modells eine Symmetrie ver-letzt, bezuglich derer der Hamiltonoperator invariant ist. Furbestimmte Werte der Systemparameter (Niveauaufspaltung,Oszillatorfrequenzen und Kopplungsstarken) existiert nur eineinziger Grundzustand, der bezuglich dieser Symmetrie inva-riant ist, wahrend fur andere Parameterwerte zwei Grund-zustande auftreten, die die Symmetrie verletzen. In letzteremFall besteht zwischen den beiden Grundzustanden eine Su-perauswahlregel, d. h. Superpositionen zwischen den beidenZustanden sind verboten. Superauswahlregeln gibt es nur beiSystemen mit unendlich vielen Freiheitsgraden. Mit dem soge-nannten Variationsprinzip kann man mit einem soliden theo-retischen Hintergrund derartige Systeme im Rahmen der tra-ditionellen Hilbertraum-Quantenmechanik untersuchen. Die-ses Variationsprinzip wird mit einem Ansatz durchgefuhrt,bei welchem der Zustand des Bosonensystems eine Superposi-tion von koharenten Zustanden ist. Diejenigen Parameterbe-reiche werden untersucht, bei welchen Grundzustande auftre-ten, die die Symmetrie verletzen, und die Moglichkeiten undSchwachen des benutzten Ansatzes werden diskutiert.

Contents

1 Introduction and survey 1

2 Some heuristic backgrounds of the spin–boson model 92.1 The paradox of optical isomers . . . . . . . . . 102.2 The spin–boson Hamiltonian as a

model Hamiltonian for a moleculecoupled to the electromagnetic field . . . . . . . 17

2.3 Conditions for symmetry breaking . . . . . . . 25

3 Applying the variation principle to thespin–boson Hamiltonian 313.1 The variation principle for systems

with infinitely many degrees of freedom . . . . 323.2 When are superpositions of ground states

forbidden? . . . . . . . . . . . . . . . . . . . . . 423.3 Ground states of the spin–boson model approxi-

mated by a superposition of coherent states . . 503.4 Minimising the energy expectation value . . . . 563.5 The infinite-mode limit . . . . . . . . . . . . . . 663.6 The phase diagram of the spin–boson model

for a representative ohmic example . . . . . . . 70

References 87

Chapter 1

Introduction and Survey

The superposition principle is one of the most puzzling fea-tures of quantum mechanics. It claims that any two possiblestates Ψ1 and Ψ2 of a system may be ‘added together’ to formanother possible state, say, c1Ψ1 + c2Ψ2 (where the complexnumbers c1 and c2 may be arbitrarily chosen with the provisothat the superposition is properly normalised). For almost allmicroscopic systems, e.g., elementary particles, atoms, andsmall molecules, this seemingly counterintuitive principle hasbeen and still is overwhelmingly well-confirmed by a wealth ofexperiments. However, there remain some systems where thenaıve application of the superposition principle would allowstates that seem absurd from a common-sense point of viewand, much worse, that have eluded any empirical confirmationup to now. The most prominent examples for such systems aremacroscopic objects. Everyday-life items like billiard balls orplanets, for instance, have never been observed to be in a su-perposition of two states localised at different positions. Oneof the most grotesque examples of this kind is Schrodinger’s

1

2 CHAPTER 1. INTRODUCTION AND SURVEY

cat paradox (Schrodinger ): one can devise a machine thatwould allow to prepare a cat being in a superposition of thetwo states ‘cat alive’ and ‘cat dead.’

To account for the fact that such states have never been ob-served, one might claim that macroscopic systems should bedescribed in terms of classical mechanics rather than quantummechanics; classical mechanics does not contain a universallyvalid superposition principle. But then the question arises atwhich scale quantum mechanics breaks down, and, much moreseriously, what a universally valid theory containing classi-cal and quantum mechanics as special, limiting cases wouldbe like. There are still other more empirical objections. Mostmolecules, even small ones—hence systems clearly within therange of validity of quantum mechanics—seem to exhibit someobservables, such as handedness, that always have a definitevalue. Thus one single molecule of, say, an amino acid hasnever been observed to be in a superposition of the right-handed and left-handed states, although such a superpositionought to be the ground state of the isolated molecule. Thisparadox of optical isomers will be extensively dealt with inthe next chapter.

Another possible account for the non-observability of somesuperpositions can be found within the ‘orthodox’ Copen-hagen interpretation of quantum mechanics. The so-called re-duction postulate claims that, whenever an observable of aquantum-mechanical system is measured, this system imme-diately turns into a state where this observable has a def-inite value. No matter in which state a cat in a hermeticallyclosed Schrodinger apparatus happens to be, an observationof the cat by opening the apparatus constitutes a measure-

CHAPTER 1. INTRODUCTION AND SURVEY 3

ment, which transforms the state of the cat into either thestate ‘cat alive’ or the state ‘cat dead.’ Thus grotesque su-perpositions as the one of the states ‘cat alive’ and ‘cat dead’can in principle never be observed directly. This view arousesserious philosophical problems, since common sense demandsthat the cat must be either dead or alive, no matter whetherwe measure something or not.1

The purpose of this work is to explain that for some ‘sim-ple’ examples, the interaction of the system with the environ-ment can be shown to be responsible for a breakdown of thesuperposition principle. The system can never be completelyprevented from interacting with its surroundings, so it is notastonishing that the application of quantum mechanics to, say,an isolated cat or an isolated molecule may sometimes lead towrong results. It turns out that the key property of such anenvironment is to have a huge, if not infinite, number of de-grees of freedom. The previously mentioned paradoxes wouldbe resolved by showing that for systems coupled to a largeenvironment, certain ‘unwanted’ superpositions are unstableand decay quickly into some components.2 This approach, al-beit appealing, is not the one chosen in this work. Our inter-est will be restricted to the somewhat pathological situationwhere the number of degrees of freedom is infinite. Systemswith infinitely many degrees of freedom exhibit a qualitatively

1. For an extensive critique of the reduction postulate, see Primas(), chapter ; a more thorough discussion of Schrodinger’s catparadox may be found in Primas ().

2. See Amann () for a detailed description of this programme.


different behaviour, which in many respects may be simplerthan the one of systems with a finite but large number ofdegrees of freedom.

One important property of systems with infinitely many de-grees of freedom is that there are infinitely many inequivalentways of describing them within the framework of traditionalHilbert-space quantum mechanics. There exist, however, someformalisms, among them algebraic quantum mechanics,3 thatallow to deal with infinite systems in a manner independent ofthese inequivalent descriptions. Algebraic quantum mechanicsprovides an elegant tool for discussing the new features arisingonly in infinite systems, the most striking of which being su-perselection rules and classical observables. A superselectionrule is a rule forbidding superpositions between certain classesof states. If two pure states Ψ1 and Ψ2 are separated by asuperselection rule, then the state c1Ψ1 + c2Ψ2 is not a purestate any more, but a mixture. For a chiral molecule, e.g., amixture would correspond to a racemate. Superselection rulesentail the emergence of classical observables, i.e., observablesthat always have a definite value if the system is in a purestate. (Mathematically speaking, classical observables corres-pond to operators that commute with any other operator.)The handedness of a chiral molecule would be such a classicalobservable. Other good candidates for being shown to be clas-sical observables are, for instance, the charge of a particle, the

3. For an overview of algebraic quantum mechanics see, for instance,the introduction of Bratteli and Robinson () or chapter . ofPrimas ().


knot type of a dna strand, the temperature, or the chemicalpotential.4

Strictly speaking, the environment of a system consists inthe rest of the universe. There are therefore most sensiblereasons to replace it with a tractable ‘toy’ environment. Theenvironment dealt with in this work consists of a so-calledbosonic bath, i.e., infinitely many harmonic oscillators. It isone of the simplest systems with infinitely many degrees offreedom and is—more or less—appropriate for modelling thefollowing physical environments:

• the electromagnetic radiation field, which has been pro-posed to be responsible for the emergence of handednessas a classical observable for a molecule (see next chapter),

• phonons, e.g., for modelling an atom or a molecule embed-ded in a crystal,

• a ‘heat bath,’ for the discussion of dissipation effects,5

• the gravitational field, which is conjectured to be responsi-ble for the breakdown of delocalised states of macroscopicobjects.6

For the system itself, utmost simplification is used here, too:the simplest non-trivial quantum-mechanical system is a two-level system. Since a two-level system is completely equivalent

4. For further details and references, see Amann (), p. –,.

5. See, for instance, Leggett et al. ().

6. For an ad-hoc approach see, e.g., Diosi (); for a detailed reviewof the localisation problem and its relationship to gravitation, seeKasemir (), p. –, and references therein.


to a spin- 12 particle in a constant magnetic field, the system

is often referred to as ‘spin,’ and a two-level system coupledto a bosonic bath is thus called a spin–boson model. Whilewe shall take it here mainly as a model for a chiral moleculecoupled to an electromagnetic field, the spin–boson model hasa wealth of other applications, some of which will be brieflydiscussed in the next chapter.

We shall restrict our interest to the ground state of thisspin–boson system. Under certain circumstances the modelexhibits only one ground state, which is invariant with re-spect to some inversion symmetry (corresponding to a chi-rality inversion for a handed molecule). Under other circum-stances, there are two ground states that the inversion sym-metry transforms one into the other. The symmetry of theground state is then said to be broken, and the two groundstates turn out to be separated by a superselection rule. Thetwo parameters that decide whether the symmetry is brokenor not are the energy-level difference of the spin system andthe coupling strength between spin and bosonic system.

The main part of this work will be devoted to the Rayleigh–Ritz variation principle applied to the spin-boson model. Thisprocedure permits to investigate the symmetry properties ofthe ground state in a mathematically elementary way usingtraditional Hilbert-space quantum mechanics. Loosely speak-ing, it consists in choosing some particular state space (as pre-viously mentioned, there are infinitely many inequivalent pos-sibilities to do so), and then ‘approximating’ the ground stateby a sequence of states in this chosen state space, no matterwhether the latter does or does not contain the ground state.This variation principle has a rigorous foundation within the


framework of algebraic quantum mechanics. At least for somespecial cases there are some hints whether, for some givenvalues of the level splitting and the coupling strength, thesymmetry is broken or not. The variation principle may thusbe considered as an illustrative example of how to discuss sys-tems with infinitely many degrees of freedom using traditionalHilbert-space quantum mechanics.

Chapter 2

Some Heuristic Backgrounds of the

Spin–Boson Model

The spin–boson model, i.e., a two-level system (a ‘spin’) coup-led to a bath of infinitely many harmonic oscillators, may beconsidered as a prototype for the coupling of a ‘small’ sys-tem to its environment. A wealth of physical systems canbe considered as a heuristically sensible background for it.Here the paradox of optical isomers will be taken as the mainparadigm for discussing symmetry-breaking ground states ofthis model; other applications will only be mentioned briefly.It will be shown how the spin–boson model might be relevantfor modelling a molecule interacting with the electromagneticfield; it must, however, be kept in mind that it does by nomeans provide a rigorous formalisation of this problem. Fi-nally, we discuss the criteria a given ground state has to fulfillfor breaking the inversion symmetry, and we present somerigorous results obtained by Spohn () telling for whichvalues of the system parameters symmetry-breaking groundstates occur.

9

10 CHAPTER 2. HEURISTIC BACKGROUNDS

2.1 The Paradox of Optical Isomers

STATEMENT OF THE PARADOX

We consider two prototypes for ‘chiral’ molecules: the aminoacid alanine on one hand and isotopically substituted ammo-nia, NHDT, on the other hand. Let us assume that the nuclearmotion be governed by a Born–Oppenheimer-type Hamilto-nian where the nuclei move in an effective potential. If forthe sake of simplicity we restrict the nuclear motion to someinversion coordinate r , we obtain a double-minimum poten-tial like the one sketched in figure .. The energy spectrumconsists of a series of ‘doublets,’ at least for the lower energylevels. The higher the potential barrier V0 , the smaller will bethe level splitting of a doublet. For instance, the energy dif-ference (E2 − E1)/hc between the ground state and the firstexcited state of NHDT may be roughly estimated1 to be be-tween 0.01 cm−1 and 0.1 cm−1 . For alanine on the other hand,where V0 is much higher, (E2 − E1)/hc has been estimatedto be as small as 2 ·10−65 cm−1 (Pfeifer , p. ), so thatthese two levels may be considered as quasi-degenerate. Foran infinitely high potential barrier (V0 → ∞) the level split-ting E2 − E1 vanishes, resulting thus in energy levels with atwofold degeneracy.

1. This estimate is based on the experimental values for 14NH3

(0.7934 cm−1 ), 14ND3 (0.0531 cm−1 ), and 14NT3 (0.0102 cm−1 ),taken from Papousek and Spirko (), table .

2.1 THE PARADOX OF OPTICAL ISOMERS 11

The ground state and the first excited state correspond to asymmetric wave function Ψ1 and an antisymmetric wave func-tion Ψ2 , respectively. Both can be chosen to be real and then

V (r)

V0

r

E1E2

C

HOOC

H2N

H3CH

C

COOH

NH2

H3C

H

C

COOH

NH2

CH3H

N

H

DT

N

H

DT

N

H

DT

Figure .: The Born–Oppenheimer potential energy V (r) of a chiralmolecule plotted vs. the inversion coordinate r , resulting in a dou-ble-minimum potential, and the six lowest-lying energy levels


have the shape sketched in figure .. Since they ‘sit’ in bothminima of the potential, they do not correspond to a nuclearframe, i.e., the involved nuclei do not have an unambiguousposition. In figure . an attempt is made to visualise the dis-tribution of the nuclei. A molecule in one of these states isinvariant under space inversion and hence not chiral.

The chiral, i.e., the ‘left-handed’ and ‘right-handed’ forms ofalanine or NHDT, however, do have an approximate ‘nuclearframe’ defined by bond lengths and angles. They correspond

V (r)

r

2ε

Ψ2(r)

Ψ1(r)

ΨL(r)

ΨR(r)

r

r

r

r

Figure .: Wave functions Ψ1 and Ψ2 of a chiral molecule in its groundstate and in its first excited state, respectively. The non-stationary statesΨL and ΨR correspond to a well-defined nuclear frame, whereas thestationary states Ψ1 and Ψ2 do not.


Figure .: Distribution of the nuclei in the ground state Ψ1 of an am-monia-type molecule. Though still drawn, the nitrogen–hydrogen bondslose their meaning in this context.

to ‘localised’ wave functions that are peaked around one ofthe potential minima and may be constructed by appropriatesuperpositions of Ψ1 and Ψ2 :

ΨL := 1√2

(Ψ1 + Ψ2) and ΨR := 1√2

(Ψ1 − Ψ2) . (.)

Of course ΨL and ΨR are not stationary states any more sinceΨ1 and Ψ2 are eigenstates of the Hamiltonian with respectto—albeit slightly—different eigenvalues. Actually if for t = 0the molecule is in the state corresponding to, say, ΨL , the wave


function Ψt(r) along the inversion coordinate evolves in thefollowing way:

Ψt(r) = 1√2

[Ψ1(r) eiE1t/ + Ψ2(r) eiE2t/

]

= e−iE1+E2

2t [ΨL(r) cos εt+ i ΨR(r) sin εt] , (.)

with ε := E2−E12

, so the time evolution of the probabilitydensity is given by

|Ψt(r)|2 = ΨL(r)2 cos2 εt+ ΨR(r)2 sin2 εt . (.)

Hence the molecule ‘tunnels’ back and forth between thestates ΨL and ΨR (see figure .) with a beat period ofτ := π/ε , which may be considered as twice the mean lifeof the l- or r-form of the molecule.

Up to now, alanine has never been observed in the stateΨ1 or Ψ2 but only in the non-stationary superpositions ΨL

and ΨR , whereas microwave experiments suggest that NHDTnormally is in a non-localised state like Ψ1 or Ψ2 . This in-triguing situation is referred to as the paradox of opticalisomers.

Quack () as well as Cina and Harris (, ) have proposed someexperiments to prepare superpositions of handed states. These are basedon the fact that for most chiral molecules, there are electronically excitedstates that show a considerably reduced, if not absent, potential barrier.By trickily performed transitions from the electronic ground state tosuch an electronically excited state and back again, a state like Ψ1 mayin principle be prepared. There are even procedures for testing whetherthe obtained state is really achiral or not. Sadly, this experiment hasnot yet been carried out up to now; its result, whichever it may be, willsurely be highly interesting.


|Ψt(r)|2

r

t = 0 or t = π/ε

t = 18π/ε

t = 14π/ε

t = 38π/ε

t = 12π/ε

t = 58π/ε

t = 34π/ε

t = 78π/ε

Figure .: Time evolution of the probability density |Ψt(r)|2 along theinversion coordinate r of a chiral molecule. For t = 0 the molecule is inthe localised state ΨL .

Various attempts to account for this paradox have been madesince its discovery by Hund (); for an excellent review seePfeifer (), p. . One of the most popular ways to dealwith it is the explanation by Hund himself, stating that oncethe handedness of the molecule has been measured, it is in oneof the states ΨR or ΨL ; using the estimate from page foralanine these states have a mean life around 1047 years andmay therefore be considered as stable for all practical pur-poses. This point of view, however, does not really resolve theparadox: all chemical experience indicates that molecules likealanine always are in one of the chiral states ΨL or ΨR (and


not in the states Ψ1 or Ψ2 , which for an isolated molecule arestable as well), no matter whether we are measuring anythingor not. So the problem is only shifted to the question whoperforms or has performed the handedness measurement.

A SHORT DIGRESSION ON SOME RELATED PROBLEMS

There are some more physical settings based on a double-or multiple-minimum potential, where the occurrence or non-occurrence of ‘localised’ states as well as the symmetry prop-erties of the ground state(s) play a prominent role:

• Tunnelling Defects in Metals. Transition metals that con-tain a small amount of impurities may contain trappingcentres for dissolved hydrogen atoms. The tunnelling of ahydrogen atom between two neighbouring sites is thoughtto be influenced by lattice phonons (Vitali ).

• Macroscopic Quantum Superpositions in a SQUID Ring.A squid (Superconducting QUantum Interference Device)ring is a superconducting ring containing a ‘point-like’Josephson junction. The energy of the current through thisring varies periodically in function of the flux through thering. It is conjectured that there may be superpositions ofstates with different fluxes. Up to now, it is the only knowndevice that promises to prepare macrocsopic superpositions(see Leggett (a, b, ) for further details).

For both problems (as well as for paradox of optical isomers,see below), the spin–boson model is conjectured to be of somerelevance.

2.2 THE SPIN–BOSON HAMILTONIAN 17

2.2 The Spin–Boson Hamiltonianas a Model Hamiltonian for a MoleculeCoupled to the Electromagnetic Field

There is some evidence that the paradox of optical isomerscannot be resolved if, as in Hund’s approach, the molecule isregarded as an isolated system. The energy difference betweenthe ground state Ψ1 and the first excited state Ψ2 is indeed sosmall that even the tiniest external perturbations may havedramatic consequences. So there remains the question whichexternal medium can be chosen as an adequate model environ-ment. The electromagnetic field seems to be the most suitablecandidate for a molecular system (Pfeifer , p. ).

‘DERIVATION’ OF THE HAMILTONIAN

Let HMF = HM+HF be the usual nonrelativistic Hamiltonianof a molecule coupled to a quantised electromagnetic field,HM being the molecular part (inclusive the coupling to thefield) and HF being the part referring to the free radiationfield. Using si units and the Coulomb gauge, the molecularpart is given by

HM :=∑

j

12mj

[Pj⊗1l − zjeA(Qj)]2

+∑ ∑

j > j′

zjzj′e2

4πε0|Qj − Qj′ |⊗1l , (.)

where Pj , Qj , mj , and zje denote the momentum operator,the position operator, the mass, and the charge of the j-th


particle (assumed to be spinless), respectively. A is the vectorpotential of the magnetic field. If the zero-point energy of thefree radiation field is omitted, the Hamiltonian of the freeradiation field is given by

HF := 1l ⊗∑

σ=1,2

∫

R3

d3k ωk a∗kσakσ , (.)

where k is the wave vector of the electromagnetic radiationand σ = 1, 2 an index for its polarisation direction. Here ωk ,akσ and a∗kσ are the frequency, the annihilation and the cre-ation operator of the mode with wave vector k and polarisa-tion σ , respectively. The boson operators akσ and a∗kσ fulfillthe commutation relations

[akσ, ak′σ′ ] = 0

[akσ, a∗k′σ′ ] = δσσ′ δ(k − k′) .

(.)

The transverse vector potential A(r) is a function of akσ

and a∗kσ (see for example Cohen-Tannoudji et al. ):

A(r) =√

16π3ε0

∑

σ=1,2

∫

R3

d3kekσ√ωk

(akσ eik·r + a∗kσ e

−ik·r) ,

(.)where ek1 and ek2 denote the polarisation vectors, i.e., a setof two mutually perpendicular vectors such that k·ekσ = 0and |ekσ| = 1 for σ = 1, 2. Inserting (.) into (.) yieldsa rather formidable expression for HMF that will not be re-produced here. It can, however, be considerably simplified by


restricting the molecular part HM to the subspace spanned bythe two lowest-lying quasi-degenerate energy eigenstates2 Ψ1

and Ψ2 . This approximation can be reasonable if the energylevels E1 and E2 are sufficiently well separated from the restof the spectrum of HM so that the radiative coupling betweenΨ1 , Ψ2 and the higher-lying eigenstates of HM is negligible.For later convenience, we will use the two-level basis given bythe chiral states

Ψ+ := 1√2(Ψ1 + iΨ2) and Ψ− := 1√

2(Ψ1 − iΨ2) (.)

rather than Ψ1 and Ψ2 . The two-level approximation will thenbe implemented by the projector Π defined by

ΠHMFΠ :=(〈Ψ+|HMFΨ+〉〈Ψ+|HMFΨ−〉〈Ψ−|HMFΨ+〉〈Ψ−|HMFΨ−〉

), (.)

where the matrix elements 〈Ψ±|HMFΨ±〉 are operators actingon the Hilbert space of the radiation field, i.e., they are tobe understood as partial inner products with respect to themolecular states Ψ± . The two-level approximation HMF :=ΠHMFΠ of the Hamiltonian HMF then reads, up to an ad-

2. Note that here Ψ1, Ψ2 are in L2(R3N ) , where N is the number of par-ticles of the molecule, in contradistinction to the one-dimension andone-‘particle’ wavefunctions Ψ1, Ψ2 ∈ L2(R) referred to in page .


ditive constant (for details, see Pfeifer , p. ):

HMF

= ε σ1⊗1l + 1l ⊗

∑

σ=1,2

∫

R3

d3kωk a∗kσakσ

+ (2π)−32 σ3 ⊗

∑

σ=1,2

∫

R3

d3k λkσ(akσ + a∗kσ)

+ (2π)−3∑

σ=1,2

∑

σ′=1,2

∫

R3

∫

R3

d3k d3k′ ekσ·ek′σ′√ωkωk′

·

· (γ1⊗akσak′σ′ + γ2⊗akσa∗k′σ′

+ γ3⊗a∗kσak′σ′ + γ4⊗a∗kσa∗k′σ′) .

(.)Here σ1 is one of the Pauli matrices

σ1 :=(

0 11 0

), σ2 :=

(0 −ii 0

), σ3 :=

(1 00 −1

),

which describe the two-level ‘molecule’ (the ‘spin’). The so-called coupling constants λkσ are given by

λkσ :=√

12ε0ωk

ekσ ·∑

j

zje

mj〈Ψ2|i cos(k·Qj) Pj Ψ1〉 , (.)

and the two-level molecular operators γ1 , γ2 , γ3 , and γ4 are2×2-matrices whose elements have a form similar to (.).The derivation of (.) relies on several assumptions aboutthe wavefunctions Ψ1 and Ψ2 , the most important of which


being that

Ψ1(−q1, . . . ,−qN ) = Ψ1(q1, . . . , qN )Ψ2(−q1, . . . ,−qN ) = −Ψ2(q1, . . . , qN ) .

(.)

These symmetry properties can be derived rigourously (Pfei-fer , p. ), but are already heuristically plausible uponinspection of the energy eigenstates of a double-minimum po-tential (figure .).

The Hamiltonian (.) is still untractable because of thelast term, i.e., the one quadratic in the boson operators, whichstems from the so-called A2-term of the full molecular Hamil-tonian (.). There are several ways to deal with it:

• First approach. Because the A2-term is believed to havesignificant effects only in extreme strong-field situations,the practice of simply dropping it altogether from (.) iscustomary, albeit questionable. This is the approach chosenby Pfeifer ().

• Second approach. If one uses the standard dipole approxi-mation, which consists in setting r = 0 = const in (.)(and hence, for instance, setting the cosine in (.) tounity), the A2-term in the Hamiltonian (.) simplifies to

(2π)−3∑

σ=1,2

∑

σ′=1,2

∫

R3

∫

R3

d3k d3k′ ekσ·ek′σ′√ωkωk′

·

·(akσ + a∗kσ)(ak′σ′ + a∗k′σ′) (.)

and may then be incorporated into the other terms of theHamiltonian (.) by redefining the boson operators with a


so-called Bogoliubov transformation (for details, see Amann, p. ); this transformation also changes the couplingconstants in a way difficult to investigate. Furthermore, thedipole approximation has the drawback that it might leadto divergences like

∑

σ=1,2

∫

R3

d3k|λkσ|2ωk

= ∞ (.)

(with the consequence that the Hamiltonian is not boundedbelow any more) that must be repaired by an appropriateultraviolet cutoff.

Both approaches lead to the same form of the Hamiltonian,

˜HMF

= ε σ1⊗1l + 1l ⊗

∑

σ=1,2

∫

R3

d3kωk a∗kσakσ

+ (2π)−32 σ3 ⊗

∑

σ=1,2

∫

R3

d3k λkσ(akσ + a∗kσ) , (.)

but nevertheless to different coupling constants λkσ . We shallsee later that the infrared singularity

∑

σ=1,2

∫

R3

d3k|λkσ|2ω2

k

= ∞ (.)

will finally decide whether symmetry-breaking ground statesarise or not. This relation can be shown to hold if one recurs tothe first approach, whereas little is known about the infrared


behaviour of the coupling constants resulting from the sec-ond approach. It is even conjectured that the A2-term mightregularise the problem by cancelling the infrared singularity(Amann , p. ).

THE DISCRETE-MODE SPIN–BOSON HAMILTONIAN

In chapter , we are going to apply the variational principleto the ‘discrete-mode’ spin–boson Hamiltonian

H

= ε σ1⊗1l

︸︷︷︸HS

+ 1l ⊗∞∑

n=1ωna

∗nan

︸︷︷︸HB

+σ3 ⊗∞∑

n=1λn(an + a∗n)

︸︷︷︸HI

,

(.)which is thought to approximate the ‘continuous-mode’ spin–boson Hamiltonian (.) reasonably well.3 The term HS

refers to the isolated two-level system, the ‘spin,’ with an en-ergy difference of 2ε between the two levels. The term HB

refers to the isolated boson field, and the last term HI refersthe coupling between spin and boson field.

The canonical commutation relations of the annihilationand creation operators an and a∗n of the boson field now read

[an, an′ ] = 0 , [an, a∗n′ ] = δnn′ , n, n′ = 1, 2, 3, . . . (.)

3. Note that, in our case, switching to a discrete spin–boson Hamilto-nian does not correspond to confining the electromagnetic field intoa cavity. Although this indeed results in a discrete set of field modes,it also introduces an infrared cutoff that cancels the crucial infraredsingularity.


The ‘level splitting’ ε and the angular frequencies ωn of thebosonic modes numbered by n are assumed to be strictlypositive. The coupling constants λn between the spin and then-th bosonic mode are assumed to be real numbers fulfilling

Λ :=∞∑

n=1

λ2n

ωn< ∞ and

∞∑

n=1λ2

n < ∞ . (.)

The first condition guarantees that the Hamiltonian H isbounded below (Amann b, p. , proposition A). Notethat ε , the ωn ’s, and the λn ’s all have the dimension of anangular frequency.

The infrared singularity, which will turn out to be crucialfor the emergence of symmetry-breaking ground states, nowreads ∞∑

n=1

λ2n

ω2n

= ∞ . (.)

Its strength is determined by the function

W (t) :=∞∑

n=1λ2

n e−ωnt (.)

introduced by Spohn (). One can show that there exists aconstant ζ > 0 such that the following proportionality holds:

W (t) ∼ t−ζ for large t . (.)

The ir singularity holds true if 1 < ζ 2. In the terminol-ogy of Leggett et al. (), the case ζ = 2 is referred to asohmic; the situation 1 < ζ < 2, where the ir singularity iseven stronger, is called subohmic. If ζ > 2 no ir singularityat all arises. This situation is called superohmic.

2.3 CONDITIONS FOR SYMMETRY BREAKING 25

2.3 Conditions for Symmetry Breaking

SYMMETRY-BREAKING STATES

The space-inversion symmetry is defined by

Q → −Q and P → −P

and, according to (.), leaves Ψ1 untouched and transformsΨ2 into −Ψ2 . If a ground state Ψ of the system molecule +field is—in contradistinction to the ground state Ψ1 of theisolated molecule—not invariant under the space inversion,then the space-inversion symmetry is said to be broken withrespect to the ground states.

The space-inversion symmetry transforms Ψ+ into Ψ− andΨ− into Ψ+ . For the two-level system the space inversion istherefore implemented by the transformation matrix

(0 11 0

),

which transforms the Pauli matrices σ1 , σ2 and σ3 in thefolllowing way:

σ1 → σ1

σ2 → −σ2

σ3 → −σ3 .(.)

The transformation properties of the boson operators akσ

and a∗kσ depend on the transformation properties of the po-larisation vectors ekσ , which are a matter of convention. Herethe choice

akσ → −akσ , (.)

according to (.), has been made.


Thus, for the finite-mode Hamiltionian H from equa-tion (.), we can define a ‘space-inversion’ symmetry ι byputting

ι(σ1⊗1l) := σ1⊗1lι(σ2⊗1l) := −σ2⊗1lι(σ3⊗1l) := −σ3⊗1l

ι(1l⊗an) := −1l⊗an

ι(1l⊗a∗n) := −1l⊗a∗nn = 1, 2, 3, . . .

(.)

One can easily verify that H is invariant under ι . Hence, ifsome ground state of H is implemented by an expectation-value functional ϕ: T → ϕ(T ) (where T denotes an arbitraryobservable of the spin–boson system, i.e., an operator built upfrom Pauli matrices and boson operators), then the functional

ϕ˚ι: T → ϕ(ι(T )

)(.)

is another ground state. If ϕ˚ι = ϕ , the symmetry ι issaid to be broken with respect to the ground states. Weshall focus our interest to the special case where σ3 has dif-ferent expectation values with respect to ϕ and ϕ˚ι , i.e.,ϕ(σ3⊗1l) = ϕ˚ι(σ3⊗1l). Obviously this is the case whenϕ(σ3⊗1l) is non-zero. The quantity ϕ(σ3⊗1l) is called the or-der parameter of the model.

Using the expectation value of σ3⊗1l for comparing ϕ and ϕ˚ι is theonly reasonable choice if one restricts oneself to observables of the two-level system alone. Indeed, σ1 always has the same expectation valuewith respect to ϕ and ϕ˚ι ; on the other hand, the expectation valueof σ2 with respect to a stationary state ϕ is always zero, since one has

0 = ϕ(σ3) = ϕ

(i

[H, σ3]

)= 2ε ϕ(σ2) (.)

in the Heisenberg picture.


RIGOROUS RESULTS FOR THE PHASE DIAGRAMOF THE SPIN–BOSON MODEL

It will prove useful to introduce a parameter , called couplingstrength, which allows to scale simultaneously all the couplingconstants λn . It is defined to be twice the proportionalityconstant in equation (.), viz.

W (t) =N∑

n=1λ2

n e−ωnt ≈ 1

2 t−ζ for large t . (.)

Note that has the dimension [time]ζ−2 ; it is dimensionlessin the ohmic case (ζ = 2). The coupling strength and thelevel splitting ε will turn out to be the only important pa-rameters for deciding whether the symmetry is broken or not.A chart where the symmetry-breaking properties are sketchedon a (, ε)-plane is called the phase diagram of the spin–bosonmodel.

The pioneering study of the conditions for symmetry-breaking ground states of a spin–boson model has been madeby Pfeifer (). He investigated the ohmic case and re-stricted the class of allowed states to product states. Thephase diagram thus obtained (figure .) will turn out to ex-hibit at least the correct asymptotic behaviour for large .

The first rigorous results were derived by Spohn (). Theground states investigated there are obtained as temperature-to-zero limits of thermic equilibrium states. Furthermore, a


coupling strength

level splittingε

slope Λ

Figure .: Phase diagram of the spin–boson model obtained by Pfeifer.The space-inversion symmetry is broken for parameters in the shadedregion.

perturbation (‘bias’) hσ3⊗1l has been added to the Hamilto-nian; then two different ground states of the original Hamil-tonian may be obtained by performing the limit h → 0. Theresults are:

• In the superohmic case (ζ > 2) the symmetry remains un-broken.

• In the ohmic case (ζ = 2), the line ε = 0 belongs to thebroken-symmetry region. Furthermore, for any level split-


ting ε = 0 there exists a critical coupling strength cr(ε)such that the symmetry is broken for cr(ε) and thatthe symmetry is unbroken for > cr(ε). This critical coup-ling strength fulfills (cf. figure .)

cr(ε) 1 for all ε > 0 and lim supε→0

cr(ε) 2 (.)

cr(ε)ε

1Λ

for all ε > 0 and limε→∞

cr(ε)ε

=1Λ

. (.)

1 2

ε

Figure .: A typical phase diagram respecting Spohn’s conditionsfor the ohmic spin–boson model. The dashed line refers to Pfeifer’sphase-transition line.


• In the subohmic case (1 < ζ < 2), for any, even small coup-ling strength the inversion symmetry is broken if only thelevel splitting is small enough. Furthermore, the criticalcoupling strength is of the order ε2−ζ for small ε (cf. fig-ure .).

ε

Figure .: A typical phase diagram respecting Spohn’s conditions forthe subohmic spin–boson model. The dashed line refers to Pfeifer’sphase-transition line.

Chapter 3

Applying the Variation Principle

to the Spin–Boson Hamiltonian

When applying the Rayleigh–Ritz variation principle to somegiven Hamiltonian, one normally assumes that the underlyingHilbert space actually contains the ground-state vector lookedfor. This is in general not the case for systems with infinitelymany degrees of freedom; the ground-state vector may ‘live’in some ‘different’ Hilbert space belonging to a new represen-tation of the system’s observables. But even when working inthe ‘wrong’ representation (here the Fock representation), thevariation principle can yield the right ground-state energy andcan provide some information about the symmetry-breakingproperties of the ground state. It will be carried out here usinga fairly general ansatz where the state of the bosonic system isassumed to be in a superposition of coherent states. For somespecial ranges of the level splitting and the coupling strength,some hints whether the symmetry is broken or not will beobtained.

31

32 CHAPTER 3. APPLYING THE VARIATION PRINCIPLE

3.1 The Variation Principlefor Systems with Infinitely ManyDegrees of Freedom

INEQUIVALENT REPRESENTATIONSOF COMMUTATION RELATIONS

If the spin–boson Hamiltonian H from equation (.) is trun-cated to a finite-mode ‘approximation,’

H (N) := ε σ1⊗1l︸︷︷︸

H(N)S

+ 1l ⊗N∑

n=1ωna

∗nan

︸︷︷︸H

(N)B

+σ3 ⊗N∑

n=1λn(an + a∗n)

︸︷︷︸H

(N)I

,

(.)(we set = 1 from now on), it is, by virtue of the Stone–von-Neumann theorem (Reed and Simon , theorem viii.),uniquely defined up to trivialities; this means that, up to uni-tary equivalence and multiplicity, the bosonic annihilation andcreation operators an and a∗n can be uniquely represented asoperators acting on a Hilbert space H(N) so that the canonicalcommutation relations

[an, an′ ] = 0 , [an, a∗n′ ] = δnn′ , n, n′ = 1, 2, 3, . . . (.)

are fulfilled. If, however, the number of degrees of freedom(in our case the number N of bosonic modes) is infinite, theStone–von-Neumann theorem no longer holds true: there existuncountably many physically inequivalent representations ofthe commutation relations (.) on Hilbert spaces.

3.1 INFINITELY MANY DEGREES OF FREEDOM 33

Once a particular representation has been chosen for thebosonic system alone, the Hilbert space for the full spin–bosonmodel (spin and boson field) then is the tensor product ofa two-dimensional Hilbert space with the Hilbert space Hthe chosen representation acts on, namely C

2⊗HF . Since theproblem of inequivalent representations only occurs for thebosonic system—the spin system having only one degree offreedom—no distinction will be made between the represen-tations for the bosonic system and the representations for thefull spin–boson system.

Even if one imposes on the representation that the infinite-mode limit N → ∞ of the finite-mode Hamiltonian H (N)

exist—a reasonable condition for a representation to be ofphysical interest—the number of inequivalent representationsstill remains infinite. An example for a representation meet-ing this requirement is the so-called Fock representation; theHilbert space the bosonic operators act on is the (bosonic)Fock space HF (Reed and Simon , p. ). One of themain features of the Fock representation is that the bosonpart HS of the (infinite-mode) Hamiltonian H admits aground state ΨF referred to as Fock state.

The Fock space does, in general, not contain a ground statefor the full Hamiltonian H . So another physically sound con-dition to be imposed on a representation of the bosonic com-mutation relations is that the infinite-mode Hamiltonian Hadmit a ground state. This means that the underlying Hilbertspace H must contain an eigenvector Ψ0 of H ,

HΨ0 = E0Ψ0 , (.)


with an eigenvalue E0 at the lower end of the Hamiltonian’sspectrum, i.e., the inequality

〈Ψ |HΨ〉〈Ψ |Ψ〉 E0 (.)

must hold for every non-zero vector Ψ in the underlyingHilbert space. If one happens to use the ‘wrong’ represen-tation, the infimum

E0 := infΨ =0

〈Ψ |HΨ〉〈Ψ |Ψ〉 (.)

still exists if the Hamiltonian is bounded below, but the un-derlying Hilbert space does not contain a vector Ψ0 fulfill-ing (.), which ‘lives’ in a different representation.

To be more explicit, consider a sequence of normalised vec-tors Φk (k = 1, 2, . . .) in the original (‘wrong’) Hilbert space,such that 〈Φk|HΦk〉 converges to E0 defined by (.), andsuch that the expectation values 〈Φk|TΦk〉 converge to ϕ(T )for arbitrary ‘observables’ T of the system. Then one cannotexpect that this Hilbert space contains some vector Ψ0 whichfulfills ϕ(T ) = 〈Ψ0|TΨ0〉 . For every given state1 T → ϕ(T ),one can, however, construct a representation where the un-derlying ‘new’ Hilbert space contains a vector Ψ0 that ‘im-plements’ the state ϕ in the sense that ϕ(T ) = 〈Ψ0|TΨ0〉 .

1. Here a state ϕ is taken to be a linear functional that maps an observ-able T (here an operator acting on a Hilbert space) onto its expecta-tion value ϕ(T ) . The state ϕ is considered to be physically significantif it is possible to implement it by a density matrix, i.e., a positivetrace-class operator D with trace one, such that ϕ(T ) = tr(DT ) .


This representation is called the Gel’fand–Naımark–Segal rep-resentation (in short: gns representation) with respect to thestate ϕ (Bratteli and Robinson , ..−−).

Note that, since all infinite-dimensional Hilbert spaces witha countable basis are isomorphic, not the Hilbert spaces them-selves ‘differ’ from one representation to another, but ratherthe way the operators of the relevant observables act on them.

One possible way to circumvent the difficulties arising frominequivalent representation is using the formalism of algebraicquantum mechanics (see references in footnote on page ).Here the context-independent observables, i.e., the observ-ables referring to finitely, but arbitrary many degrees of free-dom, are treated abstractly without any reference to a partic-ular Hilbert-space representation. Once such a representationis chosen, context-dependent observables can be introducedas appropriate limits of the context-independent ones.

A VARIATION PRINCIPLE DERIVED FROMAN ALGEBRAIC VERSION OF THE SPIN–BOSON MODEL

Consider the functions u → Q(N)(u) and u → Q(u) suchthat, heuristically speaking, Q(N)(u) and Q(u) are the ‘low-est achievable’ energy expectation value of the HamiltoniansH (N) and H , respectively, under the constraint that the cor-responding expectation value of σ3⊗1l be equal to u . Morerigorously, the functions

Q(N) : [−1,+1] → R

Q : [−1,+1] → R


are defined by

Q(N)(u) := infω(H (N))

∣∣∣ ω is a normal state on B(C2⊗H(N))and ω(σ3⊗1l) = u

Q(u) := infω(H)

∣∣∣ ω is a normal state on B(C2⊗H)and ω(σ3⊗1l) = u

.

(.)A state is called normal if it is σ-weakly continuous; this isthe case if and only if it is implemented by a trace-class opera-tor acting on the underlying Hilbert space. Here B(C2⊗H(N))denotes the set of bounded operators acting on the Hilbertspace C

2⊗H(N) of the finite-mode system and B(C2⊗H) de-notes the set of bounded operators acting on the Hilbertspace C

2⊗H of the finite-mode system within a particularrepresentation where H is the underlying Hilbert space forthe boson system. Note that Q must a priori be consideredrepresentation-dependent.

The following theorem can be proved rigorously (Amannb, section ) within the framework of algebraic quantummechanics:

Theorem 3.1 The functions u → Q(N)(u) and u → Q(u)are convex, continuous on the open interval ]−1,+1[ and evenwith respect to the transformation u → −u on [−1,+1] . Forevery given u ∈ [−1,+1] the sequence

(Q(N)(u)

)N=1,2,...

ismonotonously decreasing, i.e., N M implies

Q(N)(u) Q(M)(u) . (.)

The infimumQ(u) := inf

N=1,2,...Q(N)(u) (.)


coincides with Q(u) up to normalisation, i.e., up to a constantfunction of u .

The theorem states that the function Q does not depend,up to normalisation, on the particular representation chosen.Thus, if ϕ is a ground state of the spin–boson model (withrespect to an arbitrary representation), then Q(u) takes itsminimal value at u = ϕ(σ3⊗1l).

Conversely, if one has the situation sketched in figure .,i.e., if Q(u) admits a unique minimum (this minimum canonly be taken at u = 0 since Q is even), then the groundstate ϕ fulfills ϕ(σ3⊗1l) = 0, and the symmetry is not broken.This ground state even turns out to be the Fock ground state,according to the following theorem (Amann c, theorem p. ):

u

Q(u)

−1 +1

Figure .: Sketch of the function u → Q(u) as defined by equation (.)if the space-inversion symmetry ι is unbroken


Theorem 3.2 The spin–boson model admits a ground-statevector in the Fock representation if and only if the functionu → Q(u) has precisely one minimum at u = 0 .

If, on the other hand, the function u → Q(u) behavesas sketched in figure ., i.e., if Q(u) is not only minimal

u

Q(u)

−1 −u0 +u0 +1

Figure .: Sketch of the function u → Q(u) as defined by equation (.)if the space-inversion symmetry ι is broken

for a unique value of u , but for a whole range (here foru ∈ [−u0,+u0]), the situation is a little bit more delicate.Heuristically one might expect that there are different groundstates (i.e., states with minimal energy) ϕ+ and ϕ− fulfilling

ϕ±(σ3⊗1l) = ±u0 , (.)

which implies that the inversion symmetry is broken. It turnsout that this can indeed be proven rigorously, according tothe following theorem (Amann b, section ):


Theorem 3.3 Let u0 be the supremum of all u ∈ [−1, 1]such that Q(u) takes its minimal value, i.e.,

u0 := supu ∈ [−1,+1] | Q(u) is minimal . (.)

Then there exist ground states ϕ+ and ϕ− of the spin–bosonmodel fulfilling equation (.).

For a given set of the parameters ε , ωn , and λn , one canthus decide whether the inversion symmetry ι is broken ornot by examining at which values of u the quantity Q(u) isminimal. The representation the calculation of Q(u) is madewithin can be chosen at random; one may use, e.g., the Fockrepresentation, even if the underlying Hilbert space does notnecessarily contain the ground state looked for. Hence the de-cision whether the symmetry is broken can be made withoutany reference to algebraic quantum mechanics: algebraic con-siderations only provide, behind the scenes, a rigorous justifi-cation of the recipe used for the calculations, putting it intoa proper frame and showing that it is not completely empty.

Since the computation of Q(u) is by no means a simpletask, one must look for further simplifications for the aboveprocedure. A ‘variant’ of the minimal-energy function whereone restricts oneself to pure vector states will turn out to beuseful as well. For the finite-mode Hamiltonian, it is definedby

R(N)(u) := inf〈Ω|H (N)Ω〉

∣∣∣∣Ω ∈ C

2⊗H(N) , 〈Ω|Ω〉 = 1 ,and 〈Ω|σ3⊗1lΩ〉 = u

.

(.)


One can easily show that this function u → R(N)(u), like u →Q(N)(u), is even with respect to the transformation u → −u .Furthermore, it has the following properties:

Theorem 3.4 For every given u ∈ [−1,+1] the sequence(R(N)(u)

)N=1,2,...

is bounded below, namely

R(N)(u) −(

N∑

n=1

λ2n

ωn+ ε

)

(.)

and monotonously decreasing, i.e., N M implies

R(N)(u) R(M)(u) . (.)

Proof: The first assertion is a consequence of equation (.); see below.For the second assertion recall that

H(M) = H(N) + 1l ⊗M∑

n=N+1ωna

∗nan + σ3 ⊗

M∑

n=N+1λn(an + a

∗n) (.)

and consider product vectors Ω′ of the form Ω′ = Ω ⊗ |0〉⊗ · · · ⊗ |0〉 ,with the |0〉 being the ground-state vectors of the bosons N + 1through M , respectively. Because of an|0〉 = 0 one has

〈Ω′|H(M)Ω′〉 = 〈Ω|H(N)Ω〉 , (.)

from which equation (.) follows immediately.

For infinitely many modes of the boson field one can put forevery u ∈ [−1,+1]:

R(u) := limn→∞

R(N)(u) . (.)

The existence of this limit is a consequence of theorem ..Since for Q(N) and Q the class of states used for calculating


the infimum of the energy expectation value is larger than theone used for R(N) and R (not every state can be implementedby a vector), one expects that R(N)(u) Q(N)(u) and R(u) Q(u) for every u ∈ [+1,−1]. The following theorem makes aneven stronger assertion:

Theorem 3.5 The functions Q(N) and Q are the largestconvex functions that are smaller than R(N) and R , respec-tively.

Proof: Every density operator is a mixture of vector states.

Thus, if the inversion symmetry is broken, it might happenthat R(u) has several minima, e.g., at u = ±u0 as in fig-ure . (solid line), whereas Q(u) is minimal for a whole rangeof u , namely for u ∈ [−u0,+u0] . Nevertheless the shape of

u

R(u)

−1 −u0 +u0 +1

Figure .: Sketch of the function u → R(u) if the space-inversion sym-metry ι is broken (solid line). The function u → Q(u) is the largestconvex function that is smaller than R , i.e., the graph of Q is the lowerborder of the grey area.


the function u → R(u) allows to decide as well whether thesymmetry is broken or not.

3.2 When Are Superpositions ofGround States Forbidden?

TRANSFORMING AWAY THE INTERACTION TERMIN THE FINITE-MODE HAMILTONIAN

First of all, we shall apply a unitary transformation Uto the Hamiltonian H (N) that ‘eliminates’ the interactionterm H (N)

I , namely

U :=(W (ω−1λ) 00 W (−ω−1λ)

). (.)

Here the coupling constants and the oscillator frequencies havebeen grouped together into a vector and into a diagonal ma-trix, respectively,

λ :=

λ1

...λN

, ω :=

ω1 0. . .

0 ωN

, (.)

and W ( · ) denotes the unitary Weyl operator defined by

W (z) := exp

N∑

n=1(a∗nzn − z∗nan)

for z :=

z1...zN

∈ C

N .

(.)

3.2 WHEN ARE SUPERPOSITIONS FORBIDDEN? 43

In (.) an obvious matrix notation has been used instead ofthe tensor-product notation, i.e., one has made the identifica-tion (

aT bTcT dT

):=

(a bc d

)⊗ T (.)

for arbitrary scalars a, b, c, d ∈ C and arbitrary observables Tfor the boson field alone.

Using the following well-known properties of the Weyl op-erators,

W (z)W (z′) = W (z + z′) e−i(z|z′)

W (−z) an W (z) = an + zn

W (−z) a∗n W (z) = a∗n + z∗n ,

(.)

where ( · ) denotes the imaginary part and ( · | · ) denotesthe standard scalar product in C

2 , namely

(x|y) :=N∑

n=1x∗nyn and ‖x‖ :=

√(x|x) , (.)

a somewhat tedious but straightforward calculation yields

U H(N)S U−1 = ε

(0 W (2ω−1λ)W (−2ω−1λ) 0

)(.)

U (H (N)B +H (N)

I )U−1 = 1l ⊗N∑

n=1ωna

∗nan −

N∑

n=1

λ2n

ωn1l . (.)

Hence the entire transformed Hamiltonian reads

H (N) := U H (N) U−1

= ε

(0 W (2ω−1λ)W (−2ω−1λ) 0

)+ 1l ⊗

N∑

n=1ωna

∗nan − Λ1l ,

(.)


with Λ as usually defined by

Λ :=N∑

n=1

λ2n

ωn= (λ|ω−1λ) . (.)

GROUND STATES OF THE SPIN–BOSON HAMILTONIANWITH ZERO LEVEL SPLITTING

As a preliminary example we shall investigate the groundstates of the spin–boson model for the special case ε = 0. Aninspection of these ground states will turn out to be heuris-tically useful for finding a suitable ansatz for the variationprinciple when we are discussing the general case ε = 0.

Let H (N)0 be the finite-mode Hamiltonian with zero level

splitting. The corresponding transformed Hamiltonian H (N)0

is given by equation (.), namely

H (N)0 = U H(N)

0 U−1 = 1l ⊗N∑

n=1ωna

∗nan − Λ1l . (.)

The ground-state vectors of this operator are known to be ofthe form

Ψ0 =(α

β

)⊗ Ψ (N)

F =(αΨ (N)

F

β Ψ (N)F

), α, β ∈ C , (.)

where Ψ (N)F denotes the ground state of a system of N un-

coupled harmonic oscillators, namely the N -fold tensor prod-uct |0〉⊗ · · · ⊗|0〉 with |0〉 being the ground-state vector ofone harmonic oscillator. It is characterised by

anΨ(N)F = 0 for N = 1, 2, . . . , N (.)


or, equivalently, by

〈Ψ (N)F |W (z)Ψ (N)

F 〉 = e−12‖z‖2

for every z ∈ CN . (.)

The corresponding eigenvalue is E0 := −Λ . This implies thatthe ground states of the untransformed zero-level-splittingHamiltonian H0 are of the form

Ψ0 = U−1 Ψ0 =(W (−ω−1λ) 00 W (ω−1λ)

) (αΨ (N)

F

β Ψ (N)F

). (.)

Hence the eigenspace of E0 is spanned by the two vectors

Ψ (N)+ =

(W (−ω−1λ)Ψ (N)

F

0

)

Ψ (N)− =

(0

W (+ω−1λ)Ψ (N)F

),

(.)

and arbitrary superpositions within (and outside) that eigen-space are allowed. The bosonic states implemented by vectorsof the form W (z)Ψ (N)

F , with z ∈ CN , are referred to as co-

herent states of the N -boson system.

What happens now if one goes to the limit N → ∞? Withinthe Fock representation, the ground state of the transformedzero-level-splitting Hamiltonian H0 is the Fock vector ΨF

mentioned on page , which ‘lives’ in the underlying FockHilbert space HF . The Fock vector ΨF may be consideredas defined by equation (.). However, things become muchmore delicate if one considers the corresponding ground-state‘vectors’ for the untransformed Hamiltonian. Actually the


transformation U itself is ill-defined, since for a Weyl op-erator W (z) the norm ‖z‖ must be finite (i.e., z must besquare-integrable) because of equation (.). This is not thecase for z = ω−1λ because of the infrared singularity. Hencethe vectors

Ψ+ =(W (−ω−1λ)ΨF

0

)

Ψ− =(

0W (+ω−1λ)ΨF

),

(.)

do not ‘live’ in the Fock space. Fortunately, by abuse of no-tation, they can be regarded as defining states in the sense ofexpectation-value functionals. More precisely, let us define

ϕ(N)+ : T →

⟨Ψ (N)

+

∣∣T Ψ (N)

+

⟩

ϕ(N)− : T →

⟨Ψ (N)

−

∣∣T Ψ (N)

−

⟩ (.)

for arbitrary observables T . The expectation values of (.)can be shown to converge to finite values for N → ∞ (Amann, p. ):

limN→∞

⟨Ψ (N)

+

∣∣T Ψ (N)

+

⟩=: ϕ+(T ) < ∞

limN→∞

⟨Ψ (N)

−

∣∣T Ψ (N)

−

⟩=: ϕ−(T ) < ∞

The ‘pseudo-vectors’ Ψ+ and Ψ− are simply meant to ‘rep-resent’ the states ϕ+ and ϕ− , even if these states cannot beimplemented as vectors in the Fock space. They may, however,be thought of as a limit of a sequence of vectors. To constructsuch a sequence, it is sufficient to ‘embed’ the finite-mode vec-tors Ψ (N)

+ and Ψ (N)− into the infinite-mode Fock space: Ψ (N)

+


and Ψ (N)− are understood to behave as before for the first

N degrees of freedom and as Fock vectors for the remainingdegrees of freedom (from N + 1 through infinity). The expec-tation values of these vectors then converge precisely to theexpectation values ϕ+(T ) and ϕ−(T ).

This suggests the following recipe for applying the variationprinciple to the Hamiltonian with non-zero level splitting: Forfinite N , start with a suitable ansatz for the ‘test vectors’ Ωoccurring in the definition of R(u) (equations . and .)and compute the lowest achievable energy expectation value;then—at the level of the expectation values—switch to thelimit N → ∞ . This limit can now be performed quite safelysince it refers to mere numbers rather than Hilbert-space vec-tors or operators.

For both zero-level-splitting Hamiltonians H (N)0 and H0 the

inversion symmetry must be considered to be broken since,say, for the finite-mode case one has

〈Ψ (N)+ |σ3⊗1lΨ (N)

+ 〉 = +1〈Ψ (N)

− |σ3⊗1lΨ (N)− 〉 = −1

(.)

and R(N)(u) has the constant value −Λ . (The same holdstrue for the infinite-mode case.) Nevertheless in the finite-mode case the situation is not very interesting since arbitrarysuperpositions between Ψ (N)

+ and Ψ (N)− are allowed and yield

other pure ground states. So the question is now: what hap-pens if one adds together the two ground-state ‘vectors’ of theinfinite-mode Hamiltonian?


A SHORT DIGRESSION ON SUPERSELECTION RULES

By an informal calculation—and there are very sound heuris-tic arguments that this calculation can be transformed intoa rigorous proof (Amann , p. )—, one can show thatthe transition probability between Ψ+ and Ψ− vanishes withrespect to all observables:

〈Ψ+|T Ψ−〉 = 0 for ‘every’ T . (.)

Here ‘every T ’ means every observable built up from bosonoperators and spin operators. Any two vectors fulfilling thiscondition, like Ψ+ and Ψ− , are said to be disjoint or separatedby a superselection rule. Note that the ir singularity (.)plays a crucial role in the derivation of (.), so that thesuperselection rule certainly does not hold for a finite numberof bosons. Actually superselection rules are features that onecan only encounter in systems with infinitely many degrees offreedom.

A subspace of the state space in which the superpositionprinciple holds unrestrictedly is called a sector. Hence twovectors Ψ1 and Ψ2 are disjoint if and only if they are in dif-ferent sectors. If these vectors Ψ1 and Ψ2 correspond to purestates, then the state implemented by the vector Ψ1 + Ψ2 isnot a pure state any more but a mixture. One then says thatthe superposition between Ψ1 and Ψ2 is ‘forbidden.’

Recall that an expectation-value functional T → ϕ(T ) issaid to be a pure state if

ϕ(T ) = c1ϕ1(T ) + c2ϕ2(T ) for all observables T (.)


implies that either c1 = 0 or c2 = 0 or ϕ1 = cϕ2 with c ∈ C .Otherwise the state ϕ is said to be mixed. The assertion abovethat a superposition of two disjoint vectors corresponds to amixed state may be elucidated as follows: Let the state ϕ beimplemented by a vector Ψ , and let Ψ = Ψ1 + Ψ2 . Then theexpectation value of an observable T with respect to Ψ isgiven by

ϕ(T ) = |c1|2〈Ψ1|TΨ1〉 + c∗1c2〈Ψ1|TΨ2〉

+ c∗2c1〈Ψ2|TΨ1〉 + |c2|2〈Ψ2|TΨ2〉 .

(.)If Ψ1 and Ψ2 are separated by a superselection rule, then theinterference terms 〈Ψ1|TΨ2〉 and 〈Ψ2|TΨ1〉 vanish and one has

ϕ(T ) = |c1|2ϕ1(T ) + |c2|2ϕ2(T ) , (.)

with ϕi(T ) := 〈Ψi|TΨi〉 for i = 1, 2. Hence ϕ is a mixture.

The commutation-relation representations most chemists or physicistsare used to—those where, say, the state of a spin system is implementedby a vector in C

n , or where the state of a particle is described by aSchrodinger-type wavefunction—happen to have the feature that vectorsalways correspond to pure states, whereas mixtures have to be imple-mented by density matrices. Thus it might seem a little bit bewilderingthat one has to deal here with systems where a vector may correspond toa mixture. In fact, every quantum system admits a representation whereall states, pure or mixed, may be implemented by a vector.

Note that for arbitrary disjoint vectors Ψ1 and Ψ2 there existsan observable S that commutes with all other observables ofthe system in question, such that Ψ1 and Ψ2 are eigenvectorsof S with different eigenvalues. Such observables are referred


to as classical observables. Classical observables must be ac-cessible as an appropriate limit2 of observables Tk (for thespin–boson model, the Tk are observables built up from bo-son operators and spin operators):

S = limk→∞

Tk . (.)

3.3 Ground States of the Spin–boson modelApproximated by a Superpositionof Coherent States

Coherent states are good candidates for building up an ex-pression for the vectors Ω occurring in equation (.). Onone hand they are easy to handle; on the other hand they arerelatively stable under perturbations and enjoy a widespreaduse in quantum optics, so the assumption that the bosonicsystem be in a coherent state seems to be physically sound.In that spirit, the variation principle has already been carriedout by Amann (a) with the ansatz

Ω =(αW (−ω−1λ)W (ξ) Ψ (N)

F

β W (+ω−1λ)W (η)Ψ (N)F

), (.)

2. The so-called σ-weak limit: the expectation values tr(TkD) have toconverge to tr(SD) for every positive trace-class operator D .

3.3 GROUND STATES APPROXIMATED BY COHERENT STATES 51

which is a generalisation of the ansatz of Silbey and Harris():

Ω =(αW (−ω−1λ)W (+ξ)Ψ (N)

F

β W (+ω−1λ)W (−ξ)Ψ (N)F

). (.)

Here α and β are complex numbers, and

ξ =

ξ1...ξN

and η =

η1

...ηN

(.)

are N -dimensional complex vectors. The factors W (±ω−1λ)are present because it is convenient to work with the trans-formed version H of the Hamiltonian; they disappear afterapplying the unitary transformation U to Ω . The main draw-back of restricting Ω to vectors described by equations (.)and (.) is that only a quite small subspace of the Fockspace is covered up. Therefore it is not astonishing that thismay (and in fact does) lead to incorrect phase diagrams (seebelow), as compared with the rigorous bounds derived bySpohn ().

Since the calculation of R(u) is quite tedious, it is crucialto start with an ansatz for Ω where the number of parametersinvolved is kept to a minimum. Hence it is wise to restrict one-self to vectors Ω that implement pure states, keeping in mindthat the corresponding function u → Q(u), which involvesmixed states, too, can be inferred very easily from R . (Re-call that Q is the largest convex function smaller than R .)The following theorem is very helpful for deciding whethercoherent bosonic states are disjoint or not.


Theorem 3.6 Bosonic state ‘vectors’ W (ξ)ΨF and W (η)ΨF

are disjoint if and only if ‖ξ − η‖ = ∞ .

Note that here ‘vectors’ has been put between quotes sinceξ and η are not assumed to be square-integrable (were theysquare-integrable, ‖ξ−η‖ = ∞ could never be infinite). ThusW (ξ)ΨF and W (η)ΨF are thought to implement expectation-value functionals in the sense of page .

‘Proof’ (Amann a, p. ): Any operator of the boson system can bewritten as a linear combination of Weyl operators W (z) with vectors zof the type

z =

z1...

zM

0...

, (.)

corresponding to finitely, but arbitrarily many modes of the field. Thus,for the mentioned ‘vectors’ to be separated by a superselection rule, it issufficient that

〈W (ξ) ΨF|W (z) W (η) ΨF〉 = 0 (.)

holds for arbitrary z of the form (.). This is the same to say

〈ΨF|W (−ξ + z + η) ΨF〉 = e−12 ‖−ξ+z+η‖2

= 0 (.)

for arbitrary z . Equivalently, ‖−ξ + z + η‖2 = ∞ for arbitrary z , or‖−ξ + η‖2 = ∞ .

It might first be tempting to use the following ansatz:

Ω =∑

j

(αj W (−ω−1λ)W (ξj) Ψ

(N)F

βj W (+ω−1λ)W (ηj)Ψ(N)F

), (.)


with αj , βj ∈ C and ξj ,ηj ∈ CN . Every vector in the Fock

space may be expressed as a linear combination of coherentstates and hence be written in the form (.). However, af-ter performing the limit N → ∞ , there are reasons to assumethat several of the ξj or ηj will not be square-integrable. Forinstance, this turns out to be the case (under certain con-ditions) for the ξ and η in the ansatz (.). If, say, forsome j and j′ the ‘distance’ ‖ξj − ξj′‖ happens to be in-finite, the resulting vector Ω corresponds to a mixed state.Thus the Ω ’s from equation (.) will encompass a myriadof—redundant—mixed states.

Therefore we are going to use the following ansatz instead:

Ω =∑

j

(αj W (−ω−1λ)W (ζj)W (ξ) Ψ (N)

F

βj W (+ω−1λ)W (ζj)W (η)Ψ (N)F

), (.)

with αj , βj ∈ C and ζj , ξ,η ∈ CN . The vectors ζj are consid-

ered to be fixed and chosen so that the family W (ζj)ΨF ofcoherent states spans the whole Fock space. Thus the ζj ’s aremust be square-integrable after performing the limit N → ∞ ,i.e.,

‖ζj‖ < ∞ . (.)

Any ‘non-square-integrability’ in the arguments of the Weyloperators is thus intended to be shifted over to W (±ω−1λ)and to W (ξ) or W (η), respectively. The two componentsof Ω always correspond to pure bosonic states, even when ‖ξ‖and ‖η‖ diverge. If the norm ‖ξ − η − 2ω−1λ‖ is finite forinfinite N , the whole vector Ω corresponds to a pure state for


the spin–boson system. If on the contrary this norm diverges,Ω corresponds to a mixed state; pure states can neverthelessbe constructed as follows:

Ω+ :=∑

j

(αjW (−ω−1λ)W (ζj)W (ξ)ΨF

0

)

Ω− :=∑

j

(0

βjW (+ω−1λ)W (ζj)W (η)ΨF

),

(.)

in a way analogous to the zero-level-splitting case.

For the one-boson system with frequency ω , the most ob-vious choice for a complete system of coherent states wouldbe the von-Neumann grid (Perelomov , p. ), i.e., thefamily W (ζj)ΨF , where the ζj ∈ C are on the vertices ofa lattice whose elementary cell has the area π , for instance3

j = (j′, j′′) ∈ Z2 and ζj := j′ + +iπj′′ . (.)

For the N -boson system, additional complications arise. Con-sider a family W (ζj)ΨF , with ζj ∈ C

N for j ∈ Z2N , where

each of the components ζjn of ζj has a form analogous toequation (.). Then, in the limit N → ∞ , all the ζjn ’s mustbe zero for every n greater than some arbitrary high modenumber n0 , otherwise ζj would not be square-integrable any

3. Note that the coherent-state family defined by (.) is slightly over-complete and may be transformed into a basis by taking one (j′, j′′)out at random. For our discussion, however, this is irrelevant.


more. Although we are not going to use explicit expressionsfor the ζjn , we can exploit this fact as a heuristical hint forrestricting ourselves to ζj ’s where only a finite, but arbitraryhigh number n0 of modes are present:

ζj =

ζj1...

ζjn0

n0

0...0

N

(.)

By the way the application of the variation principle to thespin–boson model will turn out to be so tedious that this cut-off seems to be crucial if we want to infer any result, howevertiny it may be. Nevertheless there is also a physical rationalebehind the cutoff: Since large mode numbers n correspond tolow frequencies (infrared singularity), only the modes refer-ring to high frequencies are left in the ζj ’s. The number n0

will therefore be referred to as infrared ζj-cutoff. So only thelow-frequency bosons (the ‘infrared’ bosons) are believed tocontribute to the symmetry breaking for the ground states.One should, however, keep in mind that the condition (.)restricts the set of all Ω ’s in a way that the whole Fock spaceis not covered up any more.


3.4 Minimising the Energy Expectation Value

To calculate 〈Ω|HΩ〉 , it is convenient to use the transformedHamiltonian (.):

R(N) := 〈Ω|H (N)Ω〉 = 〈Ω|H (N)Ω〉 (.)

with

Ω := UΩ =∑

j

(αj W (ζj)W (ξ) ΨF

βj W (ζj)W (η)ΨF

). (.)

(To avoid unnecessary clumsiness of the notation, the argu-ment of R(N)(u) will henceforth be omitted.) Using the ab-breviations

ajk := α∗jαk e

− 12‖ζj−ζk‖2

eiϕjk

bjk := β∗j βk e

− 12‖ζj−ζk‖2

eiχjk

cjk := α∗j βk e

− 12‖ζj−ζk+ξ−η−2ω−1λ‖2

eiϑjk

xjk := (ξ + ζj |ω |ζk + ξ)yjk := (η + ζj |ω |ζk + η) ,

(.)

where

ϑjk := (ζj |ξ) + (η|ζk) + (ζj + ξ|ζk + η)

+ (ζj + ξ|2ω−1λ) − (2ω−1λ|ζk + η)

ϕjk := 2(ζj |ξ) + 2(ξ|ζk) + (ζj |ζk)χjk := 2(ζj |η) + 2(η|ζk) + (ζj |ζk) ,

(.)

3.4 MINIMISING THE ENERGY EXPECTATION VALUE 57

one can—after some lengthy but simple calculations—expressthe expectation value of energy in the following way:

R(N) = 2ε ∑

j

∑

k

cjk

+

∑

j

∑

k

ajk xjk +∑

j

∑

k

bjk yjk − Λ .

(.)

Our task is now to minimise R(N) with respect to the pa-rameters αj , βk , ξ , η under the constraints

〈Ω|Ω〉 = 1 (.)〈Ω|σ3⊗1lΩ〉 = u , (.)

which, using

〈Ω|Ω〉 =∑

j

∑

k

ajk +∑

j

∑

k

bjk (.)

〈Ω|σ3⊗1lΩ〉 =∑

j

∑

k

ajk −∑

j

∑

k

bjk , (.)

are equivalent to

a :=∑

j

∑

k

ajk − 12 (1 + u) = 0 (.)

b :=∑

j

∑

k

bjk − 12 (1 − u) = 0 . (.)


Introducing Lagrange multipliers µ and ν for the two condi-tions above leads to the following expression to be minimised:

S(N) := R(N) + µa+ νb

= 2ε ∑

j

∑

k

cjk

+∑

j

∑

k

ajk (xjk + µ) +∑

j

∑

k

bjk (yjk + ν)

− Λ− 12 (1 + u)µ− 1

2 (1 − u)ν . (.)

The quantities ajk do not change if all the αj are multipliedby the same phase factor eiγa ; the same goes for the bjk ifall the βj are multiplied by the same phase factor eiγb . Thusthe phase of the expression

∑j

∑k cjk may be chosen ar-

bitrarily without affecting the second and the third term ofequation (.). Obviously S(N) is minimal if the phases ofαj and βj are chosen so that

∑

j

∑

k

cjk =∣∣∣∑

j

∑

k

cjk

∣∣∣ eiπ . (.)

So eventually we are to minimise

S(N) = −2ε

√(∑

j

∑

k

c∗jk

)(∑

j

∑

k

cjk

)

+∑

j

∑

k

ajk (xjk + µ) +∑

j

∑

k

bjk (yjk + ν)

− Λ− 12 (1 + u)µ− 1

2 (1 − u)ν (.)


with respect to the parameters αj , βk , ξ , η , µ , and ν . Thisimplies that the partial derivatives

∂S(N)

∂αj

,∂S(N)

∂α∗j

,∂S(N)

∂ξm,

∂S(N)

∂ξ∗m

,∂S(N)

∂µ

∂S(N)

∂βj

,∂S(N)

∂β∗j

,∂S(N)

∂ηm

,∂S(N)

∂η∗m

,∂S(N)

∂ν

must vanish for all j and m . The partial derivatives of S(N)

with respect to any complex parameter z are defined to be

∂S(N)

∂z= 1

2

(∂S(N)

∂ z − i∂S(N)

∂z

)

∂S(N)

∂z∗= 1

2

(∂S(N)

∂ z + i∂S(N)

∂z

).

(.)

Since S(N) is always real (as a linear combination of expecta-tion values with real coefficients), one has

∂S(N)

∂z=

(∂S(N)

∂z∗

)∗. (.)

This almost halves the number of derivatives to be computedand set to zero.

Tedious but straightforward computations lead to the fol-lowing system of equations:


0 =∂S(N)

∂ξ∗n= −ε

∑

j

∑

k

cjk

(ζjn − ζkn + ξn − ηn − 2λn

ωn

)

+∑

j

∑

k

ajk[ωnξn − (xjk + µ)ζjn

+ (xjk + µ+ ωn)ζkn] (.)

0 =∂S(N)

∂η∗n= ε

∑

j

∑

k

c∗jk

(ζjn − ζkn + ξn − ηn − 2λn

ωn

)

+∑

j

∑

k

bjk[ωnηn − (yjk + ν)ζjn

+ (yjk + ν + ωn)ζkn] (.)

0 =∂S(N)

∂α∗j

= ε∑

k

βk e− 1

2‖ζj−ζk+ξ−η−2ω−1λ‖2eiϑjk

+∑

k

αk e− 1

2‖ζj−ζk‖2eiϕjk(xjk + µ) (.)

0 =∂S(N)

∂βk= ε

∑

j

α∗j e

− 12‖ζj−ζk+ξ−η−2ω−1λ‖2

eiϑjk

+∑

j

β∗j e

− 12‖ζj−ζk‖2

eiχjk(xjk + ν) . (.)

The Lagrange multipliers µ and ν must be chosen so that theinitial constraints

∑

j

∑

k

ajk = 12 (1 + u) (.)

∑

j

∑

k

bjk = 12 (1 − u) (.)


are fulfilled, which by the way also follow from setting thederivatives ∂S(N)

∂µ and ∂S(N)

∂ν to zero.

Using the relationships ϕkj = −ϕjk and xkj = x∗jk , equa-

tion (.) may be rewritten in the form

0 = −ε∑

j

∑

k

cjk

(ζjn + ξn − ζkn − ηn − 2λn

ωn

)

−∑

j

ζjnα∗j

[∑

k

αk e− 1

2‖ζj−ζk‖2eiϕjk(xjk + µ)

]

+∑

j

ζjnαj

[∑

k

αk e− 1

2‖ζj−ζk‖2eiϕjk(xjk + µ)

]∗

+∑

j

∑

k

ajkζkn + 12 (1 + u)ωnξn .

If one now replaces the expression in brackets by

−ε∑

k

βk e− 1

2‖ζj−ζk+ξ−η−2ω−1λ‖2eiϑjk ,

according to equation (.), the Lagrange multiplier µ willbe eliminated:

0 = −ε∑

j

∑

k

c∗jkζjn − ε

∑

j

∑

k

cjk

(ξn−ζkn−ηn−

2λn

ωn

)

+∑

j

∑

k

ajkζjn + 12 (1 + u)ωnξn . (.)


With an analogous procedure, ν may be eliminated fromequation (.), yielding

0 = −ε∑

j

∑

k

cjkζkn + ε∑

j

∑

k

c∗jk

(ζjn+ξn−ηn−

2λn

ωn

)

+∑

j

∑

k

bjkζkn + 12 (1 − u)ωnηn . (.)

Equations (.) and (.) can immediately be solved withrespect to ξn and ηn . Using the abbreviations

σ :=4ε

1 − u2

∣∣∣∑

j

∑

k

cjk

∣∣∣ (.)

and

An := ε∑

j

∑

k

(c∗jkζjn − cjkζkn) − ωn

∑

j

∑

k

ajkζkn

Bn := −ε∑

j

∑

k

(c∗jkζjn − cjkζkn) − ωn

∑

j

∑

k

bjkζkn ,(.)

the solutions can be written as follows:

ξn =(1 − u)λn +An +Bn

ωn

σ

ωn + σ+

2An

(1 + u)(ωn + σ)

ηn =−(1 + u)λn +An +Bn

ωn

σ

ωn + σ+

2Bn

(1 − u)(ωn + σ).

(.)

As announced above (equation .), all the componentsζjn with n > n0 are assumed to vanish, so that An = Bn = 0for n > n0 . It will prove convenient to introduce a ‘partial’


scalar product and norm, which refer to the the ‘lower’ modes1 through n0 only,

(x|y)↓ :=n0∑

n=1x∗nyn and ‖x‖↓ :=

√(x|x)↓ , (.)

and a corresponding scalar product and norm referring to the‘higher’ modes n0 + 1 through N only:

(x|y)↑ :=

N∑

n=n0+1x∗nyn if N > n0

0 if N = n0

‖x‖↑ :=√

(x|x)↑ .

(.)

For n > n0 , equations (.) then simplify to

ξn = (1 − u)λn

ωn

σ

ωn + σ∈ R

ηn = −(1 + u)λn

ωn

σ

ωn + σ∈ R .

(.)

Thus σ may be split into two factors, the first one—whichwill be denoted by C(σ) below—depending only on the com-ponents ξn and ηn with n n0 , and the other one dependingonly on the components ξn and ηn with n > n0 :


σ = − 4ε1 − u2

∑

j

∑

k

cjk

= − 4ε1 − u2

∑

j

∑

k

α∗jβke

− 12‖ζj−ζk+ξ−η−2ω−1λ‖2

eiϑjk

= −4εC(σ)1 − u2

e−12‖ξ−η−2ω−1λ‖2

↑ ei(ξ|η)↑+(ξ+η|2ω−1λ)↑︸︷︷︸

= 1

.(.)

The last exponential factor in equation (.) is unity sinceξn and ηn are real for n > n0 . The quantity C(σ) is definedto be

C(σ) :=∑

j

∑

k

α∗jβk e

− 12‖ξ−η+ζj−ζk−2ω−1λ‖2

↓

× ei(ξ+η|ζj−ζk)↓+(ξ|η)↓+(ζj |ζk)↓

× ei(ξ+η+ζj+ζk|2ω−1λ)↓ . (.)

Note that C(σ) has the same sign as∑

j

∑k cjk and thus—

by virtue of equation (.)—is negative. Furthermore, it isimportant to keep in mind that C(σ) not only depends on σ ,but also on the coupling constants λn and on u , the orderparameter.4 Fortunately enough, the function σ → C(σ) isindependent of the number N of bosonic modes. Its argu-ment σ , on the other hand, does depend on N .

4. For further information about the origin, the purpose, and the deepersignificance of C(σ) , see von Doderer (), p. .


By inserting equations (.) into equation (.), a shortcomputation yields

σ =4ε|C(σ)|1 − u2

e−2F Nn0

(σ) , (.)

where FNn0

(σ) is defined to be

FNn0

(σ) :=N∑

n=n0+1

(λn

ωn + σ

)2

. (.)

Equation (.) must be regarded as the defining equationfor σ .

In a similar way, the minimal energy expectation value maybe split into a ‘lower-mode’ and a ‘higher-mode’ term:

R(N) = − 12 (1 − u2)σ

+ 12 (1 + u)(ξ|ωξ)↑ + 1

2 (1 − u)(η|ωη)↑ +D(σ) − Λ

(.)where D(σ) is defined as follows

D(σ) :=∑

j

∑

k

ajk (ξ + ζj |ω |ζk + ξ )↓

+∑

j

∑

k

bjk (η + ζj |ω |ζk + η)↓ . (.)

D(σ) depends only on ξn and ηn with n n0 and is thusindependent of the number N of bosonic modes. Nastilyenough, it does depend on the parameter u . By insertingequations (.) into equation (.), one obtains

R(N) = (1 − u2)[− 12σ + LN

n0(σ)] +D(σ) − Λ , (.)


with

LNn0

(σ) :=N∑

n=n0+1

λ2n

ωn

(σ

σ + ωn

)2

. (.)

Note that Amann (a) has obtained relationships for σand R(N) that are very similar to the ones in equation (.)and (.), respectively. Therefore we are now going to followthe lines of argumentation of that paper quite closely.

3.5 The Infinite-Mode Limit

All the discussions below refer to the case ε = 0. The spin–boson model with ε = 0 has already been shown to exhibitsymmetry-breaking ground states (see page f.).

For finite N the function σ → FNn0

(σ) is monotonouslydecreasing and fulfills

FNn0

(0) < ∞ and limσ→∞

FNn0

(σ) = 0 , (.)

For σ > 0 one has∞∑

n=n0+1

(λn

ωn + σ

)2

<∞∑

n=n0+1

(λn

σ

)2

< ∞ ; (.)

hence the infinite-mode limit

Fn0(σ) := limN→∞

FNn0

(σ) =∞∑

n=n0+1

(λn

ωn + σ

)2

(.)

3.5 THE INFINITE-MODE LIMIT 67

exists for every σ > 0 and fulfills

limσ→0

Fn0(0) = +∞ and limσ→∞

Fn0(σ) = 0 , (.)

the divergence for σ → 0 being a consequence of the ir singu-larity (.). The function Fn0 is positive and monotonouslydecreasing. For simplicity we shall assume that F is continu-ous, so that the convergence in (.) is uniform on boundedintervals not containing σ = 0.

Equation (.) either admits σ = 0 as a solution for all Nor does not admit σ = 0 as a solution for any N . If we assumethat the latter be the case, we can rewrite (.) as follows:

fNn0

(σ) := lnσ + 2FNn0

(σ) = ln4ε|C(σ)|1 − u2

. (.)

Using (.) we can deduce that

limσ→0

fNn0

(σ) = −∞ and limσ→∞

fNn0

(σ) = +∞ . (.)

The above assumption that σ = 0 must not fulfill equation(.) also implies that C(0) = 0, hence ln 4ε|C(σ)|

1−u2 > −∞ .Furthermore we assume that (i) the function σ → C(σ) becontinuous and (ii) C(0) be finite. Thus, since fN

n0is continu-

ous, equation (.) has at least one strictly positive solutionfor N large enough (see figure .). If it happens to havemore than one solution, one has to select the one that pro-vides the smallest value of R(N) in equation (.). Note thata necessary but not sufficient condition for σ to yield a globalminimum of R(N) is that the expression − 1

2σ + LNn0

(σ) benegative.


Since Fn0(σ) and lnσ diverge in different directions forσ → 0, we cannot a priori tell how

fn0(σ) := lnσ + 2Fn0(σ) (.)

will behave for σ → 0. We must therefore distinguish threecases:

• First possibility: Fn0(σ) diverges quickly enough so thatone has limσ→0 fn0(σ) = +∞ . Then one of the solutions ofequation (.) converges to zero for N → ∞ , as figure .suggests. Interestingly enough, σ = 0 is not a solution ofthe infinite-mode limit of equation (.), namely

fn0(σ) = ln4ε|C(σ)|1 − u2

; (.)

it sort of gets ‘lost’ when N goes to infinity. This phe-nomenon has to be kept in mind when one is investigatingthe infinite-mode behaviour of the energy R(N) .

• Second possibility: Fn0(σ) diverges exactly as fast as lnσ ,so that limσ→0 fn0(σ) is finite. At least for some values of4ε|C|/(1 − u2), one has to take account of the ‘improper’solution σ = 0 disappearing in the limit N → ∞ .

• Third possibility: Fn0(σ) diverges more slowly than lnσso that one has limσ→0 fn0(σ) = −∞ . Then the solu-tion σ = 0 always arises, for finite N as well for N → ∞ .

We conclude that we always have to keep in mind the caseσ = 0, no matter whether it is a solution of (.) or not.

3.5 THE INFINITE-MODE LIMIT 69

If for the sake of simplicity we assume that the infinite-modelimit of LN

n0(σ), namely

Ln0(σ) := limN→∞

LNn0

(σ) =N∑

n=n0+1

λ2n

ωn

(σ

σ + ωn

)2

, (.)

fn0 (σ)

fNn0

(σ)

fn0n0 (σ) = ln σ

ln4ε|C(σ)|1 − u2

ln4ε|C(0)|1 − u2

σ′N σ′′

N σ′′ σ′′′ σ′′′N σ

Figure .: Behaviour of the solutions of equation (.) when perform-ing the limit N → ∞ . The grey arrows indicate the direction of increas-

ing N . Note that the shape of σ → ln 4ε|C(σ)|1−u2 is a mere offspring of the

author’s fantasy.


is continuous then the convergence of the above limit is uni-form on bounded intervals. This ensures that for a sequence(σN )N=1,2,... converging to σ for N → ∞ one has

limN→∞

LNn0

(σN ) = Ln0(σ) . (.)

Proof: |Ln0 (σ)−LNn0

(σN )| |Ln0 (σ)−Ln0 (σN )|+|Ln0 (σN )−LNn0

(σN )|converges to zero for N → ∞ , the second absolute value converging tozero by virtue of the uniform convergence mentioned above.

Hence the minimal energy of the spin–boson model is givenby

R(N) = (1 − u2)[− 12σ + Ln0(σ)] +D(σ) − Λ , (.)

where either σ is a solution of equation (.) or σ = 0,whichever yields the smaller value of E . In the latter case,one simply has R(N) = D(σ) − Λ .

3.6 The Phase Diagramof the Spin–Boson Modelfor a Representative Ohmic Example

EXPLICIT EXPRESSIONS FOR THE FREQUENCIES ANDCOUPLING CONSTANTS

In a dispersion-free medium, the wave vector k and the an-gular frequency ω are related by ω = c|k| , where c denotesthe velocity of light. If we keep considering a molecule in an

3.6 THE PHASE DIAGRAM OF THE SPIN–BOSON MODEL 71

electromagnetic field as an appropriate paradigm, then equa-tion (.) suggests the following choice for the frequenciesand the coupling constants (we set c = 1):

ωk := |k|

λk :=

√

21

√4π|k|

for |k| K

0 for |k| > K

(.)

Here again denotes the coupling strength and can be shownto be consistent with the definition of page ; the functionW (t) exhibits ohmic behaviour. Note that k , which corres-ponds to n in the previous calculations, is now a contin-uous parameter, as in the Hamiltonian of equation (.).The mathematical difficulties arising from switching betweencontinuous-mode and discrete-mode Hamiltonians are hope-fully minor and will not be discussed further here.

Since the wavelength increases with increasing n , theλk-cutoff K is an ultraviolet cutoff. It is necessary for prevent-ing the integrals over k from diverging. It especially guaran-tees that the quantity Λ is finite:

Λ :=∞∑

n=1

λ2n

ωn=

∫

|k|K

d3kλ2

k

ωk= 1

2K . (.)

On the other hand, the norm of the wave vector correspondingto the ζj-cutoff mode n0 (cf. equation .) will be denoted


0 k0 K ∞ → |k|‘infrared’

.

.

....

.

.

. ‘ultraviolet’n ← ∞ N n0 1

Figure .: Correspondence of parameters in the continuous-mode andthe discrete-mode Hamiltonian

by k0 . Figure . is intended to give an overview over the dif-ferent parameters in the continuous-mode and discrete-modepicture, respectively.

Now explicit expressions for F (σ) and L(σ), and (λ|ω−1λ)may be derived (the index k0 will now be omitted to makethe notation not unnecessarily clumsy):

F (σ) :=N∑

n=n0+1

(λn

ωn + σ

)2

=∫

|k|k0<K

d3k

(λk

ωk + σ

)2

= 12

[lnk0 + σ

σ− k0

k0 + σ

](.)

L(σ) :=N∑

n=n0+1

λ2n

ωn

(σ

σ + ωn

)2

=∫

|k|k0<K

d3kλ2

k

ωk

(σ

σ + ωk

)2

= 12

σk0

σ + k0(.)

Note that C(σ) contains integrations over expressions with λk

and ωk , namely ‘lower’ scalar products ( · | · )↓ , and ‘whole’scalar products ( · | · ), corresponding to integrals of the types


∫k0|k|K

d3k and∫0|k|K

d3k , respectively. The quantityC(σ) therefore depends on k0 , K , and . On the other hand,D(σ) only contains integrations over expressions with ωk ,namely ‘lower’ scalar products, and thus only depends on Kand k0 , but not on .

SOLVING THE EQUATION FOR σ

For the discussion of below it will turn out to be importantthat σ must a priori be in a certain range. Let us recall thedefinition of σ

σ =4ε

1 − u2

∣∣∣∑

j

∑

k

cjk

∣∣∣ ; (.)

upon inspection of equation (.) we see that

〈Ω|σ1⊗1lΩ〉 = 2ε ∑

j

∑

k

cjk

= −2

∣∣∣∑

j

∑

k

cjk

∣∣∣ . (.)

On the other hand the constraints 〈Ω|σ3⊗1lΩ〉 = u and〈Ω|Ω〉 = 1 imply

〈Ω|σ1⊗1lΩ〉 ∈[−√

1−u2, +√

1−u2]

. (.)

Proof: Let us put

Ω =(a Ψa

b Ψb

)

with a and b being complex numbers and Ψa and Ψb being normalisedbosonic state vectors. The constraints 〈Ω|Ω〉 = 1 and 〈Ω|σ3⊗1l Ω〉 = uimply that

a = eiϕ√

12(1 + u) and b = eiχ

√12(1 − u)


with ϕ, χ ∈ R . It follows from the Cauchy–Schwarz inequality that|〈Ψa|Ψb〉| 1 , hence 〈Ψa|Ψb〉 = r eiϑ with 0 r 1 and ϑ ∈ R . Thus

〈Ω|σ1⊗1l Ω〉 = 2a∗b 〈Ψa|Ψb〉 = r√

1 − u2 cos(ϕ − χ − ϑ) , (.)

from which the assertion (.) follows immediately.

Hence the range of σ is given by

σ ∈[0,

2ε√1 − u2

]. (.)

Let us now focus our attention to the minimal-energy func-tion R(N) . It turns out to be

R(N) = 12 (1 − u2)σ

(k0

σ + k0− 1

)+D(σ) − 1

2K . (.)

where either σ is a solution of

f(σ) := (1 − ) lnσ + ln(k0 + σ) − k0

k0 + σ= ln

4ε|C(σ)|1 − u2

(.)or σ = 0, whichever yields the smaller value of R(N) . Thusthe sign of k0

σ+k0−1 will decide which one of the two cases will

be realised. If it is negative, one has σ = 0, if it is positive,one has σ = 0. It turns out that one can stick to a solutionσ = 0 of (.) if and only if this solution fulfills

σ > k0(− 1) . (.)

On the other hand f(σ) has a unique minimum, namely at

σ = k0(√− 1) . (.)


This suggests that we must distinguish the three cases > 1, = 1 and < 1. The following shortcut

G(σ) :=4ε|C(σ)|1 − u2

(.)

will turn out to be convenient.

• Case > 1 . The part of f that is relevant for a solutionσ = 0 is a monotonously increasing function. Obviously thecondition that the intersection between G(σ) and the relev-ant part of f(σ) occurs in the ‘allowed’ range (.) of σreads 2ε/

√1 − u2 < k0( − 1); see figure .. For u = 0

this condition for a solution σ = 0 specialises to

2ε < k0(− 1) . (.)

If (.) is fulfilled, there is at least a small range u ∈[−u0,+u0] where the solutions are of a σ = 0 type. Forlarger |u| , the quantity 2ε/

√1 − u2 increases and there

may occur solutions σ = 0.

If on the contrary one has 2ε/√

1 − u2 k0( − 1), noassertion can be made since the graph of G may or maynot cross the graph of f in the relevant part (figure .).


σ

f(σ)

G(σ)

ln σ

k0(√

−1) 2ε√1−u2

1 k0( −1)

Figure .: Solutions of (.) in the case > 1 . Only the solid part ofthe graph of f is relevant. If the graph of G crosses the graph of f inthe dashed part, the solution σ = 0 has to be selected instead. Here, inthe case 2ε/

√1 − u2 < k0( − 1) , no solution σ = 0 is possible.


σ

f(σ)G(σ)

ln σ

k0(√

−1) 2ε√1−u2

1 k0( −1)

Figure .: Solutions of (.) in the case > 1 , just as in figure .,but for the case 2ε/

√1 − u2 k0(− 1) . Two possible graphs of G are

shown, the upper one crosses the relevant part of the graph of f andhence yields a solution σ = 0, the other one does not. Thus no decisionis possible.


• Case = 1 . The condition (.) is no restriction anymore since σ is always positive. The graph of f turns outto cross the ordinate at a finite value. No decision is possiblesince the shape of the graph of G is not known (figure .).

• Case < 1 . It turns out that f(σ) tends to −∞ for σ → 0.We shall now exploit the fact that the graph of G must crossthe graph of σ → lnσ for some σ < 2ε/

√1 − u2 .

Proof: Since an energy-minimising state must have a physically sen-sible value for the expectation value of σ1⊗1l , the finite-mode equa-tion (.) for σ must admit a solution σ < 2ε/

√1 − u2 , no matter

how small the number of degrees of freedom is. Translated into thecontinuous-mode picture this means that equation (.) must ad-mit a solution even in the case k0 = 0. In this case one has F (σ) ≡ 0 ,hence f(σ) = ln σ .

Thus there is always a solution σ = 0 in this case (fig-ure .).


σ

f(σ)

G(σ)

ln σ

2ε√1−u2

1

Figure .: Solutions of (.) in the case = 1. The graph of G mayor may not cross the graph of f , hence no decision is possible.


σ

f(σ)

G(σ)

ln σ

2ε√1−u2

1

Figure .: Solutions of (.) in the case < 1 . Since the graph of Gmust cross the logarithm curve, there is always a solution σ = 0.


1

ε

slope 12k0

σ = 0

σ = 0

?

Figure .: ‘Phase diagram’ of the spin–boson model, according to theassertions that can be inferred from the variation principle. Only deci-sions with respect to the occurrence of solutions of type σ = 0 or σ = 0can be made. The dark-grey and light-grey areas designate the values of(, ε) where σ = 0 and σ = 0, respectively. In the white area no decisionseems to be possible.

The above results can be summarised in the ‘phase diagram’of figure .. The significance of the fact that σ = 0 will bediscussed below. Disappointingly enough, the border of thearea where one knows for sure that σ = 0 ‘melts down’ to the-axis when the ζj-cutoff k0 becomes arbitrarily small, i.e.,moves towards the infrared.


For a derivation of the full phase diagram the behaviour ofthe function D(σ) will be crucial; calculating this function ex-plicity, however, seems to be an utterly formidable task with aquite uncertain outcome. Therefore some heuristical thoughtswill be given below about what one might expect from theresults of a variation principle applied successfully.

WHEN IS THE SYMMETRY BROKEN?

The question arises what the physical significance of the solu-tions σ = 0 might be. According to equations (.) this im-plies that ξn = ηn = 0 for n > n0 (we have now switched backto the discrete-mode picture). The corresponding ground-state vector has the form

Ψ0 =(Φ+

Φ−

)(.)

with respect to the transformed Hamiltonian H . Here Φ+

and Φ− denote some vectors in the Fock space, since ξ and ηturn out to be square-integrable. With respect to the untrans-formed Hamiltonian H the ground-state ‘vector’ reads

Ψ0 = U−1Ψ =(W (−ω−1λ)Φ+

W (+ω−1λ)Φ−

). (.)

Since Φ+ and Φ− are in the Fock space, they can be ex-pressed as a linear combination of coherent states W (z)ΨF ,with ‖z‖ < ∞ . Thus an argumentation similar to the proof oftheorem . shows that the ir singularity implies that the two


components W (∓ω−1λ)Φ± of Ψ0 are separated by a super-selection rule, hence Ψ0 corresponds to a mixed state. How-ever, since Ψ0 is not a product vector any more—in contradis-tinction to the zero-level-splitting case—, it remains an openquestion whether Ψ can be ‘split up,’ in a way analogous tothe case ε = 0, to form two pure disjoint ground states Ψ+

and Ψ− .

Digression on a Solvable Expample for the Variation Principle

That the case σ = 0, i.e., 〈Ψ0|σ1⊗1l Ψ0〉 = 0, could signify that the inver-sion symmetry is broken might be plausible upon inspection of Amann(a). Here the variation principle has been applied to the much morerestricted class (.), which allows to calculate explicitly the boundsof the phase diagram. The form of the minimal-energy expression is al-most identical to the one derived here (equation .), with the onlydifference that the spurious quantity D(σ) is missing. Thus σ = 0 sim-ply implies that R(N) = −Λ and the inversion symmetry is broken. Theground-state vector can then be shown to be identical to the one for thezero-level-splitting case.

The occurrence of a solution σ = 0 is, however, only a sufficient con-dition for symmetry breaking. There exists a region in the phase diagram(figure .) where one has σ = 0 and symmetry breaking; the minimal-energy function turns out to have two distinct minima, corresponding totwo ground-state vectors that are no product vectors any more.

CONCLUDING REMARKS

It is disappointing that the results obtained are so meagre.The parameters ξn and ηn have been successfully eliminatedfor n > n0 in the sense that they could all be expressed witha single parameter σ . Unfortunately, this has not yet beendone for the ξn and ηn for n n0 , and this is, in the au-thor’s opinion, the procedure to try out in the future. If the


12

1

ε

Λ

σ = 0

σ = 0

symmetry unbroken

−1 +1

−1 +1

−1 +1

R(N)

R(N)

R(N)

u

u

u

−Λ

−Λ

−Λ

Figure .: Phase diagram of the spin–boson model obtained with thevariational ansatz (.). In the dark-grey region the symmetry is bro-ken and one has solutions of type σ = 0. In the light-grey region thesymmetry is broken with solutions of type σ = 0. In the white regionthe symmetry remains unbroken. For each region the shape of the cor-responding minimal-energy function is drawn.

‘lower-mode’ ξn and ηn could be expressed with a few ‘univer-sal,’ i.e., mode-independent parameters like σ , the variationalprinciple would become tractable.


In spite of all these technical difficulties, the variation prin-ciple exhibits several appealing and unique features:

• systems with infinitely many degrees of freedom can betreated within the framework of traditional Hilbert quan-tum mechanics, but nevertheless with a rigorous algebraicbackground;

• the mathematical tools required are only elementary alge-bra and calculus;

• The ground states of the spin–boson model could be dis-cussed without any reference to time dependence, and with-out any reference to non-zero temperature states, whileother methods heavily rely on time-evolution and thermo-dynamical considerations.

Thus it may perhaps seem worthwile to keep on trying to inferdefinitive results from the variation principle.

References

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Amann, A. (1991b)Ground states of a spin–boson model, Ann. Phys. , –

Amann, A. (1991c)Molecular superselection rules generated by a bosonic environ-ment. Unpublished manuscript

Amann, A. (1995)Structure, dynamics, and spectroscopy of single molecules: achallenge to quantum mechanics. Preprint, to be published inJ. Math. Chem.

Bratteli, O., and Robinson, D.W. (1987)Operator algebras and quantum statistical mechanics, Vol. 1.Second, revised edition, Springer, New York

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von Doderer, H. (1962)Die Merowinger oder Die totale Familie. Biederstein Verlag,Munchen

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Acknowledgments

It is a pleasure to thank Prof. Hans Primas for support, manyfruitful discussions, and patience. Furthermore, I am hopelesslyindebted to pd Dr Anton Amann for introducing me to the subjectof this thesis, for never-ending encouragement, and for countlessinvaluable pieces of advice.

I am very grateful to Prof. Dr Ulrich Muller-Herold for havingemployed me as a teaching assistant during my dissertation.

Warm thanks I also give to all the other members of the‘G-Stock,’ i.e., Marco Berg, Dr Bernd Kasemir, Dr Jurgen Pot-tinger, Dr Guy Poupart (whose idea for the epigraph of this workshall herewith be gratefully acknowledged), Martin Scheringer, andWolfgang Zaoral, for the good time we had together.

Curriculum Vitæ

. . geboren in Luxemburg-Stadt

– Primarschule in Differdingen und Esch/Alzette(Luxemburg)

– Lycee Hubert-Clement in Esch/Alzette,Matura Typ

”B classique“ (Mathematik und Latein)

– Mathematik-Propadeutikum am Centre Universitairede Luxembourg, Abschluß mit dem

”Certificat

d’Etudes Scientifiques“

– Studium an der eth Zurich, AbteilungNaturwissenschaften, physikalisch-chemischeTeilrichtung (X B)Abschluß als Diplom-Naturwissenschafter

– Dissertation bei Prof. H. Primas am Laboratorium furPhysikalische Chemie der eth Zurich

seit Assistenz bei Prof. Dr. U. Muller-Herold an derAbteilung fur Umweltnaturwissenschaften der ethZurich

seit Gesangsstudium bei Prof. Richard Wistreich an derStaatlichen Hochschule fur Musik Trossingen(Deutschland)

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