P. Zanardi USC

Ground State and out of equilibrium fidelities: from quantum phase transitions to equilibration dynamics

KITPC April 2011

(Quantum) Phase Transition: dramatic change of the (ground) state properties of a quantum system with respectA smooth change of some control parameter e.g., temperature, external field, coupling constant,…

Characterization of PTs:“Traditional”:Local Order parameter (OP), symmetry breaking (SB)(correlation functions/length) Landau-Ginzburg frameworkSmoothness of the (GS) Free-energy (I, II,.. order QPTs)Quantum Information views:Quantum Entanglement (concurrence, block entanglement,..)Geometrical Phases

Question: How to map out the phase diagram of a system with no a priori knowledge of its symmetries and ops?Question: How to map out the phase diagram of a system with no a priori knowledge of its symmetries and ops?

Ground State of

F is a universal (Hilbert space) geometrical quantity:No a priori understanding of the SB pattern or OP is needed

PZ, N. Paunkovic, PRE 74 031123 (2006) PZ, N. Paunkovic, PRE 74 031123 (2006)

At the quantum critical points (QCPs) F should have a sharp drop i.e., the induced metric d should have a sharp increase. Weak topology vs norm topology: a METRIC APPROACH

At the quantum critical points (QCPs) F should have a sharp drop i.e., the induced metric d should have a sharp increase. Weak topology vs norm topology: a METRIC APPROACH

THE KEY (naive) IDEA

Answer(?): distinguishability =distance in the information spaceAnswer(?): distinguishability =distance in the information space

THE XY Model (I)

=anysotropy parameter, =external magnetic field

QCPs:

XX line II-order QPT

Ferro/para-magnetic II order QPT

Jordan-Wigner mapping H Free-Fermion system: EXACTLY SOLVABLE!

Quasi-particle spectrum: zeroes in the TDL in all the QCPs Gaplessness of the many-body spectrum

THE XY Model (II)

Ground State

Fidelity

Strong L-dependent peak at universality features in the TDL

Orthogonalization in the TDL: enhanced @ QCPs!

PZ, N. Paunkovic, Phys. Rev E (2006)PZ, N. Paunkovic, Phys. Rev E (2006)

XY Model (III): Overlap Functions

Differential Geometry of QPTs:QGTHamiltonians

Ground states

Control parameters

Hermitean metric over the projective space

Riemannian metric

2-form (=Berry curvature!)

Metric scaling behaviour: the XY model

Ricci Scalar

QGT: Spectral representations

Local operator (trans inv)

Spectral gap

(Hastings 06)

Gapped system

For gapped systems the QGT entries scales at most extensively

Superextensive scaling implies gaplessness

QGT Critical scaling

Continuum limit

Scaling transformations

Proximity of the critical point

At the critical points

Scal dim of QGT: the smaller the faster the orthogonalization rateSuper-extensivity

Criticality it is not sufficient, one needs enough relevance….

Dynamical Fidelity:Loschmidt EchoDynamical Fidelity:Loschmidt Echo

Spectral resolution Probability distribution(s)

Different Time-Scales & Characteristic quantities

•Relaxation Time (to get to a small value by dephasing and oscillate around it)

•Revivals Time (signal strikes back due to re-phasing)

Q1: how all these depend on H, , and system size?Q1: how all these depend on H, , and system size?

Q2: how the global statistical features of L(t) depend on H, and system size N?Q2: how the global statistical features of L(t) depend on H, and system size N?

Typical Time Pattern of L(t)

Transverse Ising (N=100)

For a given initial state L-echo is a RV over the time line [0,∞) with Prob Meas

Characteristic function of

Probability distribution of L-echo

Goal: study P(y) to extract global information about theEquilibration process Goal: study P(y) to extract global information about theEquilibration process

1 Moments of P(y)

Each moment is a RV over the unit sphere (Haar measure) of initial states

Mean:

Long time average of L(t) is the purity of the time-averagedensity matrix (or 1 -Linear Entropy)

Question: How about the other moments e.g., variance andinitial state dependence? Are there “typical” values?Question: How about the other moments e.g., variance andinitial state dependence? Are there “typical” values?

Remark Is a projection on the algebra of the fixed pointsOf the (Heisenberg) time-evolution generated by H

Remark dephased state min purity given the constraints I.e., constant of motion

H.T Quan et al, Phys. Rev. Lett. 96, 140604 (2006)

Ising in transverse field:

Large size limit (TDL)==> spec(H) quasi-continuous ==> Large t limits exist (R-L Lemma) =time averages

Inverting limits I.e., 1st t-average, 2nd TDL

•g and s are qualitatively the same but when we consider different phases

(m=band min, M=band max

First Two Moments

P(L=y) Different Regimes

• Large ==>L for (moderately) large (quasi) exponential

• Small and close to criticality a) Exponentialb) Quasi critical I.e., universal “Batman Hood”

• Small and off critical a) Exponential b) Otherwise Gaussian

L=18, h(1)=0.3, h(2)=1.4 L=20, h(1)=0.1, h(2)=0.11

L=40, h(1)=0.99, h(2)=1.1

+ n-body spectrum contributions

Different regimes depend on how manyfrequencies have a non-negligible weight

L=20,30,40,60,120

h(1)=0.2, h(2)=0.6

L=10,20,30,40

h(1)=0.9, h(2)=1.2

Approaching exp for large sizes

Joint work with

Nikola PaunkovicMarco Cozzini Paolo GiordaRadu IonicioiuL. Campos-Venuti

and now @USC

Damian AbastoToby JacobosonSilvano GarneroneStephan Haas

THANKS!

Summary Part I•We analyze the phase diagram of a systems in terms

of the induced geometric tensor Q on the parameter space

•Boundaries bewteen different phases can be detected by singularities of Q (real & immaginary parts)

•Q and related functions show critical scaling behaviour andUniversality (response function representation)

Quantum Tensor(s) approach to (Q)PTs: Geometrical, QI-theoretic, UniversalQuantum Tensor(s) approach to (Q)PTs: Geometrical, QI-theoretic, Universal

•Extensions to finite temperature and to classical systems

Summary Part IISummary Part II

•Unitary equilibrations: measure convergence/concentration

•Moments of Probability distribution of LE (large time)

•Transverse Ising chain: mean, variance, regimes for P(L)

•Universal content of the short-time behaviorof L(t) and criticality

Phys. Rev. A 81,022113 (2010)Phys. Rev. A 81,022113 (2010) Phys. Rev. A 81, 032113 (2010)Phys. Rev. A 81, 032113 (2010)

Short-time behavior

Square of a characteristic function --> cumulant expansion

H sum of N local operator in the TDL N-> ∞ one expects CLT to hold I.e.,

Relaxation timeOff critical (or large quench)

Critical (& small quench)

•0 for

for

Gaussian Non Gaussian

(universal)Non Gaussian & non universal)

Quantum Fidelity and Thermodynamics

Uhlmann fidelity between Gibbs states of the same Hamiltonianat different temperatures =partition function

Singularities of the specific heat shows up at the fidelity level:Divergencies e.g., lambda points, results in a singular drop of fidelity

Temperature driven phase transitions can be detected by mixed state fidelity in both classical and quantum systems!

Temperature driven phase transitions can be detected by mixed state fidelity in both classical and quantum systems!

The “Hey dude! If you have the GS you have EVERYTHING…” Slide

The “Hey dude! If you have the GS you have EVERYTHING…” Slide

Ok cool. Let’s suppose you’re somehow given the GS e.g.,a MPS

Please, now tell me where the QCPs are….

“Great! Now I compute the OP and its correlations! ….”Well… you have infinitely many candidates: Which one?!? and perhaps none of them is gonna work out (e.g., TOPorder)…

“Ok, fair enough, then I compute the energy!”I see, so you want also the H? You didn’t ask for that so far…Anyways, here we go: identically vanishing E(g)=0….

You may answer:

Qualitative: the difference between the means should be(much) bigger than the finite size variances….

Quantitative: Fisher Information metric

Problem: Quantumly each Q-preparation defines infinitelymany prob distributions, one for each observable: one has to maximize over all possible experiments!

d(P,Q)= number of (asymptotically) distinguishable preparations between P and Q: ds=dX/Variance(X)d(P,Q)= number of (asymptotically) distinguishable preparations between P and Q: ds=dX/Variance(X)

Distinguishability of Exps Distance in Prob space

Surprise!

Projective Hilbert space distance!

(Wootters 1981)

Statistical distance and geometrical one collapse:

Hilbert space geometry is (quantum) information geometry…..Question: What about non pure preparations ?!?

Answer: Bures metric & Uhlmann fidelity!(Braunstein Caves 1994)

Remark: In the classical case one has commuting objects, simplification ….

Q-fidelity!