Correlations after quantum quenches in the XXZ spin chain ...€¦ · Correlations after quantum...

Correlations after quantum quenches in the XXZspin chain: Failure of the Generalized Gibbs

Ensemble

Gábor Takács

Department of Theoretical PhysicsBudapest University of Technology and Economics

andMTA-BME “Momentum” Statistical Field Theory Research Group

B. Pozsgay, M. Mestyán, M. A. Werner, M. Kormos, G. Zaránd and G. Takács:Correlations after quantum quenches in the XXZ spin chain: Failure of the

Generalized Gibbs EnsemblePhys. Rev. Lett. 113 (2014) 117203

28th April, 2015HAS Institute for Nuclear Research, Debrecen

Outline

1 Introduction

2 Quantum quenches in the XXZ spin chain

3 Moment of truth: comparing to real time evolution

4 The overlap TBA

5 Follow-up

6 Conclusions

Outline

1 IntroductionThermalization in isolated classical systemsThermalization in isolated quantum systemsIntegrable systems and the GGE hypothesis

4 The overlap TBA

5 Follow-up

6 Conclusions

Outline

4 The overlap TBA

5 Follow-up

6 Conclusions

Thermalization in isolated classical systems

Isolated classical system: N degrees of freedom, energy E, volume V

−→ microcanonical ensemble

Phase space Γ : (q1, . . . , qN , p1, . . . , pN)Dynamics: Hamiltonian H

qi =∂H∂pi

pi = −∂H∂qi

Fundamental postulate of statistical physicsThe probability density in equilibrium is

ρ(q, p) =

const. (q, p) ∈ Γ(N,E ,V )

0 otherwise

qi =∂H∂pi

pi = −∂H∂qi

ρ(q, p) =

0 otherwise

qi =∂H∂pi

pi = −∂H∂qi

ρ(q, p) =

0 otherwise

qi =∂H∂pi

pi = −∂H∂qi

ρ(q, p) =

0 otherwise

Boltzmann ergodic hypothesis

Boltzmann (1871): ergodic hypothesisFor an observable A(q, p)

time average: A = limT→∞

0dtA(q(t), p(t))

mc average: 〈A〉mc =1

vol(Γ(N,E ,V ))

ˆΓ(N,E ,V )

dNqdNpA(q, p)

Ergodicity:

A = 〈A〉mc

justifies use of microcanonical ensemble.It results from dynamical chaos:

individual trajectory homogeneously spreads over Γ(N,E ,V )

0dtA(q(t), p(t))

vol(Γ(N,E ,V ))

ˆΓ(N,E ,V )

dNqdNpA(q, p)

Ergodicity:

A = 〈A〉mc

0dtA(q(t), p(t))

vol(Γ(N,E ,V ))

ˆΓ(N,E ,V )

dNqdNpA(q, p)

Ergodicity:

A = 〈A〉mc

Outline

4 The overlap TBA

5 Follow-up

6 Conclusions

Time evolution in isolated quantum systems

Quantum systems differ fundamentally from classical ones:if thermalization occurs, it must have a fundamentally differentexplanation as time evolution is linear!

|ψ(t = 0)〉 =∑α

Cα|ψα〉 Cα : overlaps

H|ψα〉 = Eα|ψα〉

Time evolution|ψ(t)〉 =

Cαe−i~Eαt |ψα〉

Observables

〈A(t)〉 = 〈ψ(t)|A|ψ(t)〉 =∑α

C ∗αCβe− i

~ (Eα−Eβ)tAαβ

Aαβ = 〈ψα|A|ψβ〉

|ψ(t = 0)〉 =∑α

Observables

〈A(t)〉 = 〈ψ(t)|A|ψ(t)〉 =∑α

C ∗αCβe− i

~ (Eα−Eβ)tAαβ

|ψ(t = 0)〉 =∑α

Observables

〈A(t)〉 = 〈ψ(t)|A|ψ(t)〉 =∑α

C ∗αCβe− i

~ (Eα−Eβ)tAαβ

|ψ(t = 0)〉 =∑α

Observables

〈A(t)〉 = 〈ψ(t)|A|ψ(t)〉 =∑α

C ∗αCβe− i

~ (Eα−Eβ)tAαβ

The diagonal ensembleMicrocanonical setting: start with narrow energy distribution

〈E 〉 =∑α

|Cα|2Eα

∆E =

√∑α

|Cα|2 (Eα − 〈E 〉)2 〈E 〉

Time average

〈A(t)〉 =∑α

C ∗αCβe− i

~ (Eα−Eβ)tAαβ

A = limT→∞

0dt 〈A(t)〉 =

|Cα|2Aαα

Diagonal ensemble

A = Tr ρDEA

ρDE =∑α

|Cα|2|ψα〉〈ψα|

〈E 〉 =∑α

|Cα|2Eα

∆E =

√∑α

|Cα|2 (Eα − 〈E 〉)2 〈E 〉

Time average

〈A(t)〉 =∑α

C ∗αCβe− i

~ (Eα−Eβ)tAαβ

A = limT→∞

0dt 〈A(t)〉 =

|Cα|2Aαα

Diagonal ensemble

A = Tr ρDEA

ρDE =∑α

〈E 〉 =∑α

|Cα|2Eα

∆E =

√∑α

|Cα|2 (Eα − 〈E 〉)2 〈E 〉

Time average

〈A(t)〉 =∑α

C ∗αCβe− i

~ (Eα−Eβ)tAαβ

A = limT→∞

0dt 〈A(t)〉 =

|Cα|2Aαα

Diagonal ensemble

A = Tr ρDEA

ρDE =∑α

Eigenstate thermalization hypothesisStationary state: for large t

A(t) =∑α

C∗αCβe−i~ (Eα−Eβ )tAαβ → A =

|Cα|2Aαα

Microcanonical average

〈A〉mc =1

N(〈E〉 ,∆E)

∑α:Eα∈I

Aαα I = [〈E〉 −∆E , 〈E〉+ ∆E ]

Deutsch (1991): “quantum ergodicity”∑α

|Cα|2Aαα =1

N(〈E 〉 ,∆E )

∑α:Eα∈I

Eigenstate thermalization hypothesis, ETH (Srednicki, 1994)

Aαα = 〈A〉mc (Eα) ∀α

Supposed to hold for large systems (thermodynamic limit – TDL)

A(t) =∑α

|Cα|2Aαα

〈A〉mc =1

N(〈E〉 ,∆E)

∑α:Eα∈I

Aαα I = [〈E〉 −∆E , 〈E〉+ ∆E ]

|Cα|2Aαα =1

N(〈E 〉 ,∆E )

∑α:Eα∈I

A(t) =∑α

|Cα|2Aαα

〈A〉mc =1

N(〈E〉 ,∆E)

∑α:Eα∈I

Aαα I = [〈E〉 −∆E , 〈E〉+ ∆E ]

|Cα|2Aαα =1

N(〈E 〉 ,∆E )

∑α:Eα∈I

A(t) =∑α

|Cα|2Aαα

〈A〉mc =1

N(〈E〉 ,∆E)

∑α:Eα∈I

Aαα I = [〈E〉 −∆E , 〈E〉+ ∆E ]

|Cα|2Aαα =1

N(〈E 〉 ,∆E )

∑α:Eα∈I

A(t) =∑α

|Cα|2Aαα

〈A〉mc =1

N(〈E〉 ,∆E)

∑α:Eα∈I

Aαα I = [〈E〉 −∆E , 〈E〉+ ∆E ]

|Cα|2Aαα =1

N(〈E 〉 ,∆E )

∑α:Eα∈I

Thermalization in closed quantum systems

1 ETH is time independent: every eigenstate of Hamiltonian isalready thermal. It is masked by quantum coherence, butrevealed by dephasing effect of time evolution.

2 Quantum ergodic theorem (Neumann): for a typical finitefamily of commuting observables, every initial wave functionfrom a microcanonical shell evolves so that for most times theaverages of the given observables is close to theirmicrocanonical value.Rigol & Srednicki, 2012:

Neumann’s main technical assumption is equivalentto the Eigenstate Thermalization Hypothesis

3 What observables?1 local2 few-body

Outline

4 The overlap TBA

5 Follow-up

6 Conclusions

Integrable systems violate ergodicity

Fermi-Pasta-Ulam (1953; Los Alamos technical report 1955)

mxj = k(xj+1 + xj−1 − 2xj) [1 + α(xj+1 − xj−1)] j = 1, . . . ,N

Exciting a vibration mode: thermalization expected.Found instead: complicated quasi-periodic behaviour!Reason: system is integrable! (discretized version of KdV)Integrability: action-angle variables exist

[Qi ,Qj ] = 0 H ∈ Qii=1,...,N

The quantities are conserved: motion along tori inphase space −→Lissajous orbits with frequencies ω1, . . . , ωN→ may be ergodic on the tori→ ? microcanonical ensemble on the tori ?

[Qi ,Qj ] = 0 H ∈ Qii=1,...,N

Generalized Eigenstate Thermalization Hypothesis

Ergodicity on tori

⇒ at quantum level:

Generalized Eigenstate Thermalization Hypothesis (GETH)

ETH holds on Q-shells i.e. subspaces determined by fixing the Qi

Generalized Eigenstate Thermalization Hypothesis

Ergodicity on tori

⇒ at quantum level:

Generalized Eigenstate Thermalization Hypothesis (GETH)

ETH holds on Q-shells i.e. subspaces determined by fixing the Qi

Gibbs and generalized Gibbs ensemblesNonintegrable system: steady state is thermalDE is equivalent to a Gibbs ensemble for relevant observables

Tr ρDEA = Tr ρGEA

ρGE =1Z

e−βH Z = Tr e−βH

Determining the temperature: Tr ρGEH = 〈ψ(0)|H|ψ(0)〉 → β

Generalized Gibbs Ensemble (GGE)

natural assumption: in integrable systems quantum ergodicity holdson common eigensubspaces of Qi

ρGGE =1Z

e−∑

i βiQi Z = Tr e−∑

i βiQi

Generalized temperatures/chemical potentials

Tr ρGGEQi = 〈ψ(0)|Qi |ψ(0)〉 i = 1, . . . ,N → βii=1,...,N

GE/GGE: follow from conditional maximum entropy principle.

Tr ρDEA = Tr ρGEA

ρGE =1Z

ρGGE =1Z

e−∑

i βiQi

Tr ρDEA = Tr ρGEA

ρGE =1Z

ρGGE =1Z

e−∑

i βiQi

Tr ρDEA = Tr ρGEA

ρGE =1Z

ρGGE =1Z

e−∑

i βiQi

Tr ρDEA = Tr ρGEA

ρGE =1Z

ρGGE =1Z

e−∑

i βiQi

Closed quantum systems in reality

Can be realized with cold atoms...

Can be realized with cold atoms... ...which sometimes do not thermalize

T. Kinoshita, T. Wenger and D.S. Weiss:A quantum Newton’s cradle, Nature 440 (2006) 900-903.

Outline

1 Introduction

2 Quantum quenches in the XXZ spin chainIntegrability, Bethe Ansatz and thermodynamicsQuantum quenchGGE in the XXZ spin chain

4 The overlap TBA

5 Follow-up

6 Conclusions

Outline

1 Introduction

4 The overlap TBA

5 Follow-up

6 Conclusions

The Heisenberg XXZ spin chainA paradigmatic integrable model: chain of 1/2 spins

. . . ↑↓↓↓↑↑↓ . . . −→ H =⊗

C2 ↓

HXXZ =L∑

i σxi+1 + σy

i σyi+1 + ∆σz

i σzi+1)

PBC: σkL+1 ≡ σk

∆ > 1: gapped phase, ∆→∞: quantum Ising chain−1 < ∆ < 1: gapless phase (Luttinger liquid)

The chain is integrable and can be diagonalized by Bethe Ansatz:

|λ1, . . . , λN〉 = B(λ1) . . .B(λM)|Ω〉|Ω〉 = | . . . ↑↑↑ . . . 〉 pseudovacuum(

sin (λj + iη/2)

sin (λj − iη/2)

sin (λj − λk + iη)

sin (λj − λk − iη)∆ = cosh η

Qn|λ1, . . . , λN〉 =n∑

qn(λi )|λ1, . . . , λN〉

The conserved charges Qn are local → additivity!

. . . ↑↓↓↓↑↑↓ . . . −→ H =⊗

C2 ↓

HXXZ =L∑

i σxi+1 + σy

i σyi+1 + ∆σz

i σzi+1)

PBC: σkL+1 ≡ σk

sin (λj + iη/2)

sin (λj − iη/2)

Qn|λ1, . . . , λN〉 =n∑

qn(λi )|λ1, . . . , λN〉

. . . ↑↓↓↓↑↑↓ . . . −→ H =⊗

C2 ↓

HXXZ =L∑

i σxi+1 + σy

i σyi+1 + ∆σz

i σzi+1)

PBC: σkL+1 ≡ σk

sin (λj + iη/2)

sin (λj − iη/2)

Qn|λ1, . . . , λN〉 =n∑

qn(λi )|λ1, . . . , λN〉

Excitations and string hypothesis

Sz =L2−M

Ground state: unique solution with M = L/2 and all λi real

|0〉 = B(λ1) . . .B(λL/2)|Ω〉 λi ∈ R

Excitations: can be holes in the Dirac sea, or

n-string configurations

λj = λ+iη2

(n + 1− 2j)

+O(1/L)

j = 1, . . . , n

String hypothesisThermodynamics can be described by string degrees of freedom

Sz =L2−M

|0〉 = B(λ1) . . .B(λL/2)|Ω〉 λi ∈ R

λj = λ+iη2

(n + 1− 2j)

+O(1/L)

j = 1, . . . , n

Sz =L2−M

|0〉 = B(λ1) . . .B(λL/2)|Ω〉 λi ∈ R

λj = λ+iη2

(n + 1− 2j)

+O(1/L)

j = 1, . . . , n

Sz =L2−M

|0〉 = B(λ1) . . .B(λL/2)|Ω〉 λi ∈ R

λj = λ+iη2

(n + 1− 2j)

+O(1/L)

j = 1, . . . , n

Thermodynamic limit of Bethe Ansatz

Number of n-strings centered in an interval [λ, λ+ dλ]: Lρn(λ)dλNumber of n-string holes in an interval [λ, λ+ dλ]: Lρh

n(λ)dλ

Thermodynamic limit of BA: Bethe-Takahashi equations

ρn(λ) + ρhn(λ) = an(λ)−

∞∑m=1

(Tnm ∗ ρm)(λ)

an(λ) =1π

sinh nηcosh nη − cos 2λ

Tnm(λ) =

a|n−m|(λ) + 2a|n−m|+2(λ) + . . . m 6= n

+2an+m−2(λ) + an+m(λ)

2a2(λ) + · · ·+ 2a2n−2(λ) + a2n(λ) m = n

(f ∗ g)(λ) =

ˆ π/2

−π/2dλ′f (λ− λ′)g(λ′)

n(λ)dλ

∞∑m=1

(Tnm ∗ ρm)(λ)

an(λ) =1π

Tnm(λ) =

a|n−m|(λ) + 2a|n−m|+2(λ) + . . . m 6= n

+2an+m−2(λ) + an+m(λ)

2a2(λ) + · · ·+ 2a2n−2(λ) + a2n(λ) m = n

(f ∗ g)(λ) =

ˆ π/2

−π/2dλ′f (λ− λ′)g(λ′)

n(λ)dλ

∞∑m=1

(Tnm ∗ ρm)(λ)

an(λ) =1π

Tnm(λ) =

a|n−m|(λ) + 2a|n−m|+2(λ) + . . . m 6= n

+2an+m−2(λ) + an+m(λ)

2a2(λ) + · · ·+ 2a2n−2(λ) + a2n(λ) m = n

(f ∗ g)(λ) =

ˆ π/2

−π/2dλ′f (λ− λ′)g(λ′)

Outline

1 Introduction

4 The overlap TBA

5 Follow-up

6 Conclusions

Selection of initial states: the quantum quench paradigm

Quantum quench

H(g0) −→t=0

ground state −→ time evolution −→large t

steady state

Role of locality in our considerationsStart from ground state of local HamiltonianEvolved by a local HamiltonianRelevant observables: local operatorsExpect local charges to play a role

Example starting states for XXZ quenches:Neel: | ↑↓↑↓ . . . ↑↓〉 – ground state at ∆ =∞Dimer:(

(| ↑↓〉 − | ↓↑〉))⊗(

(| ↑↓〉 − | ↓↑〉))⊗· · ·⊗

(1√2

(| ↑↓〉 − | ↓↑〉))

ground state of Majumdar-Ghosh Hamiltonian

Quantum quench

H(g0) −→t=0

steady state

(| ↑↓〉 − | ↓↑〉))⊗(

(| ↑↓〉 − | ↓↑〉))⊗· · ·⊗

(1√2

(| ↑↓〉 − | ↓↑〉))

Quantum quench

H(g0) −→t=0

steady state

(| ↑↓〉 − | ↓↑〉))⊗(

(| ↑↓〉 − | ↓↑〉))⊗· · ·⊗

(1√2

(| ↑↓〉 − | ↓↑〉))

Outline

1 Introduction

4 The overlap TBA

5 Follow-up

6 Conclusions

Truncated GGE

Previously: GGE only for theories equivalent to free particles.Pozsgay (2013): truncated GGE for XXZ chain– the first GGE realization in a genuinely interacting modelIdea:

use the quantum transfer method (QTM) to solve the chainkeep first few (up to 6) even chargesQ2(= H), Q4, Q6, Q8, Q10, Q12

use QTM formula to predict correlatorsObservations:

converges fast in truncation level (improved by extrapolation)truncated GGE is “the GGE” as a matter of principle

Fagotti & Essler (2013): construct full/untruncated GGEFagotti, Collura, Essler & Calabrese (2013):

compare GGE to tDMRG → some puzzles:translational invariance is not restored for dimer

Truncated GGE

Outline

1 Introduction

3 Moment of truth: comparing to real time evolutionPrevious results: problems with dimer caseCompare to iTEBD (Werner & Zaránd)

4 The overlap TBA

5 Follow-up

6 Conclusions

Outline

1 Introduction

4 The overlap TBA

5 Follow-up

6 Conclusions

Problems with the dimer state

M. Fagotti, M. Collura, F.H.L. Essler and P. Calabrese:Relaxation after quantum quenches in the spin-1/2 HeisenbergXXZ chain, Phys. Rev. B89 (2014) 125101

0 1 2 3 4 5 6-1

0|MG⟩ ⟶ Δ = 2

⟨σz jσ

z j+k⟩

0 1 2 3 4 5 6-1

0.2 |MG⟩ ⟶ Δ = 4

⟨σz jσ

z j+k⟩

0 1 2 3 4 5 6-1

0.2 |MG⟩ ⟶ Δ = 8

⟨σz jσ

z j+k⟩

0 1 2 3 4 5 6-1

0|MG⟩ ⟶ Δ = 2

⟨σx jσ

x j+k⟩

0 1 2 3 4 5 6-1

0.2 |MG⟩ ⟶ Δ = 4

⟨σx jσ

x j+k⟩

0 1 2 3 4 5 6-1

0.2 |MG⟩ ⟶ Δ = 8

⟨σx jσ

x j+k⟩

Translation invariance seems broken: no relaxation at all?

Problems with the dimer state

M. Fagotti, M. Collura, F.H.L. Essler and P. Calabrese:Relaxation after quantum quenches in the spin-1/2 HeisenbergXXZ chain, Phys. Rev. B89 (2014) 125101

0 1 2 3 4 5 6-1

0|MG⟩ ⟶ Δ = 2

⟨σz jσ

z j+k⟩

0 1 2 3 4 5 6-1

0.2 |MG⟩ ⟶ Δ = 4

⟨σz jσ

z j+k⟩

0 1 2 3 4 5 6-1

0.2 |MG⟩ ⟶ Δ = 8

⟨σz jσ

z j+k⟩

0 1 2 3 4 5 6-1

0|MG⟩ ⟶ Δ = 2

⟨σx jσ

x j+k⟩

0 1 2 3 4 5 6-1

0.2 |MG⟩ ⟶ Δ = 4

⟨σx jσ

x j+k⟩

0 1 2 3 4 5 6-1

0.2 |MG⟩ ⟶ Δ = 8

⟨σx jσ

x j+k⟩

Translation invariance seems broken: no relaxation at all?

Outline

1 Introduction

4 The overlap TBA

5 Follow-up

6 Conclusions

Result of iTEBD

0 1 2 3

Time0.4

<σz i

σz i+2>

Neel ∆ = 3

Result of iTEBD compared to thermal

0 1 2 3

Time0.4

<σz i

σz i+2>

Neel ∆ = 3

Result of iTEBD compared to thermal and GGE

0 1 2 3

Time0.4

<σz i

σz i+2>

Neel ∆ = 3

Result of iTEBD

0 1 2 3Time

<σz i

σz i+2 >

Dimer ∆ = 3

Result of iTEBD compared to GGE

0 1 2 3Time

<σz i

σz i+2 >

Dimer ∆ = 3

Outline

1 Introduction

4 The overlap TBAQuench action methodInitial states and exact overlapsOverlap TBASpin-spin correlations

5 Follow-up

6 Conclusions

Outline

1 Introduction

5 Follow-up

6 Conclusions

The quench action

A = limT→∞

0dt 〈A(t)〉 =

|Cα|2Aαα Cα = 〈ψ(0)|α〉

Replace sum:∑α

−→ˆ ∞∏

Dρn(λ)eLs[ρn(λ)]

eLs[ρn(λ)]: number of Bethe states scaling to ρn(λ) in the TDL.Supposing that Cα and Aαα only depend on ρn(λ) in the TDL:

ˆ ∞∏n=1

Dρn(λ)e−L(− 2LRe ln〈ψ(0)|ρn(λ)〉−s[ρn(λ)])〈ρn(λ)|A|ρn(λ)〉

In TDL: determined by saddle point of quench action functional

S[ρn(λ)] = −2LRe ln〈ψ(0)|ρn(λ)〉 − s[ρn(λ)]

A = 〈ρ∗n(λ)|A|ρ∗n(λ)〉

J.-S. Caux & F.H.L. Essler, 2013

The quench action

A = limT→∞

0dt 〈A(t)〉 =

Replace sum:∑α

−→ˆ ∞∏

ˆ ∞∏n=1

A = 〈ρ∗n(λ)|A|ρ∗n(λ)〉

The quench action

A = limT→∞

0dt 〈A(t)〉 =

Replace sum:∑α

−→ˆ ∞∏

ˆ ∞∏n=1

A = 〈ρ∗n(λ)|A|ρ∗n(λ)〉

The quench action

A = limT→∞

0dt 〈A(t)〉 =

Replace sum:∑α

−→ˆ ∞∏

ˆ ∞∏n=1

A = 〈ρ∗n(λ)|A|ρ∗n(λ)〉

Outline

1 Introduction

5 Follow-up

6 Conclusions

Initial states and overlapInitial states: translation invariant Neel and dimer states(T is translation by one site)

|ΨN〉 =1 + T√

⊗L/2

| ↑↓〉 |ΨD〉 =1 + T√

⊗L/2

(| ↑↓〉 − | ↓↑〉√

)Only overlaps with Sz = 0 states built upon “Cooper pairs”:

|λk〉C = |λ1,−λ1, λ2,−λ2, . . . , λL/4,−λL/4〉

Neel overlaps (Brockmann, De Nardis, Wouters & Caux, 2014)

〈ΨN |λk〉C√C 〈λk|λk〉C

L/4∏j=1

√tan(λj + iη/2) tan(λj − iη/2)

2 sin(2λj)×(Fredholm det)

Dimer overlaps (Pozsgay, 2014)

〈ΨD |λk〉 = 〈ΨN |λk〉∏k

(1− sin(λj − iη/2)

sin(λj + iη/2)

|ΨN〉 =1 + T√

⊗L/2

| ↑↓〉 |ΨD〉 =1 + T√

⊗L/2

(| ↑↓〉 − | ↓↑〉√

|λk〉C = |λ1,−λ1, λ2,−λ2, . . . , λL/4,−λL/4〉

L/4∏j=1

(1− sin(λj − iη/2)

sin(λj + iη/2)

|ΨN〉 =1 + T√

⊗L/2

| ↑↓〉 |ΨD〉 =1 + T√

⊗L/2

(| ↑↓〉 − | ↓↑〉√

|λk〉C = |λ1,−λ1, λ2,−λ2, . . . , λL/4,−λL/4〉

L/4∏j=1

(1− sin(λj − iη/2)

sin(λj + iη/2)

|ΨN〉 =1 + T√

⊗L/2

| ↑↓〉 |ΨD〉 =1 + T√

⊗L/2

(| ↑↓〉 − | ↓↑〉√

|λk〉C = |λ1,−λ1, λ2,−λ2, . . . , λL/4,−λL/4〉

L/4∏j=1

(1− sin(λj − iη/2)

sin(λj + iη/2)

Overlaps in the thermodynamic limit

Integral form for L→∞

−2Re ln〈ψ(0)|ρn(λ)〉 =∞∑

ˆ π/2

−π/2dλρn(λ)gψn (λ)

gψn (λ) =n∑

(n + 1− 2j))

For our translation invariant Neel and dimer states

gN1 (λ) = − ln

(tan(λ+ iη/2) tan(λ− iη/2)

4 sin2(2λ)

)gN1 (λ) = − ln

(sinh4(η/2) cot(λ)

sin(2λ+ iη) sin(2λ− iη)

Overlaps in the thermodynamic limit

Integral form for L→∞

−2Re ln〈ψ(0)|ρn(λ)〉 =∞∑

ˆ π/2

−π/2dλρn(λ)gψn (λ)

gψn (λ) =n∑

(n + 1− 2j))

For our translation invariant Neel and dimer states

gN1 (λ) = − ln

(tan(λ+ iη/2) tan(λ− iη/2)

4 sin2(2λ)

)gN1 (λ) = − ln

(sinh4(η/2) cot(λ)

sin(2λ+ iη) sin(2λ− iη)

Outline

1 Introduction

5 Follow-up

6 Conclusions

The overlap TBA equationsEntropy: 1/2 of Yang-Yang expression (only paired states!)

s[ρn(λ)] =12

∞∑n=1

ˆ π/2

−π/2dλ[ρn(λ) ln

ρhn(λ)

ρn(λ)

)+ ρh

n(λ) ln(

1 +ρn(λ)

ρhn(λ)

)]Compute saddle point of quench action

S[ρn(λ)] = − 2L

Re ln〈ψ(0)|ρn(λ)〉 − s[ρn(λ)]

with the condition that the Bethe-Takahashi equations must hold

∞∑m=1

(Tnm ∗ ρm)(λ)

Result: the overlap thermodynamic Bethe Ansatz equations

log ηn(λ) = gψn (λ)− µn +∞∑

Tnm ? ln(1 + η−1m ) ηn(λ) =

ρhn(λ)

ρn(λ)

µ: chemical potential for Sz (external magnetic field).Independently obtained for the Neel case by Caux et al. before us.

s[ρn(λ)] =12

∞∑n=1

ˆ π/2

ρhn(λ)

ρn(λ)

)+ ρh

n(λ) ln(

1 +ρn(λ)

ρhn(λ)

S[ρn(λ)] = − 2L

∞∑m=1

(Tnm ∗ ρm)(λ)

Tnm ? ln(1 + η−1m ) ηn(λ) =

ρhn(λ)

ρn(λ)

s[ρn(λ)] =12

∞∑n=1

ˆ π/2

ρhn(λ)

ρn(λ)

)+ ρh

n(λ) ln(

1 +ρn(λ)

ρhn(λ)

S[ρn(λ)] = − 2L

∞∑m=1

(Tnm ∗ ρm)(λ)

Tnm ? ln(1 + η−1m ) ηn(λ) =

ρhn(λ)

ρn(λ)

Outline

1 Introduction

5 Follow-up

6 Conclusions

Correlations on the XXZ chainWe want observables: spin-spin correlators

〈σiσi+n〉

But: it was only known how to get these in the QTM formalism!

1st idea: get from oTBA to QTM.May 2014: conference lecture by J-S Caux in New York:discrepancy between GGE and oTBA densities ρn(λ)−→ first idea killed :(

2nd idea: Hellmann-Feynman theorem

HXXZ =L∑

i σxi+1 + σy

i σyi+1 + ∆σz

i σzi+1)

⇒ ∂EΨ

∂∆= L〈Ψ|(σz

1σz2 − 1)|Ψ〉

〈σiσi+n〉

HXXZ =L∑

i σxi+1 + σy

i σyi+1 + ∆σz

i σzi+1)

⇒ ∂EΨ

∂∆= L〈Ψ|(σz

1σz2 − 1)|Ψ〉

〈σiσi+n〉

HXXZ =L∑

i σxi+1 + σy

i σyi+1 + ∆σz

i σzi+1)

⇒ ∂EΨ

∂∆= L〈Ψ|(σz

1σz2 − 1)|Ψ〉

〈σiσi+n〉

HXXZ =L∑

i σxi+1 + σy

i σyi+1 + ∆σz

i σzi+1)

⇒ ∂EΨ

∂∆= L〈Ψ|(σz

1σz2 − 1)|Ψ〉

Correlations on the XXZ chain

3rd idea: a real stroke of genius (B. Pozsgay)

Hellmann-Feynman result analogous to QTM correlator formulas−→ can be extended to a conjecture for all 〈σiσi+n〉

M. Mestyán and B. Pozsgay,J. Stat. Mech. (2014) P09020.

How does it work? (without formulas)

1 Solve oTBA for the ηn functions.2 Solve Bethe-Takahashi for the ρn and ρh

n.3 Check that 〈Qi 〉 agree with initial state

(+ overlap sum rule S[ρ∗n(λ)] = 0. GGE gives −∞!)4 Solve auxiliary integral equations for some auxiliary functions5 Substitute auxiliary functions into the place of their analogues

in the QTM correlator formulas6 Compute correlator

Result of iTEBD compared to GGE and oTBA

0 1 2 3Time

<σz i

σz i+2 >

Dimer ∆ = 3

This is very strong evidence for breakdown of GGE

1. n = 2: sublattice independent (projection is immaterial)2. Agreement with oTBA shows system has relaxed

Dimer correlators as function of ∆

1 2 3 4 5 6 7 8 9∆

<σz 1

σz 2>

iTEBDoTBAGGEtGGE

1 2 3 4 5 6 7 8 9∆

<σz 1

σz 3>

iTEBDoTBAGGEtGGE

Remarks:1. oTBA6=GGE for Neel as well (observed also by Caux et al.)... but difference is too small for iTEBD to resolve2. xx correlators show same patterns, but iTEBD is less accurate

Dimer correlators as function of ∆

1 2 3 4 5 6 7 8 9∆

<σz 1

σz 2>

iTEBDoTBAGGEtGGE

1 2 3 4 5 6 7 8 9∆

<σz 1

σz 3>

iTEBDoTBAGGEtGGE

Remarks:1. oTBA6=GGE for Neel as well (observed also by Caux et al.)... but difference is too small for iTEBD to resolve2. xx correlators show same patterns, but iTEBD is less accurate

Outline

1 Introduction

4 The overlap TBA

5 Follow-upRole of bound statesBreakdown of the GETH

6 Conclusions

Outline

1 Introduction

4 The overlap TBA

6 Conclusions

Role of bound statesXXZ spectrum: n ≥ 2 strings are bound states of n = 1 magnonsin such a case the Qi do not uniquely determine the ρGGE: maximum entropy with given Qi... but that is irrelevant (S[k] = −∞)!)

Ii0 !GGE

! Ii0 !GGE k

G. Goldstein and N. Andrei: Phys. Rev. A 90, 043625

Outline

1 Introduction

4 The overlap TBA

6 Conclusions

Breakdown of GETHHowever: if GETH held, any state (including GGE) on the shell

Γ(ψ(0)) = |ρn(λ)〉 : 〈ρn(λ)|Qi |ρn(λ)〉 = 〈ψ(0)|Qi |ψ(0)〉

would give the same expectation value for local operators!Indeed, this is how non-integrable systems thermalize.

However: GETH is broken!B. Pozsgay: Failure of the generalized eigenstate thermalizationhypothesis in integrable models with multiple particle species,J. Stat. Mech. (2014) P09026.This also shows that GGE is broken for almost all initial states!

GGE seems to work in systems without bound statesB. Pozsgay: Quantum quenches and Generalized Gibbs Ensemble ina Bethe Ansatz solvable lattice model of interacting bosons,J. Stat. Mech., in press.

Outline

1 Introduction

4 The overlap TBA

5 Follow-up

6 Conclusions

Conclusions

1 The GGE description is not generally valid for genuinelyinteracting integrable systems.

2 Failure of GGE is caused by failure of thegeneralized eigenstate thermalization hypothesis.(Classical version: ergodicity on action-angle tori.)

3 Failure of GGE seems to be tied with existence of bound states.

4 The quench action correctly captures the steady state.However, it is a microscopic description!

Open issues1 Are there more local charges? Very-very likely: NO!

2 Non-local charges?Known to exist for −1 < ∆ < 1, recently also for ∆ > 1!

Must be careful (!) under what condition they can play a role:

derivation of canonical from microcanonical ensemble assumessubsystem additivity of conserved quantities (e.g. energy)!

Additivity is needed in TDL only: quasi-local charges?

3 Any other ideas?

The quest is open:the ensemble describing the steady state of integrable systems ispresently unknown.

Open issues1 Are there more local charges? Very-very likely: NO!

2 Non-local charges?Known to exist for −1 < ∆ < 1, recently also for ∆ > 1!

Must be careful (!) under what condition they can play a role:

derivation of canonical from microcanonical ensemble assumessubsystem additivity of conserved quantities (e.g. energy)!

Additivity is needed in TDL only: quasi-local charges?

3 Any other ideas?

The quest is open:the ensemble describing the steady state of integrable systems ispresently unknown.

Pozsgai Balázs: Junior Prima Díj 2014

Correlations after quantum quenches in the XXZ spin chain ...€¦ · Correlations after quantum...

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