Jean-Sébastien Caux Universiteit van Amsterdam

Work done in collaboration with (among others):

Dynamics and relaxation in integrable

quantum systemsWEH Seminar

Isolated Quantum Many-Body Quantum Systems Out Of EquilibriumBad Honnef, 30 November 2015

A’dam gang: R. van den Berg, R. Vlijm, S. Eliens, J. De Nardis, B. Wouters,M. Brockmann, D. Fioretto, O. El Araby,

F.H.L. Essler, R. Konik, N. Robinson, M. Haque, E. Ilievski, T. Prosen, …

Plan of the talk

Out-of-equilibrium dynamics

Summary & perspectives

Interaction quench in Lieb-LinigerThe

Quench Action

Anisotropy quench in XXZ

Quasisoliton dynamics in XXZ

Quantum Newton’s cradle: TG limit

Applications of integrability in many-body physics

Ultracold atomsQuantum magnetism

Atomic nucleiQuantum dots,

NV centers

Heisenberg spin-1/2 chain

Models discussed in this talk:

HN = −

N!

j=1

∂2

∂x2j

+ 2c!

1≤j<l≤N

δ(xj − xl)

Interacting Bose gas (Lieb-Liniger)

H =N

!

j=1

"

J(Sxj S

xj+1+S

yj S

yj+1+∆S

zj S

zj+1)−HzS

zj

#

Bethe Ansatz (1931)

July 2, 1906 – March 6, 2005

H =Z L

0dx H(x)

Integrable Hamiltonian:

‘Reference state’:

‘Particles’:

vacuum, FM state,...

atoms, down spins, ...

Exact many-body wavefunctions (in N-particle sector):

N

({x}|{�}) =X

P

(�1)[P ]A

P

({�})eixjk(�Pj )

... made up of free waves ...... parametrized by rapidities...... with specified relative amplitudes...... and obeying some form of Pauli principle

The Bethe Wavefunction

The Bethe Wavefunction

The Bethe W

avefunction

Michel Gaudin Translated by Jean-Sébastien Caux

Gaudin and C

aux

Michel Gaudin’s book La fonction d’onde de Bethe is a uniquely influential masterpiece on exactly solvable models of quantum mechanics and statistical physics. Available in English for the first time, this translation brings his classic work to a new generation of graduate students and researchers in physics. It presents a mixture of mathematics interspersed with powerful physical intuition, retaining the author’s unmistakably honest tone.

The book begins with the Heisenberg spin chain, starting from the coordinate Bethe Ansatz and culminating in a discussion of its thermodynamic properties. Delta-interacting bosons (the Lieb-Liniger model) are then explored, and extended to exactly solvable models associated with a reflection group. After discussing the continuum limit of spin chains, the book covers six- and eight-vertex models in extensive detail, from their lattice definition to their thermodynamics. Later chapters examine advanced topics such as multicomponent delta-interacting systems, Gaudin magnets and the Toda chain.

MICHEL GAUDIN is recognized as one of the foremost experts in this field, and has worked at Commissariat à l’énergie atomique (CEA) and the Service de Physique Théorique, Saclay. His numerous scientific contributions to the theory of exactly solvable models are well known, including his famous formula for the norm of Bethe wavefunctions.

JEAN-SÉBASTIEN CAUX is a Professor in the theory of low-dimensional quantum condensed matter at the University of Amsterdam. He has made significant contributions to the calculation of experimentally observable dynamical properties of these systems.

Cover illustration: a representation of the Yang-Baxter relation by John Collingwood.

Cover designed by Hart McLeod Ltd

9781

1070

4585

9 G

AU

DIN

& C

AU

X –

TH

E B

ETH

E W

AV

EFU

NC

TIO

N C

M Y

K

|{�}i

The general idea, simply stated:Start with your favourite quantum state

(expressed in terms of Bethe states)

OApply some operator on it

Reexpress the result in the basis of Bethe states:

O|{�}i =X

{µ}FO{µ},{�}|{µ}i

FO{µ},{�} = h{µ}|O|{�}iusing ‘matrix elements’

Quantum spin chainsCorrelations, experiments (INS, RIXS), prefactors, ...

Sr CuO32

Walters, Perring, Caux, Savici, Gu, Lee, Ku, Zaliznyak,

NATURE PHYSICS 2009

(C D N) CuBr125 2 4

Thielemann, Rüegg, Rønnow, Läuchli, Caux, Normand, Biner, Krämer, Güdel, Stahn, Habicht, Kiefer, Boehm, McMorrow, Mesot, PRL 2009

Lake, Tennant, Caux, Barthel, Schollwöck, Nagler, Frost, PRL 2013

KCuF3

Schlappa, Wohlfeld, Zho, Mourigal, Haverkort, Strocov, Hozoi, Monney,

Nishimoto, Singh, Revcolevschi, Caux, Patthey Rønnow, van den

Brink, Schmitt,NATURE 2012

Sr CuO32(RIXS)

Out-of-equilibrium dynamics

from integrability

Pyotr L. Kapitza (8/7/1894-8/4/1984)

Kapitza pendulum, 1951

The simple pendulum on its head

The Kapitza pendulum

Out-of-equilibrium using integrability

Interaction quench in RichardsonDomain wall release in HeisenbergGeometric quenchInteraction turnoff in Lieb-LinigerRelease of trapped Lieb-Liniger

The super Tonks-Girardeau gasSplit Fermi sea in Lieb-Liniger

BEC to Lieb-Liniger quench

Highly excited initial (eigen)states:

Quenched states:

Néel to XXZ quenchDriven systems: Spin echo in quantum dots

Quasisolitons

Quantum Newton’s cradle: TG

Quasisoliton dynamics

in spin chains

Solitons (classical)John Scott Russell:

wave of translation (1834)

(Herriot-Watt University)

Solitons (classical)

Further simulations: Zabusky & Kruskal 1965concept of a soliton

(Boussinesq)Korteweg-de Vries equation

@t

u+ u@x

u+ �2@3x

u = 0

Classical inverse scattering

First simulations: Fermi-Pasta-Ulam-(Tsingou)

absence of ergodicity

“Particle content” of XXZ: nontrivialSolution of Bethe equations: rapidities + strings

} iζ

Single particle (bound state of magnons)

Classification of strings: Bethe, Takahashi, Suzuki, ...

λj,aα = λ

jα + i

ζ

2(nj + 1 − 2a) + iδ

j,aα

O(e−(cst)N )

String wavepacketstJ

j

�2

�

0

40

80

20 40 60 80

0.2

0.3

0.4

0.5

hSz j(t)i

� = 0.9 Two-string

Three-string

10

�6

10

�4

0.01

1

0 2 4 6 8 10

| (t)i = e�iHt| (0)i =X

{�}

e�iE{�}tC{�}|{�}i

In the eigenbasis, time evolution of a generic state: simple!

Localized wavepacket: | (0)i = N0

X

p

e�ipx�↵2

4 (p�p)2 |�(n)(p)i

�x(t) =

r↵

2

4+

�

2nt

2

↵

2

�2n = J2

✓�2(0)

�2n(0)

◆2

cos

2(p)

Dispersion: width ~ t:

function of anisotropy:

Vlijm, Ganahl, Fioretto, Brockmann, Haque, Evertz and Caux, 2015

Quasisoliton scattering (quantum)

tJ

j j

0

20

40

25 50 75

0

50

100

150

20025 50 75

0.2

0.3

0.4

0.5

hSz j(t)i

� = 2

0

100

200

20 40 60 80

tJ

j

0

2

4

6

0 1 2 3 4

Displacement

�

Measured av.

Linear fits

Single magnons

Bound magnons

��(1,1)

��(2,2)Displacement as a function of anisotropy (fixed incoming momenta)

1str-1str 2str-2str

‘Worldlines’of colliding

wavepackets:

Vlijm, Ganahl, Fioretto, Brockmann, Haque, Evertz and Caux, 2015

Quantum quenches

Quenches (more generally: ‘prepare and release’)

The problem: considering a generic initial state, what is the time evolution of the system?

| (t)i = e�iHt| (t = 0)i

initial state is NOT eigenstate of H

O(t) ⌘ h (t)|O| (t)ih (t)| (t)i

Hamiltonian driving time evolution

{Observable expectation values depend on time:

First question: what is the steady state long after the quench?

Conjecture: steady state is described by a generalized Gibbs ensemble (GGE)

Rigol, Dunjko, Yurovsky, Olshanii, PRL 2007 see also Jaynes, Phys. Rev. 1957

Crucial point: time evolution in the presence of myriads of constraints (due to integrability) is special

Fundamental issue: does the system relax? thermalize?

limt!1

O(t) = hOiGGE =Tr{Oe�

Pn �nQn}

Tr{e�Pn �nQn}

�Qm⇥ = Trn

Qme�P

n �nQn

o

/ZGGE m = 0, 1, 2, . . .

Generalized inverse temperatures to be set using the initial conditions on conserved charges

In practice: implementable for free theories onlySelf-consistency problem difficult to solve

In reality, three major difficulties: Conserved charges are generically nontrivial

ZGGE = Tre�P

n �nQnwhere

For interacting cases: not understood in general.

GGE implementation

(charges: momentum occupation modes)

Finding a complete set of charges is an issue

Quantum Newton’s cradle

Quantum quenches:

Ergodicity in interacting quantum systems close to an integrable model

David Weiss’s quantum Newton’s cradle experiment

Atoms do not thermalize during the experimental time

scale (about 50 cycles)

Experimentally possible to ‘break’ integrability in different ways, to test

relaxation and ergodicity

Does there exist a quantum KAM theorem ?

Nonequilibrium & quench physics experimentally accessible

Quantum Newton’s cradle:strongly-interacting limit

Kapitza-Dirac pulse:

ˆ

UB(q,A) = exp

� iA

Zdx cos(qx)

ˆ

†(x)

ˆ

(x)

!

Tonks-Girardeau limit: bosonic wavefns from fermionic ones

B(x; t) =Y

1i<jN

sgn(xi � xj) F (x; t)

Slater determinant of single-particle states

van den Berg, Wouters, Eliëns, De Nardis, Konik and Caux, 2015

Observables:

hn(k, t)i = 1

2⇡

Zdxdye

i(x�y)kh †(x, t) (y, t)i

h⇢(x, t)i = h †(x, t) (x, t)iLocal density

Momentum dist fn

Two geometries:

Ring (periodic):

Harmonic trap:

Single-particle wavefunctions after pulse of momentum q:

j

(x; t) =1X

�=�1I

�

(�iA)1pL

e

�i(�j+�q)xe

�i(�j+�q)2t/2m

j

(x; t) =1X

�=�1I

�

(�iA)e�i�q cos(!t)(x+ �q2m! sin(!t))

j

(x+ �q

m!

sin(!t))e�i!(j+

12 )t

Overlaps of post-Bragg pulse states with eigenstates: from matrix elements of Bragg pulse operator

hµ|UB(q, A)|�iLN

= detN

I�j�µk

q

(�iA) �(q)�j ,µk

�

Stationary state distribution: (Quench Action and GGE give same)

⇢spq,A(�) =

1

2⇡

X

�2Z

⇥✓(�� �q + �F )� ✓(�� �q � �F )

⇤|I�(iA)|2

Local density: lim

th

h q,A(t)|⇢(x)| q,A(t) =nm

q�F t⇥

1X

�=�1J�(�2A sin(q

2�t/2m)) cos(xq�)

sin(q�F�t/m)

�

Quantum Newton’s cradle:TG limit, trap geometry

q = 3⇡, A = 1.5, ! = 10/N (N = 50)

Local density

Quantum Newton’s cradle:TG limit, trap geometry

q = 3⇡, A = 1.5, ! = 10/N (N = 50)

Momentum distn fn extremely rapid relaxation (much faster than

trap oscillation)

Impurity in a Lieb-Liniger gas:arrested expansion/quantum stutter

Neil Robinson, J.-S. C. and Robert Konik

BEC to repulsive Lieb-Liniger

quench

Quantum quenches:

Quench from BEC to repulsive gas

Turn repulsive interactions on from t=0 onwards:

particles ‘repel away’ from each other, system heats up, momentum distribution broadens, ...

Start from GS of noninteracting theory,

|0N i ⌘ 1pLNN !

⇣ †k=0

⌘N|0i

This is a difficult problem to treat...

Davies 1990; Davies and Korepin

Qn({�}N ) =NX

j=1

�njQn : Qn|{�}N i = Qn|{�}N i

Conserved charges:Kormos, Shashi, Chou and Imambekov, arxiv:1204.3889

1) Generalized Gibbs ensemble logic

GGE inapplicable, charges take infinite values!J-S C + J. Mossel, unpublished

2) GGE on lattice, q-deformed model

Works, partial results only (using a few charges)Kormos, Shashi, Chou, JSC, Imambekov, PRA 2014

The ‘quench action’ approachJ-SC & F.H.L. Essler, PRL 2013

in pictures...

Initial state:

in pre-quench Hilbert space basis

in post-quench Hilbert space basis

The ‘quench action’ approachJ-SC & F.H.L. Essler, PRL 2013

Quench action landscape: determined by overlaps & entropy

‘Generalized thermodynamic Bethe Ansatz’J. Mossel and J-SC, JPA 2012; J-SC & R. Konik, PRL 2012,

see also Fioretto & Mussardo NJP 2010, Pozsgay JSTAT 2011

Variational approach, implemented by a

The ‘quench action’ approachJ-SC & F.H.L. Essler, PRL 2013

Saddle-point state: determines steady state

States around saddle-point:determine relaxation towards steady state

limTh

O(t) = limTh

1

2

X

{e}

he��S{e}[⇢sp]�i!{e}[⇢sp]th⇢sp|O|⇢sp; {e}i

+e��S⇤{e}[⇢sp]+i!{e}[⇢sp]th⇢sp; {e}|O|⇢spi

i

Main message: the *full* time dependence is recoverable using a minimal amount of data

saddle-point distribution (from GTBA)excitations in vicinity of sp state (easy)differential overlapsselected matrix elements

The ‘quench action’ approachGeneric time-dependent expectation values:

J-SC & F.H.L. Essler, PRL 2013

GQjk = �jk

⇣L+

N/2X

l=1

KQ(lj , ll)⌘�KQ(lj , lk)

h{�j}N/2j=1 , {��j}N/2

j=1 |0i =s

(cL)�NN !

detNj,k=1 Gjk

detN/2j,k=1 G

Qjk

N/2Y

j=1

�j

c

s�2j

c2+

1

4

KQ(�, µ) = K(�� µ) +K(�+ µ) K(�) =2c

�2 + c2

J. De Nardis, B. Wouters, M. Brockmann & J-SC, PRA 89, 2014Explicit result:

with matrix

(reminiscent of Gaudin formula)

Back to BEC-LL quench

M. Brockmann JPA 2014

Quench action approach to BEC-LL quench

Need thermodynamic limit form of overlaps:

lim

Thh�,��|0i = exp

⇣�L

2

n⇣log

c

n+ 1

⌘⌘

⇥ exp

⇢�L

2

Z 1

0d�⇢(�) log

�2

c2

✓�2

c2+

1

4

◆�+O(L0

)

�

Quench action now defined, saddle-point solution via generalized thermodynamic Bethe ansatz

We are now in position to apply the quench action logic!

J. De Nardis, B. Wouters, M. Brockmann & J-SC, PRA 89, 2014

Quench action solution to BEC-LL quench

⇢(�) = � �

2⇡

@a(�)

@�(1 + a(�))�1

a(�) =2⇡/�

�c sinh

�2⇡�c

�I1�2i�c

✓4p�

◆I1+2i�

c

✓4p�

◆

It is in fact possible to give a closed form solution of the GTBA for the saddle-point state,

for any value of the interaction:

J. De Nardis, B. Wouters, M. Brockmann & J-SC, PRA 89, 2014

⇢s(x) =p�⇢(c

p�x/2)/2

Subplot: scaled fn

Quench action solution to BEC-LL quench

2⇡⇢(�) ⇠ n4�2

�4+

n6�3(24� �)

4�6+ . . .Asymptotics as from q-bosons:

⇢(�) =1

2⇡

4n2

�2 + 4n2

⇢(�) ⇠ 1

⇡p�

s

1� �2

4�n2

Large c:

Small c: semicircle

Tail explains divergences of evalues of conserved charges

0.0

0.1

0.2

0.3

0.4

-9 -6 -3 0 3 6 9 12

�/n

⇢

sp(�)

0.0

0.2

0.4

0.6

0.8

-1 0 1

⇢s(x) ⇢

ths (x)

x

� = 0� = 0.08� = 0.8� = 8� = 1

J. De Nardis, B. Wouters, M. Brockmann & J-SC, PRA 89, 2014

Néel to XXZ quench

Quantum quenches:

Quench from Néel to XXZStart from Néel state:

From t=0 onwards, evolve with XXZ Hamiltonian

Can one treat this problem exactly?

H =N

!

j=1

"

J(Sxj S

xj+1+S

yj S

yj+1+∆S

zj S

zj+1)−HzS

zj

#

Quench action approachto Néel-XXZ quench

First step: exact overlaps of Néel state with XXZ eigenstates

h 0|{±�j}M/2j=1 i

k{±�j}M/2j=1 k

=p2

2

4M/2Y

j=1

ptan(�j + i⌘/2) tan(�j � i⌘/2)

2 sin(2�j)

3

5

vuutdetM/2(G+jk)

detM/2(G�jk)

G±jk = �jk

0

@NK⌘/2(�j)�M/2X

l=1

K+⌘ (�j ,�l)

1

A+K±⌘ (�j ,�k)

K⌘(�) =sinh(2⌘)

sin(�+ i⌘) sin(�� i⌘)K±⌘ (�, µ) = K⌘(�� µ)±K⌘(�+ µ)

M. Brockmann, J. De Nardis, B. Wouters & J-SC JPA 2014Gaudin-like form again!Tsuchiya JMP1998; Kozlowski & Pozsgay JSTAT 2012

Quench action approachto Néel-XXZ quench

Second step: generalized TBA

Wn(�) =

8><

>:

1

2

n+1sin

22�

coshn⌘�cos 2�coshn⌘+cos 2�

Qn�12

j=1

⇣cosh(2j�1)⌘�cos 2�

(cosh(2j�1)⌘+cos 2�)(cosh 4⌘j�cos 4�)

⌘2

if n odd,

tan

2 �2

ncoshn⌘�cos 2�coshn⌘+cos 2�

1

Qn2j=1(cosh 2(2j�1)⌘�cos 4�)2

Qn�22

j=1

⇣cosh 2j⌘�cos 2�cosh 2j⌘+cos 2�

⌘2

if n even.

ln ⌘n(�) = �2hn� lnWn(�) +1X

m=1

anm ⇤ ln�1 + ⌘�1

m

�(�)

⌘n(�) ⌘ ⇢n,h(�)/⇢n(�)

Solution of this GTBA gives steady-state(analytically!)

where

B. Wouters, J. De Nardis, M. Brockmann, D. Fioretto, M.Rigol & J-SC, PRL 2014

and the effective driving terms (pseudo-energies) are

an(�) =1

⇡

sinn⌘

coshn⌘ � cos 2�

Quench action approachto Néel-XXZ quench

Equivalent form of generalized TBA:

ln(⌘n) = dn + s ⇤⇥ln(1 + ⌘n�1) + ln(1 + ⌘n+1)

⇤

dn(�) =X

k2Ze�2ik� tanh(⌘k)

k

�(�1)n � (�1)k

�with driving terms

d1(�) = � 1

⇡

1X

m=1

�2m

X

k2Ze�2ik� k2m�2

cosh k⌘

GGE with local charges: same form of coupled equations, but driving term only for n=1:

unknown

Néel-XXZ quench: conserved charges

X

k2Zk2m�2

⇣e�|k|⌘ � ⇢h1 (k)

2 cosh k⌘

⌘= hQ2mi m 2 N

Initial expectation value of local charges:

Remarkable correspondence between 1-string hole density and local charges:

Fagotti & Essler JSTAT 2013

Quench action nontrivially reproduces this; GGE also of course, but only by definition

⇢Neel1,h (�) =

⇡2a31(�) sin2(2�)

⇡2a21(�) sin2(2�) + cosh

2(⌘)

which fixes

B. Wouters, J. De Nardis, M. Brockmann, D. Fioretto, M. Rigol & J-SC, PRL 2014

lim

N!1

1

N

hNeel|Qn+1|Neeli = ��

2

@

n�1

@x

n�1

1��

2

cosh[

p1��

2x]��

2

����x=0

The steady state: Néel to XXZ

Solid lines: quench action

Dashed lines: GGE (local charges)

QA and (local)GGE have different saddle-

point densities

⇢GGE1 � ⇢sp1 =

1

4⇡�2+O(��3),

⇢GGE2 � ⇢sp2 =

1� 3 sin2(�)

3⇡�2+O(��3).

Large Delta expansion:

0

0.2

0.4

0.6

0.8

0 ⇡4

⇡2

�

⇢sp1 (�)

(a)

0

0.005

0.010

0.015

⇢sp2 (�)

(b)

0 ⇡4

⇡2

-0.0020

0.004

�

⇢GGE1 (�)� ⇢sp1 (�)

(c)

� = 1.2� = 1.5� = 2� = 5� = 1

Difference in distribution: impact on correlations

Large Delta expansions:

h�z1�

z2iQA = �1 +

2

�2� 7

2�4+

77

16�6+ . . .

h�z1�

z2iGGE = �1 +

2

�2� 7

2�4+

43

8�6+ . . .

0

0.2

0.4

0.6

0.8

1

1 3 5 7 9

�

h�z1�

z3i

(a)

-0.09

00.02

1 2

�h�z1�

z3i

0.11

0.12

0.13

1 1.01 1.02 1.03

�

h�z1�

z3i

(b) h�z1�

z3isp

h�z1�

z3iGGE

h�z1�

z3iNLCE Numerical

verification using NLCE (M. Rigol)

Not convinced?Look at other results by Budapest group

B. Pozsgay, M. Mestyán, M. A. Werner, M. Kormos, G. Zaránd, G. Takács, PRL 2014

reobtain our Néel results also consider initial dimer state obtain numerical (iTEBD) evidence for

correlations being different in dimer case

There remains no doubt about the correctness of the quench action results because…

‘Revalidating’ the GGE

for Néel to XXZ

Quasilocal charges in XXZ

L(z, s) =1

sinh ⌘

⇣sinh(z) cosh (⌘szs)⌦ �0

+ cos(z) sinh (⌘szs)⌦ �z+ sinh(⌘)(s�s ⌦ �+

+ s+s ⌦ ��)

⌘

[s+s , s�s ] = [2szs ]q [szs , s

±s ] = ±s±s

Auxiliary spins obey q-deformed su(2)

szs |ki = k|ki, s±s |ki =q[s+ 1± k]q[s⌥ k]q|k ± 1i

[x]q = sinh (⌘x)/ sinh(⌘)

in 2s+1-dim irrep

Starting point: q-deformed L-operator

Ilievski, De Nardis, Wouters, Caux, Essler, Prosen 2015Ilievski, Medenjak and Prosen, arXiv:1506.05049

Pereira, Pasquier, Sirker and Affleck, JSTAT 2014

Prosen 2011; Prosen and Ilievski 2013; Ilievski and Prosen 2013; Prosen 2014

Mierzejewski, Prelovšek and Prosen 2015

Previously discovered in XXX, XXZ(gapless)

Here : need generalization to XXZ(gapped)

Quasilocal charges in XXZ(gpd)

s =1

2, 1,

3

2, ...

Ts(z) = Tra [La,1(z, s) . . . La,N (z, s)]

Higher-spin transfer matrices:

Xs(�) = ⌧�1s (�)Ts(z

�� )T 0

s(z+� ), z±� = ±⌘

2+ i�

f(z) = (sinh (z)/ sinh (⌘))N

bXs(�) := T (�)s (z�� )T (+)0

s (z+� )

H(n+1)s =

1

n!@n�bXs(�)

����=0

L(±)(z, s) = L(z, s) sinh (⌘)/[sinh (z ± s⌘)]

lead to spin-s conserved charges

in which ⌧s(�) = f(�(s+ 12 )⌘ + i�)f((s+ 1

2 )⌘ + i�)

More convenient for ThLim:

built from transfer matrix with

Families of quasilocal charges:

A complete GGE for XXZ

%GGE =

1

Zexp

"�

1X

n,s=1

�snH

(n)s/2

#

Ilievski, De Nardis, Wouters, Caux, Essler, Prosen 2015

Throughout the gapped regime (including XXX limit), the GGE density matrix is given by

Steady state: fixed by initial conditions through the generalized remarkable ‘string-charge’ correspondence

⇢ 02s,h(�) = a2s(�) +

1

2⇡

⇥⌦ 0

s (�+ i⌘2 ) + ⌦ 0

s (�� i⌘2 )

⇤

s =1

2, 1,

3

2, ...⌦ 0

s (�) = limth

h 0| bXs(�)| 0iN

Fixing the Néel-to-XXZ GGE

0.11

0.115

0.12

0.125

1.0 1.01 1.02 1.03 1.04�

h�z1�

z3i

1.0 1.01 1.02 1.03 1.04

10�4

10�3

10�2

10�1

�

|�h�z1�z3i|

QA

GGE1/2

GGE1

GGE3/2

GGE2

Implementing the construction for the Néel-to-XXZ quench makes the GGE converge to correct QA answer

Effect on some simple steady-state correlations:

Ilievski, De Nardis, Wouters, Caux, Essler, Prosen PRL 2015

Summary & perspectives

Quench action logic new approach to out-of-equilibrium problems gives access to full time evolution with minimal data

BEC to LL: exact solution from QA (inaccessible to GGE) Néel to XXZ: exact solution from QA GGE with local charges gives different steady state! GGE needs to include quasilocal charges to reproduce QA

Integrability out of equilibrium real-time dynamics in experimentally accessible setups quasisoliton scattering pulsed systems

Food for thought for GGE users

Take-home message: there is more to equilibration than meets the eye