Dynamics and relaxation in integrable quantum … and relaxation in integrable quantum systems ......
Transcript of Dynamics and relaxation in integrable quantum … and relaxation in integrable quantum systems ......
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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, …
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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
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Applications of integrability in many-body physics
Ultracold atomsQuantum magnetism
Atomic nucleiQuantum dots,
NV centers
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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
#
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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
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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
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|{�}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’
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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)
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Out-of-equilibrium dynamics
from integrability
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Pyotr L. Kapitza (8/7/1894-8/4/1984)
Kapitza pendulum, 1951
The simple pendulum on its head
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The Kapitza pendulum
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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
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Quasisoliton dynamics
in spin chains
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Solitons (classical)John Scott Russell:
wave of translation (1834)
(Herriot-Watt University)
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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
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“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 )
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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
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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
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Quantum quenches
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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:
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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}
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�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
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Quantum Newton’s cradle
Quantum quenches:
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Ergodicity in interacting quantum systems close to an integrable model
David Weiss’s quantum Newton’s cradle experiment
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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
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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
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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
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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)
�
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Quantum Newton’s cradle:TG limit, trap geometry
q = 3⇡, A = 1.5, ! = 10/N (N = 50)
Local density
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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)
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Impurity in a Lieb-Liniger gas:arrested expansion/quantum stutter
Neil Robinson, J.-S. C. and Robert Konik
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BEC to repulsive Lieb-Liniger
quench
Quantum quenches:
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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
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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
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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
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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
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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
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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
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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
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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
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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
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⇢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
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Néel to XXZ quench
Quantum quenches:
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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
#
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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
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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�
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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
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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
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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
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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)
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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…
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‘Revalidating’ the GGE
for Néel to XXZ
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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)
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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:
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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
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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
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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