Jean-Sébastien Caux Universiteit van Amsterdam Work done in collaboration with (among others): Dynamics and relaxation in integrable quantum systems WEH Seminar Isolated Quantum Many-Body Quantum Systems Out Of Equilibrium Bad 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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Dynamics and relaxation in integrable quantum … and relaxation in integrable quantum systems ... Pyotr L. Kapitza (8/7/1894-8/4/1984) Kapitza pendulum, ... Kapitza-Dirac pulse: U
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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, …
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:
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