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Timetable (QIMW01)

Quantum integrable models in and out of equilibrium

Monday 11th January 2016 to Friday 15th January 2016

Monday 11th January 2016
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from John Toland (INI Director) INI 1
10:00 to 11:00 Marton Kormos
Quantum quenches in the sine-Gordon model: a semiclassical approach
Co-author: Gergely Zarand (Budapest University of Technology and Economics)

I will report results on the time evolution of correlation functions after quantum quenches in the sine-Gordon model computed using a semiclassical approach which is expected to yield accurate results for small and slow quenches producing slow quasiparticles with low density. I will demonstrate this by showing that the method reproduces the results of a recent form factor calculation of the relaxation of expectation values. Extending these results, the expectation values of most vertex operators will be shown not to decay to zero. I will give analytic expressions for the relaxation of dynamic correlation functions to find that they have diffusive behavior for large timelike separation. At the end of my talk I will show preliminary results obtained with an improved "semi-semiclassical" version of the method.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Joel Moore
Expansion potentials for exact far-from-equilibrium spreading of energy in the XXZ model
Co-authors: Romain Vasseur (UC Berkeley), Christoph Karrasch (UC Berkeley; FU Berlin)

The rates at which energy and particle densities move to equalize arbitrarily large temperature and chemical potential differences in a closed quantum system have an emergent thermodynamical description whenever energy or particle current commutes with the Hamiltonian. Concrete examples include the energy current in the 1D spinless fermion model with nearest-neighbor interactions (XXZ spin chain), energy current in Lorentz-invariant theories or particle current in interacting Bose gas in arbitrary dimension. Even far from equilibrium, these rates are controlled by state functions (generalized pressures or "expansion potentials"), expressed as integrals of equilibrium Drude weights. This relation between nonequilibrium quantities and linear response implies non-equilibrium Maxwell relations for the Drude weights which explain some old XXZ model identities. We verify our results via DMRG calculations for the XXZ chain, which suggest some interesting additional fe atures beyond the analytical results.
12:30 to 13:30 Lunch @ Wolfson Court
14:00 to 15:00 Maurizio Fagotti
Stationary states and (quasi)local charges in a semi-infinite chain
Motivated by the remarkable fact that a local defect can have global effects on the dynamics of local observables after global quenches, I revisit the conserved quantities of the quantum XY model with open boundary conditions. I present two approaches to determine the local conservation laws. A direct method that relies on the diagonalization of the model and a more abstract procedure based on a correspondence between quadratic forms of fermions and block-Toeplitz-plus-Hankel operators. Specifically, I establish the conditions that the symbol of a particular class of block-Toeplitz-plus-Hankel operators must satisfy in order to commute with a given block-Toeplits. I exhibit new families of (quasi)local conservation laws and discuss some physical consequences of the results. 
15:00 to 16:00 Romain Vasseur
Thermalization, integrability and localization
Co-authors: Andrew Potter (UC Berkeley), Sid Parameswaran (UC Irvine)

In this talk, I will review the physics of many-body localized systems that fail to self-thermalize in isolation. I will argue that such systems can be thought of as being generically “integrable”, and discuss their universal dynamics after a quantum quench. I will also describe the new types of nonequilibrium phase transitions that can arise in the presence of many-body localization, and introduce new effective renormalization group approaches to capture their universal critical properties.
16:00 to 16:30 Afternoon Tea
16:30 to 17:30 Francisco Alcaraz
Shannon mutual information of critical quantum chains
Associated to the equilibrium Gibbs state of a given critical classical system in d dimensions we can associate a special quantum mechanical eigenfunction defined in a Hilbert space with the dimension given by the number of configurations of the classical system and components given by the Boltzmann weights of the equilibrium probabilities of the critical system. This class of eigenfunctions are generalizations of the Rokhsar-Kivelson, initially proposed for the dimer problem in 2 dimensions. In particular in two dimensions, where most of the critical systems are conformal invariant, such functions exhibit quite interesting universal features. The entanglement entropy of a line of contiguous variables (classical spins), is given by the shannon entropy of d=1 quantum chains, and the entanglement spectrum of the two dimensional system are given by the amplitudes of the ground-state eigenfunction of the quantum chain. We present a conjecture showing that the Shannon mutual information of the quantum chains in some appropriate basis (we called conformal basis) show a universal behavior with the size of the line of the entangled spins (subsystem size). This dependence allow us to identify the conformal charge of the associated classical critical system (used to define the d=2 quantum eigenfunction) or the quantum critical chain. Tests of this conjecture for integrable and non integrable quantum chains will be presented. We also consider numerical results for two distinct generalizations of the Shannon mutual information: the one based in the concept of the R\'enyi entropy and the one based on the R\'enyi divergence. A numerical test of the extension of this conjecture for critical random chains (not conformal invariant) is also presented. 
17:30 to 18:30 Welcome Wine Reception
Tuesday 12th January 2016
10:00 to 11:00 Ulrich Schneider
Dynamical Quasicondensation of Hard-Core Bosons at Finite Momenta
We experimentally study the expansion of initially localized ultracold bosons in homogeneous optical lattices in real and momentum space and find that both dimensionality and interaction strength crucially influence these out-of-equilibrium dynamics. While the atoms expand ballistically in all integrable limits, deviations from these limits dramatically suppress the expansion and lead to the appearance of almost bimodal real-space density distributions. For strongly interacting bosons, we observe a dimensional crossover of the dynamics from ballistic in the one-dimensional hard-core case to diffusive in two dimensions, as well as a similar crossover when higher occupancies are introduced into the system. Studying the same expansion in momentum space, we observe the onset of quasicondensation of expanding hard-core bosons at finite momenta in a high energy far-from-equilibrium situation, even though long-range order is usually associated with low-temperature equilibrium situations. In particular, we observe the emergence of peaks at finite momenta that corresponds to the spontaneous formation of coherence with a phase order that differs from the ground-state order.

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11:00 to 11:30 Morning Coffee
11:30 to 12:30 Vladimir Gritsev
Integrability and supersymmetry in quantum optics
First I will give an overview of integrable structures in quantum optics. Then I will discuss some new methods recently developed by us for solving quantum dynamics in various nonlinear models of quantum optics. Finally I am going to talk about supersymmetry in quantum optics and spin-orbit coupled systems.
12:30 to 13:30 Lunch @ Wolfson Court
14:00 to 15:00 Jean-Sébastien Caux
Dynamics and relaxation in quantum integrable systems
This talk will review a number of integrability-based results on the dynamics of quantum spin chains and interacting atomic gases. In equilibrium, space- and time-dependent correlations will be reviewed, together with corresponding experimental observations. Integrability will also be used to describe the dynamics of (quasi)solitonic excitations and their scattering properties. For more generic out-of-equilibrium situations, a number of recent developments will be explained, including the Quench Action method for explicitly calculating the relaxation of observables after a quantum quench. The exact solution to the Néel-to-XXZ quench using the QA will be presented, together with recent results on the necessity to include quasilocal charges in the Generalized Gibbs Ensemble in order to properly describe post-quench steady-state properties.
15:00 to 16:00 Pasquale Calabrese
Real time confinement following a quench to a non-integrable model
Light cone spreading of correlations after a quantum quench is a key phenomenon characterising integrable models, which has also been measured in cold atomic experiments. Guided by numerical results for the Ising chain, we'll show that when integrability is broken in such a way that the spectrum consists of only bound states, correlations at large distance get frozen soon after the quench as a consequence of the confinement of the elemetary excitations, implying a breakdown of the light-cone scenario.
16:00 to 16:30 Afternoon Tea & Poster Session
Wednesday 13th January 2016
10:00 to 11:00 Ignacio Cirac
Tensor Network Techniques and systems out of equilibrium
Co-authors: mari carmen banuls (MPQ), matt Hastings (Microsoft), Hyungwon Kim (Rutgers), David Huse (Princeton), Nicola Pancotti (MPQ)

Tensor networks can efficiently describe many-body quantum systems with local interactions in thermal equilibrium. Howevever, as a consequence of the violation of the area law, they cannot describe their dynamics, in general. Still, they may provide useful information about several physical aspects of many-body systems out of equilibrium. In this talk I will mention few of the applications of tensor networks realted to that problem: the computation of quasi-constants of motion, and the characterization of many-body localized states.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Benjamin Doyon
Thermalization and pseudolocality in extended quantum systems
The age-old questions of when and how thermalization occurs in extended systems have received a large amount of attention recently. It has been found that many integrable models, because of the presence of infinitely many conservation laws, display generalized thermalization: generalized Gibbs ensembles (GGE) describe equilibration states after long time evolutions with short-range Hamiltonians. It was also found that extended concepts of locality, such as those of pseudolocal and quasilocal conserved charges, were sometimes involved in GGEs. In this talk I will discuss my recent results in this subject. Using precise definitions of pseudolocal charges and GGEs, I will describe simple conditions on the time-evolved state and the dynamical response functions that guarantee generalized thermalization. I will explain in which cases this implies ordinary (Gibbs) thermalization. The results are based on studying a family of states defined by paths whose tangents are pseudolocal ch arges. The framework does not involve integrability, and all results are mathematically rigorous.

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12:30 to 13:30 Lunch @ Wolfson Court
14:00 to 15:00 John Cardy
Revivals and Entanglement Propagation after a Quantum Quench
We consider the evolution from a state with area-law entanglement by a gapless quantum hamiltonian. Do sub-systems thermalize? How fast does the entanglement spread? In a finite system, are there quantum revivals? We attempt to answer these questions in simple solvable field theory models.
15:00 to 16:00 Tomaz Prosen
Integrability of a deterministic cellular automaton driven by stochastic boundaries
We propose an interacting many-body space-time-discrete Markov chain model, which is composed of an integrable deterministic and reversible cellular automaton (the rule 54 of [Bobenko et al, CMP 158, 127 (1993)]) on a finite one-dimensional lattice Z_2^n, and local stochastic Markov chains at the two lattice boundaries which provide chemical baths for absorbing or emitting the solitons. Ergodicity and mixing of this many-body Markov chain is proven for generic values of bath parameters, implying existence of a unique non-equilibrium steady state. The latter is constructed exactly and explicitly in terms of a particularly simple form of matrix product ansatz which is termed a patch ansatz. This gives rise to an explicit computation of observables and k-point correlations in the steady state as well as the construction of a nontrivial set of local conservation laws. Feasibility of an exact solution for the full spectrum and eigenvectors (decay modes) of the Markov matrix is sug gested as well. We conjecture that our ideas can pave the road towards a theory of integrability of boundary driven classical deterministic lattice systems.
16:00 to 16:30 Afternoon Tea
16:30 to 17:30 Jacopo Viti
Inhomogeneous quenches and arctic curves in fermionic systems
Co-authors: Nicola Allegra (Universite de Lorraine), Jerome Dubail (Universite de Lorraine and CNRS), Masud Haque (MPIPKS), Jean-Marie Stephan (MPIPKS)

In this talk I will consider two fermionic chains at different densities joined together and evolved unitarily. I will present many exact results concerning the large-time limit of correlation functions and the Loschmidt echo. I will show that this type of quench is deeply related, via analytic continuation in imaginary time, to the arctic curve phenomena in statistical mechanics: interfaces separating fluctuating and frozen regions, a celebrated example of which is the arctic circle theorem for dimers on the aztec diamond.

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19:30 to 22:00 Conference Dinner at Trinity College
Thursday 14th January 2016
10:00 to 11:00 Alessandro Silva
Dynamical phase transitions and statistics of excitations
Co-authors: Pietro Smacchia (Rutgers), Eugene Demler (Harvard), Michael Knap (Munich), Anna Maraga (SISSA)

Dynamical phase transitions can occur in isolated quantum systems that are brought out of equilibrium by either a sudden or a gradual change of their parameters. Theoretical examples range from the behaviour of the O(N) model as well as spin-models with long range interactions both showing dynamical criticality in their prethermal steady-states. In this talk I will start by discussing the characterization of such dynamical phase transitions based on the statistics of produced excitations. I will also discuss the role of fluctuations as well as on the differences between sudden and gradual changes of the system parameters. Finally, I will discuss the emergence of zeroes in the Loschmidt amplitude attempting to elucidate their physical meaning in connection with the dynamical transitions above.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Jörg Schmiedmayer
High order correlations and what we can learn about the solution for many body problems from experiment
The knowledge of all correlation functions of a system is equivalent to solving the corresponding quantum many-body problem. If one can identify the relevant degrees of freedom, the knowledge of a finite set of correlation functions is in many cases sufficient to determine a sufficiently accurate solution of the corresponding field theory. Complete factorization is equivalent to identifying the relevant degrees of freedom where the Hamiltonian becomes diagonal. I will give examples how one can apply this powerful theoretical concept in experiment.

A detailed study of non-translation invariant correlation functions reveals that the pre-thermalized state a system of two 1-dimensional quantum gas relaxes to after a splitting quench [1], is described by a generalized Gibbs ensemble [2]. This is verified through phase correlations up to 10th order.

Interference in a pair of tunnel-coupled one-dimensional atomic super-fluids, which realize the quantum Sine-Gordon / massive Thirring models, allows us to study if, and under which conditions the higher correlation functions factorize [3]. This allowed us to characterize the essential features of the model solely from our experimental measurements: detecting the relevant quasi-particles, their interactions and the different topologically distinct vacuum-states the quasi-particles live in. The experiment thus provides a comprehensive insights into the components needed to solve a non-trivial quantum field theory.

Our examples establish a general method to analyse quantum systems through experiments. It thus represents a crucial ingredient towards the implementation and verification of quantum simulators.

Work performed in collaboration with E.Demler (Harvard), Th. Gasenzer und J. Berges (Heidelberg). Supported by the Wittgenstein Prize, the Austrian Science Foundation (FWF): SFB FoQuS: F40-P10 and the EU: ERC-AdG QuantumRelax

[1] M. Gring et al., Science, 337, 1318 (2012);
[2] T. Langen et al., Science 348 207-211 (2015). [3] T. Schweigler et al., arXiv:1505.03126
12:30 to 13:30 Lunch @ Wolfson Court
14:00 to 15:00 Robert Konik
Studies of the Loschmidt Echo in Two Dimensional Coupled Arrays of Quantum Ising Chains and Luttinger Liquids
Co-authors: Andrew James (LCN), Andrew Hallam (LCN)

We describe a method for simulating the real time evolution of extended quantum systems in two dimensions. The method combines the benefits of integrability and matrix product states in one dimension. In particular it can be extended to systems whose geometry is that of an infinitely long cylinder. As example applications we present results for quantum quenches in both arrays of coupled quantum Ising chains and coupled Luttinger liquids. In quenches that cross a phase boundary we find that the return probability shows non-analyticities in time.
15:00 to 16:00 Veronique Terras
Heisenberg spin chains by separation of variables: recent advances
Co-authors: G. Niccoli (ENS Lyon), N. Kitanine (Univ. de Bourgogne), J.M. Maillet (ENS Lyon)

During the last decades, important progresses have been made concerning the computation of form factors and correlation functions of simple models solvable by algebraic Bethe Ansatz (ABA) such as the XXZ spin-1/2 chain or 1D Bose gas with periodic boundary conditions. However, the generalization of these results to more complicated models or different types of integrable boundary conditions is for the moment limited by the range of applicability of ABA or by some difficulties of the method.

In this talk, we discuss the solution of Heisenberg spin chains (XXX, XXZ or XYZ) in the framework of a complementary approach, Sklyanin's quantum Separation of Variables approach. This enables us notably to consider for these models various types of boundary conditions (quasi-periodic, open...) not directly solvable by Bethe ansatz. More precisely, we discuss in this framework some new results and open problems concerning the description of the spectrum by means of solutions of a functional T-Q equation (or equivalently in terms of Bethe-type equations). We also discuss the problem of the computation of the eigenstates scalar products and of the form factors of local operators.
16:00 to 16:30 Afternoon Tea & Poster Session
16:30 to 17:30 Evgeny Sklyanin
Quantisation of Kadomtsev-Petviashvili equation
Co-authors: Karol Kozlowski (Institut de mathematiques de Bourgogne, Dijon, France), Alessandro Torrielli (University of Surrey)

A quantisation of the KP equation on a cylinder is proposed that is equivalent to an infinite system of one-dimensional bosons carrying masses m=1,2,... The Hamiltonian is Galilei-invariant and includes the cubic split/merge (m1,m2)(m1+m2) terms for all combinations of particles with masses m1, m2 and m1+m2, with a special choice of coupling constants. The Bethe eigenfunctions for the model are constructed. The consistency of the coordinate Bethe ansatz, and therefore, the quantum integrability of the model is verified for the sectors up to the total mass M=8. 
Friday 15th January 2016
10:00 to 11:00 Andreas Ludwig
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Gabor Takacs
Initial states in integrable quantum field theory quenches from an integral equation hierarchy
Co-authors: David X. Horvath (Budapest University of Technology and Economics), Spyros Sotiriadis (SISSA)

We consider the problem of determining the initial state of integrable quantum field theory quenches in terms of the post-quench eigenstates. The corresponding overlaps are a fundamental input to most exact methods to treat integrable quantum quenches. We construct and examine an infinite integral equation hierarchy based on the form factor bootstrap, proposed earlier as a set of conditions determining the overlaps. Using quenches of the mass and interaction in Sinh-Gordon theory as a concrete example, we present theoretical arguments that the state has the squeezed coherent form expected for integrable quenches, and supporting an Ansatz for the solution of the hierarchy. Moreover we also develop an iterative method to solve numerically the lowest equation of the hierarchy. The iterative solution along with extensive numerical checks performed using the next equation of the hierarchy provides a strong numerical evidence that the proposed Ansatz gives a very good approximation for the solution.

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12:30 to 13:30 Lunch @ Wolfson Court
14:00 to 15:00 Austen Lamacraft
z-measures and the non-linear Luttinger liquid
15:00 to 16:00 Jens Eisert
Out of equilibrium dynamics and a unifying view on many-body localisation
The phenomenon of many-body localisation received a lot of attention recently, both for its implications in condensed-matter physics of allowing systems to be an insulator even at non-zero temperature as well as - maybe most importantly - in the context of the foundations of quantum statistical mechanics, providing examples of systems showing the absence of thermalisation following out-of-equilibrium dynamics. Still, it seems fair to say that many aspects of it are still unsatisfactorily understood.

In this talk, following an introduction into recent progress on thermalisation of closed quantum systems, I will make the attempt to bring together several aspects of the phenomenology of many-body localisation, attaining new insights into the connections between seemingly unrelated features. Ideas of entanglement area laws, Lieb-Robinson bounds, filter functions, approximately local constants of motion, transport, and tensor networks will feature strongly. We will discuss experimentally accessible witnesses of many-body localisation in cold atomic quantum simulators.

[1] Quantum many-body systems out of equilibrium Nature Physics 11, 124 (2015) [2] Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems Phys. Rep., in press (2015) [3] Many-body localization implies that eigenvectors are matrix-product states Phys. Rev. Lett. 114, 170505 (2015) [4] Local constants of motion imply information propagation New J. Phys. 17, 113054 (2015) [5] Absence of thermalization in non-integrable systems Phys. Rev. Lett. 106, 040401 (2011) [6] Total correlations of the diagonal ensemble herald the many-body localisation transition Phys. Rev. B 92, 180202(R) (2015) [7] Experimentally accessible witnesses of many-body localisation In preparation (2015)
16:00 to 16:30 Afternoon Tea & Poster Session
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons