We consider the non-equilibrium time evolution of piecewise homogeneous states in the XXZ spin-1/2 chain, a paradigmatic example of an interacting integrable model. The initial state can be thought as the result of joining chains with different global properties. Through dephasing, at late times the state becomes locally equivalent to a stationary state which explicitly depends on position and time. We propose a kinetic theory of elementary excitations and derive a continuity equation which fully characterizes the thermodynamics of the model. We restrict ourselves to the gapless phase and consider cases where the chains are prepared: 1) at different temperatures; 2) in the ground state of two different models; 3) in the "domain wall" state. We find excellent agreement (any discrepancy is within the numerical error) between theoretical predictions and numerical simulations of time evolution based on tebd algorithms. As a corollary, we unveil an exact expression for the expectation values of the charge currents in a generic stationary state.During the last decade, the study of non-equilibrium dynamics in quantum many-body systems has experienced a golden age. The experimental possibility for investigating almost purely unitary time evolution [1] sparked off a diffuse theoretical excitement [2][3][4][5][6][7]. The challenge was to understand in which sense unitarily evolving systems can relax to stationary states, and, if this happens, how to determine the stationary values of the observables. The main focus has been on translationally invariant systems. There, a clear theoretical construction has been developed: while the full system can not relax, in the thermodynamic limit finite subsystems can, as the rest of the system acts as an unusual bath. It was argued that the stationary values of local observables are determined by local and quasi-local conservation laws [2,4,8]. It is then convenient to distinguish between generic models, where the Hamiltonian is the only local conserved quantity, and integrable models, where the number of local charges scales with the systems's size. It was conjectured that in the former case stationary values of local observables are described by Gibbs ensembles (ge) [9] while in the latter by socalled generalised Gibbs ensembles (gge) [10]. Importantly, traces of the underlying integrability remain even in the presence of small integrability-breaking perturbations: at intermediate times the expectation values of local observables approach quasi-stationary plateaux retaining infinite memory of the initial state [11][12][13][14].In the absence of translational invariance the situation gets more complicated. In this context a variety of different settings have been considered, which can be cast into two main classes. The first consists of dynamics governed by translationally invariant Hamiltonians on inhomogeneous states. Relevant examples are the sudden junction of two chains at different temperature [15][16][17][18][19][20][21][22], with different magnetizations [23,24], or with othe...

In integrable many-particle systems, it is widely believed that the stationary state reached at late times after a quantum quench can be described by a generalized Gibbs ensemble (GGE) constructed from their extensive number of conserved charges. A crucial issue is then to identify a complete set of these charges, enabling the GGE to provide exact steady-state predictions. Here we solve this long-standing problem for the case of the spin-1=2 Heisenberg chain by explicitly constructing a GGE which uniquely fixes the macrostate describing the stationary behavior after a general quantum quench. A crucial ingredient in our method, which readily generalizes to other integrable models, are recently discovered quasilocal charges. As a test, we reproduce the exact postquench steady state of the Néel quench problem obtained previously by means of the Quench Action method. Introduction.-Understanding and describing the equilibration of isolated many-particle systems is one of the main current challenges of quantum physics. The presence of higher conserved charges (above the Hamiltonian) is linked to the absence of full relaxation to a thermalized state; the conjectured appropriate framework to characterize the steady-state properties in such a situation is the generalized Gibbs ensemble (GGE) [1], in which all available charges are ascribed an individual "chemical potential" set by the initial conditions, and the steady state is the maximal entropy state fulfilling all the constraints associated to the conserved charges . The basic idea underlying the GGE is as follows. Let H ≃ H ð1Þ be the Hamiltonian of an integrable model, and fH ðnÞ g a set of conserved charges fulfilling ½H ðnÞ ; H ðmÞ ¼ 0. The situation we are interested in is that of a quantum quench, where we initially prepare our system in the ground state jΨð0Þi of a local Hamiltonian H 0 and then consider unitary time evolution with respect to our integrable Hamiltonian

Solution for an interaction quench in the Lieb-Liniger Bose gasDe Nardis, J.; Wouters, B.M.; Brockmann, M.; Caux, J.S. General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. We study a quench protocol where the ground state of a free many-particle bosonic theory in one dimension is let unitarily evolve in time under the integrable Lieb-Liniger Hamiltonian of δ-interacting repulsive bosons. By using a recently proposed variational method, we here obtain the exact nonthermal steady state of the system in the thermodynamic limit and discuss some of its main physical properties. Besides being a rare case of a thermodynamically exact solution to a truly interacting quench situation, this interestingly represents an example where a standard implementation of the generalized Gibbs ensemble fails.

General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. We study quenches in integrable spin-1=2 chains in which we evolve the ground state of the antiferromagnetic Ising model with the anisotropic Heisenberg Hamiltonian. For this nontrivially interacting situation, an application of the first-principles-based quench-action method allows us to give an exact description of the postquench steady state in the thermodynamic limit. We show that a generalized Gibbs ensemble, implemented using all known local conserved charges, fails to reproduce the exact quench-action steady state and to correctly predict postquench equilibrium expectation values of physical observables. This is supported by numerical linked-cluster calculations within the diagonal ensemble in the thermodynamic limit. [6] all the way to atomic-scale isolated quantum systems [7]. Much recent experimental and theoretical activity has been focused on the latter, raising fundamental questions as to whether, how, and to what state such systems relax under unitary time evolution following a sudden quantum quench . From this work, two scenarios for equilibration have emerged, one applicable to models having only a few local conserved quantities, the other relevant to integrable models characterized by an infinite number of local conserved charges. In the former, thermalization to a Gibbs ensemble is the rule [11], while in the latter, equilibration to a so-called generalized Gibbs ensemble (GGE) [9,10] is generally thought to occur, in particular for lattice spin systems [12][13][14][15][16][17][18][19][20].In this Letter, we study a quench in which the second scenario breaks down. Our initial state, defined as a purely antiferromagnetic (spin-1=2 Néel) state, is let to evolve unitarily in time according to the XXZ spin chain Hamiltonian. This is a physically meaningful quench protocol, which can, in principle, be implemented using cold atoms [43][44][45][46][47]. We provide a thermodynamically exact solution for the steady state reached long after the quench, derived directly from microscopics using the recently proposed quench-action method [48]. The solution takes the form of a set of distributions of quasimomenta that completely characterizes the macrostate representing the steady state, from which observables ...

We extend beyond the Euler scales the hydrodynamic theory for quantum and classical integrable models developed in recent years, accounting for diffusive dynamics and local entropy production. We review how the diffusive scale can be reached via a gradient expansion of the expectation values of the conserved fields and how the coefficients of the expansion can be computed via integrated steadystate two-point correlation functions, emphasising that PT -symmetry can fully fix the inherent ambiguity in the definition of conserved fields at the diffusive scale. We develop a form factor expansion to compute such correlation functions and we show that, while the dynamics at the Euler scale is completely determined by the density of single quasiparticle excitations on top of the local steady state, diffusion is due to scattering processes among quasiparticles, which are only present in truly interacting systems. We then show that only two-quasiparticle scattering processes contribute to the diffusive dynamics. Finally we employ the theory to compute the exact spin diffusion constant of a gapped XXZ spin−1/2 chain at finite temperature and half-filling, where we show that spin transport is purely diffusive.

We show that hydrodynamic diffusion is generically present in many-body interacting integrable models. We extend the recently developed generalised hydrodynamic (GHD) to include terms of Navier-Stokes type which lead to positive entropy production and diffusive relaxation mechanisms. These terms provide the subleading diffusive corrections to Euler-scale GHD for the large-scale nonequilibrium dynamics of integrable systems, and arise due to two-body scatterings among quasiparticles. We give exact expressions for the diffusion coefficients. Our results apply to a large class of integrable models, including quantum and classical, Galilean and relativistic field theories, chains and gases in one dimension, such as the Lieb-Liniger model describing cold atom gases and the Heisenberg quantum spin chain. We provide numerical evaluations in the Heisenberg spin chain, both for the spin diffusion constant, and for the diffusive effects during the melting of a small domain wall of spins, finding excellent agreement with tDMRG numerical simulations.

Nonergodic dynamical systems display anomalous transport properties. Prominent examples are integrable quantum systems, whose exceptional properties are diverging dc conductivities. In this Letter, we explain the microscopic origin of ideal conductivity by resorting to the thermodynamic particle content of a system. Using group-theoretic arguments we rigorously resolve the long-standing controversy regarding the nature of spin and charge Drude weights in the absence of chemical potentials. In addition, by employing a hydrodynamic description, we devise an efficient computational method to calculate exact Drude weights from the stationary currents generated in an inhomogeneous quench from bipartitioned initial states. We exemplify the method on the anisotropic Heisenberg model at finite temperatures for the entire range of anisotropies, accessing regimes that are out of reach with other approaches. Quite remarkably, spin Drude weight and asymptotic spin current rates reveal a completely discontinuous (fractal) dependence on the anisotropy parameter.

We present an identification of the spectra of local conserved operators of integrable quantum lattice models and the density distributions of their thermodynamic particle content. This is derived explicitly for the Heisenberg XXZ spin chain. As an application we discuss a quantum quench scenario, in both the gapped and critical regimes. We outline an exact technique which allows for an efficient implementation on periodic matrix product states. In addition, for certain simple product states we obtain closed-form expressions for the density distributions in terms of solutions to Hirota difference equations. Remarkably, no reference to a maximal entropy principle is invoked.ArXiv ePrint: 1512.04454