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...

Light cone spreading of correlations and entanglement is a key feature of the non-equilibrium quench dynamics of many-body quantum systems. First proposed theoretically [1], it has been experimentally revealed in cold-atomic gases [2,3] and it is expected to be a generic characteristic of any quench in systems with shortrange interactions and no disorder. Conversely, here we propose a mechanism that, through confinement of the elementary excitations, strongly suppresses the light-cone spreading. Confinement is a celebrated concept in particle physics, but it also exists in condensed matter systems, most notably in one spatial dimension where it has been experimentally observed [4]. Our results are obtained for the Ising spin chain with transverse and longitudinal magnetic field, but the proposed mechanism is of general validity since it is based on the sole concept of confinement and it should be easily observed in cold atom experiments.

We consider the time evolution after quantum quenches in the spin-1/2 Heisenberg XXZ quantum spin chain with Ising-like anisotropy. The time evolution of short-distance spin-spin correlation functions is studied by numerical tensor network techniques for a variety of initial states, including Néel and Majumdar-Ghosh states and the ground state of the XXZ chain at large values of the anisotropy. The various correlators appear to approach stationary values, which are found to be in good agreement with the results of exact calculations of stationary expectation values in appropriate generalized Gibbs ensembles. In particular, our analysis shows how symmetries of the post-quench Hamiltonian that are broken by particular initial states are restored at late times.