We numerically investigate Heisenberg XXZ spin-1 / 2 chain in a spatially random static magnetic field. We find that time-dependent density-matrix renormalization group simulations of time evolution can be performed efficiently, namely, the dimension of matrices needed to efficiently represent the time evolution increases linearly with time and entanglement entropies for typical chain bipartitions increase logarithmically. As a result, we show that for large enough random fields, infinite temperature spin-spin correlation function displays exponential localization in space indicating insulating behavior of the model.

Fidelity serves as a benchmark for the relieability in quantum information processes, and has recently atracted much interest as a measure of the susceptibility of dynamics to perturbations. A rich variety of regimes for fidelity decay have emerged. The purpose of the present review is to describe these regimes, to give the theory that supports them, and to show some important applications and experiments. While we mention several approaches we use time correlation functions as a backbone for the discussion. Vanicek's uniform approach to semiclassics and random matrix theory provides an important alternative or complementary aspects. Other methods will be mentioned as we go along. Recent experiments in micro-wave cavities and in elastodynamic systems as well as suggestions for experiments in quantum optics shall be discussed.Comment: Review article with some original results in integrable systems and random matrix models; 133 pages, 53 figure

The Lindblad master equation for an arbitrary quadratic system of n fermions is solved explicitly in terms of diagonalization of a 4n × 4n matrix, provided that all Lindblad bath operators are linear in the fermionic variables. The method is applied to the explicit construction of non-equilibrium steady states and the calculation of asymptotic relaxation rates in the far from equilibrium problem of heat and spin transport in a nearest neighbor Heisenberg XY spin 1/2 chain in a transverse magnetic field.

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

We review recent progress in understanding the notion of locality in integrable quantum lattice systems. The central concept are the so-called quasilocal conserved quantities, which go beyond the standard perception of locality. Two systematic procedures to rigorously construct families of quasilocal conserved operators based on quantum transfer matrices are outlined, specializing on anisotropic Heisenberg XXZ spin-1/2 chain. Quasilocal conserved operators stem from two distinct classes of representations of the auxiliary space algebra, comprised of unitary (compact) representations, which can be naturally linked to the fusion algebra and quasiparticle content of the model, and non-unitary (non-compact) representations giving rise to charges, manifestly orthogonal to the unitary ones. Various condensed matter applications in which quasilocal conservation laws play an essential role are presented, with special emphasis on their implications for anomalous transport properties (finite Drude weight) and relaxation to non-thermal steady states in the quantum quench scenario.