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

Time-dependent density matrix renormalization group method with a matrix product ansatz is employed for explicit computation of non-equilibrium steady state density operators of several integrable and non-integrable quantum spin chains, which are driven far from equilibrium by means of Markovian couplings to external baths at the two ends. It is argued that even though the time-evolution can not be simulated efficiently due to fast entanglement growth, the steady states in and out of equilibrium can be typically accurately approximated, so that chains of length of the order n ≈ 100 are accessible. Our results are demonstrated by performing explicit simulations of steady states and calculations of energy/spin densities/currents in several problems of heat and spin transport in quantum spin chains. Previously conjectured relation between quantum chaos and normal transport is re-confirmed with high acurracy on much larger systems.

We analytically and numerically study spin transport in a one-dimensional Heisenberg model in linear-response regime at infinite temperature. It is shown that as the anisotropy parameter Δ is varied spin transport changes from ballistic for Δ<1 to anomalous at the isotropic point Δ=1, to diffusive for finite Δ>1, ending up as a perfect isolator in the Ising limit of infinite Δ. Using perturbation theory for large Δ a quantitative prediction is made for the dependence of diffusion constant on Δ.

We study high temperature spin transport in a disordered Heisenberg chain in the ergodic regime. By employing a density matrix renormalization group technique for the study of the stationary states of the boundary-driven Lindblad equation we are able to study extremely large systems (400 spins). We find both a diffusive and a subdiffusive phase depending on the strength of the disorder and on the anisotropy parameter of the Heisenberg chain. Studying finite-size effects we show numerically and theoretically that a very large crossover length exists that controls the passage of a clean-system dominated dynamics to one observed in the thermodynamic limit. Such a large length scale, being larger than the sizes studied before, explains previous conflicting results. We also predict spatial profiles of magnetization in steady states of generic nondiffusive systems.Introduction.-There are ever increasing technological capabilities in simulating isolated quantum systems through cold atomic gases [1] and, recently, through coupled, controlled superconducting qubits [2]. While there is a commensurately good theoretical handle on capturing ground state properties of such systems [3], understanding their dynamical properties, especially away from the ground state, is fraught with analytical and numerical challenges.Despite this, in the recent years we have witnessed a change in paradigm in the study of isolated quantum systems, in particular with regard to the role that disorder plays in such systems. The turning point came about from the study of Anderson localization [4] in interacting, many-body quantum systems [5]. The observation that disorder and quantum effects can hinder transport (of energy, charge or spin) even at an infinite temperature and in the presence of interactions [6] opened the door to a new phenomenology of a so-called manybody localized (MBL) phase exhibiting many unique and interesting properties. Slow growth of entanglement [7,8], emergent integrability [9], protection of symmetries [10], and change in the properties of eigenstates [11][12][13] are a few of the peculiar properties of this newly identified phase; see review [14] for a comprehensive list. The implications of the new MBL physics, being inherently robust, are far reaching, going from fundamental physics to the theory of quantum computation [15], some of which have already been experimentally probed [16].While the deep MBL region (in one dimensional systems) is well understood, much remains to be said about the conducting regime and the transition to it. Although both aspects are important, here we focus on characterizing the conducting phase, in particular its transport properties, in, what is by now, an archetypal model that harbors the MBL phase, i.e., the one dimensional anisotropic Heisenberg model.Generic arguments and numerical evidence on very small systems (about 20 spins) have been put forward for the existence of subdiffusive transport of spin [17][18][19][20] and energy [18,21,22]. A number of recent works have analyzed its sp...

Generalized hydrodynamics predicts universal ballistic transport in integrable lattice systems when prepared in generic inhomogeneous initial states. However, the ballistic contribution to transport can vanish in systems with additional discrete symmetries. Here we perform large scale numerical simulations of spin dynamics in the anisotropic Heisenberg XXZ spin 1/2 chain starting from an inhomogeneous mixed initial state which is symmetric with respect to a combination of spin reversal and spatial reflection. In the isotropic and easy-axis regimes we find non-ballistic spin transport which we analyse in detail in terms of scaling exponents of the transported magnetization and scaling profiles of the spin density. While in the easy-axis regime we find accurate evidence of normal diffusion, the spin transport in the isotropic case is clearly super-diffusive, with the scaling exponent very close to 2/3, but with universal scaling dynamics which obeys the diffusion equation in nonlinearly scaled time.

We derive a simple and general relation between the fidelity of quantum motion, characterizing the stability of quantum dynamics with respect to arbitrary static perturbation of the unitary evolution propagator, and the integrated time auto-correlation function of the generator of perturbation. Surprisingly, this relation predicts the slower decay of fidelity the faster decay of correlations is. In particular, for non-ergodic and non-mixing dynamics, where asymptotic decay of correlations is absent, a qualitatively different and faster decay of fidelity is predicted on a time scale ∝ 1/δ as opposed to mixing dynamics where the fidelity is found to decay exponentially on a time-scale ∝ 1/δ 2 , where δ is a strength of perturbation. A detailed discussion of a semi-classical regime of small effective values of Planck constanth is given where classical correlation functions can be used to predict quantum fidelity decay. Note that the correct and intuitively expected classical stability behavior is recovered in the classical limith → 0, as the two limits δ → 0 andh → 0 do not commute. In addition we also discuss non-trivial dependence on the number of degrees of freedom. All the theoretical results are clearly demonstrated numerically on a celebrated example of a quantized kicked top.

We calculate a nonequilibrium steady state of a quantum XX chain in the presence of dephasing and driving due to baths at chain ends. The obtained state is exact in the limit of weak driving while the expressions for one- and two-point correlations are exact for an arbitrary driving strength. In the steady state the magnetization profile and the spin current display diffusive behavior. The spin–spin correlation function, on the other hand, has long-range correlations which, though, decay to zero in either the thermodynamical limit or for equilibrium driving. At zero dephasing a nonequilibrium phase transition occurs from a ballistic transport having short-range correlations to a diffusive transport with long-range correlations.