Tomaz Prosen

Many-body localization in the Heisenberg XXZ magnet in a random field

We numerically investigate Heisenberg XXZ spin-1/2 chain in a spatially random static magnetic field. We find that tDMRG 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.

Open XXZ spin chain: nonequilibrium steady state and a strict bound on ballistic transport.

An explicit matrix product ansatz is presented, in the first two orders in the (weak) coupling parameter, for the nonequilibrium steady state of the homogeneous, nearest neighbor Heisenberg XXZ spin 1/2 chain driven by Lindblad operators which act only at the edges of the chain. The first order of the density operator becomes, in the thermodynamic limit, an exact pseudolocal conservation law and yields-via the Mazur inequality-a rigorous lower bound on the high-temperature spin Drude weight. Such a Mazur bound is a nonvanishing fractal function of the anisotropy parameter Δ for |Δ|<1.

Quasilocal charges in integrable lattice systems

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.

Stability of quantum motion and correlation decay

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 the decay of correlations. 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 timescale 1/δ as opposed to mixing dynamics where the fidelity is found to decay exponentially on a timescale 1/δ2, where δ is the strength of perturbation. A detailed discussion of a semiclassical regime of small effective values of Planck constant is given where classical correlation functions can be used to predict quantum fidelity decay. Note that the correct and intuitively expected classical stability behaviour is recovered in the classical limit → 0, as the two limits δ → 0 and → 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 the celebrated example of a quantized kicked top.

Exact nonequilibrium steady state of a strongly driven open XXZ chain.

An exact and explicit ladder-tensor-network ansatz is presented for the nonequilibrium steady state of an anisotropic Heisenberg XXZ spin-1/2 chain which is driven far from equilibrium with a pair of Lindblad operators acting on the edges of the chain only. We show that the steady-state density operator of a finite system of size n is-apart from a normalization constant-a polynomial of degree 2n - 2 in the coupling constant. Efficient computation of physical observables is facilitated in terms of a transfer operator reminiscent of a classical Markov process. In the isotropic case we find cosine spin profiles, 1/n(2) scaling of the spin current, and long-range correlations in the steady state. This is a fully nonperturbative extension of a recent result [Phys. Rev. Lett. 106, 217206 (2011)].

Families of quasilocal conservation laws and quantum spin transport.

For fundamental integrable quantum chains with deformed symmetries we outline a general procedure for defining a continuous family of quasilocal operators whose time derivative is supported near the two boundary sites only. The program is implemented for a spin 1/2 XXZ chain, resulting in improved rigorous estimates for the high temperature spin Drude weight.

Fourier law in the alternate-mass hard-core potential chain

We study energy transport in a one-dimensional model of elastically colliding particles with alternate masses m and M. In order to prevent total momentum conservation, we confine particles with mass M inside a cell of finite size. We provide convincing numerical evidence for the validity of Fourier law of heat conduction in spite of the lack of exponential dynamical instability. Comparison with previous results on similar models shows the relevance of the role played by total momentum conservation.

PT-symmetric wave Chaos.

We study a new class of chaotic systems with dynamical localization, where gain or loss mechanisms break the Hermiticity, while allowing for parity-time (PT) symmetry. For a value gamma{PT} of the gain or loss parameter the spectrum undergoes a spontaneous phase transition from real (exact phase) to complex values (broken phase). We develop a one parameter scaling theory for gamma{PT}, and show that chaos assists the exact PT phase. Our results have applications to the design of optical elements with PT symmetry.

General relation between quantum ergodicity and fidelity of quantum dynamics.

A general relation is derived, which expresses the fidelity of quantum dynamics, measuring the stability of time evolution to small static variation in the Hamiltonian, in terms of ergodicity of an observable generating the perturbation as defined by its time correlation function. Fidelity for ergodic dynamics is predicted to decay exponentially on time scale proportional to delta(-2), delta approximately strength of perturbation, whereas faster, typically Gaussian decay on shorter time scale proportional delta(-1) is predicted for integrable, or generally nonergodic dynamics. This result needs the perturbation delta to be sufficiently small such that the fidelity decay time scale is larger than any (quantum) relaxation time, e.g., mixing time for mixing dynamics, or averaging time for nonergodic dynamics (or Ehrenfest time for wave packets in systems with chaotic classical limit). Our surprising predictions are demonstrated in a quantum Ising spin-(1/2) chain periodically kicked with a tilted magnetic field where we find finite parameter-space regions of nonergodic and nonintegrable motion in the thermodynamic limit.

Quasilocal Conserved Operators in the Isotropic Heisenberg Spin-1/2 Chain.

Composing higher auxiliary-spin transfer matrices and their derivatives, we construct a family of quasilocal conserved operators of isotropic Heisenberg spin-1/2 chain and rigorously establish their linear independence from the well-known set of local conserved charges.