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arXiv:1810.03640 · Physical review. B./Physical review. B

High-temperature coherent transport in the XXZ chain in the presence of an impurity

Marlon Brenes, Eduardo Mascarenhas, Marcos Rigol, John Goold

Published Dec 12, 2018• 68 citations
arXivHTMLPDF

Abstract

Numerical simulations show that the spin-\textonehalf{} XXZ chain in the presence of an impurity displays coherent (ballistic) spin transport at high temperature despite the fact that the model is quantum chaotic. Transport is studied on the nonequilibrium steady state, using a combination of tensor networks and open quantum system theory within Kubo's linear response theory.

DOI
10.1103/physrevb.98.235128

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Numerical simulations show that the spin-\textonehalf{} XXZ chain in the presence of an impurity displays coherent (ballistic) spin transport at high temperature despite the fact that the model is quantum chaotic. Transport is studied on the nonequilibrium steady state, using a combination of tensor networks and open quantum system theory within Kubo's linear response theory.
Numerical and analytic results have been used to characterize quantum transport in spin chains, showing the existence of both ballistic and diffusive motion. Experiments have shown that heat transfer is surprisingly always diffusive. The scattering from phonons and impurities have been postulated to be the two factors critical in causing the diffusive transport. In this work, we evaluate the transport process by incorporating a bath of phonons and impurities in order to understand the role played by each of the factors. While methods like time-dependent density matrix renormalization group (tDMRG) can be used to simulate isolated spin chains, the coupling with phonons make simulations significantly more challenging. The recently developed multisite tensor network path integral (MS-TNPI) method builds a framework for simulating the dynamics in extended open quantum systems by combining ideas from tDMRG and Feynman-Vernon influence functional. This MS-TNPI is used to characterize dynamics in open, extended quantum systems. Simulations are done with the commonly used sub-Ohmic, Ohmic and super-Ohmic spectral densities describing the phononic bath. We show that while the transport in presence of impurities eventually becomes diffusive, the exact details are dependent on the specifics of the interactions and amount of impurities. In contrast, the presence of a bath makes the transport diffusive irrespective of the parameters characterizing the bath.
In the study of quantum transport, much has been known about dynamics near thermal equilibrium. However, quantum transport far away from equilibrium is much less well understood---the linear response approximation does not hold for physics far out of equilibrium in general. In this work, motivated by recent cold atom experiments on probing quantum many-body dynamics of a one-dimensional XXZ spin chain, where a transition from ballistic to diffusive dynamics has been established by increasing the interaction strengths, we study the strong interaction limit of the one-dimensional spinless fermion model, which is dual to the XXZ spin chain. We develop a highly efficient computation algorithm for simulating the nonequilibrium dynamics of this system exactly, and examine the nonequilibrium dynamics starting from a density modulation quantum state. We find ballistic transport in this strongly correlated setting and show that a plane-wave description emerges at long-time evolution. We also observe a sharp distinction between transport velocities in short and long times as induced by interaction effects and provide a quantitative interpretation for the long-time transport velocity. We expect our results to shed light on the understanding of the dynamics of the XXZ spin chain in the strong interaction regime.
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State-of-the-art approaches to extract transport coefficients of many-body quantum systems broadly fall into two categories: (i) they target the linear-response regime in terms of equilibrium correlation functions of the closed system; or (ii) they consider an open-system situation typically modeled by a Lindblad equation, where a nonequilibrium steady state emerges from driving the system at its boundaries. While quantitative agreement between (i) and (ii) has been found for selected model and parameter choices, also disagreement has been pointed out in the literature. Studying magnetization transport in the spin-1/21/21/2 XXZ chain, we here demonstrate that at weak driving, the nonequilibrium steady state in an open system, including its buildup in time, can remarkably be constructed just on the basis of correlation functions in the closed system. We numerically illustrate this direct correspondence of closed-system and open-system dynamics, and show that it allows the treatment of comparatively large open systems, usually only accessible to matrix product state simulations. We also point out potential pitfalls when extracting transport coefficients from nonequilibrium steady states in finite systems.
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.
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.