Diffusive and Subdiffusive Spin Transport in the Ergodic Phase of a Many-Body Localizable System

Abstract

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...

Rare‐region effects and dynamics near the many‐body localization transition

“…Ultimately, these rare regions do thermalize due to being surrounded by thermal regions, which act as a "bath," but their slow dynamics can dominate the long-time or low-frequency dynamics of such a system [58][59][60]. These effects are most dramatic in one dimension, where such inclusions of the MBL phase can bottleneck transport, making the diffusion constant vanish even within the thermal phase, as has been seen in some numerical studies [61,62]. On the other hand, in the MBL phase of a system with quenched randomness there may be rare regions that are less disordered and thus locally thermalizing [41].…”

“…The qualitative behavior of transport on the thermal side of the transition at least is mostly settled: both Griffiths estimates [58,59] and large-system numerical approaches [62] indicate that there is a diffusive regime at weak disorder as well as a subdiffusive regime at stronger disorder. However, various questions, concerning the growth of entanglement and the typical behavior of correlation functions in one dimension, are still somewhat poorly understood.…”

“…A resolution to this issue was recently provided in the work ofŽnidarič, et al [62], who were able to explore much larger system sizes by an elegant application of matrix product operator techniques. This work considers a finite XXZ spin chain, initially prepared in a fully mixed infinite temperature state, that is connected to simple baths at different magnetic fields at its two ends, such that in this system's nonequilibrium steady state there is (a) a spin density gradient and (b) a spin current along the chain.…”

Many-body-localization transition: sensitivity to twisted boundary conditions

“…is the most studied Many-Body-Localization model where numerical results for many observables are available [42][43][44][45][46][47][48][49][50][51][52][53] As explained in the Introduction, besides the usual periodic boundary conditions σ L+1 = σ 1 , it is interesting to consider twisted boundary conditions with some angle φ for the spin operators [21] …”

Interaction-induced weakening of localization in few-particle disordered Heisenberg chains

“…This disorder-induced transition has been under active scrutiny and different probes have been suggested 28 such as subdiffusive power laws in the vicinity of the critical disorder 20 (which may or may not exist 21,[29][30][31][32][33][34][35] ). So far, a full understanding of the MBL transition is still lacking.…”