Density propagator for many-body localization: finite size effects, transient subdiffusion, and exponential decay

TL;DR

This study examines charge relaxation in disordered quantum wires, revealing transient subdiffusive behavior, questioning the existence of many-body mobility edges, and highlighting the impact of strong disorder and Griffiths effects on transport properties.

Contribution

It provides new insights into the finite size effects, transient subdiffusion, and the nature of decay in many-body localized systems, challenging previous claims of mobility edges.

Findings

Subdiffusive behavior observed in a large time window.

No definitive evidence for many-body mobility edges.

Subdiffusion coexists with slow decay of return probability.

Abstract

We investigate charge relaxation in quantum-wires of spin-less disordered fermions ($t{-}V$-model). Our observable is the time-dependent density propagator, $\Pi_{\varepsilon}(x,t)$, calculated in windows of different energy density, $\varepsilon$, of the many-body Hamiltonian and at different disorder strengths, $W$, not exceeding the critical value $W_\text{c}$. The width $\Delta x_\varepsilon(t)$ of $\Pi_\varepsilon(x,t)$ exhibits a behavior $d\ln \Delta x_\varepsilon(t) / d\ln t {=} \beta_\varepsilon(t)$, where the exponent function $\beta_\varepsilon(t){\lesssim}1/2$ is seen to depend strongly on $L$ at all investigated parameter combinations. (i) We confirm the existence of a region in phase space that exhibits subdiffusive dynamics in the sense that $\beta_\varepsilon{<}1/2$ in large window of times. However, subdiffusion might possibly be transient, only, finally giving way to a conventional diffusive behavior with $\beta_{\varepsilon}{=}1/2$. (ii) We cannot confirm the existence of many-body mobility edges even in regions of the phase-diagram that have been reported to be deep in the delocalized phase. (iii) (Transient) subdiffusion $0<\beta_\varepsilon(t)\lesssim 1/2$, coexists with an enhanced probability for returning to the origin, $\Pi_\varepsilon(0,t)$, decaying much slower than $1/\Delta x_\varepsilon (t)$. Correspondingly, the spatial decay of $\Pi_\varepsilon(x,t)$ is far from Gaussian being exponential or even slower. On a phenomenological level, our findings are broadly consistent with effects of strong disorder and (fractal) Griffiths regions.