Sub-diffusion in the Anderson model on random regular graph

TL;DR

This paper investigates the sub-diffusive spreading of a wave-packet in the Anderson model on random regular graphs, revealing four regimes of propagation and linking the dynamics to many-body localization phenomena.

Contribution

It provides the first detailed analysis of sub-diffusive wave-packet dynamics on RRG, identifying regimes and connecting the exponent to relaxation rates, with implications for MBL.

Findings

Wave-packet spreads sub-diffusively over a broad disorder range.

Four distinct propagation regimes are identified in space-time.

Wave-front moves as $X_{front}(t) \,\sim\, t^{\beta}$ with $\beta<1$, indicating sub-diffusion.

Abstract

We study the finite-time dynamics of an initially localized wave-packet in the Anderson model on the random regular graph (RRG). Considering the full probability distribution $\Pi(x,t)$ of a particle to be at some distance $x$ from the initial state at time $t$, we give evidence that $\Pi(x,t)$ spreads sub-diffusively over a range of disorder strengths, wider than a putative non-ergodic phase. We provide a detailed analysis of the propagation of $\Pi(x,t)$ in space-time $(x,t)$ domain, identifying four different regimes. These regimes in $(x,t)$ are determined by the position of a wave-front $X_{\text{front}}(t)$, which moves sub-diffusively to the most distant sites $X_{\text{front}}(t) \sim t^{\beta}$ with an exponent $\beta < 1$. We support our numerical results by a self-consistent semiclassical picture of wavepacket propagation relating the exponent $\beta$ with the relaxation rate of the return probability $\Pi(0,t) \sim e^{-\Gamma t^\beta}$. Importantly, the Anderson model on the RRG can be considered as proxy of the many-body localization transition (MBL) on the Fock space of a generic interacting system. In the final discussion, we outline possible implications of our findings for MBL.