Subdiffusion in the Anderson model on the random regular graph

Abstract

We study the finite-time dynamics of an initially localized wave packet in the Anderson model on the random regular graph (RRG) and show the presence of a subdiffusion phase coexisting both with ergodic and putative nonergodic phases. The full probability distribution $\mathrm{\ensuremath{\Pi}}(x,t)$ of a particle to be at some distance $x$ from the initial state at time $t$ is shown to spread subdiffusively over a range of disorder strengths. The comparison of this result with the dynamics of the Anderson model on ${\mathbb{Z}}^{d}$ lattices, $d>2$, which is subdiffusive only at the critical point implies that the limit $d\ensuremath{\rightarrow}\ensuremath{\infty}$ is highly singular in terms of the dynamics. A detailed analysis of the propagation of $\mathrm{\ensuremath{\Pi}}(x,t)$ in space-time $(x,t)$ domain identifies four different regimes determined by the position of a wave front ${X}_{\text{front}}(t)$, which moves subdiffusively to the most distant sites ${X}_{\text{front}}(t)\ensuremath{\sim}{t}^{\ensuremath{\beta}}$ with an exponent $\ensuremath{\beta}<1$. 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.