Sub-diffusion in the Anderson model on random regular graph

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 $Π(x,t)$ of a particle to be at some distance $x$ from the initial state at time $t$, we give evidence that $Π(x,t)$ 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 $Π(x,t)$ 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 $X_{\text{front}}(t) \sim t^β$ with an exponent $β< 1$. We support our numerical results by a self-consistent semiclassical picture of wavepacket propagation relating the exponent $β$ with the relaxation rate of the return probability $Π(0,t) \sim e^{-Γt^β}$. 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.

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