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.

Highlights

  • The common belief that generic, isolated quantum systems thermalize as a result of their own dynamics has been challenged by a recent line of works showing that strong enough disorder can prevent them reaching thermal equilibrium [1,2]

  • The dynamics on the regular graph (RRG) is typically not isotropic, the time scale tTh at which the wave front reaches the diameter could be seen as a natural choice for the Thouless time analogous to the time that a charge needs to propagate through a diffusive conductor [82]

  • We provide evidence of the existence of a subdiffusive phase for a finite range of parameters by probing the dynamics of an initially localized particle on the RRG via the probability distribution (x, t ) to detect it at distance x at time t

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Summary

Subdiffusion in the Anderson model on the random regular graph

Giuseppe De Tomasi,1,2,* Soumya Bera ,3 Antonello Scardicchio, and Ivan M. 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 (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 Zd lattices, d > 2, which is subdiffusive only at the critical point implies that the limit d → ∞ is highly singular in terms of the dynamics. We outline possible implications of our findings for MBL

Introduction
Published by the American Physical Society
The Anderson model on the RRG is defined as
Results
Conclusion and discussions

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