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
Summary
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
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