Noise-Induced Subdiffusion in Strongly Localized Quantum Systems

We consider the dynamics of strongly localized systems subject to dephasing noise with arbitrary correlation time. Although noise inevitably induces delocalization, transport in the noise-induced delocalized phase is subdiffusive in a parametrically large intermediate-time window. We argue for this intermediate-time subdiffusive regime both analytically and using numerical simulations on single-particle localized systems. Furthermore, we show that normal diffusion is restored in the long-time limit, through processes analogous to variable-range hopping. With numerical simulations based on Lanczos exact diagonalization, we demonstrate that our qualitative conclusions are also valid for interacting systems in the many-body localized phase.

Figure 1

Noise-induced delocalization. (a) We consider strongly localized fermions in a random potential that are weakly coupled to an environment. (b) In that limit, the environment can be modeled by classical noise ${\xi }_i(t)$ that couples locally to the density. (c) We study the noise-induced transport, by preparing the system in a wave packet and computing its spread $\sigma (t)$ in time. For finite coupling to the environment three regimes can be distinguished: (i) a short-time ballistic expansion $\sigma (t)\sim t$, (ii) a parametrically large regime of subdiffusive transport $\sigma (t)\sim t^{\beta }$ with a continuously increasing power $\beta$ that approaches (iii) a diffusive regime $\sigma (t)\sim \sqrt{t}$ at late times. Numerical data taken as direct (solid) and inverse (dashed) average of the spread over individual realizations are shown for disorder $W=16J$, noise strength $\mathrm{\Lambda }=20J$, and noise correlation time $\tau J=100$.

Figure 2

Time evolution of a localized wave packet. The numerically evaluated, inverse-averaged spread $\sigma (t)$ of an initially localized wave packet for systems of size $L=400$ is shown for (a) fixed $\mathrm{\Lambda }=2J$, $\tau J=100$, (b) $\mathrm{\Lambda }=2J$, $W=2J$, and (c) $W=8J$, $\tau J=100$. Dotted lines indicate diffusive expansion $\sigma (t)\sim \sqrt{t}$. Error bars are obtained from the sample average of 250 noise and disorder realizations.

Figure 3

Short-time subdiffusion exponent. The power-law exponent $\beta$ characterizing the initial subdiffusive transport $\sigma (t)\sim t^{\beta }$ is extracted from fitting the numerical data in the range $1<tJ<\tau J$ for $\tau J=10$. In the weak noise limit $\mathrm{\Lambda }\lesssim J$, the exponent depends strongly on the disorder strength $W$ approaching 0 with increasing $W$ whereas it is constant for strong noise $\mathrm{\Lambda }=20J$. For weak disorder strength $W\to 0$, $\beta$ universally approaches within the error bars the limit of diffusive transport with $\beta =1/2$. Solid lines indicate our estimate $\beta ={\mathrm{\Lambda }}^2/({\mathrm{\Lambda }}^2+W^2)$.

Figure 4

Asymptotic diffusion constant. The diffusion constant $D$ evaluated from the asymptotic spread of the wave packet $\sigma (t\to \infty )=\sqrt{2Dt}$ is shown as a function of the noise strength $\mathrm{\Lambda }$ for different values of the disorder strength $W$ and fixed noise correlation time $\tau J=1$, symbols. The numerical data are compared to the single-hop model valid for $\mathrm{\Lambda }\gtrsim W$, solid lines, and the variable-range hopping model of Eq. (7) valid for $\mathrm{\Lambda }\lesssim W$, dashed lines.