Anderson and many-body localization in the presence of spatially correlated classical noise

We study the effect of spatially correlated classical noise on both Anderson and many-body localization of a disordered fermionic chain. By analyzing the evolution of the particle density imbalance following a quench from an initial charge density wave state, we find prominent signatures of localization also in the presence of the time-dependent noise, even though the system eventually relaxes to the infinite temperature state. In particular, for sufficiently strong static disorder, we observe the onset of metastability, which becomes more prominent the stronger the spatial correlations of the noise. In this regime, we find that the imbalance decays as a stretched-exponential—a behavior characteristic of glassy systems. We identify a simple scaling behavior of the relevant relaxation times in terms of the static disorder and of the noise correlation length. We discuss how our results could be exploited to extract information about the localization length in experimental setups.

Figure 1

Model. Sketch of the model of fermions on a lattice with a random onsite potential uniformly distributed between $-2h$ and $2h$, hopping amplitude $J$, interaction between neighboring sites $V$ and spatially correlated noise with correlation matrix $C_{\ell m}$.

Figure 2

Metastability of the imbalance. (a) Imbalance vs time (note a logarithmic scale on time axis), for a fixed dephasing rate ($\gamma =1$), and different values of $h$ and $\xi$. The metastable plateau is due to localization effects and it is higher (i.e., more memory about the initial condition is retained) for larger disorder strength. The correlation length $\xi$ controls the duration of the plateau, with a longer plateau for larger $\xi$. The Hamiltonian case (stable plateau) is recovered for $\xi \to \infty$. (b) The quantity $-ln\left(I\right(t\left)\right)$ is plotted for $h=10$ in a log-log scale to highlight the stretched exponential decay of the imbalance. For the different values of $\xi$, the exponent $\alpha$ is approximately the same, $\alpha \approx 0.42$. (Results for $L=80$, averaged over 80 disorder samples.)

Figure 3

(a) Scaling of the relaxation time (no interaction). Time $t^{*}$ to reach imbalance $I=0.05$, plotted as a function of $\xi$ for different values of $h$. Rescaling $t^{*}$ by a factor $h^2$ results in collapse onto a single curve. The behavior is flat for values of $\xi$ too small to see the lattice spacing, while it is power-law ($\sim {\xi }^{0.9}$) for large $\xi$. The collapse of curves shows that one has $t^{*}\sim h^2{\xi }^{0.9}$ at large $\xi$. (b) Metastability with interactions. Imbalance vs time, for fixed dephasing rate ($\gamma =1$), fixed disorder strength ($h=8$), and interaction ($V=1$), for different values of $\xi$. (c) Relaxation time with interactions. The relaxation time $t^{*}$ (with threshold imbalance $I=0.25$) is plotted against $\xi$ for different values of $h$ and $V=1$. All curves, after rescaling by $h^2$, collapse onto a single one, thus showing that at large $\xi$ one has $t^{*}\sim h^2{\xi }^{0.9}$.