Time evolution of many-body localized systems with the flow equation approach

The interplay between interactions and quenched disorder can result in rich dynamical quantum phenomena far from equilibrium, particularly when many-body localization prevents the system from full thermalization. With the aim of tackling this interesting regime, here we develop a semianalytical flow equation approach to study the time evolution of strongly disordered interacting quantum systems. We apply this technique to a prototype model of interacting spinless fermions in a random on-site potential in both one and two dimensions. Key results include (i) an explicit construction of the local integrals of motion that characterize the many-body localized phase in one dimension, ultimately connecting the microscopic model to phenomenological descriptions, (ii) calculation of these quantities in two dimensions, and (iii) an investigation of the real-time dynamics in the localized phase which reveals the crucial role of $l$-bit interactions for enhancing dephasing and relaxation.

Figure 1

(a) Real-space exponential decay of fixed-point couplings ${\tilde{\mathrm{\Delta }}}_{ij}$, describing mutual interactions among $l$-bits, for increasing values of disorder (top to bottom) in $d=1$. (b) The same quantity for $d=2$, plotted to the same scale. (c) Localization length scale extracted from ${\tilde{\mathrm{\Delta }}}_{ij}$ (see main text) as a function of disorder from large to small. The system size is $L=100$ for $d=1$ and $L=12\times 12$ for $d=2$, and the number of disorder realizations for each is $N_S=100$.

Figure 2

Real-time evolution of the disorder-averaged density of fermions in the middle of the chain, starting from a CDW initial state. (a) Effect of disorder on the melting of CDW correlations, which suppresses the fluctuations but does not appear to change the relaxation time scale. The frequencies of the oscillations at early times are set by the interaction strength. (b) Interaction effects between $l$-bits strongly quench their dynamics and induce decoherence. Parameters: $L=64, N_S=500$.