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

Abstract

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 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 for the first time 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.

Describing many-body localized systems in thermal environments

“…Finally, at 'high' temperatures (T J  ), I decreases as a function of T. Note that the locations of the three regions depend on various parameters, such as the disorder strength W (as shown in figure 5(a)), the coupling to the leads κ, and so on. The independence of the current on temperature in the 'low' temperature regime is due to the suppression of heat exchange, which can be inferred from the good agreement of the results with that without coupling to the global thermal bath (equation (48), dotted lines). Nevertheless, the remaining current T 0  ( ) is a finite system size effect [71].…”

“…Here we propose to exploit the l-bit representation [43][44][45][46][47][48] of MBL systems to address the diagonalization problem. The existence of an extensive number of quasi-local integrals of motion in MBL systems makes them special in the sense that they are interacting quantum systems whose full many-body spectrum can be accessed efficiently.…”

“…The dotted line is the result in the absence of the global thermal bath, i.e. equation(48). The disorder strength is W=2J.…”

Dynamics of disordered quantum systems using flow equations

“…For non-interacting systems, this can be accomplished exactly. For interacting systems, which we shall discuss only briefly at the end of this manuscript, the situation is typically more complicated and additional approximations are necessary in order to obtain an analytic set of flow equations [50,37,33].…”

“…We can use the flow equation formalism to compute the dynamics of any given observable, as in Refs. [52,53,54,33]. Consider the case where we have a complicated, offdiagonal microscopic Hamiltonian H which generates unitary time evolution.…”