Critical behavior near the many-body localization transition in driven open systems

In a many-body localized (MBL) system, the coupling to an external bath typically breaks local integrals of motion. Thus the system relaxes to a unique thermal steady state. When the bath is non-thermal or when the system is weakly driven out of equilibrium, local conservation laws can be excited far from any thermal equilibrium value. I will show how this property can be used to study the MBL phase transition in weakly open systems. Here, the strength of the coupling to the non-thermal bath plays a similar role as a finite temperature in a T=0 quantum phase transition. By tuning this parameter, we can detect key features of the MBL transition: the divergence of dynamical exponent due to Griffiths effects and the critical disorder strength.

We propose a new order parameter, based on the fluctuations in local temperatures. For vanishing strength of coupling to the bath, fluctuations vanish on the ergodic side, while they are large on the MBL side. By increasing the coupling strength, fluctuations grow with a fractional exponent related to the inverse dynamical exponent on the ergodic side, while they decrease monotonically on the MBL side. This paves the way for studies of the MBL transition with new numerical approaches and, importantly, also with solid-state experiments.