Entanglement, dynamics and <i>breakdown</i> of many-body localization in current driven system

What is the fate of a many-body localized system under a voltage bias between two ends? Can the system undergo a transition to a current carrying non-equilibrium steady state and how the entanglement properties of the quantum states change across the transition? Motivated by these questions, we model a current driven interacting disorder system through a non-Hermitian Hamiltonian and study the entanglement properties of its eigenstates. We also discuss the dynamics, entanglement growth and long-time fate of a generic initial state under an appropriate time-evolution of the system governed by the non-Hermitian Hamiltonian. Our study reveals rich entanglement structures of current driven states and multiple dynamical transitions as a function of disorder and the strength of the non-Hermitian term, that is related to the external bias.