Power-law approach to steady state in open lattices of noninteracting electrons

We address the question of how a nonequilibrium steady state (NESS) is reached in the Lindbladian dynamics of an open quantum system. We develop an expansion of the density matrix in terms of the NESS excitations, each of which has its own (exponential) decay rate. However, when the decay rates tend to zero for many NESS excitations (the spectral gap of the Liouvillian is closed in the thermodynamic limit), the long-time dynamics of the system can exhibit a power-law behavior. This relaxation to NESS expectation values is determined by the density of states close to zero spectral gap and the value of the operator in these states. We illustrate this main idea on the example of the lattice of noninteracting fermions coupled to Markovian leads at infinite bias voltage. The current comes towards its NESS value starting from a typical initial state as tau(-3/2). This behavior is universal and independent of the space dimension.