The steady states of Markovian processes can be written as extremal eigenvectors of a matrix (e.g., a quantum channel) that generates the dynamics. Unlike ground states of Hamiltonians, however, the gap of the channel does not necessarily control relaxation to the ground state. I will argue that this discrepancy is precisely what allows nontrivial gapped phases of Lindbladians to exist. I will discuss an alternative criterion for deciding whether two Lindbladians (and their steady states) are in the same phase. I will show that this criterion implies many of the properties one would naturally demand of a phase, such as the persistence of any long-range order and the analytic evolution of correlation functions within a phase.

Rakovszky, SG, Von Keyserlingk, arxiv:2308.15495