Disorder in dissipation-induced topological states: Evidence for a different type of localization transition

The quest for nonequilibrium quantum phase transitions is often hampered by the tendency of driving and dissipation to give rise to an effective temperature, resulting in classical behavior. Could this be different when the dissipation is engineered to drive the system into a nontrivial quantum coherent steady state? In this work we shed light on this issue by studying the effect of disorder on recently introduced dissipation-induced Chern topological states, and examining the eigenmodes of the Hermitian steady-state density matrix or entanglement Hamiltonian. We find that, similarly to equilibrium, each Landau band has a single delocalized level near its center. However, using three different finite-size scaling methods we show that the critical exponent ν describing the divergence of the localization length upon approaching the delocalized state is significantly different from equilibrium if disorder is introduced into the nondissipative part of the dynamics. This indicates a different type of nonequilibrium quantum critical universality class accessible in cold-atom experiments.