  Thermal noise can destroy topological insulators (TI). However we demonstrate
how TIs can be made stable in dissipative systems. To that aim, we introduce
the notion of band Liouvillian as the dissipative counterpart of band
Hamiltonian, and show a method to evaluate the topological order of its steady
state. This is based on a generalization of the Chern number valid for general
mixed states (referred to as density matrix Chern value), which witnesses
topological order in a system coupled to external noise. Additionally, we study
its relation with the electrical conductivity at finite temperature, which is
not a topological property. Nonetheless, the density matrix Chern value
represents the part of the conductivity which is topological due to the
presence of quantum mixed edge states at finite temperature. To make our
formalism concrete, we apply these concepts to the two-dimensional Haldane
model in the presence of thermal dissipation, but our results hold for
arbitrary dimensions and density matrices.
