Super-operator structures and no-go theorems for dissipative quantum phase transitions

In the thermodynamic limit, the steady states of open quantum many-body systems can undergo nonequilibrium phase transitions due to a competition between Hamiltonian and dissipative terms. Here, we consider Markovian systems and elucidate structures of the Liouville super-operator that generates the dynamics. In many cases of interest, a non-orthogonal basis transformation can bring the Liouvillian into block-triangular form, making it possible to assess its spectrum. The spectral gap sets the asymptotic decay rate. The super-operator structure can be used to bound gaps from below, showing that, in a large class of systems, dissipative phase transitions are actually impossible and that the convergence to steady states is exponential. Furthermore, when the blocks on the diagonal are Hermitian, the Liouvillian spectra obey Weyl ordering relations. The results are exemplified by various spin models.