Symmetry Classes of Open Fermionic Quantum Matter

We present a full symmetry classification of fermion matter in and out thermal equilibrium. Our approach starts from first principles, the ten different classes linear anti-linear state transformations fermionic Fock spaces, symmetries defined via invariance properties dynamical equation for density matrix. The object are then generators reversible dynamics, dissipation fluctuations featuring generally irreversible interacting equations. A sharp distinction between equilibrium respectively, arises role played by `time' these two cases: In unitary quantum mechanics as well `micro-reversible' equilibrium, combined with an inversion time define reversal symmetry. However, becomes meaningless, while anti--linear space remain physically significant, hence must be considered autonomy. practical consequence this dichotomy is novel realization antilinear (six fundamental classes) non-equilibrium dynamics that fundamentally established rules At large times, thus classified determine steady distributions arbitrary systems. To illustrate principle, we consider fixation protected topological phase system lattice fermions. More generally, practically important class mean field systems, represented Gaussian states. This naturally described language non-Hermitian matrices, which allows us to compare previous schemes literature.