Nonnormality-induced spontaneous symmetry breaking in open quantum systems

Recently, there has been a surge in theoretical efforts aimed at defining and classifying phases of matter in open quantum systems. In particular, for open quantum systems described by the Lindblad equation, it is a challenging problem to clarify and classify what kind of non-equilibrium steady state (NESS) the system relaxes to after a sufficiently long time. In this talk, we consider open quantum spin systems on infinite lattices described by the Lindblad equation and present a definition of the NESS in such infinite systems. The NESS in infinite systems is not necessarily equivalent to the thermodynamic limit of the NESS in finite systems; the former corresponds in finite systems to a metastable state, not an NESS. We see this in a solvable model. We also find sufficient conditions for equivalence of them, one of which relates to the spectral gap of the Liouvillian and the other to nonnormality, i.e., nonorthogonality among eigenmodes of the Liouvillian. The enhancement of nonnormality can cause a spontaneous symmetry breaking (SSB) for the NESS in infinite systems, which we dub the nonnormality-induced SSB. Since a Liouvillian is normal in unitary dynamics, such a type of SSB is unique to open quantum systems. We provide a method to evaluate the nonnormality based on non-Hermitian skin effects. This work uncovers a novel phase of matter in open quantum systems and establishes a way to classify it by focusing on nonnormality.