Protection of parity-time symmetry in topological many-body systems: Non-Hermitian toric code and fracton models

In the study of P T -symmetric quantum systems with non-Hermitian perturbations, one of the most important questions is whether eigenvalues stay real or whether P T symmetry is spontaneously broken when eigenvalues meet. A particularly interesting set of eigenstates is provided by the degenerate ground-state subspace of systems with topological order. In this paper, we present simple criteria that guarantee the protection of P T symmetry and, thus, the reality of the eigenvalues in topological many-body systems. We formulate these criteria in both geometric and algebraic form and demonstrate them using the toric code and several different fracton models as examples. Our analysis reveals that P T symmetry is robust against a remarkably large class of non-Hermitian perturbations in these models; this is particularly striking in the case of fracton models due to the exponentially large number of degenerate states.