Symmetry-Protected Multifold Exceptional Points and Their Topological Characterization

We investigate the occurrence of n-fold exceptional points (EPs) in non-Hermitian systems, and show that they are stable in n-1 dimensions in the presence of antiunitary symmetries that are local in parameter space, such as, e.g., parity-time (PT) or charge-conjugation parity (CP) symmetries. This implies in particular that threefold and fourfold symmetry-protected EPs are stable, respectively, in two and three dimensions. The stability of such multofold exceptional points (i.e., beyond the usual twofold EPs) is expressed in terms of the homotopy properties of a resultant vector that we introduce. Our framework also allows us to rephrase the previously proposed Z_{2} index of PT and CP symmetric gapped phases beyond the realm of two-band models. We apply this general formalism to a frictional shallow water model that is found to exhibit threefold exceptional points associated with topological numbers ±1. For this model, we also show different non-Hermitian topological transitions associated with these exceptional points, such as their merging and a transition to a regime where propagation is forbidden, but can counterintuitively be recovered when friction is increased furthermore.