Topological Localisation in Time from PT Symmetry

Time has entered the domain of topological phases in the field of non-Hermitian physics. Previous studies have relied on periodic modulation in time to make an intuitive connection to established spatial topological invariants, albeit with energy and momentum exchanged. This connection has revealed the potential for topological interface states along the time axis, analogous to those in spatial models. In this work, we uncover a theoretical framework describing such topological interface states along the time axis, with no underlying connection to spatial models nor need for periodic driving. This new framework uncovers that this phenomenon -- the robust localisation of waves at an interface -- appears in every system that has parity-time symmetry and two coupled modes or bands, regardless of its spatial dimensionality. The topological nature of this localisation is understood by the identification of certain topological phases that are specific to parity-time-symmetric models of two coupled modes. Our theoretical framework can be applied to all existing experimental observations, notably including photonic time crystals, and serves as a foundation for future experiments in areas in which the topological localisation of waves in time has yet to be studied.