Emergent phenomena with broken parity-time symmetry: Odd-order versus even-order effects

Symmetry often governs the laws of nature, and breaking symmetry accompanies a new order parameter and emergent observable phenomena. Herein, we focus on broken Parity (P)-Time (T) symmetry, which lifts the Kramers' degeneracy, and thus, guarantees non-trivial spin textures in excitation spectra. To attain non-zero measurables, we use the concept of symmetry operational similarity (SOS), which consider the symmetry relationship between a specimen and an experimental setup, rather than the symmetry of specific coupling terms. Even without specific coupling terms, this SOS approach can tell if the relevant phenomenon is a zero, non-zero odd-order or non-zero even-order effect. We discuss systematically numerous steady-state physical phenomena, in which breaking P-T symmetry is a necessary condition. These phenomena include Odd-order or Even-order Anomalous Hall Effect, Optical activities, Directional nonreciprocity in transverse magnetic field, Diagonal or Off-diagonal current-induced magnetization (current can be associated with electrons, phonons, or light), Diagonal or Off-diagonal piezomagnetism and piezoelectricity. Some of these phenomena turn out to be conjugate to each other through P to T. Our findings unveil numerous new non-traditional candidate materials for various exotic physical phenomena, many of which have never been realized in the standard coupling term/tensorial approaches, and are a transformative and unconventional avenue for symmetry-guided materials designs and discoveries.