Degeneracies and symmetry breaking in pseudo-Hermitian matrices

Real eigenvalues of pseudo-Hermitian matrices, such as real matrices and $\mathcal{PT-}$symmetric matrices, frequently split into complex conjugate pairs. This is accompanied by the breaking of certain symmetries of the eigenvectors and, typically, also a drastic change in the behaviour of the system. In this work, we classify the eigenspace of pseudo-Hermitian matrices and show that such symmetry breaking occurs if and only if modes of opposite kinds collide on the real axis. In particular we clarify that the degeneracy involved is not always an exceptional point. This enables a classification of the disconnected regions in parameter space where all eigenvalues are real -- which correspond, physically, to the stable phases of the system. We also discuss how our work relates to conserved quantities and to degeneracies caused by external symmetries. We illustrate our results with examples from photonics, condensed matter physics, and mechanics.Comment: 13 pages, 4 figures. Version 2 clarifies the language, expands on the examples discussed, and contains more figures. Submitted to Physical Review Research. Comments and feedback welcome

Space-time symmetry and parametric resonance in dynamic mechanical systems

Linear mechanical systems with time-modulated parameters can harbor oscillations with amplitudes that grow or decay exponentially with time due to the phenomenon of parametric resonance. While the resonance properties of individual oscillators are well understood, identifying the conditions for parametric resonance in systems of coupled oscillators remains challenging. Here, we identify internal symmetries that arise from the real-valued and symplectic nature of classical mechanics and determine the parametric resonance conditions for periodically time-modulated mechanical metamaterials using these symmetries. Upon including external symmetries, we find additional conditions that prohibit resonances at some modulation frequencies for which parametric resonance would be expected from the internal symmetries alone. In particular, we analyze systems with space-time symmetry where the system remains invariant after a combination of discrete translation in both space and time. For such systems, we identify a combined space-time translation operator that provides more information about the system than the Floquet operator does, and use it to derive conditions for one-way amplification of traveling waves. Our results establish an exact theoretical framework based on symmetries to engineer exotic responses such as nonreciprocal transport and one-way amplification in space-time modulated mechanical systems, and can be generalized to all physical systems that obey space-time symmetry.Comment: 14 pages, 4 figure