Degeneracies and symmetry breaking in pseudo-Hermitian matrices
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Date
2023-04-18
Authors
Melkani, Abhijeet
Journal Title
Journal ISSN
Volume Title
Publisher
APS
Abstract
Real eigenvalues of pseudo-Hermitian matrices, such as real matrices and 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 behavior of the system. In this paper, we classify the
eigenspace of pseudo-Hermitian matrices and show that such symmetry breaking occurs if and only if eigenvalues
of opposite kinds collide on the real axis of the complex eigenvalue plane. 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. These disconnected regions are surrounded by exceptional surfaces, which comprise
all the real-valued exceptional points of pseudo-Hermitian matrices. The exceptional surfaces, together with the
diabolic points created by their intersections, comprise all points of pseudo-Hermiticity breaking. In particular,
this clarifies that the degeneracy involved in symmetry breaking is not necessarily an exceptional point. We also
discuss how our study relates to conserved quantities and derive the conditions for when degeneracies caused by
external symmetries are susceptible to thresholdless pseudo-Hermiticity breaking. We illustrate our results with
examples from photonics, condensed matter physics, and mechanics.
Description
15 pages
Keywords
Photonics, Classical mechanics, Classical physical mechanics
Citation
Melkani, A. (2023). Degeneracies and symmetry breaking in pseudo-Hermitian matrices. Physical Review Research, 5, 1—15. https://doi.org/10.1103/PhysRevResearch.5.023035