We consider the role of degeneracy in Parity-Time (PT) symmetry breaking for non-hermitian wave equations beyond one dimension. We show that if the spectrum is degenerate in the absence of T-breaking, and T is broken in a generic manner (without preserving other discrete symmetries), then the standard PT-symmetry breaking transition does not occur, meaning that the spectrum is complex even for infinitesimal strength of gain and loss. However the realness of the entire spectrum can be preserved over a finite interval if additional discrete symmetries X are imposed when T is broken, if X decouple all degenerate modes. When this is true only for a subset of the degenerate spectrum, there can be a partial PT transition in which this subset remains real over a finite interval of T-breaking. If the spectrum has odd-degeneracy, a fraction of the degenerate spectrum can remain in the symmetric phase even without imposing additional discrete symmetries, and they are analogous to dark states in atomic physics. These results are illustrated by the example of different T-breaking perturbations of a uniform dielectric disk and sphere, and a group theoretical analysis is given in the disk case. Finally, we show that multimode coupling is capable of restoring the T-symmetric phase at finite T-breaking. We also analyze these questions when the parity operator is replaced by another spatial symmetry operator and find that the behavior can be qualitatively different.
Degeneracy beyond the parity-symmetry protection in one-dimensional spinless models: The parity-violating Kerr parametric oscillator
Universidad de Córdoba（科尔多瓦大学）；Universidad de Huelva（韦尔瓦大学）；Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada（格拉纳达大学卡洛斯一世理论与计算物理研究所）
Degeneracy beyond the parity-symmetry protection in the Lipkin-Meshkov-Glick model
Universidad de Córdoba（科尔多瓦大学）；Universidad de Huelva（韦尔瓦大学）；CSIC-UHU（西班牙国家研究委员会-韦尔瓦大学联合单位）；Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada（格拉纳达大学卡洛斯一世理论与计算物理研究所）
On the role of true and false chirality in producing parity violating energy differences
Fundacion Humanismo y Ciencia（人文主义与科学基金会）；Departamento de Física Aplicada, Universidad de Alicante（阿利坎特大学应用物理系）；Instituto de Física Fundamental, Consejo Superior de Investigaciones Científicas（西班牙国家科学研究委员会基础物理研究所）
Topological degeneracy (Majorana zero-mode) and 1+1D fermionic topological order in a magnetic chain on superconductor via spontaneous Z2 symmetry breaking
