Degeneracies and symmetry breaking in pseudo-Hermitian matrices
Degeneracies and symmetry breaking in pseudo-Hermitian matrices
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DOI：
10.1103/physrevresearch.5.023035
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发表时间：
2022-09
影响因子：
4.2
通讯作者：
Abhijeet Melkani
中科院分区：
文献类型：
--
作者：
Abhijeet Melkani
伪厄米特矩阵的真实的特征值，如真实的矩阵和$\mathcal{PT-}$对称矩阵，经常分裂成复共轭对。这伴随着本征向量的某些对称性的破坏，并且通常还伴随着系统行为的急剧变化。本文对伪厄米特矩阵的特征空间进行了分类，并证明了当且仅当两类特征值在复特征值平面的真实的轴上发生碰撞时，对称性破缺才发生.这使得能够对参数空间中的断开区域进行分类，其中所有特征值都是真实的-其在物理上对应于系统的稳定相。这些不连通区域被例外曲面包围，这些例外曲面包含伪厄米特矩阵的所有实值例外点。例外的表面，连同由它们的交叉点创建的恶魔点，包括所有的点的伪厄米性打破。特别是，这澄清了对称性破缺所涉及的简并不一定是一个例外点。我们还讨论了我们的研究如何涉及到守恒量，并推导出外部对称性引起的简并时容易受到无阈值伪厄米性破坏的条件。我们用光子学、凝聚态物理学和力学的例子来说明我们的结果。
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 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.