Dynamical Stability Phase Transitions

We present an in-depth analysis of the dynamical stability phase diagrams of QBHs. The first key observation is that boundaries separating dynamically stable and dynamically unstable phases are defined via two distinct flavors of spectral degeneracies: EPs and Krein collisions (KCs). These two flavors of degeneracy are distinguished by whether or not the relevant dynamical matrix is diagonalizable. This difference of origin will entail tangible differences in the dynamical and algebraic features of the normal modes at the transition point. Despite their differences, we are able to unify them through the lens of spontaneous generalized PT-symmetry (GPT) breaking. We connect this unifying perspective to the long-established, and rapidly growing, field of PT-symmetric quantum mechanics. Moreover, we prove equivalence between GPT-symmetry and pseudo-Hermiticity, a property intrinsic to the dynamical matrices of QBHs. Our dissection of dynamical stability phase boundaries utilizes the mathematical techniques of Krein stability theory, which we describe along the way as necessary. Putting these tools to use, we introduce a numerical indicator for dynamical stability phase transition known as Krein phase rigidity (KPR). Our development of this indicator, along with the understanding of its behavior, is inspired by the previously established algebraic features of bosonic normal modes in the vicinity of stability phase transition. Despite this boson-centric perspective, we find that it is, in fact, an extension of phase rigidity (a quantity relevant to the study of EPs in certain non-Hermitian systems) to the pseudo-Hermitian realm. Three example Hamiltonians are studied in detail: a single-mode toy model, a two-mode cavity QED model, and a bosonic Kitaev chain (BKC) under a family of BCs. In all cases, the stability phase diagrams are computed and boundaries analyzed. The GPT-symmetry breaking and behavior of the KPR are studied in detail. Further general features regarding phase-dependent transport in QBHs are elucidated by means of the BKC.