Toward a classification of PT-symmetric quantum systems: From dissipative dynamics to topology and wormholes

Studies of many-body non-Hermitian parity-time (PT)-symmetric quantum systems are attracting a lot of interest due to their relevance in research areas ranging from quantum optics and continuously monitored dynamics to Euclidean wormholes in quantum gravity and dissipative quantum chaos. While a symmetry classification of non-Hermitian systems leads to 38 universality classes, we show that, under certain conditions, PT-symmetric systems are grouped into 24 universality classes. We identify 14 of them in a coupled two-site Sachdev-Ye-Kitaev (SYK) model and confirm the classification by spectral analysis using exact diagonalization techniques. Intriguingly, in 4 of these 14 universality classes, AIII$_\nu$, BDI$^\dagger_\nu$, BDI$_{++\nu}$, and CI$_{--\nu}$, we identify a basis in which the SYK Hamiltonian has a block structure in which some blocks are rectangular, with $\nu \in \mathbb{N}$ the difference between the number of rows and columns. We show analytically that this feature leads to the existence of $\nu$ robust purely \emph{real} eigenvalues, whose level statistics follow the predictions of Hermitian random matrix theory for classes A, AI, BDI, and CI, respectively. We have recently found that this $\nu$ is a topological invariant, so these classes are topological. By contrast, nontopological real eigenvalues display a crossover between Hermitian and non-Hermitian level statistics. Similarly to the case of Lindbladian dynamics, the reduction of universality classes leads to unexpected results, such as the absence of Kramers degeneracy in a given sector of the theory. Another novel feature of the classification scheme is that different sectors of the PT-symmetric Hamiltonian may have different symmetries.