Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding

TL;DR

This paper presents a solvable quantum model with non-Hermitian Hamiltonians, using symbolic computation to analyze exceptional points and their unfoldings, providing insights into phase transitions and metric reconstruction.

Contribution

It introduces a symbolic-manipulation approach to construct metrics for non-Hermitian Hamiltonians and studies the topology of exceptional point unfoldings in a specific matrix model.

Findings

Constructed metrics via polynomial equations using symbolic computation.

Analyzed the structure of exceptional points in a discrete square well model.

Outlined classification methods for unfoldings of phase transition points.

Abstract

The practical use of non-Hermitian (i.e., typically, PT-symmetric) phenomenological quantum Hamiltonians is discussed as requiring an explicit reconstruction of the {\em ad hoc} Hilbert-space metrics which would render the time-evolution unitary. Just the N-dimensional matrix toy models Hamiltonians are considered, therefore. For them, the matrix elements of alternative metrics are constructed via solution of a coupled set of polynomial equations, using the computer-assisted symbolic manipulations for the purpose. The feasibility and some consequences of such a model-construction strategy are illustrated via a discrete square well model endowed with multi-parametric close-to-the-boundary real bidiagonal-matrix interaction. The degenerate exceptional points marking the phase transitions are then studied numerically. A way towards classification of their unfoldings in topologically non-equivalent dynamical scenarios is outlined.