Exceptional points in quantum and classical dynamics

TL;DR

This paper explores the connection between exceptional points in quantum systems with non-Hermitian Hamiltonians and the resulting singular classical dynamics, highlighting structural changes in trajectories and analogies with supersymmetric Yang-Mills theory.

Contribution

It demonstrates the classical behavior associated with quantum exceptional points and analyzes a specific crypto-Hermitian Hamiltonian with complex classical dynamics.

Findings

Classical trajectories restructure at quantum exceptional points

The studied Hamiltonian exhibits hyper-elliptic classical dynamics

Analogies with supersymmetric Yang-Mills dynamics are established

Abstract

We notice that, when a quantum system involves exceptional points, i.e. the special values of parameters where the Hamiltonian loses its self-adjointness and acquires the Jordan block structure, the corresponding classical system also exhibits a singular behaviour associated with restructuring of classical trajectories. The system with the crypto-Hermitian Hamiltonian H = (p^2+z^2)/2 -igz^5 and hyper-ellictic classical dynamics is studied in details. Analogies with supersymmetric Yang-Mills dynamics are elucidated.