Non-hermitian time evolution: from static to parametric instability

TL;DR

This paper explores the complex dynamics of non-hermitian time evolution, especially around exceptional points, using a classification based on the Möbius group, and reveals new phenomena in Floquet systems related to parametric resonance.

Contribution

It introduces a classification of two-level non-hermitian Hamiltonians via the Möbius group and applies this to analyze dynamical encircling of exceptional points in Floquet systems, uncovering richer physics.

Findings

Floquet non-hermitian systems show complex behaviors beyond simple EP-encircling rules.

EPs can occur without encircling, and vice versa, in Floquet systems.

Parametric resonance explains the interplay between non-hermitian and modulation instabilities.

Abstract

Eigenmode coalescence imparts remarkable properties to non-hermitian time evolution, culminating in a purely non-hermitian spectral degeneracy known as an exceptional point (EP). Here, we revisit time evolution at the EP and classify two-level non-hermitian Hamiltonians in terms of the M\"obius group. We then leverage that classification to study dynamical EP encircling, by applying it to periodically-modulated (Floquet) Hamiltonians. This reveals that Floquet non-hermitian systems exhibit rich physics whose complexity is not captured by an EP-encircling rule. For example, Floquet EPs can occur without encircling and vice-versa. Instead, we show that the elaborate interplay between non-hermitian and modulation instabilities is better understood through the lens of parametric resonance.