$\mathcal{PT}$-like phase transitions from square roots of supersymmetric Hamiltonians

TL;DR

This paper presents a novel framework for realizing $\\mathcal{PT}$-like phase transitions in non-Hermitian systems by constructing Hamiltonians as square roots of supersymmetric partner Hamiltonians, enabling controlled spectral transitions without explicit $\\mathcal{PT}$-symmetry.

Contribution

It introduces a general supersymmetry-based approach to engineer non-Hermitian systems with tunable spectral properties and higher-order exceptional points, expanding beyond traditional $\\mathcal{PT}$-symmetry constraints.

Findings

Real spectra occur when the passive Hamiltonian has no zero mode.

Spectral transitions happen at second-order exceptional points as gain/loss increases.

The framework applies to models like Hatano--Nelson and cSSH lattices.

Abstract

We introduce a general framework for realizing $\mathcal{PT}$-like phase transitions in non-Hermitian systems without imposing explicit parity--time ($\mathcal{PT}$) symmetry. The approach is based on constructing a Hamiltonian as the square root of a supersymmetric partner energy-shifted by a constant. This formulation naturally leads to bipartite dynamics with balanced gain and loss and can incorporate non-reciprocal couplings. The resulting systems exhibit entirely real spectra over a finite parameter range precisely when the corresponding passive Hamiltonian lacks a zero mode. As the non-Hermitian parameter representing gain and loss increases, the spectrum undergoes controlled real-to-complex transitions at second-order exceptional points. We demonstrate the versatility of this framework through several examples, including well-known models such as the Hatano--Nelson (HN) and complex Su--Schrieffer--Heeger (cSSH) lattices. Extending the formalism to $q$-commuting matrices further enables the systematic realization of higher-order exceptional points in systems with unidirectional couplings. Overall, this work uncovers new links between non-Hermitian physics and supersymmetry, offering a practical route to engineer photonic arrays with tunable spectral properties beyond what is achievable with explicit $\mathcal{PT}$-symmetry.