# Effects of detuning on $\mathcal{PT}$-symmetric, tridiagonal, tight-binding models

## TL;DR

This paper explores how detuning affects the $ ext{PT}$-symmetry breaking thresholds and exceptional point structures in non-Hermitian, tight-binding models, providing analytical insights and explicit operators for complex quantum extensions.

## Contribution

It introduces the dependence of $ ext{PT}$-thresholds on detuning in non-Hermitian models and derives explicit formulas for intertwining operators, advancing understanding of $ ext{PT}$-symmetric systems.

## Key findings

- EP curves often have cusp points where the order increases
- Explicit analytical expressions for positive-definite intertwining operators
- Detuning influences $ ext{PT}$-thresholds and exceptional point structures

## Abstract

Non-Hermitian, tight-binding $\mathcal{PT}$-symmetric models are extensively studied in the literature. Here, we investigate two forms of non-Hermitian Hamiltonians to study the $\mathcal{PT}$-symmetry breaking thresholds and features of corresponding surfaces of exceptional points (EPs). They include one-dimensional chains with uniform or 2-periodic tunnelling amplitudes, one pair of balanced gain and loss potentials $\Delta\pm\i\gamma$ at parity-symmetric sites, and periodic or open boundary conditions. By introducing a Hermitian detuning potential, we obtain the dependence of the $\mathcal{PT}$-threshold, and therefore the exceptional-point curves, in the parameter space of detuning and gain-loss strength. By considering several such examples, we show that EP curves of a given order generically have cusp-points where the order of the EP increases by one. In several cases, we obtain explicit analytical expressions for positive-definite intertwining operators that can be used to construct a complex extension of quantum theory by re-defining the inner product. Taken together, our results provide a detailed understanding of detuned tight-binding models with a pair of gain-loss potentials.