Solutions of PT symmetric tight-binding chain and its equivalent Hermitian counterpart

TL;DR

This paper provides an exact solution for a PT-symmetric tight-binding chain with imaginary potentials, revealing phase transitions characterized by real and complex eigenvalues, and constructs an equivalent Hermitian model with long-range hoppings.

Contribution

It offers an exact analytic solution for a PT-symmetric chain and constructs its Hermitian counterpart, advancing understanding of non-Hermitian quantum systems.

Findings

Identifies unbroken and broken PT-symmetry phases with real and complex spectra.

Shows the phase boundary exhibits exceptional point characteristics.

Constructs an equivalent Hermitian Hamiltonian with long-range hopping terms.

Abstract

We study the Non-Hermitian quantum mechanics for the discrete system. This paper gives an exact analytic single-particle solution for an $N$-site tight-binding chain with two conjugated imaginary potentials $\pm i\gamma $ at two end sites, which Hamiltonian has parity-time symmetry ($\mathcal{PT}$ symmetry). Based on the Bethe ansatz results, it is found that, in single-particle subspace, this model is comprised of two phases, an unbroken symmetry phase with a purely real energy spectrum in the region $\gamma \prec \gamma_{c}$ and a spontaneously-broken symmetry phase with $N-2$ real and 2 imaginary eigenvalues in the region $\gamma \succ \gamma_{c}$. The behaviors of eigenfunctions and eigenvalues in the vicinity of $\gamma_{c}$ are investigated. It is shown that the boundary of two phases possesses the characteristics of exceptional point. We also construct the equivalent Hermitian Hamiltonian of the present model in the framework of metric-operator theory. We find out that the equivalent Hermitian Hamiltonian can be written as another bipartite lattice model with real long-range hoppings.