Interference of non-Hermiticity with Hermiticity at exceptional points

TL;DR

This paper investigates how non-Hermitian, PT-symmetric Hamiltonians with complex substructures behave at exceptional points, revealing decoupling, degeneracies, and symmetry breaking phenomena in a family of toy models.

Contribution

It introduces a class of parametrized non-Hermitian Hamiltonians with coupled submatrices, analyzing their spectral properties and exceptional points, which advances understanding of PT-symmetry and non-Hermitian degeneracies.

Findings

Submatrices decouple at specific parameter values.

Hamiltonians exhibit Kato's exceptional-point degeneracies.

PT-symmetry is spontaneously broken at certain parameter thresholds.

Abstract

A family of non-Hermitian but ${\cal PT}-$symmetric $2J$ by $2J$ toy-model tridiagonal-matrix Hamiltonians $H^{(2J)}=H^{(2J)}(t)$ with $J=K+M=1,2,\ldots$ and $t<J^2$ is studied, for which a real but non-Hermitian $2K$ by $2K$ tridiagonal-submatrix component $C(t)$ of the Hamiltonian is assumed coupled to its other two complex but Hermitian $M$ by $M$ tridiagonal-submatrix components $A(t)$ and $B(t)$. By construction, (i) all of the submatrices get decoupled at $t=t_M=M\,(2J-M)$ with $M=1,2,\ldots,J$; (ii) at all of the parameters $t=t_M$ with $M=J-K=0,1,\ldots,J-1$ the Hamiltonian ceases to be diagonalizable exhibiting the Kato's exceptional-point degeneracy of order $2K$; (iv) the system's ${\cal PT}-$symmetry gets spontaneously broken when $t\leq t_{J-1}=J^2-1$.