# Generalized Bose-Hubbard Hamiltonians exhibiting a complete non-Hermitian degeneracy

## TL;DR

This paper introduces a method to construct a broader class of non-Hermitian Bose-Hubbard Hamiltonians with complete exceptional point degeneracies, overcoming previous technical obstacles and enabling detailed spectral analysis.

## Contribution

It presents a novel approach to circumvent technical challenges in modeling non-Hermitian Bose-Hubbard systems with EPN degeneracies, expanding the class of eligible models.

## Key findings

- Broader class of Hamiltonians with EPN degeneracies identified
- Method demonstrated through symbolic computations for larger N
- Spectral analysis near and far from EPN regimes performed

## Abstract

The conventional toy-model constructions of phase diagrams often use various versions of the standard Hermitian Bose-Hubbard Hamiltonians $H$. These studies were recently extended to cover several non-Hermitian PT-symmetric versions of the model. A key technical merit of the extensions has been found in the existence of Kato's phase-transition-related exceptional-point spectral degeneracies of the $N$th order (EPN). In this setting, an ill-conditioned nature of $H$ near its EPN singularity represented a decisive technical obstacle. In our paper we show how the obstacle can be circumvented, leading to a broader class of the eligible models with tridiagonal and symmetric complex Hamiltonians $H^{(N)}(z)$ exhibiting the EPN degeneracies. The implementation of the method (requiring the use of computer-assisted symbolic manipulations at the larger integers $N$) is illustrated by sampling the real or complex spectra of energies near as well as far from the EPN dynamical regime.