Unitary unfoldings of Bose-Hubbard exceptional point with and without particle number conservation

TL;DR

This paper explores the structure of generalized Bose-Hubbard Hamiltonians at exceptional points, analyzing how particle number conservation affects unitarity and the physical properties of non-Hermitian bosonic systems.

Contribution

It introduces two families of Hamiltonians at exceptional points, with and without particle number conservation, revealing their distinct perturbation structures and physical implications.

Findings

Two different Hamiltonian families constructed at exceptional points.

Particle number conservation influences the structure of admissible perturbations.

Physical Hilbert space anisotropy affects system unitarity and realizability.

Abstract

Non-Hermitian but ${\cal PT}-$symmetric quantum system of an $N-$plet of bosons described by the three-parametric Bose-Hubbard Hamiltonian $H(\gamma,v,c)$ is picked up, in its special exceptional-point limit $c \to 0$ and $\gamma \to v$, as an unperturbed part of the family of generalized Bose-Hubbard-like Hamiltonians $\mathfrak{H}(\lambda)=H(v,v,0)+\lambda\,{\cal V}$ for which the unitarity of the perturbed system is required. This leads to the construction of two different families of Hamiltonians $\mathfrak{H}(\lambda)$. In the first one the number $N$ of bosons is assumed conserved while in the second family such an assumption is relaxed. In both cases the anisotropy of the related physical Hilbert space is shown reflected by a highly counterintuitive but operationally realizable structure of admissible perturbations $\lambda\,{\cal V}$.