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

Miloslav Znojil

TL;DR

This work investigates unitary unfoldings of Bose-Hubbard exceptional points in non-Hermitian, PT-symmetric settings by constructing two families of perturbed Hamiltonians of the form $\mathfrak{H}(\lambda)=H(v,v,0)+\lambda\,\mathcal{V}$, with and without fixed particle number $N$. A central tool is a real, structured fundamental matrix $C^{(K)}$ whose real, nondegenerate spectrum guarantees leading-order unitarity and yields a nonempty corridor ${\cal D}$ of admissible perturbations near the exceptional-point regime $\gamma\to1$. The paper develops both conservative (block-diagonal) and nonconservative (partitioned) perturbation schemes, analyzing their spectral reality via finite-dimensional, sparse, partitioned Hamiltonians and their associated secular problems. It then provides explicit, worked examples for the elementary $(2+3)$ partition and a more complex $(2+3+4)$ partition, showing that a real, nondegenerate spectrum can be achieved and partially classifying the corresponding boundary structures ${\partial\cal D}$ of unitarity. The results establish that closed-system BH-type models can be tuned to retain unitarity near degenerate EPs, offering a framework to study dynamics near EPs with and without particle-number conservation, with potential implications for photonic and ultracold-atom platforms.

Abstract

Non-Hermitian but ${\cal PT}-$symmetric quantum system of an $N-$plet of bosons described by the three-parametric Bose-Hubbard Hamiltonian $H(γ,v,c)$ is picked up, in its special exceptional-point limit $c \to 0$ and $γ\to v$, as an unperturbed part of the family of generalized Bose-Hubbard-like Hamiltonians $\mathfrak{H}(λ)=H(v,v,0)+λ\,{\cal V}$ for which the unitarity of the perturbed system is required. This leads to the construction of two different families of Hamiltonians $\mathfrak{H}(λ)$. 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 $λ\,{\cal V}$.

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

TL;DR

This work investigates unitary unfoldings of Bose-Hubbard exceptional points in non-Hermitian, PT-symmetric settings by constructing two families of perturbed Hamiltonians of the form , with and without fixed particle number . A central tool is a real, structured fundamental matrix whose real, nondegenerate spectrum guarantees leading-order unitarity and yields a nonempty corridor of admissible perturbations near the exceptional-point regime . The paper develops both conservative (block-diagonal) and nonconservative (partitioned) perturbation schemes, analyzing their spectral reality via finite-dimensional, sparse, partitioned Hamiltonians and their associated secular problems. It then provides explicit, worked examples for the elementary partition and a more complex partition, showing that a real, nondegenerate spectrum can be achieved and partially classifying the corresponding boundary structures of unitarity. The results establish that closed-system BH-type models can be tuned to retain unitarity near degenerate EPs, offering a framework to study dynamics near EPs with and without particle-number conservation, with potential implications for photonic and ultracold-atom platforms.

Abstract

Non-Hermitian but symmetric quantum system of an plet of bosons described by the three-parametric Bose-Hubbard Hamiltonian is picked up, in its special exceptional-point limit and , as an unperturbed part of the family of generalized Bose-Hubbard-like Hamiltonians for which the unitarity of the perturbed system is required. This leads to the construction of two different families of Hamiltonians . In the first one the number 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 .

Paper Structure

This paper contains 17 sections, 3 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Conventional non-Hermitian Bose-Hubbard model
  3. Matrix representation of Hamiltonian
  4. Exceptional point
  5. Unitarity-breaking perturbations at a fixed $N=K-1$
  6. Modified non-Hermitian Bose-Hubbard models
  7. Perturbations conserving the number of bosons
  8. Perturbations which do not violate the unitarity
  9. Perturbations not conserving the number of bosons
  10. Models with one off-diagonal pair of perturbation submatrices
  11. Leading-order block-non-diagonal Hamiltonians
  12. Examples
  13. Models with the bosonic pairs coupled to triples
  14. The most elementary case
  15. Boundary $\partial{\cal D}$ of the corridor of unitarity
  16. ...and 2 more sections

Key Result

Lemma 1

After the most elementary Kronecker-delta choice of $a_k^{(j)}=\delta_{j,1}$, $k=0, 1, \ldots, K-1$ the set of eigenvalues $\varepsilon_n$ of matrix (uzmatko) becomes defined in terms of roots of classical orthogonal Chebyschev polynomials which are all real and non-degenerate cheby.

Theorems & Definitions (5)

  • Lemma 1
  • Theorem 2
  • Conjecture 3
  • Conjecture 4
  • Lemma 5