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Non-Hermitian Bose-Hubbard-like quantum models

Miloslav Znojil

TL;DR

This work studies non-Hermitian Bose-Hubbard-like quantum models built from analytically continued, non-self-adjoint Hamiltonians. It develops a practical toolkit based on Hermitization to compute singular values and on matrix continued fractions to represent Green's functions, enabling efficient analysis of resonances and exceptional points in large or infinite tridiagonal systems. The paper provides both conceptual foundations and concrete demonstrations, including elementary and few-state models, and extends the framework to standard and doubly-infinite matrix models. The resulting approach offers user-friendly, numerically efficient methods for probing spectral properties of non-Hermitian, Bose-Hubbard-like quantum systems with potential applicability to open- and closed-system interpretations.

Abstract

Among all of the non-Hermitian large-tridiagonal-matrix quantum Hamiltonians we choose a subclass with the structure resembling the ``benchmark'' realistic Bose-Hubbard model. We demonstrate that this choice can be declared user-friendly in the sense that the underlying singular values can be specified via a ``Hermitized'' Schrödinger-like equation. In particular, the related ``Hermitized'' Green's functions is shown given the two alternative compact and numerically efficient matrix continued fraction forms.

Non-Hermitian Bose-Hubbard-like quantum models

TL;DR

This work studies non-Hermitian Bose-Hubbard-like quantum models built from analytically continued, non-self-adjoint Hamiltonians. It develops a practical toolkit based on Hermitization to compute singular values and on matrix continued fractions to represent Green's functions, enabling efficient analysis of resonances and exceptional points in large or infinite tridiagonal systems. The paper provides both conceptual foundations and concrete demonstrations, including elementary and few-state models, and extends the framework to standard and doubly-infinite matrix models. The resulting approach offers user-friendly, numerically efficient methods for probing spectral properties of non-Hermitian, Bose-Hubbard-like quantum systems with potential applicability to open- and closed-system interpretations.

Abstract

Among all of the non-Hermitian large-tridiagonal-matrix quantum Hamiltonians we choose a subclass with the structure resembling the ``benchmark'' realistic Bose-Hubbard model. We demonstrate that this choice can be declared user-friendly in the sense that the underlying singular values can be specified via a ``Hermitized'' Schrödinger-like equation. In particular, the related ``Hermitized'' Green's functions is shown given the two alternative compact and numerically efficient matrix continued fraction forms.

Paper Structure

This paper contains 17 sections, 1 theorem, 40 equations.

Table of Contents

  1. Bose-Hubbard-like Hamiltonians. .
  2. Unconventional non-Hermitian model. .
  3. Example. .
  4. Physical appeal of the models. .
  5. Conventional non-Hermitian Bose-Hubbard system. .
  6. Generalized non-Hermitian Bose-Hubbard-like models. .
  7. Green's functions. .
  8. Tridiagonal Hamiltonians and analytic continued fractions. .
  9. Proofs of convergence using the fixed-point-attraction technique. .
  10. Singular values. .
  11. The trick of Hermitization. .
  12. The most elementary example. .
  13. Further instructive few-state models. .
  14. Matrix continued fractions. .
  15. Standard (though, possibly, infinite-dimensional) matrix models. .
  16. ...and 2 more sections

Key Result

Lemma 1

. The Green's function associated with the Hermitian quasi-Hamiltonian (bimatrix) can be defined by formula ${G}(z)=\det F_0(z)$, with

Theorems & Definitions (1)

  • Lemma 1