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arXiv:2512.07250 · Journal of Physics Conference Series

Non-Hermitian Bose-Hubbard-like quantum models

Miloslav Znojil

quant-phmath.NASubmitted Dec 8, 2025• Announced Dec 9, 2025• 0 citations
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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.

Journal reference
J. Phys.: Conf. Ser. 3152 (2025) 012023
DOI
10.1088/1742-6596/3152/1/012023
Comments
talk presented to the XXIX Int. Conf. on Integrable Systems and Quantum Symmetries (Prague, CTU, July 7 - 11, 2025)

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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.
In the conventional Schrödinger's formulation of quantum mechanics the unitary evolution of a state ψψψ is controlled, in Hilbert space L{\cal L}L, by a Hamiltonian h\mathfrak{h}h which must be self-adjoint. In the recent, ``quasi-Hermitian'' reformulation of the theory one replaces h\mathfrak{h}h by its isospectral but non-Hermitian avatar H=Ω−1hΩH = Ω^{-1}\mathfrak{h}ΩH=Ω−1hΩ with Ω†Ω=Θ≠IΩ^\daggerΩ= Θ\neq IΩ†Ω=Θ=I. Although acting in another, manifestly unphysical Hilbert space H{\cal H}H, the amended Hamiltonian H≠H†H \neq H^\daggerH=H† can be perceived as self-adjoint with respect to the amended inner-product metric ΘΘΘ. In our paper motivated by a generic technical ``user-unfriendliness'' of the non-Hermiticity of HHH we introduce and describe a specific new family of Hamiltonians HHH for which the metrics ΘΘΘ become available in closed form.
Abstract In the conventional Schrödinger’s formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space <?CDATA L{\mathcal{L}}L?> L , by a Hamiltonian ɧ which must be self-adjoint. In the recent, “quasi-Hermitian” reformulation of the theory one replaces ɧ by its isospectral but non-Hermitian avatar H = Ω−1 𝖍Ω with Ω†Ω = Θ ≠ I. Although acting in another, manifestly unphysical Hilbert space <?CDATA H{\mathcal{H}}H?> H , the amended Hamiltonian H ≠ H † can be perceived as self-adjoint with respect to the amended inner-product metric Θ. In our paper motivated by a generic technical “user-unfriendliness” of the non-Hermiticity of H we introduce and describe a specific new family of Hamiltonians H for which the metrics Θ become available in closed form.
For the two-mode and $(N\ensuremath{-}1)\ensuremath{-}\mathrm{bosonic}$ Bose-Hubbard quantum system a less usual phase transition controlled by the parameter $\ensuremath{\varepsilon}$ representing the on-site energy difference is studied. In the literature the parameter is considered either real ($\ensuremath{\varepsilon}>0$) or purely imaginary (with, say, $\ensuremath{\gamma}=\mathrm{Im}\phantom{\rule{4pt}{0ex}}\ensuremath{\varepsilon}>0$), so the phase transition is analyzed here at the interface $\ensuremath{\varepsilon}=\ensuremath{\gamma}=0$. The evolution in the $\ensuremath{\gamma}\ensuremath{-}\mathrm{controlled}$ phase is required unitary so that the main task for the theory is found in the (quasi-)Hermitization of the Hamiltonian, achieved by a suitable amendment of the inner product in Hilbert space, $\ensuremath{\langle}\ifmmode\cdot\else\textperiodcentered\fi{}|\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\rangle}\ensuremath{\rightarrow}\ensuremath{\langle}\ifmmode\cdot\else\textperiodcentered\fi{}|\mathrm{\ensuremath{\Theta}}|\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\rangle}$. In the most relevant domain of small $\ensuremath{\gamma}$ the linearized Hilbert-space metric $\mathrm{\ensuremath{\Theta}}(\ensuremath{\gamma})$ [constrained by the requirement ${lim}_{\ensuremath{\gamma}\ensuremath{\rightarrow}0}\mathrm{\ensuremath{\Theta}}(\ensuremath{\gamma})=I$ of the smoothness of the change of the Hilbert space at the phase transition] is constructed in closed form. Beyond the phase-transition instant, several forms of the systematic non-numerical recurrent construction of the exact metrics $\mathrm{\ensuremath{\Theta}}(\ensuremath{\gamma})$ are also shown user friendly and feasible, at the not too large matric dimensions NNN at least.
An NNN-level quantum model is proposed in which the energies are represented by an NNN-plet of zeros of a suitable classical orthogonal polynomial. The family of Gegenbauer polynomials G(n,a,x)G(n,a,x)G(n,a,x) is selected for illustrative purposes. The main obstacle lies in the non-Hermiticity (aka crypto-Hermiticity) of Hamiltonians $H\ensuremath{\ne}{H}^{\ifmmode\dagger\else\textdagger\fi{}}$. We managed to (i) start from elementary secular equation G(N,a,En)=0G(N,a,{E}_{n})=0G(N,a,En​)=0, (ii) keep our HHH, in the nearest-neighbor-interaction spirit, tridiagonal, (iii) render it Hermitian in an ad hoc, nonunique Hilbert space endowed with metric $\ensuremath{\Theta}\ensuremath{\ne}I$, (iv) construct eligible metrics in closed forms ordered by increasing nondiagonality, and (v) interpret the model as a smeared NNN-site lattice.