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arXiv:1810.05771 · Physical Review A

Broken-Hermiticity phase transition in the Bose-Hubbard model

Miloslav Znojil

Published Nov 5, 2018• 6 citations
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Abstract

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 $N$ at least.

DOI
10.1103/physreva.98.052102

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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.
The description of unitary evolution using non-Hermitian but ‘hermitizable’ Hamiltonians H is feasible via an ad hoc metric Θ = Θ ( H ) and a (non-unique) amendment 〈 ψ 1 | ψ 2 〉 → 〈 ψ 1 | Θ | ψ 2 〉 of the inner product in Hilbert space. Via a proper fine-tuning of Θ ( H ) this opens the possibility of reaching the boundaries of stability (i.e. exceptional points) in many quantum systems sampled here by the fairly realistic Bose–Hubbard (BH) and discrete anharmonic oscillator (AO) models. In such a setting, it is conjectured that the EP singularity can play the role of a quantum phase-transition interface between different dynamical regimes. Three alternative ‘AO ↔ BH’ implementations of such an EP-mediated dynamical transmutation scenario are proposed and shown, at an arbitrary finite Hilbert-space dimension N , exact and non-numerical.