Generalized Bose–Hubbard Hamiltonians exhibiting a complete non-Hermitian degeneracy

Znojil, Miloslav

doi:10.1016/j.aop.2019.03.022

Cited by 10 publications

(7 citation statements)

References 37 publications

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“…Moreover, the recommended transition from the "unacceptable" differential operators [sampled here by Eq. ( 1)] to non-Hermitian matrices encountered a number of technical obstacles [23] and revealed multiple manifestations of a user-unfriendlines of the models [24,25].…”

Section: Discussion and Summarymentioning

confidence: 99%

Exceptional points and domains of unitarity for a class of strongly non-Hermitian real-matrix Hamiltonians

Znojil

2021

Preprint

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A family of non-Hermitian real and tridiagonal-matrix candidates H (N ) for a hiddenly Hermitian (a.k.a. quasi-Hermitian) quantum Hamiltonian is proposed and studied. Fairly weak assumptions are imposed upon the unperturbed matrix (the square-well-simulating spectrum of H (N ) 0is not assumed equidistant) as well as upon its maximally non-Hermitian N−parametric antisymmetric-matrix perturbations (matrix W (N ) (λ) is not even required to be PT −symmetric). In spite of that, the "physical" parametric domain D [N ] is (constructively!) shown to exist, guaranteeing that in its interior the spectrum remains real and non-degenerate, rendering the quantum evolution unitary. Among the non-Hermitian degeneracies occurring at the boundary ∂D [N ] of the domain of stability our main attention is paid to their extreme version corresponding to the Kato's exceptional point of order N (EPN). The localization of the EPNs and, in their vicinity, of the quantum-phase-transition boundaries ∂D [N ] is found feasible, at the not too large N, using computer-assisted symbolic manipulations including, in particular, the Gröbner basis elimination and the high-precision arithmetics.

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Section: Discussion and Summarymentioning

confidence: 99%

Exceptional points and domains of unitarity for a class of strongly non-Hermitian real-matrix Hamiltonians

Znojil

2021

Preprint

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“…Marginally, let us add that in such a branched-evolution setting one could find applications even for some results on non-unitary, spectral-realityviolating evolutions. An illustration may be found in papers (sampled by [53]) where just the search for the EP degeneracies has been performed without any efforts of guaranteeing the reality of the spectrum.…”

Section: Further Phenomenological Challengesmentioning

confidence: 99%

Paths of unitary access to exceptional points

Znojil

2021

Preprint

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With an innovative ideas of acceptability and usefulness of the non-Hermitian representations of Hamiltonians for the description of unitary quantum systems (dating back to the Dyson's papers), the community of quantum physicists was offered a new and powerful tool for the building of models of quantum phase transitions. In this paper the mechanism of such transitions is discussed from the point of view of mathematics. The emergence of the direct access to the instant of transition (i.e., to the Kato's exceptional point) is attributed to the underlying split of several roles played by the traditional single Hilbert space of states L into a triplet (viz., in our notation, spaces K and H besides the conventional L). Although this explains the abrupt, quantum-catastrophic nature of the change of phase (i.e., the loss of observability) caused by an infinitesimal change of parameters, the explicit description of the unitarity-preserving corridors of access to the phenomenologically relevant exceptional points remained unclear. In the paper some of the recent results in this direction are summarized and critically reviewed.

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“…At the higher matrix dimensions N one may encounter serious technical difficulties even for the BH-related complex-symmetric matrices. The reasons were explained in [48]. In essence, these difficulties may only partly be attributed to the above-mentioned ill-conditioned nature of equation (2.6).…”

Section: Non-numerical Construction (Bh Case)mentioning

confidence: 99%

Passage through exceptional point: case study

Znojil

2020

Proc. R. Soc. A.

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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.

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Generalized Bose–Hubbard Hamiltonians exhibiting a complete non-Hermitian degeneracy

Cited by 10 publications

References 37 publications

Exceptional points and domains of unitarity for a class of strongly non-Hermitian real-matrix Hamiltonians

Exceptional points and domains of unitarity for a class of strongly non-Hermitian real-matrix Hamiltonians

Paths of unitary access to exceptional points

Passage through exceptional point: case study

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