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arXiv:2104.11016 · Journal of Mathematical Physics

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

Miloslav Znojil

Published May 1, 2021• 10 citations

Abstract

A family of non-Hermitian real and tridiagonal-matrix candidates H(N)(λ)=H0(N)+λW(N)(λ) 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 H0(N) is not assumed equidistant)] and upon its maximally non-Hermitian N-parametric antisymmetric-matrix perturbations [matrix W(N)(λ) is not even required to be PT-symmetric]. Despite 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 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.

DOI: 10.1063/5.0041185

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A family of non-Hermitian real and tridiagonal-matrix candidates H(N)(λ)=H0(N)+λW(N)(λ) 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 H0(N) is not assumed equidistant)] and upon its maximally non-Hermitian N-parametric antisymmetric-matrix perturbations [matrix W(N)(λ) is not even required to be PT-symmetric]. Despite 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 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.

Quantum phase transition is interpreted as an evolution, at the end of which a parameter-dependent Hamiltonian

H(g)

loses its observability. In the language of mathematics, such a quantum catastrophe occurs at an exceptional point of order

N

(EPN). Although the Hamiltonian

H(g)

itself becomes unphysical in the limit of

g \to g^{EPN}

, it is shown that it can play the role of an unperturbed operator in a perturbation-approximation analysis of the vicinity of the EPN singularity. Such an analysis is elementary at

N\leq 3

and numerical at

N\geq 5

, so we pick up

N=4

. We demonstrate that the specific EP4 degeneracy becomes accessible via a unitary evolution process realizable inside a parametric domain

{\cal D}_{\rm physical}

, the boundaries of which are determined non-numerically. Possible relevance of such a mathematical result in the context of non-Hermitian photonics is emphasized.

The phenomenon of degeneracy of an N-plet of bound states is studied in the framework of the quasi-Hermitian (a.k.a. PT-symmetric) formulation of quantum theory of closed systems. For a general non-Hermitian Hamiltonian H=H(λ) such a degeneracy may occur at a real Kato's exceptional point λ^{(EPN)} of order N and of the geometric multiplicity alias clusterization index K. The corresponding unitary process of collapse (loss of observability) can be then interpreted as a generic quantum phase transition. The dedicated literature deals, predominantly, with the non-numerical benchmark models of the simplest processes where K=1. In our present paper it is shown that in the "anomalous" dynamical scenarios with 1<K≤N/2 an analogous approach is applicable. A multiparametric anharmonic-oscillator-type exemplification of such systems is constructed as a set of real-matrix N by N Hamiltonians which are exactly solvable, maximally non-Hermitian, and labeled by specific ad hoc partitionings R(N) of N.

The conventional non-Hermitian but P T -symmetric three-parametric Bose–Hubbard Hamiltonian H ( γ , v , c ) represents a quantum system of N bosons, unitary only for parameters γ , v and c in a domain D . Its boundary ∂ D contains an exceptional point of order K (EPK; K = N + 1) at c = 0 and γ = v , but even at the smallest non-vanishing parameter c ≠ ~0 the spectrum of H ( v , v , c ) ceases to be real, i.e. the system ceases to be observable. In this paper, the question is inverted: all of the stable, unitary and observable Bose–Hubbard quantum systems are sought which would lie close to the phenomenologically most interesting EPK-related dynamical regime. Two different families of such systems are found. Both of them are characterized by the perturbed Hamiltonians H ( λ ) = H ( v , v , 0 ) + λ V for which the unitarity and stability of the system is guaranteed. In the first family the number N of bosons is assumed conserved while in the second family such an assumption is relaxed. Attention is paid mainly to an anisotropy of the physical Hilbert space near the EPK extreme. We show that it is reflected by a specific, operationally realizable structure of perturbations λ V which can be considered small.

The phenomenon of degeneracy of an

N-

plet of bound states is studied in the framework of quantum theory of closed (i.e., unitary) systems. For an underlying Hamiltonian

H=H(λ)

the degeneracy occurs at a Kato's exceptional point

λ^{(EPN)}

of order

N

and of the spectral geometric multiplicity

K2

and

K>1

benchmark models of the process of degeneracy becomes feasible and non-numerical for a broad class of specific, maximally non-Hermitian anharmonic-oscillator toy-model Hamiltonians. An exhaustive classification of non-equivalent processes is given by a partitioning of the unperturbed spectrum into equidistant and centered unperturbed subspectra.

Via several toy-model quantum Hamiltonians H(λ) of a non-tridiagonal low-dimensional matrix form the existence of unusual observability horizons is revealed. At the corresponding limiting values of parameter λ=λ(critical) these new types of quantum phase transitions are interpreted as the points of confluence of several decoupled Kato's exceptional points of equal or different orders. Such a phenomenon of degeneracy of non-Hermitian degeneracies seems to ask for a reclassification of the possible topologies of the complex energy Riemann surfaces in the vicinity of branch points.