Acta Polytechnica

PT-symmetric Quantum Chain Models

Miloslav Znojil

Published Jan 2, 2007• 5 citations

Abstract

A review is given of certain tridiagonal N-dimensional non-Hermitian J-parametric real-matrix quantum Hamiltonians H(N). The domains Ɗ(N) of reality of their spectra of energies are studied, with particular attention paid to their exceptional-point boundaries ∂Ɗ(N). The strongest admissible couplings are specified in closed form for all N.

DOI: 10.14311/918

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Acta Polytechnica

PT-symmetric Quantum Chain Models

Miloslav Znojil

Published Jan 2, 2007• 5 citations

Publisher

Abstract

DOI: 10.14311/918

Related papers

Maximal couplings in -symmetric chain models with the real spectrum of energies

Miloslav Znojil

arXiv:math-ph/0703070Journal of Physics A Mathematical and TheoreticalApr 17, 200755 citations

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

Miloslav Znojil

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.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.

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Stable bound states living on a discrete one-dimensional N-point lattice are assumed generated by a certain two-parametric PT symmetric potential $V=V(A,B)$ . At any $N$ , the coordinates $A=A^{(EP)}$ and $B=B^{(EP)}$ of the Kato's exceptional point (marking the phase-transition instant of the loss of stability) are defined, exactly, via roots of a certain elementary quartic polynomial.Stable bound states living on a discrete one-dimensional N-point lattice are assumed generated by a certain two-parametric PT symmetric potential $V=V(A,B)$. At any $N$, the coordinates $A=A^{(EP)}$ and $B=B^{(EP)}$ of the Kato's exceptional point (marking the phase-transition instant of the loss of stability) are defined, exactly, via roots of a certain elementary quartic polynomial.

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Stable bound states living on a discrete one-dimensional N-point lattice are assumed generated by a certain two-parametric PT symmetric potential

V=V(A,B)

. At any

N

, the coordinates

A=A^{(EP)}

and

B=B^{(EP)}

of the Kato's exceptional point (marking the phase-transition instant of the loss of stability) are defined, exactly, via roots of a certain elementary quartic polynomial.