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Exceptional points, non-normal matrices, hierarchy of spin matrices and an eigenvalue problem

Willi-Hans Steeb, Yorick Hardy

Published Jan 1, 2014• 5 citations
OpenAlex

Abstract

Exceptional points of a class of non-hermitian Hamilton operators Ĥ of the form Ĥ = Ĥ0 + iĤ1 are studied, where Ĥ0 and Ĥ1 are hermitian operators. Finite dimensional Hilbert spaces are considered. The linear operators Ĥ0 and Ĥ1 are given by spin matrices for spin s = 1/2, 1, 3/2, …. Since the linear operators studied are non-normal, properties of such operators are described.

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Exceptional points of a class of non-hermitian Hamilton operators Ĥ of the form Ĥ = Ĥ0+ iĤ1are studied, where Ĥ0and Ĥ1are hermitian operators. Finite dimensional Hilbert spaces are considered. The linear operators Ĥ0and Ĥ1are given by spin matrices for spin s = 1/2, 1, 3/2, …. Since the linear operators studied are non-normal, properties of such operators are described.

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W.J. Caspers

Degeneracy of the eigenvalues of hermitian matrices is analyzed in terms of algebraic relations between the matrix elements. Only finite-dimensional matrices are considered, representing Hamiltonians of spin systems in particular.The analysis is in terms of linear relations in the space of 'diagonal' operators, i.e. a complete set of powers of the Hamiltonian. A relation is found with the theorem formulated by Von Neumann and Wigner that only the variation of three parameters of the Hamiltonian may result in a 'crossing' or degeneracy of two eigenvalues.

Journal of Physics Conference Series•Mar 1, 2008•4 citations
Exceptional points of a class of non-hermitian Hamilton operators Ĥ of the form Ĥ = Ĥ0 + iĤ1 are studied, where Ĥ0 and Ĥ1 are hermitian operators. Finite dimensional Hilbert spaces are considered. The linear operators Ĥ0 and Ĥ1 are given by spin matrices for spin s = 1/2, 1, 3/2, …. Since the linear operators studied are non-normal, properties of such operators are described.
Pseudo-Hermitian operators generalize the concept of Hermiticity. This class of operators includes the quasi-Hermitian operators, which reformulate quantum theory while retaining real-valued measurement outcomes and unitary time evolution. This thesis is devoted to the study of locality in quasi-Hermitian theory, the symmetries and conserved quantities associated with non-Hermitian operators, and the perturbative features of pseudo-Hermitian matrices. In addition to the presented original research, scholars will appreciate the lengthy introduction to non-Hermitian physics. Local quasi-Hermitian observable algebras are examined. Expectation values of local quasi-Hermitian observables equal expectation values of local Hermitian observables. Thus, quasi-Hermitian theories do not increase the values of nonlocal games set by Hermitian theories. Furthermore, Bell's inequality violations in quasi-Hermitian theories never exceed the Tsirelson bound of Hermitian quantum theory. Exceptional points, which are branch points in the spectrum, are a perturbative feature unique to non-Hermitian operators. Cusp singularities of algebraic curves are related to higher-order exceptional points. To exemplify novelties of non-Hermiticity, one-dimensional lattice models with a pair of non-Hermitian defect potentials with balanced loss and gain, Δ±iγΔ\pm i γΔ±iγ, are explored. When the defects are nearest neighbour, the entire spectrum becomes complex when γγγ is tuned past a second-order exceptional point. When the defects are at the edges of the chain and the hopping amplitudes are 2-periodic, as in the Su-Schrieffer-Heeger chain, the PT\mathcal{PT}PT-phase transition is dictated by the topological phase. Chiral symmetry and representation theory are used to derive large classes of pseudo-Hermitian operators with closed-form intertwining operators.
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.
The aim of the paper is to study the question whether or not equilibrium states exist in open quantum systems that are embedded in at least two environments and are described by a non-Hermitian Hamilton operator H\cal HH. The eigenfunctions of H\cal HH contain the influence of exceptional points (EPs) as well as that of external mixing (EM) of the states via the environment. As a result, equilibrium states exist (far from EPs). They are different from those of the corresponding closed system. Their wavefunctions are orthogonal although the Hamiltonian is non-Hermitian.
Degeneracy of the eigenvalues of hermitian matrices is analyzed in terms of algebraic relations between the matrix elements. Only finite-dimensional matrices are considered, representing Hamiltonians of spin systems in particular.The analysis is in terms of linear relations in the space of 'diagonal' operators, i.e. a complete set of powers of the Hamiltonian. A relation is found with the theorem formulated by Von Neumann and Wigner that only the variation of three parameters of the Hamiltonian may result in a 'crossing' or degeneracy of two eigenvalues.