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arXiv:0709.1569 · Journal of Physics A Mathematical and Theoretical

Tridiagonal {\cal PT} -symmetricN-by-NHamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime

Miloslav Znojil

Published Oct 9, 2007• 72 citations
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Abstract

A generic -symmetric Hamiltonian is assumed tridiagonalized and truncated to N < ∞ dimensions, H → H (chain model) , and all its up–down symmetrized special cases with J = [ N /2] real couplings are considered, H (chain model) → H ( N ) . Using symbolic manipulation and extrapolation techniques we find out that in the strongly non-Hermitian regime the secular equation gets partially factorized at all N . This enables us to reveal a fine-tuned alignment of the dominant couplings implying an asymptotically sharply spiked shape of the boundary of the J -dimensional quasi-Hermiticity domain in which all the spectrum of energies E ( N ) n remains real and observable.

DOI
10.1088/1751-8113/40/43/020

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A generic -symmetric Hamiltonian is assumed tridiagonalized and truncated to N < ∞ dimensions, H → H (chain model) , and all its up–down symmetrized special cases with J = [ N /2] real couplings are considered, H (chain model) → H ( N ) . Using symbolic manipulation and extrapolation techniques we find out that in the strongly non-Hermitian regime the secular equation gets partially factorized at all N . This enables us to reveal a fine-tuned alignment of the dominant couplings implying an asymptotically sharply spiked shape of the boundary of the J -dimensional quasi-Hermiticity domain in which all the spectrum of energies E ( N ) n remains real and observable.
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.
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudosupersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler–DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
The authors construct here an exactly solvable \ensuremath{\mathcal{P}}\ensuremath{\mathcal{T}}-symmetric non-Hermitian Heisenberg spin chain with complex-conjugate boundary Kondo couplings. They show that the model exhibits three distinct boundary phases: dissipative Kondo screening with unbroken \ensuremath{\mathcal{P}}\ensuremath{\mathcal{T}} symmetry, bound-mode phases featuring spontaneously broken \ensuremath{\mathcal{P}}\ensuremath{\mathcal{T}} symmetry and complex-conjugate eigenvalues, and an unscreened local-moment phase with a purely real spectrum. Benchmarking with non-Hermitian DMRG demonstrates quantitative agreement with Bethe ansatz, validating tensor-network methods for open quantum systems.