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arXiv:math-ph/0703070 · Journal of Physics A Mathematical and Theoretical

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

Miloslav Znojil

Published Apr 17, 2007• 55 citations
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Abstract

The domain of all the coupling strengths compatible with the reality of the energies is studied for a family of non-Hermitian N × N matrix Hamiltonians H ( N ) with tridiagonal and -symmetric structure. At all dimensions N , the coordinates are found of the extremal points at which the boundary hypersurface touches the circumscribed sphere (for odd N = 2 M + 1) or ellipsoid (for even N = 2 K ).

DOI
10.1088/1751-8113/40/18/012

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The domain of all the coupling strengths compatible with the reality of the energies is studied for a family of non-Hermitian N × N matrix Hamiltonians H ( N ) with tridiagonal and -symmetric structure. At all dimensions N , the coordinates are found of the extremal points at which the boundary hypersurface touches the circumscribed sphere (for odd N = 2 M + 1) or ellipsoid (for even N = 2 K ).
The coupling of non-Hermitian -symmetric Hamiltonians to standard Hermitian Hamiltonians, each of which individually has a real energy spectrum, is explored by means of a number of soluble models. It is found that in all cases the energy remains real for small values of the coupling constant, but becomes complex if the coupling becomes stronger than some critical value. For a quadratic non-Hermitian -symmetric Hamiltonian coupled to an arbitrary real Hermitian -symmetric Hamiltonian, the reality of the ground-state energy for small enough coupling constant is established up to second order in perturbation theory.
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.
It is know that PT-symmetric models have real spectra provided the symmetry is not spontaneously broken. Even pseudo-hermitian models have real spectra, which enlarge the the class of non-hermitian models possessing real spectra. We however consider non-hermitian models in higher dimensions which are not necessarily explicit PT-symmetric nor pseudo-hermitian. We show that the models may generate real spectra depending upon the coupling constants of the Hamiltonian. Our models thus further generalize the class of non-hermitian systems, which possess real spectra.
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.
We introduce a Hamiltonian H α, β depending on the two complex parameters α and β and develop a through analysis of the spectral properties of this Hamiltonian and its adjoint. We determine the regions in the space of the complex parameters α and β, where H α, β is (i) Hermitian, (ii) non-Hermitian with real spectrum and (iii) non-Hermitian but PT -symmetric. In the case when H α, β is non-Hermitian having real spectrum, we derive a closed formula for a family of the metric operators, depending on two arbitrary real positive parameters, which render the Hamiltonian H α, β Hermitian. In a particular case we calculate the Hermitian counterpart of H α, β .