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arXiv:1101.0351 · Journal of Physics A Mathematical and Theoretical

A physical interpretation for the non-Hermitian Hamiltonian

L. Jin, Z. Song

Published Aug 23, 2011• 30 citations
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Abstract

We explore a way of finding the link between a non-Hermitian Hamiltonian and a Hermitian one. Based on the analysis of Bethe ansatz solutions for a class of non-Hermitian Hamiltonians and the scattering problems for the corresponding Hermitian Hamiltonians, it is shown that a scattering state of an arbitrary Hermitian lattice embedded in a chain as the scattering center shares the same wavefunction with the corresponding non-Hermitian tight binding lattice, which consists of the Hermitian lattice with two additional on-site complex potentials, no matter whether the non-Hermitian is broken symmetry or even non- symmetry. Exactly solvable models are presented to demonstrate the main points of this paper.

DOI
10.1088/1751-8113/44/37/375304

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We explore an alternative way of finding the link between a PT\mathcal{P}\mathcal{T}PT non-Hermitian Hamiltonian and a Hermitian one. Based on the analysis of the scattering problem for an imaginary potential and its time-reversal process, it is shown that any real-energy eigenstate of a PT\mathcal{P}\mathcal{T}PT tight-binding lattice with on-site imaginary potentials shares the same wave function with a resonant transmission state of the corresponding Hermitian lattice embedded in a chain. It indicates that the PT\mathcal{P}\mathcal{T}PT eigenstate of a PT\mathcal{P}\mathcal{T}PT non-Hermitian Hamiltonian has a connection to the resonance transmission state of the extended Hermitian Hamiltonian.
We study the non-Hermitian quantum mechanics for the discrete system. This paper gives an exact analytic single-particle solution for an NNN-site tight-binding chain with two conjugated imaginary potentials $\ifmmode\pm\else\textpm\fi{}i\ensuremath{\gamma}$ at two end sites, which Hamiltonian has parity-time symmetry (PT\mathcal{P}\mathcal{T}PT symmetry). Based on the Bethe ansatz results, it is found that, in single-particle subspace, this model is comprised of two phases: an unbroken symmetry phase with a purely real energy spectrum in the region $\ensuremath{\gamma}\ensuremath{\prec}{\ensuremath{\gamma}}_{c}$ and a spontaneously broken symmetry phase with $N\ensuremath{-}2$ real and two imaginary eigenvalues in the region $\ensuremath{\gamma}\ensuremath{\succ}{\ensuremath{\gamma}}_{c}$. The behaviors of eigenfunctions and eigenvalues in the vicinity of ${\ensuremath{\gamma}}_{c}$ are investigated. It is shown that the boundary of two phases possesses the characteristics of exceptional point. We also construct the equivalent Hermitian Hamiltonian of the present model in the framework of metric-operator theory. We find out that the equivalent Hermitian Hamiltonian can be written as another bipartite lattice model with real long-range hoppings.
We study the scattering problem for the non-Hermitian scattering center, which consists of two Hermitian clusters with anti-Hermitian couplings between them. Counterintuitively, it is shown that it acts as a Hermitian scattering center, that is, the SSS matrix is unitary, or the Dirac probability current is conserved, ∣r∣2+∣t∣2=1{|r|}^{2}+{|t|}^{2}=1∣r∣2+∣t∣2=1, when one of the two clusters is embedded in the waveguides. This conclusion can be applied to an arbitrary parity-symmetric real Hermitian graph with additional PT\mathcal{PT}PT-symmetric potentials, which is more feasible in the experiment. An exactly solvable model is presented to illustrate the theory. Bethe ansatz solution indicates that the transmission spectrum of such a cluster displays a peculiar feature arising from the non-Hermiticity of the scattering center.
The scattering states in one-dimensional Hermitian and non-Hermitian potentials are investigated. An analytical solution for the scattering states is presented in terms of Heun functions. It is shown that for some specially chosen parameter conditions, an infinite number of the exact scattering states is obtained. In the Hermitian potentials, they correspond to the reflectionless states. In the non-Hermitian complex potentials with parity-time symmetry, they are the unidirectionally reflectionless states.
We analyze the scattering dynamics and spectrum of a quantum particle on a tight-binding lattice subject to a non-Hermitian (purely imaginary) local potential. The reflection, transmission, and absorption coefficients are studied as a function of the strength of this absorbing potential. The system is found to have an exceptional point at a certain strength of the potential. Unusually, all (or nearly all) of the spectrum pairs up into mutually coalescing eigenstate pairs at this exceptional point. At large potential strengths, the absorption coefficient decreases and the effect of the imaginary potential is similar to that of a real potential. We quantify this similarity by utilizing properties of a localized eigenstate.
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.