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arXiv:0904.1513 · Physical Review A

Solutions of PT -symmetric tight-binding chain and its equivalent Hermitian counterpart

L. Jin, Z. Song

Published Nov 11, 2009• 156 citations
arXivHTMLPDF

Abstract

We study the non-Hermitian quantum mechanics for the discrete system. This paper gives an exact analytic single-particle solution for an $N$-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 ($\mathcal{P}\mathcal{T}$ 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.

DOI
10.1103/physreva.80.052107

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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±\else\textpm\fii\ensuremathγ\ifmmode\pm\else\textpm\fi{}i\ensuremath{\gamma}\ifmmode±\else\textpm\fii\ensuremathγ 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 \ensuremathγ\ensuremath≺\ensuremathγc\ensuremath{\gamma}\ensuremath{\prec}{\ensuremath{\gamma}}_{c}\ensuremathγ\ensuremath≺\ensuremathγc​ and a spontaneously broken symmetry phase with N\ensuremath−2N\ensuremath{-}2N\ensuremath−2 real and two imaginary eigenvalues in the region \ensuremathγ\ensuremath≻\ensuremathγc\ensuremath{\gamma}\ensuremath{\succ}{\ensuremath{\gamma}}_{c}\ensuremathγ\ensuremath≻\ensuremathγc​. The behaviors of eigenfunctions and eigenvalues in the vicinity of \ensuremathγc{\ensuremath{\gamma}}_{c}\ensuremathγc​ 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.
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.
More than a decade ago, it was shown that non-Hermitian Hamiltonians with combined parity (P) and time-reversal (T ) symmetry exhibit real eigenvalues over a range of parameters. Since then, the field of PT symmetry has seen rapid progress on both the theoretical and experimental fronts. These effective Hamiltonians are excellent candidates for describing open quantum systems with balanced gain and loss. Nature seems to be replete with examples of PT -symmetric systems; in fact, recent experimental investigations have observed the effects of PT symmetry breaking in systems as diverse as coupled mechanical pendula, coupled optical waveguides, and coupled electrical circuits. Recently, PT -symmetric Hamiltonians for tight-binding lattice models have been extensively investigated. Lattice models, in general, have been widely used in physics due to their analytical and numerical tractability. Perhaps one of the best systems for experimentally observing the effects of PT symmetry breaking in a one-dimensional lattice with tunable hopping is an array of evanescently-coupled optical waveguides. The tunneling between adjacent waveguides is tuned by adjusting the width of the barrier between them, and the imaginary part of the local refractive index provides the loss or gain in the respective waveguide. Calculating the time evolution of a wave packet on a lattice is relatively straightforward in the tight-binding model, allowing us to make predictions about the behavior of light propagating down an array of PT -symmetric waveguides. In this thesis, I investigate the the strength of the PT -symmetric phase (the region over which the eigenvalues are purely real) in lattices with a variety of PT - symmetric potentials. In Chapter 1, I begin with a brief review of the postulates of quantum mechanics, followed by an outline of the fundamental principles of PT - symmetric systems. Chapter 2 focuses on one-dimensional uniform lattices with a pair of PT -symmetric impurities in the case of open boundary conditions. I find that the PT phase is algebraically fragile except in the case of closest impurities, where the PT phase remains nonzero. In Chapter 3, I examine the case of periodic boundary conditions in uniform lattices, finding that the PT phase is not only nonzero, but also independent of the impurity spacing on the lattice. In addition, I explore the time evolution of a single-particle wave packet initially localized at a site. I find that in the case of periodic boundary conditions, the wave packet undergoes a preferential clockwise or counterclockwise motion around the ring. This behavior is quantified by a discrete momentum operator which assumes a maximum value at the PT -symmetry- breaking threshold. In Chapter 4, I investigate nonuniform lattices where the parity-symmetric hop- ping between neighboring sites can be tuned. I find that the PT phase remains strong in the case of closest impurities and fragile elsewhere. Chapter 5 explores the effects of the competition between localized and extended PT potentials on a lattice. I show that when the short-range impurities are maximally separated on the lattice, the PT phase is strengthened by adding short-range loss in the broad-loss region. Consequently, I predict that a broken PT symmetry can be restored by increasing the strength of the short-range impurities. Lastly, Chapter 6 summarizes my salient results and discusses areas which can be further developed in future research.
We examine the time development of a wavepacket in a Non-Hermitian PT symmetric tight-binding chain based on the Bethe ansatz solutions. The complete orthogonal sets in both unbroken and broken PT symmetry regions are established with respect to an inner product in terms of CPT conjugation. The dynamical development of a stationary wavepacket is exhibited, showing the impacts from additional imaginary potentials within unbroken and broken symmetric regimes. Our result indicates that there are distinct dynamic characteristics for both two regions. The unbroken symmetric imaginary end potentials can be regarded as a special kind of boundary condition in the sense of Hermitian quantum mechanics.
Abstract We construct the exact solution of a non-Hermitian P T -symmetric isotropic Heisenberg spin chain with integrable boundary fields. We find that the system exhibits two types of phases named A and B . In the B type phase, the P T symmetry remains unbroken and it comprises of eigenstates with only real energies, whereas the A type phase exhibits both P T symmetry broken and unbroken sectors, comprising of eigenstates with only complex and real energies respectively. The P T -symmetry broken sector comprises of pairs of eigenstates whose energies are complex conjugates of each other. The existence of two sectors in the A type phase is associated with the exponentially localized bound states at the edges with complex energies which are described by boundary strings. We find that both A and B type phases can be further divided into sub-phases which exhibit different ground states. We also compute the bound state wavefunction in one magnon sector and find that as the imaginary value of the boundary parameter is increased, the exponentially localized wavefunction broadens thereby protruding more into the bulk, which indicates that exponentially localized bound states may not be stabilized for large imaginary values of the boundary parameter.
We study the properties of a parity- and time-reversal- (PT\mathcal{P}\mathcal{T}PT) symmetric tight-binding chain of size NNN with position-dependent hopping amplitude. In contrast to the fragile PT\mathcal{P}\mathcal{T}PT-symmetric phase of a chain with constant hopping and imaginary impurity potentials, we show that, under very general conditions, our model is always in the PT\mathcal{P}\mathcal{T}PT-symmetric phase. We numerically obtain the energy spectrum and the density of states of such a chain, and show that they are widely tunable. By studying the size dependence of inverse participation ratios, we show that although the chain is not translationally invariant, most of its eigenstates are extended. Our results indicate that tight-binding models with non-Hermitian, PT\mathcal{P}\mathcal{T}PT-symmetric hopping have a robust PT\mathcal{P}\mathcal{T}PT-symmetric phase and rich dynamics which may be explored in coupled waveguides.
Two types of non-Hermiticity found in physical systems, i.e., non-reciprocal couplings and complex potentials, have gained considerable interest recently. The former leads to the non-Hermitian skin effect, while the latter gives rise to a range of non-Hermitian symmetries such as parity-time (PTPTPT) and particle-hole symmetries. Although these two non-Hermitian forms have been incorporated in the same models, so far they have been treated as distinct families of non-Hermiticity. In this work, we establish an isospectral mapping between them using the Lanczos transformation. More specifically, we show that a generalized Hatano-Nelson model HHH with non-reciprocal and spatially varying couplings can be mapped to a tight-binding model HLH_LHL​ with real symmetric couplings and an imaginary on-site potential. This result holds even when the original HHH already has an imaginary potential and exceptional points. Our work provides a refreshed understanding of the connection between different forms of non-Hermiticity. It also offers an approach to obtaining an intuitive physical understanding of exceptional points in HLH_LHL​ that are elusive in HHH. Finally, our findings also warrant a versatile approach to constructing non-Hermitian systems with a complex potential and a real spectrum, without PTPTPT symmetry.