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arXiv:2302.13204

Effects of detuning on $\mathcal{PT}$-symmetric, tridiagonal, tight-binding models

Jacob L. Barnett, Yogesh N. Joglekar

Published Feb 26, 2023• 0 citations
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Abstract

Non-Hermitian, tight-binding $\mathcal{PT}$-symmetric models are extensively studied in the literature. Here, we investigate two forms of non-Hermitian Hamiltonians to study the $\mathcal{PT}$-symmetry breaking thresholds and features of corresponding surfaces of exceptional points (EPs). They include one-dimensional chains with uniform or 2-periodic tunnelling amplitudes, one pair of balanced gain and loss potentials $Δ\pmıγ$ at parity-symmetric sites, and periodic or open boundary conditions. By introducing a Hermitian detuning potential, we obtain the dependence of the $\mathcal{PT}$-threshold, and therefore the exceptional-point curves, in the parameter space of detuning and gain-loss strength. By considering several such examples, we show that EP curves of a given order generically have cusp-points where the order of the EP increases by one. In several cases, we obtain explicit analytical expressions for positive-definite intertwining operators that can be used to construct a complex extension of quantum theory by re-defining the inner product. Taken together, our results provide a detailed understanding of detuned tight-binding models with a pair of gain-loss potentials.

DOI
10.48550/arxiv.2302.13204

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Effects of detuning on PT\mathcal{PT}PT-symmetric, tridiagonal, tight-binding models
Non-Hermitian, tight-binding PT\mathcal{PT}PT-symmetric models are extensively studied in the literature. Here, we investigate two forms of non-Hermitian Hamiltonians to study the PT\mathcal{PT}PT-symmetry breaking thresholds and features of corresponding surfaces of exceptional points (EPs). They include one-dimensional chains with uniform or 2-periodic tunnelling amplitudes, one pair of balanced gain and loss potentials Δ±ıγΔ\pmıγΔ±ıγ at parity-symmetric sites, and periodic or open boundary conditions. By introducing a Hermitian detuning potential, we obtain the dependence of the PT\mathcal{PT}PT-threshold, and therefore the exceptional-point curves, in the parameter space of detuning and gain-loss strength. By considering several such examples, we show that EP curves of a given order generically have cusp-points where the order of the EP increases by one. In several cases, we obtain explicit analytical expressions for positive-definite intertwining operators that can be used to construct a complex extension of quantum theory by re-defining the inner product. Taken together, our results provide a detailed understanding of detuned tight-binding models with a pair of gain-loss potentials.
Parity-time (PT)(\mathcal{PT})(PT) symmetric systems are classical, gain-loss systems whose dynamics are governed by non-Hermitian Hamiltonians with exceptional-point (EP) degeneracies. The eigenvalues of a PT\mathcal{PT}PT-symmetric Hamiltonian change from real to complex conjugates at a critical value of gain-loss strength that is called the PT\mathcal{PT}PT breaking threshold. Here, we obtain the PT\mathcal{PT}PT threshold for a one-dimensional, finite Kitaev chain---a prototype for a ppp-wave superconductor---in the presence of a single pair of gain and loss potentials as a function of the superconducting order parameter, on-site potential, and the distance between the gain and loss sites. In addition to a robust, nonlocal threshold, we find a rich phase diagram for the threshold that can be qualitatively understood in terms of the band structure of the Hermitian Kitaev model. In particular, for an even chain with zero on-site potential, we find a re-entrant PT\mathcal{PT}PT-symmetric phase bounded by second-order EP contours. Our numerical results are supplemented by analytical calculations for small system sizes.
We study the PT\mathcal{PT}PT-symmetry breaking for the scattering problem in a one-dimensional non-Hermitian tight-binding lattice model with balanced gain and loss distributed on two adjacent sites. In the scattering process the system undergoes a transition from the exact PT\mathcal{PT}PT-symmetric phase to the phase with spontaneously breaking PT\mathcal{PT}PT symmetry as the amplitude of complex potentials increases. Using the S-matrix method, we derive an exact discriminant, which can be used to distinguish different symmetry phases, and determine the exceptional point for the symmetry breaking analytically. In the PT\mathcal{PT}PT-symmetry-breaking region, we also confirm the appearance of the unique feature, i.e., the coherent perfect absorption laser, in this simple non-Hermitian lattice model. The study of the scattering problem of such a simple model provides an additional way to unveil the physical effect of non-Hermitian PT\mathcal{PT}PT-symmetric potentials.
Recently, open systems with balanced, spatially separated loss and gain have been realized and studied using non-Hermitian Hamiltonians that are invariant under the combined parity and time-reversal (PT\mathcal{PT}PT) operations. Here, we model and investigate the effects of a local, two-state, quantum degree of freedom, called a pseudospin, on a one-dimensional tight-binding lattice with position-dependent tunneling amplitudes and a single pair of non-Hermitian, PT\mathcal{PT}PT-symmetric impurities. We show that if the resulting Hamiltonian is invariant under exchange of two pseudospin labels, the system can be decomposed into two uncoupled systems with tunable threshold for PT\mathcal{PT}PT symmetry breaking. We discuss implications of our results to systems with specific tunneling profiles, and open or periodic boundary conditions.
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.
PT\mathcal {PT}PT-symmetry breaking and maximal chirality in a nonuniform PT\mathcal {PT}PT symmetric ring
We study the properties of an N -site tight-binding ring with parity and time-reversal ( ) symmetric, Hermitian, site-dependent tunneling and a pair of non-Hermitian, -symmetric, loss and gain impurities ±iγ. The properties of such lattices with open boundary conditions have been intensely explored over the past 2 years. We numerically investigate the -symmetric phase in a ring with a position-dependent tunneling function t α ( k ) = [ k ( N − k )] α/2 that, in an open lattice, leads to a strengthened -symmetric phase, and study the evolution of the -symmetric phase from the open chain to a ring. We show that, generally, periodic boundary conditions weaken the -symmetric phase, although for experimentally relevant lattice sizes N ∼ 50, it remains easily accessible. We show that the chirality, quantified by the (magnitude of the) average transverse momentum of a wave packet, shows a maximum at the -symmetric threshold. Our results show that although the wavepacket intensity increases monotonically across the -breaking threshold, the average momentum decays monotonically on both sides of the threshold. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
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.