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arXiv:1107.2293 · Physical Review A

Bound states of PT -symmetric separable potentials

Carl M. Bender, Hugh F Jones

Published Sep 6, 2011• 2 citations
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Abstract

All of the $\mathcal{P}\mathcal{T}$-symmetric potentials that have been studied so far have been local. In this paper, nonlocal $\mathcal{P}\mathcal{T}$-symmetric separable potentials of the form $V(x,y)=i\ensuremath{\epsilon}[U(x)U(y)\ensuremath{-}U(\ensuremath{-}x)U(\ensuremath{-}y)]$, where $U(x)$ is real, are examined. Two specific models are examined. In each case, it is shown that there is a parametric region of the coupling strength $\ensuremath{\epsilon}$ for which the $\mathcal{P}\mathcal{T}$ symmetry of the Hamiltonian is unbroken and the bound-state energies are real. The critical values of $\ensuremath{\epsilon}$ that bound this region are calculated.

DOI
10.1103/physreva.84.032103

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All of the PT\mathcal{P}\mathcal{T}PT-symmetric potentials that have been studied so far have been local. In this paper, nonlocal PT\mathcal{P}\mathcal{T}PT-symmetric separable potentials of the form V(x,y)=i\ensuremathϵ[U(x)U(y)\ensuremath−U(\ensuremath−x)U(\ensuremath−y)]V(x,y)=i\ensuremath{\epsilon}[U(x)U(y)\ensuremath{-}U(\ensuremath{-}x)U(\ensuremath{-}y)]V(x,y)=i\ensuremathϵ[U(x)U(y)\ensuremath−U(\ensuremath−x)U(\ensuremath−y)], where U(x)U(x)U(x) is real, are examined. Two specific models are examined. In each case, it is shown that there is a parametric region of the coupling strength \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ for which the PT\mathcal{P}\mathcal{T}PT symmetry of the Hamiltonian is unbroken and the bound-state energies are real. The critical values of \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ that bound this region are calculated.
The Hamiltonian for a PT\mathcal{PT}PT-symmetric chain of coupled oscillators is constructed. It is shown that if the loss-gain parameter $\ensuremath{\gamma}$ is uniform for all oscillators, then as the number of oscillators increases, the region of unbroken PT\mathcal{PT}PT symmetry disappears entirely. However, if $\ensuremath{\gamma}$ is localized in the sense that it decreases for more distant oscillators, then the unbroken PT\mathcal{PT}PT-symmetric region persists even as the number of oscillators approaches infinity. In the continuum limit the oscillator system is described by a PT\mathcal{PT}PT-symmetric pair of wave equations, and a localized loss-gain impurity leads to a pseudobound state. It is also shown that a planar configuration of coupled oscillators can have multiple disconnected regions of unbroken PT\mathcal{PT}PT symmetry.
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.
A general formalism is worked out for the description of one-dimensional scattering in non-hermitian quantum mechanics and constraints on transmission and reflection coefficients are derived in the cases of P, T, or PT invariance of the Hamiltonian. Applications to some solvable PT-symmetric potentials are shown in detail. Our main original results concern the association of reflectionless potentials with asymptotic exact PT symmetry and the peculiarities of separable kernels of non-local potentials in connection with hermiticity, T invariance and PT invariance.
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.