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Communications in Theoretical Physics

Wavepacket Dynamics in a Non-Hermitian PT Symmetric Tight-Binding Chain

L. Jin, Song Zhi

Published Jul 1, 2010• 18 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/54/1/14

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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.
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.
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.
Based on the PT-symmetric quantum theory, the concepts of PT-frame, PT-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed. It is proved that the spectrum and point spectrum of a PT-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken PT-symmetric operator are real. For a linear operator H on C d , it is proved that H has unbroken PT-symmetry if and only if it has d different eigenvalues and the corresponding eigenstates are eigenstates of PT. Given a CPT-frame on K, a new positive inner product on K is induced and called CPT-inner product. Te relationship between the CPT-adjoint and the Dirac adjoint of a densely defined linear operator is derived, and it is proved that an operator which has a bounded CPT-frame is CPT-Hermitian if and only if it is T-symmetric, in that case, it is similar to a Hermitian operator. The existence of an operator C consisting of a CPT-frame is discussed. These concepts and results will serve a mathematical discussion about PT-symmetric 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.
Recently, much research has been carried out on Hamiltonians that are not Hermitian but are symmetric under space-time reflection, that is, Hamiltonians that exhibit PT symmetry. Investigations of the Sturm-Liouville eigenvalue problem associated with such Hamiltonians have shown that in many cases the entire energy spectrum is real and positive and that the eigenfunctions form an orthogonal and complete basis. Furthermore, the quantum theories determined by such Hamiltonians have been shown to be consistent in the sense that the probabilities are positive and the dynamical trajectories are unitary. However, the geometrical structures that underlie quantum theories formulated in terms of such Hamiltonians have hitherto not been fully understood. This paper studies in detail the geometric properties of a Hilbert space endowed with a parity structure and analyses the characteristics of a PT-symmetric Hamiltonian and its eigenstates. A canonical relationship between a PT-symmetric operator and a Hermitian operator is established. It is shown that the quadratic form corresponding to the parity operator, in particular, gives rise to a natural partition of the Hilbert space into two halves corresponding to states having positive and negative PT norm. The indefiniteness of the norm can be circumvented by introducing a symmetry operator C that defines a positive definite inner product by means of a CPT conjugation operation.
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.