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arXiv:2301.06004 · Journal of Physics A Mathematical and Theoretical

Exact solution of a non-Hermitian PT -symmetric spin chain

Pradip Kattel, Parameshwar R. Pasnoori, Natan Andrei

Published Jul 7, 2023• 20 citations
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Abstract

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.

DOI
10.1088/1751-8121/ace56e

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At large values of the anisotropy Δ, the open-boundary Heisenberg spin- chain has eigenstates displaying localization at the edges. We present a Bethe ansatz description of this ‘edge-locking’ phenomenon in the entire Δ > 1 region. We focus on the simplest spin sectors, namely the highly polarized sectors with only one or two overturned spins, i.e., one-particle and two-particle sectors. Edge-locking is associated with pure imaginary solutions of the Bethe equations, which are not commonly encountered in periodic chains. In the one-particle case, at large Δ there are two eigenstates with imaginary Bethe momenta, related to localization at the two edges. For any finite chain size, one of the two solutions become real as Δ is lowered below a certain value. For two particles, a richer scenario is observed, with eigenstates having the possibility of both particles locked on the same or different edge, one locked and the other free, or both free either as single magnons or as bound composites corresponding to ‘string’ solutions. For finite chains, some of the edge-locked spins get delocalized at certain values of Δ (‘exceptional points’), corresponding to imaginary solutions becoming real. We characterize these phenomena thoroughly by providing analytic expansions of the Bethe momenta for large chains, large anisotropy Δ, and near the exceptional points. In the large-chain limit all the exceptional points coalesce at the isotropic point (Δ = 1) and edge-locking becomes stable in the whole Δ > 1 region.
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.
The XX spin-chain with non-Hermitian diagonal boundary conditions is shown to be quasi-Hermitian for special values of the boundary parameters. This is proved by an explicit construction of a new inner product employing a ‘quasi-fermion’ algebra in momentum space, where creation and annihilation operators are not related via Hermitian conjugation. For a special example, when the boundary fields lie on the imaginary axis, we show the spectral equivalence of the quasi-Hermitian XX spin-chain with a non-local fermion model, where long-range hopping of the particles occurs as the non-Hermitian boundary fields increase in strength. The corresponding Hamiltonian interpolates between the open XX and the quantum group invariant XXZ model at the free fermion point. For an even number of sites the former is known to be related to a CFT with central charge c = 1, while the latter has been connected to a logarithmic CFT with central charge c = −2. We discuss the underlying algebraic structures and show that for an odd number of sites the superalgebra symmetry can be extended from the unit circle along the imaginary axis. We relate the vanishing of one of its central elements to the appearance of Jordan blocks in the Hamiltonian.
We study the open XXZ spin chain with a PT P T -symmetric non-Hermitian boundary field. We find an interaction-induced scale-free non-Hermitian skin effect by using the coordinate Bethe Ansatz. The steady-state and the ground state in the PT P T broken phase are constructed, and the formulas of their eigen-energies in the thermodynamic limit are obtained. The differences between the many-body scale-free states and the boundary string states are explored, and the transition between the two at the isotropic point is investigated. We also discuss an experimental scheme to verify our results.