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arXiv:2102.03299 · Physical review. B./Physical review. B

Phantom Bethe roots in the integrable open spin- 12 XXZ chain

Xin Zhang, Andreas Klümper, Vladislav Popkov

Published Mar 22, 2021• 19 citations
arXivHTMLPDF

Abstract

We investigate special solutions to the Bethe Ansatz equations (BAE) for open integrable XXZ Heisenberg spin chains containing phantom (infinite) Bethe roots. The phantom Bethe roots do not contribute to the energy of the Bethe state, so the energy is determined exclusively by the remaining regular excitations. We derive the phantom Bethe roots criterion and focus on BAE solutions for mixtures of phantom roots and regular (finite) Bethe roots. We prove that in the presence of phantom Bethe roots, all eigenstates are split between two invariant subspaces, spanned by chiral shock states. Bethe eigenstates are described by two complementary sets of Bethe ansatz equations for regular roots, one for each invariant subspace. The respective ``semiphantom'' Bethe vectors are states of chiral nature, with chirality properties getting less pronounced when more regular Bethe roots are added. For the easy plane case, ``semiphantom'' Bethe states carry nonzero magnetic current and are characterized by quasiperiodic modulation of the magnetization profile, the most prominent example being the spin helix states (SHS). We illustrate our results investigating ``semiphantom'' Bethe states generated by one regular Bethe root (the other Bethe roots being phantom), with a simple structure of the invariant subspace, in all details. We obtain the explicit expressions for Bethe vectors and calculate the simplest correlation functions, including the spin-current for all the states in the single-particle multiplet.

DOI
10.1103/physrevb.103.115435

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We investigate special solutions to the Bethe Ansatz equations (BAE) for open integrable XXZ Heisenberg spin chains containing phantom (infinite) Bethe roots. The phantom Bethe roots do not contribute to the energy of the Bethe state, so the energy is determined exclusively by the remaining regular excitations. We derive the phantom Bethe roots criterion and focus on BAE solutions for mixtures of phantom roots and regular (finite) Bethe roots. We prove that in the presence of phantom Bethe roots, all eigenstates are split between two invariant subspaces, spanned by chiral shock states. Bethe eigenstates are described by two complementary sets of Bethe ansatz equations for regular roots, one for each invariant subspace. The respective semiphantom'' Bethe vectors are states of chiral nature, with chirality properties getting less pronounced when more regular Bethe roots are added. For the easy plane case, semiphantom'' Bethe states carry nonzero magnetic current and are characterized by quasiperiodic modulation of the magnetization profile, the most prominent example being the spin helix states (SHS). We illustrate our results investigating ``semiphantom'' Bethe states generated by one regular Bethe root (the other Bethe roots being phantom), with a simple structure of the invariant subspace, in all details. We obtain the explicit expressions for Bethe vectors and calculate the simplest correlation functions, including the spin-current for all the states in the single-particle multiplet.
We construct the coordinate Bethe ansatz for all eigenstates of the open $\mathrm{spin}\text{\ensuremath{-}}\frac{1}{2}$ XXZ chain that fulfill the phantom roots criterion (PRC). Under the PRC, the Hilbert space splits into two invariant subspaces and there are two sets of homogeneous Bethe ansatz equations (BAE) to characterize the subspaces in each case. We propose two sets of vectors with chiral shocks to span the invariant subspaces and expand the corresponding eigenstates. All the vectors are factorized and have symmetrical and simple structures. Using several simple cases as examples, we present the core elements of our generalized coordinate Bethe ansatz method. The eigenstates are expanded in our generating set and show clear chirality and certain symmetry properties. The bulk scattering matrices, the reflection matrices on the two boundaries and the BAE are obtained, which demonstrates the agreement with other approaches. Some hypotheses are formulated for the generalization of our approach.
We demonstrate the existence of a special chiral ``phantom'' mode with some analogy to a Goldstone mode in the anisotropic quantum XXZXXZXXZ Heisenberg spin chain. The phantom excitations contribute zero energy to the eigenstate, but a finite fixed quantum of momentum k0{k}_{0}k0​. The mode exists not due to symmetry principles, but results from nontrivial scattering properties of magnons with momentum k0{k}_{0}k0​ given by the anisotropy via $cos{k}_{0}=\mathrm{\ensuremath{\Delta}}$. Different occupations of the phantom mode lead to energetical degeneracies between different magnetization sectors in the periodic case. This mode originates from special string-type solutions of the Bethe ansatz equations with unbounded rapidities, the phantom Bethe roots (PBRs). We derive criteria under which the spectrum contains eigenstates with PBRs, both in open and periodically closed integrable systems, for spin-1/21/21/2 and higher spins, and discuss the respective chiral eigenstates. The simplest of such eigenstates, the spin helix state, which is a periodically modulated state of chiral nature, is built up from the phantom excitations exclusively. Implications of our results for experiments are discussed.
We investigate the structure of Bethe-root configurations in the spin-1/2 XXZ chain with degenerate open boundaries. The physical solutions of the Bethe Ansatz equations are classified into three types in the fully degenerate case and two types in the partially degenerate case, for which we propose counting formulas for the number of physical solution sets of each type. Furthermore, we observe spectral degeneracies in the fully degenerate case and show that they can be naturally explained by the presence of specific phantom strings in the Bethe roots. The classification and resulting spectral degeneracies in the diagonal limit are also discussed.
We derive a criterion under which splitting of all eigenstates of an open XYZ\mathrm{XYZ}XYZ Hamiltonian with boundary fields into two invariant subspaces, spanned by chiral shock states, occurs. The splitting is governed by an integer number, which has the geometrical meaning of the maximal number of kinks in the basis states. We describe the generic structure of the respective Bethe vectors. We obtain explicit expressions for Bethe vectors, in the absence of Bethe roots, and those generated by one Bethe root, and we investigate the single-particle subspace. We also describe in detail an elliptic analog of the spin-helix state, appearing in both the periodic and the open XYZ\mathrm{XYZ}XYZ model, and we derive the eigenstate condition. The elliptic analog of the spin-helix state is characterized by a quasiperiodic modulation of the magnetization profile, governed by Jacobi elliptic functions.
We review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters {{alpha−\\alpha_-alpha−​,alpha+\\alpha_+alpha+​,beta−\\beta_-beta−​,beta+\\beta_+beta+​}} are nonzero. A generalization of the Baxter T−QT-QT−Q equation that involves more than one independent QQQ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.
We show that every regular Bethe ansatz eigenvector of the XXZ spin chain at roots of unity is a highest weight vector of the sl 2 loop algebra, for some restricted sectors with respect to eigenvalues of the total spin operator S Z , and evaluate explicitly the highest weight in terms of the Bethe roots. We also discuss whether a given regular Bethe state in the sectors generates an irreducible representation or not. In fact, we present such a regular Bethe state in the inhomogeneous case that generates a reducible Weyl module. Here, we call a solution of the Bethe ansatz equations which is given by a set of distinct and finite rapidities regular Bethe roots . We call a nonzero Bethe ansatz eigenvector with regular Bethe roots a regular Bethe state .