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arXiv:1411.5839 · Journal of Physics A Mathematical and Theoretical

Singular eigenstates in the even(odd) length Heisenberg spin chain

Pulak Ranjan Giri, Tetsuo Deguchi

Published Apr 8, 2015• 7 citations
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Abstract

We study the implications of the regularization for the singular solutions on the even(odd) length spin- 1 / 2 XXX chains in some specific down-spin sectors. In particular, the analytic expressions of the Bethe eigenstates for three down-spin sector have been obtained along with their numerical forms in some fixed length chains. For an even-length chain if the singular solutions { λ α } are invariant under the sign changes of their rapidities { λ α } = { − λ α } , then the Bethe ansatz equations are reduced to a system of ( M − 2 ) / 2 ( ( M − 3 ) / 2 ) equations in an even (odd) down-spin sector. For an odd N length chain in the three down-spin sector, it has been analytically shown that there exist singular solutions in any finite length of the spin chain of the form N = 3 ( 2 k + 1 ) with k = 1 , 2 , 3 , ⋯ . It is also shown that there exist no singular solutions in the four down-spin sector for some odd-length spin- 1 / 2 XXX chains.

DOI
10.1088/1751-8113/48/17/175207

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We investigate Bethe Ansatz equations for the one-dimensional spin-12\frac{1}{2}21​ Heisenberg XXX chain with a special interest in a finite system. Solutions for the two-particle sector are obtained. The ground state in antiferromagnetic case has been analytically studied through the logarithmic form of Bethe Ansatz equations.
We investigate symmetry properties of the Bethe ansatz wave functions for the Heisenberg XXZ spin chain. The XXZ Hamiltonian commutes simultaneously with the shift operator T and the lattice inversion operator V in the space of Ω=±1 with Ω the eigenvalue of T. We show that the Bethe ansatz solutions with normalizable wave functions cannot be the eigenstates of T and V with quantum number (Ω, ϒ)=(±1, ∓1) where ϒ is the eigenvalue of V. Therefore, the Bethe ansatz wave functions should be singular for nondegenerate eigenstates of the Hamiltonian with quantum number (Ω, ϒ)=(±1, ∓1). It is also shown that such states exist in any nontrivial down-spin number sector and that the number of them diverges exponentially with the chain length.
We examine the question of whether Bethe's ansatz reproduces all states in the periodic Heisenberg XXZ and XXX spin chains. As was known to Bethe himself, there are states for which the Bethe momenta knk_nkn​ diverge: these are in fact the simplest examples of ``string'' solutions. The coefficients of the Bethe wavefunction, too, diverge. When there are only two down spins in the system (the case considered by Bethe), we can renormalize these coefficients to get a sensible (and correct) wavefunction. We show that this is not always possible when there are more than two down spins. The Bethe equations have several such divergent solutions, and some of these correspond to genuine eigenfunctions of the Hamiltonian, but several do not. Nor do they reproduce the correct energy eigenvalues. Moreover, we point out that the algebraic Bethe ansatz, an alternative way to construct the wavefunctions proposed by Faddeev, Takhtajan et al., leads to vanishing wavefunctions for all these solutions. Thus, the Bethe ansatz solution of the Heisenberg model must be regarded as either incomplete, or inaccurate.
The authors analyse the Bethe ansatz equations for the two-particle sector of the spin 1/2 Heisenberg XXX model on a one-dimensional lattice of length N. They show that, beginning at a critical lattice length of N=21.86, new pairs of real solutions develop, whereas complex solutions start to disappear. The integers (that appear in the logarithmic form of the Bethe equations) of the new solutions do not fit into the conventional classification scheme. The total number of solutions in the two-particle sector remains unchanged and is in agreement with the claim that the SU(2) extended Bethe ansatz gives a complete set of 2 N eigenstates.
We investigate the completeness of the solutions of the Bethe equations for the integrable spin- s isotropic (XXX) spin chain with periodic boundary conditions. Solutions containing the exact string i s , i( s − 1), ..., −i( s − 1), −i s are singular. For s > 1/2, there exist also 'strange' solutions with repeated roots, which nevertheless are physical (i.e., correspond to eigenstates of the Hamiltonian). We derive conditions for the singular solutions and the solutions with repeated roots to be physical. We formulate a conjecture for the number of solutions with pairwise distinct roots in terms of the numbers of singular and strange solutions. Using homotopy continuation, we solve the Bethe equations numerically for s = 1 and s = 3/2 up to eight sites, and find some support for the conjecture. We also present several examples of strange solutions.
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