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

TL;DR

This paper investigates the properties of singular solutions in the Bethe ansatz for even and odd length Heisenberg spin chains, providing explicit eigenstates and analyzing the existence of singular solutions across different chain lengths and spin sectors.

Contribution

It derives analytic expressions for Bethe eigenstates with three down-spins and characterizes conditions for singular solutions in even and odd length chains.

Findings

Singular solutions are invariant under sign change of rapidities in even-length chains.

Existence of singular solutions in odd-length chains of specific forms N=3(2k+1).

No singular solutions in four down-spin sectors for some odd-length chains.

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 \{\lambda_\alpha\} are invariant under the sign changes of their rapidities {\lambda_\alpha\}=\{-\lambda_\alpha\} , 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\left(2k+1\right) with k=1, 2, 3, \cdots. 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.