Singular solutions, repeated roots and completeness for higher-spin chains
Abstract
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), -is 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 8 sites, and find some support for the conjecture. We also exhibit several examples of strange solutions.
Keywords
Cite
@article{arxiv.1312.2982,
title = {Singular solutions, repeated roots and completeness for higher-spin chains},
author = {Wenrui Hao and Rafael I. Nepomechie and Andrew J. Sommese},
journal= {arXiv preprint arXiv:1312.2982},
year = {2015}
}
Comments
17 pages; many tables provided as supplemental material; v2: minor changes