Singular solutions, repeated roots and completeness for higher-spin chains

Wenrui Hao, Rafael I. Nepomechie, Andrew J. Sommese

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25 Scopus citations


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 is, 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 eight sites, and find some support for the conjecture. We also present several examples of strange solutions.

Original languageEnglish (US)
Article numberP03024
JournalJournal of Statistical Mechanics: Theory and Experiment
Issue number3
StatePublished - Mar 2014

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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