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

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

Research output: Contribution to journalArticle

17 Citations (Scopus)

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 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
Volume2014
Issue number3
DOIs
StatePublished - Mar 2014

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Spin Chains
Singular Solutions
completeness
Completeness
Roots
Homotopy Continuation
Number of Solutions
Periodic Boundary Conditions
Pairwise
Strings
Distinct
eigenvectors
strings
boundary conditions
Homotopy
Boundary conditions

All Science Journal Classification (ASJC) codes

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

Cite this

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Singular solutions, repeated roots and completeness for higher-spin chains. / Hao, Wenrui; Nepomechie, Rafael I.; Sommese, Andrew J.

In: Journal of Statistical Mechanics: Theory and Experiment, Vol. 2014, No. 3, P03024, 03.2014.

Research output: Contribution to journalArticle

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