### 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 language | English (US) |
---|---|

Article number | P03024 |

Journal | Journal of Statistical Mechanics: Theory and Experiment |

Volume | 2014 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2014 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Journal of Statistical Mechanics: Theory and Experiment*,

*2014*(3), [P03024]. https://doi.org/10.1088/1742-5468/2014/03/P03024

}

*Journal of Statistical Mechanics: Theory and Experiment*, vol. 2014, no. 3, P03024. https://doi.org/10.1088/1742-5468/2014/03/P03024

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Singular solutions, repeated roots and completeness for higher-spin chains

AU - Hao, Wenrui

AU - Nepomechie, Rafael I.

AU - Sommese, Andrew J.

PY - 2014/3

Y1 - 2014/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84937031461&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84937031461&partnerID=8YFLogxK

U2 - 10.1088/1742-5468/2014/03/P03024

DO - 10.1088/1742-5468/2014/03/P03024

M3 - Article

AN - SCOPUS:84937031461

VL - 2014

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

SN - 1742-5468

IS - 3

M1 - P03024

ER -