Completeness of solutions of Bethe's equations

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

Research output: Contribution to journalArticle

35 Citations (Scopus)

Abstract

We consider the Bethe equations for the isotropic spin-1/2 Heisenberg quantum spin chain with periodic boundary conditions. We formulate a conjecture for the number of solutions with pairwise distinct roots of these equations, in terms of numbers of so-called singular (or exceptional) solutions. Using homotopy continuation methods, we find all such solutions of the Bethe equations for chains of length up to 14. The numbers of these solutions are in perfect agreement with the conjecture. We also discuss an indirect method of finding solutions of the Bethe equations by solving the Baxter T-Q equation. We briefly comment on implications for thermodynamical computations based on the string hypothesis.

Original languageEnglish (US)
Article number052113
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume88
Issue number5
DOIs
StatePublished - Nov 11 2013

Fingerprint

completeness
Completeness
Number of Solutions
roots of equations
Homotopy Continuation Method
Quantum Spin Chain
Periodic Boundary Conditions
strings
boundary conditions
Pairwise
Strings
Roots
Distinct

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics

Cite this

@article{a99fb87024894965947444a43114e0dc,
title = "Completeness of solutions of Bethe's equations",
abstract = "We consider the Bethe equations for the isotropic spin-1/2 Heisenberg quantum spin chain with periodic boundary conditions. We formulate a conjecture for the number of solutions with pairwise distinct roots of these equations, in terms of numbers of so-called singular (or exceptional) solutions. Using homotopy continuation methods, we find all such solutions of the Bethe equations for chains of length up to 14. The numbers of these solutions are in perfect agreement with the conjecture. We also discuss an indirect method of finding solutions of the Bethe equations by solving the Baxter T-Q equation. We briefly comment on implications for thermodynamical computations based on the string hypothesis.",
author = "Wenrui Hao and Nepomechie, {Rafael I.} and Sommese, {Andrew J.}",
year = "2013",
month = "11",
day = "11",
doi = "10.1103/PhysRevE.88.052113",
language = "English (US)",
volume = "88",
journal = "Physical Review E",
issn = "2470-0045",
publisher = "American Physical Society",
number = "5",

}

Completeness of solutions of Bethe's equations. / Hao, Wenrui; Nepomechie, Rafael I.; Sommese, Andrew J.

In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 88, No. 5, 052113, 11.11.2013.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Completeness of solutions of Bethe's equations

AU - Hao, Wenrui

AU - Nepomechie, Rafael I.

AU - Sommese, Andrew J.

PY - 2013/11/11

Y1 - 2013/11/11

N2 - We consider the Bethe equations for the isotropic spin-1/2 Heisenberg quantum spin chain with periodic boundary conditions. We formulate a conjecture for the number of solutions with pairwise distinct roots of these equations, in terms of numbers of so-called singular (or exceptional) solutions. Using homotopy continuation methods, we find all such solutions of the Bethe equations for chains of length up to 14. The numbers of these solutions are in perfect agreement with the conjecture. We also discuss an indirect method of finding solutions of the Bethe equations by solving the Baxter T-Q equation. We briefly comment on implications for thermodynamical computations based on the string hypothesis.

AB - We consider the Bethe equations for the isotropic spin-1/2 Heisenberg quantum spin chain with periodic boundary conditions. We formulate a conjecture for the number of solutions with pairwise distinct roots of these equations, in terms of numbers of so-called singular (or exceptional) solutions. Using homotopy continuation methods, we find all such solutions of the Bethe equations for chains of length up to 14. The numbers of these solutions are in perfect agreement with the conjecture. We also discuss an indirect method of finding solutions of the Bethe equations by solving the Baxter T-Q equation. We briefly comment on implications for thermodynamical computations based on the string hypothesis.

UR - http://www.scopus.com/inward/record.url?scp=84888147603&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84888147603&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.88.052113

DO - 10.1103/PhysRevE.88.052113

M3 - Article

AN - SCOPUS:84888147603

VL - 88

JO - Physical Review E

JF - Physical Review E

SN - 2470-0045

IS - 5

M1 - 052113

ER -