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

DataBETA

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

Dec 10, 2013

20 pages

Published in:

J.Stat.Mech. 1403 (2014) P03024

Published: Mar 26, 2014

e-Print:

1312.2982 [math-ph]

DOI:

10.1088/1742-5468/2014/03/P03024

Report number:

UMTG-278

View in:

reference search17 citations

Citations per year

Abstract: (IOP)

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.

Note:

17 pages; many tables provided as supplemental material; v2: minor changes

algebraic structures of integrable models
integrable spin chains (vertex models)
quantum integrability (Bethe ansatz)
spin: chain
boundary condition
integrability
Hamiltonian
homotopy

References(32)

Figures(0)

[1]

How algebraic Bethe ansatz works for integrable model

L.D. Faddeev
(
- Steklov Math. Inst., St. Petersburg and
- Helsinki U.
)

•
e-Print:
- hep-th/9605187

[2]

MODEL FACTORIZED S MATRIX AND AN INTEGRABLE HEISENBERG CHAIN WITH SPIN 1. (IN RUSSIAN)

A.B. Zamolodchikov
(
- Landau Inst.
)
,
V.A. Fateev
(
- Landau Inst.
)

- Sov.J.Nucl.Phys. 32 (1980) 298-303,
- Yad.Fiz. 32 (1980) 581-590

[3]

Yang-Baxter Equation and Representation Theory. 1.

- Lett.Math.Phys. 5 (1981) 393-403
•
DOI:
- 10.1007/BF02285311

[4]

QUANTUM SPECTRAL TRANSFORM METHOD. RECENT DEVELOPMENTS

- Lect.Notes Phys. 151 (1982) 61-119,
- In *Tvaerminne 1981, Proceedings, Intergrable Quantum Field Theories*, 61-119

[5]

The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins

L.A. Takhtajan
(
- Steklov Math. Inst., St. Petersburg
)

- Phys.Lett.A 87 (1982) 479-482
•
DOI:
- 10.1016/0375-9601(82)90764-2

[6]

EXACT SOLUTION OF THE ISOTROPIC HEISENBERG CHAIN WITH ARBITRARY SPINS: THERMODYNAMICS OF THE MODEL

Hratchya M. Babujian
(
- Moscow, ITEP
)

- Nucl.Phys.B 215 (1983) 317-336
•
DOI:
- 10.1016/0550-3213(83)90668-5

[7]

Critical Behavior of Spin S Heisenberg Antiferromagnetic Chains: Analytic and Numerical Results

- J.Phys.A 22 (1989) 511,
- PREPRINT - AFFLECK, I. (REC.JUL.88) 42p
•
DOI:
- 10.1088/0305-4470/22/5/015

[8]

S matrices in integrable models of isotropical magnetic chains. 1.

N. Reshetikhin
(
- Harvard U.
)

- J.Phys.A 24 (1991) 3299-3310

[9]

On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain

H. Bethe

- Z.Phys. 71 (1931) 205-226
•
DOI:
- 10.1007/BF01341708

[10]

One-dimensional Heisenberg model at finite temperature

M. Takahashi

- Prog.Theor.Phys. 46 (1971) 401-415

[11]

Combinatorial identities, and completeness of eigenstates for the Heisenberg magnet

A.N. Kirillov

- J.Sov.Math. 30 (1985) 2298

[12]

Algebro-geometric aspects of the Bethe equations

R.P. Langlands
,
Y. Saint-Aubin

[13]

Aspects combinatoires des équations de Bethe

R.P. Langlands
,
Y. Saint-Aubin

[14]

Bethe's equation is incomplete for the XXZ model at roots of unity

Klaus Fabricius
(
- Wuppertal U.
)
,
Barry M. McCoy
(
- SUNY, Stony Brook
)

- J.Statist.Phys. 103 (2001) 647-678
•
e-Print:
- cond-mat/0009279
•
DOI:
- 10.1023/A:1010380116927

[15]

Completeness of the Bethe ansatz for the six and eight vertex models

R.J. Baxter
(
- Australian Natl. U., Canberra
)

- J.Statist.Phys. 108 (2002) 1-48
•
e-Print:
- cond-mat/0111188
•
DOI:
- 10.1023/A:1015437118218

[16]

Bethe algebra of homogeneous XXX Heisenberg model has simple spectrum

E. Mukhin
,
V. Tarasov
,
A. Varchenko

- Commun.Math.Phys. 288 (2009) 1-42
•
e-Print:
- 0706.0688

[17]

ON EXCEPTIONAL SOLUTIONS OF THE BETHE ANSATZ EQUATIONS

L.V. Avdeev
(
- Dubna, JINR
)
,
A.A. Vladimirov
(
- Dubna, JINR
)

- Theor.Math.Phys. 69 (1987) 1071
•
DOI:
- 10.1007/BF01037864

[18]

Fine structure of the Bethe ansatz for the spin 1/2 Heisenberg XXX model

- J.Phys.A 25 (1992) 4115-4126

[19]

Singularities in the Bethe solution of the XXX and XXZ Heisenberg spin chains

R. Siddharthan

e-Print:
- cond-mat/9804210

[20]

Origin of the singular Bethe ansatz solutions for the Heisenberg XXZ spin chain

J.D. Noh
,
D.-S. Lee
,
D. Kim

- Physica A 287 (2000) 167

[21]

Stringing spins and spinning strings

N. Beisert
(
- Potsdam, Max Planck Inst.
)
,
J.A. Minahan
(
- Uppsala U., Inst. Theor. Phys.
)
,
M. Staudacher
(
- Potsdam, Max Planck Inst.
)
,
K. Zarembo
(
- Uppsala U., Inst. Theor. Phys. and
- Moscow, ITEP
)

- JHEP 09 (2003) 010
•
e-Print:
- hep-th/0306139
•
DOI:
- 10.1088/1126-6708/2003/09/010

[22]

A Novel long range spin chain and planar N=4 super Yang-Mills

- JHEP 07 (2004) 075
•
e-Print:
- hep-th/0405001
•
DOI:
- 10.1088/1126-6708/2004/07/075

[23]

Deformed strings in the Heisenberg model

R. Hagemans
,
J.-S. Caux

- J.Phys.A 40 (2007) 14605
•
e-Print:
- 0707.2803

[24]

Exceptional Operators in N=4 super Yang-Mills

Gleb Arutyunov
(
- Utrecht U.
)
,
Sergey Frolov
(
- Hamilton Math. Inst., Dublin
)
,
Alessandro Sfondrini
(
- Utrecht U.
)

- JHEP 09 (2012) 006
•
e-Print:
- 1205.6660
•
DOI:
- 10.1007/JHEP09(2012)006

[25]

Algebraic Bethe ansatz for singular solutions

Rafael I. Nepomechie
(
- Miami U.
)
,
Chunguang Wang
(
- Miami U.
)

- J.Phys.A 46 (2013) 325002
•
e-Print:
- 1304.7978
•
DOI:
- 10.1088/1751-8113/46/32/325002

1-25 of 32
1
2
25 / page