Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity

Gainutdinov, A.M.; Hao, Wenrui; Nepomechie, Rafael I.; Sommese, Andrew J.

doi:10.1088/1751-8113/48/49/494003

Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity

A.M. Gainutdinov
(
- DESY and
- Hamburg U., Dept. Math.
)
,
Wenrui Hao
(
- Ohio State U.
)
,
Rafael I. Nepomechie
(
- Miami U.
)
,
Andrew J. Sommese
(
- U. Notre Dame, Dept. Math.
)

May 8, 2015

38 pages

Published in:

J.Phys.A 48 (2015) 49, 494003

Published: Nov 19, 2015

e-Print:

1505.02104 [math-ph]

DOI:

10.1088/1751-8113/48/49/494003

Report number:

UMTG-282

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Abstract: (IOP)

We consider the

{U}_{q}{sl}(2)

-invariant open spin-1/2 XXZ quantum spin chain of finite length N. For the case that q is a root of unity, we propose a formula for the number of admissible solutions of the Bethe ansatz equations in terms of dimensions of irreducible representations of the Temperley–Lieb algebra, and a formula for the degeneracies of the transfer matrix eigenvalues in terms of dimensions of tilting

{U}_{q}{sl}(2)

-modules. These formulas include corrections that appear if two or more tilting modules are spectrum-degenerate. For the XX case (

q={{\rm{e}}}^{{\rm{i}}\pi /2}

), we give explicit formulas for the number of admissible solutions and degeneracies. We also consider the cases of generic q and the isotropic (

q\to 1

) limit. Numerical solutions of the Bethe equations up to N = 8 are presented. Our results are consistent with the Bethe ansatz solution being complete.

Note:

34 pages; v2: reference added; v3: two more references added and minor corrections

algebra: Temperley-Lieb
spin: chain
Bethe ansatz
numerical calculations
transfer matrix
quantum group
XXZ model

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