Algebraic Bethe ansatz for the Temperley–Lieb spin-1 chain

Nepomechie, Rafael I.; Pimenta, Rodrigo A.

doi:10.1016/j.nuclphysb.2016.04.044

Cited by 5 publications

(15 citation statements)

References 47 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is important to find proofs for these conjectures. So far, we have been able to prove only the off-shell equations for s = 1 [45].…”

Section: Discussionmentioning

confidence: 99%

“…We present here several conjectures related to the algebraic Bethe ansatz solution of the TL chain. The conjecture for the off-shell equation has been proved for s = 1 [45], while the other conjectures have been checked numerically (up to M = 3, N = 6 and s = 3 2 ). 2 1 1 1 2 3 3 8 15 4 1 5 55 209…”

Section: Algebraic Bethe Ansatzmentioning

confidence: 99%

“…We have found that the (dual) Bethe vectors are generated by the action of a single double-row operator, namely (C(u)) B(u). Indeed, let us define the Bethe vector as where [29]; for the s = 1 case a proof will be reported in a separate paper [45]. T + ij |u 1 , .…”

Section: Off-shell Equationmentioning

confidence: 99%

See 2 more Smart Citations

Universal Bethe ansatz solution for the Temperley–Lieb spin chain

Nepomechie

Pimenta

2016

Nuclear Physics B

Self Cite

View full text Add to dashboard Cite

We consider the Temperley-Lieb (TL) open quantum spin chain with "free" boundary conditions associated with the spin-s representation of quantum-deformed sl(2). We construct the transfer matrix, and determine its eigenvalues and the corresponding Bethe equations using analytical Bethe ansatz. We show that the transfer matrix has quantum group symmetry, and we propose explicit formulas for the number of solutions of the Bethe equations and the degeneracies of the transfer-matrix eigenvalues. We propose an algebraic Bethe ansatz construction of the off-shell Bethe states, and we conjecture that the on-shell Bethe states are highest-weight states of the quantum group. We also propose a determinant formula for the scalar product between an offshell Bethe state and its on-shell dual, as well as for the square of the norm. We find that all of these results, except for the degeneracies and a constant factor in the scalar product, are universal in the sense that they do not depend on the value of the spin. In an appendix, we briefly consider the closed TL spin chain with periodic boundary conditions, and show how a previously-proposed solution can be improved so as to obtain the complete (albeit non-universal) spectrum.

show abstract

“…It is important to find proofs for these conjectures. So far, we have been able to prove only the off-shell equations for s = 1 [45].…”

Section: Discussionmentioning

confidence: 99%

Section: Algebraic Bethe Ansatzmentioning

confidence: 99%

See 1 more Smart Citation

Universal Bethe ansatz solution for the Temperley–Lieb spin chain

Nepomechie

Pimenta

2016

Nuclear Physics B

Self Cite

View full text Add to dashboard Cite

show abstract

“…Furthermore, the tools used in [6,7] are very specific to their particular case, and hardly generalizable to other contexts. More specifically, the exceptional structure of the Slavnov product allowed the authors to use some Jacobi-Trudi-like identities in their study, but this feature is not expect in other cases, see the structure of [10,11]. In the current work, we also try to find some new ingredients and tools to study this duality between spin chains and classical integrable hierarchies.…”

Section: Introductionmentioning

confidence: 97%

“…First, Slavnov products are complex objects to construct, see [3], and there is no general protocol teaching us how to build them; it is safe to say that we are essentially confined to the case-by-case analysis at the time of writing. Despite these difficulties, some new results have recently appeared in the literature [10,11] allowing us to advance this program a bit further. These new results are the examples we address in the current work.…”

Section: Introductionmentioning

confidence: 99%