TY - JOUR

T1 - Universal Bethe ansatz solution for the Temperley–Lieb spin chain

AU - Nepomechie, Rafael I.

AU - Pimenta, Rodrigo A.

N1 - Funding Information:
The work of RN was supported in part by the National Science Foundation under Grant PHY-1212337 , and by a Cooper fellowship . RP thanks the São Paulo Research Foundation (FAPESP), grants # 2014/00453-8 and # 2014/20364-0 , for financial support. We also acknowledge the support by FAPESP and the University of Miami under the SPRINT grant # 2016/50023-5 .

PY - 2016/9/1

Y1 - 2016/9/1

N2 - 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 off-shell 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.

AB - 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 off-shell 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.

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U2 - 10.1016/j.nuclphysb.2016.04.045

DO - 10.1016/j.nuclphysb.2016.04.045

M3 - Article

AN - SCOPUS:84975122982

VL - 910

SP - 910

EP - 928

JO - Nuclear Physics B

JF - Nuclear Physics B

SN - 0550-3213

ER -