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arXiv:cond-mat/9505049 · Journal of Physics A Mathematical and General

Exact solutions forA-DTemperley - Lieb models

R Köberle, A Lima-Santos

Published Feb 7, 1996• 14 citations
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Abstract

We solve for the spectrum of quantum spin chains based on representations of the Temperley - Lieb algebra associated with the quantum groups for and . We employ a generalization of the coordinate Bethe ansatz developed previously for the deformed biquadratic spin-one chain. As expected, all of these models have equivalent spectra, i.e. they differ only in the degeneracy of their eigenvalues. This is true for finite length and open boundary conditions. For periodic boundary conditions the spectra of the lower-dimensional representations are contained entirely in the higher-dimensional ones. The Bethe states are highest-weight states of the quantum group, except in certain cases where the states have energy zero.

DOI
10.1088/0305-4470/29/3/007

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We solve for the spectrum of quantum spin chains based on representations of the Temperley - Lieb algebra associated with the quantum groups for and . We employ a generalization of the coordinate Bethe ansatz developed previously for the deformed biquadratic spin-one chain. As expected, all of these models have equivalent spectra, i.e. they differ only in the degeneracy of their eigenvalues. This is true for finite length and open boundary conditions. For periodic boundary conditions the spectra of the lower-dimensional representations are contained entirely in the higher-dimensional ones. The Bethe states are highest-weight states of the quantum group, except in certain cases where the states have energy zero.
We solve the spectrum of quantum spin chains based on representations of the Temperley–Lieb algebra associated with the quantum groups [Formula: see text] for Xn=A1, Bn, Cn and Dn. The tool is a modified version of the coordinate Bethe ansatz through a suitable choice of the Bethe states which give to all models the same status relative to their diagonalization. All these models have equivalent spectra up to degeneracies and the spectra of the lower-dimensional representations are contained in the higher-dimensional ones. Periodic boundary conditions, free boundary conditions and closed nonlocal boundary conditions are considered. Periodic boundary conditions, unlike free boundary conditions, break quantum group invariance. For closed nonlocal cases the models are quantum group invariant as well as periodic in a certain sense.
We solve for the spectrum of the deformed biquadratic spin-1 chain by a generalization of the coordinate Bethe ansatz. The model is invariant under the quantum group U q sl(2) and, depending on the deformation parameter q, it's Hamiltonian may not be Hermitian. We consider several boundary conditions and find in particular that for periodic ones, the complete spectrum of the spin- 1/2 XXZ model is always contained in the spectrum of the biquadratic model. For free boundary conditions the spectrum is identical to the XXZ model spectrum, only the multiplicities are, of course, different. Bethe states are highest-weight states of the quantum group, except possibly for zero-energy states.
We consider the Temperley–Lieb (TL) open quantum spin chain with “free” boundary conditions associated with the spin- s representation of quantum-deformed s l ( 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.
We determine the spectra of a class of quantum spin chains of Temperley–Lieb type by utilizing the concept of Temperley–Lieb equivalence with the S = 1/2 XXZ chain as a reference system. We consider open boundary conditions and, in particular, periodic boundary conditions. For both types of boundaries the identification with XXZ spectra is performed within isomorphic representations of the underlying Temperley–Lieb algebra. For open boundaries the spectra of these models differ from the spectrum of the associated XXZ chain only in the multiplicities of the eigenvalues. The periodic case is rather different. Here we show how the spectrum is obtained sector-wise from the spectra of globally twisted XXZ chains. As a spin-off, we obtain a compact formula for the degeneracy of the momentum operator eigenvalues. Our representative theoretical results allow for the study of the thermodynamics by establishing a TL-equivalence at finite temperature and finite field .