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Physical Review Letters

Algebraic structure of translation-invariant spin-1/2xxzandq-Potts quantum chains

Dan Lévy

Published Oct 7, 1991• 70 citations
Publisher

Abstract

Spin-1/2 xxz and q-Potts quantum chains with translation-invariant boundary conditions are analyzed as representations of the periodic Temperley-Lieb-Jones algebra. A connection with the affine Hecke algebra is established and used to find the irreducible content. This analysis provides an explanation for both the degeneracies and the overlap in the spectra of these models.

DOI
10.1103/physrevlett.67.1971

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Spin-1/2 xxz and q-Potts quantum chains with translation-invariant boundary conditions are analyzed as representations of the periodic Temperley-Lieb-Jones algebra. A connection with the affine Hecke algebra is established and used to find the irreducible content. This analysis provides an explanation for both the degeneracies and the overlap in the spectra of these models.
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 .
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 use simple diagrammatic techniques to analyse the ordinary representation theory of the Hecke algebras , and to construct modules (resp. representations) which are generically simple (irreducible) and well defined in every specialization of q, including roots of unity. We determine several physically important properties of these modules, generalizing properties of the Temperley - Lieb algebra and its diagrams which have proved useful for lattice models. We show how these results can be used to locate energy level crossings in invariant quantum spin chains, and locate a new crossing of the thermodynamic limit spin chain at as an example.
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.