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Journal of Physics A Mathematical and General

On quantum spin-chain spectra and the representation theory of Hecke algebras by augmented braid diagrams

Paul Martin, Bruce W. Westbury

Published Aug 7, 1997• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/30/15/030

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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.
Motivated by a study of the crossing symmetry of the asymmetric twin or ‘gemini’ representation of the affine Hecke algebra we give a construction for crossing tensor space representations of ordinary Hecke algebras. These representations build solutions to the Yang–Baxter equation satisfying the crossing condition (that is, integrable quantum spin chains). We show that every crossing representation of the Temperley–Lieb algebra appears in this construction, and in particular that this construction builds new representations. We extend these to new representations of the blob algebra, which build new solutions to the boundary Yang–Baxter equation (i.e. open spin chains with integrable boundary conditions). We prove that the open spin chain Hamiltonian derived from Sklyanin’s commuting transfer matrix using such a solution can always be expressed as the representation of an element of the blob algebra, and determine this element. We determine the representation theory (irreducible content) of the new representations and hence show that all such Hamiltonians have the same spectrum up to multiplicity, for any given value of the algebraic boundary parameter. (A corollary is that our models have the same spectrum as the open XXZ chain with nondiagonal boundary—despite differing from this model in having reference states.) Using these multiplicity data, and other ideas, we investigate the underlying quantum group symmetry of the new Hamiltonians. We derive the form of the spectrum and the Bethe ansatz equations.
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Lattice algebras are defined and used to study the symmetries of 2D lattice models. New and interesting examples of such algebras are provided by the affine Hecke algebra, owing to the possibility of constructing braid generators out of its generators. I propose an Ansatz for the braid generators and derive some solutions. A particular finite-dimensional quotient is shown to be a natural generalization of the Temperley-Lieb-Jones algebra. It is used to give a unified picture of known and unknown symmetries of the spin-½ xxz model with boundary terms. The Ising model in an external magnetic field is also a representation of this quotient.
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 .
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