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Corner transfer matrices and quantum affine algebras

Omar E. Foda, Tetsuji Miwa, The Netherl

Published Jan 1, 1992• 28 citations
OpenAlex

Abstract

Let H be the corner-transfer-matrix Hamiltonian for the six-vertex model in the antiferroelectric regime. It acts on the infinite tensor product W = V ⊗ V ⊗ V ⊗ · · ·, where V is the 2-dimensional irreducible representation of the quantum affine Lie algebra Uq ̂sl(2). We observe that H is the derivation of Uq ̂sl(2) , and conjecture that the eigenvectors of H form the level-1 vacuum representation of Uq

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Let H be the corner-transfer-matrix Hamiltonian for the six-vertex model in the antiferroelectric regime. It acts on the infinite tensor product W = V ⊗ V ⊗ V ⊗ · · ·, where V is the 2-dimensional irreducible representation of the quantum affine Lie algebra Uq ̂sl(2). We observe that H is the derivation of Uq ̂sl(2) , and conjecture that the eigenvectors of H form the level-1 vacuum representation of Uq
We consider corner transfer matrices of the six-vertex model and of its spin-l/2 generalizations, truncated to a finite lattice, in the anti-ferromagnetic regime. We show that, in the sense of finite size approximation, the low-lying eigenvectors approximate weight vectors of a level-l highest weight U q (sl 2 ) module, whilst the truncated corner transfer matrices approximate the role of a derivation operator. Our method is to first use the theory of the crystal base to prove the result exactly at q=0, and then extend to q not=0. We also consider the relationship between truncated corner transfer matrices, the vertex operator construction of Frenkel and Reshetikhin (1992), and Baxter's original argument for the properties of CTMs (1976).
We study the higher spin analogs of the six-vertex model on the basis of its symmetry under the quantum affine algebra [Formula: see text]. Using the method developed recently for the XXZ spin chain, we formulate the space of states, transfer matrix, vacuum, creation/ annihilation operators of particles, and local operators, purely in the language of representation theory. We find that, regardless of the level of the representation involved, the particles have spin 1/2, and that the n-particle space has an RSOS type structure rather than a simple tensor product of the one-particle space. This agrees with the picture proposed earlier by Reshetikhin.
In this paper we continue the study of Q-operators in the six-vertex model and its higher spin generalizations. In [1] we derived a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). Taking a special limit in this R-matrix we obtained new formulas for the Q-operators acting in the tensor product of representation spaces with arbitrary complex spin. Here we use a different strategy and construct Q-operators as integral operators with factorized kernels based on the original Baxter's method used in the solution of the eight-vertex model. We compare this approach with the method developed in [1] and find the explicit connection between two constructions. We also discuss a reduction to the case of finite-dimensional representations with (half-)integer spins.
The construction of auxiliary matrices for the six-vertex model at a root of unity is investigated from a quantum group theoretic point of view. Employing the concept of intertwiners associated with the quantum loop algebra U q ( 2 ) at q N = 1, a three-parameter family of auxiliary matrices is constructed. The elements of this family satisfy a functional relation with the transfer matrix allowing one to solve the eigenvalue problem of the model and to derive the Bethe ansatz equations. This functional relation is obtained from the decomposition of a tensor product of evaluation representations and involves auxiliary matrices with different parameters. Because of this dependence on additional parameters, the auxiliary matrices break in general the finite symmetries of the six-vertex model, such as spin-reversal or spin-conservation. More importantly, they also lift the extra degeneracies of the transfer matrix due to the loop symmetry present at rational coupling values. The extra parameters in the auxiliary matrices are shown to be directly related to the elements in the enlarged centre Z of the algebra U q ( 2 ) at q N = 1. This connection provides a geometric interpretation of the enhanced symmetry of the six-vertex model at rational coupling. The parameters labelling the auxiliary matrices can be interpreted as coordinates on a hypersurface Spec Z ⊂ ℂ 4 which remains invariant under the action of an infinite-dimensional group G of analytic transformations, called the quantum coadjoint action.