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arXiv:hep-th/9204068 · International Journal of Modern Physics A

CORNER TRANSFER MATRICES AND QUANTUM AFFINE ALGEBRAS

OMAR FODA, TETSUJI MIWA

Published Apr 1, 1992• 23 citations
arXivHTMLPDF

Abstract

Let ℋ be the corner-transfer-matrix Hamiltonian for the six-vertex model in the anti-ferroelectric 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 [Formula: see text]. We observe that ℋ is the derivation of [Formula: see text], and conjecture that the eigenvectors of ℋ form the level-1 vacuum representation of [Formula: see text]. We report on checks in support of our conjecture.

DOI
10.1142/s0217751x92003811

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Background of the problem The spin $1/2$ XXZ model for $\Delta <-1$ The six-vertex model in the anti-ferroelectric regime Solvability and symmetry Correlation functions-physical derivation Level one modules and bosonization Vertex operators Space of states-mathematical picture Traces of vertex operators Correlation functions and form factors The $XXX$ limit $q\rightarrow-1$ Discussions List of formulas.

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Let ℋ be the corner-transfer-matrix Hamiltonian for the six-vertex model in the anti-ferroelectric 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 [Formula: see text]. We observe that ℋ is the derivation of [Formula: see text], and conjecture that the eigenvectors of ℋ form the level-1 vacuum representation of [Formula: see text]. We report on checks in support of our conjecture.
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.
Eigenstates of the corner transfer matrix (CTM) for the six-vertex model are constructed and their relation to the Bethe ansatz eigenstates of the XXZ Hamiltonian is discussed. In the ferromagnetic regime ( Delta >1), eigenstates with any finite number of overturned arrows are constructed. They are related to the Bethe ansatz states by Fourier transformation over spectral (rapidity) variables. This exhibits the role of the CTM as a lattice boost operator. For the antiferromagnetic regime, the relation between CTM eigenstates and those of the XXZ Hamiltonian is complicated by the filling of the sea and associated Bethe ansatz integral equations. For this case, the authors construct CTM eigenstates by first showing that the Hamiltonian ground state is also an eigenstate of the CTM. Other CTM eigenstates are constructed from excited eigenstates of the Hamiltonian by Fourier transforming over the rapidity of each dressed excitation. They discuss the relationship between CTM eigenstates and a Heisenberg algebra of bosonic oscillators.
Background of the problem The spin 1/21/21/2 XXZ model for Δ<−1\Delta <-1Δ<−1 The six-vertex model in the anti-ferroelectric regime Solvability and symmetry Correlation functions-physical derivation Level one modules and bosonization Vertex operators Space of states-mathematical picture Traces of vertex operators Correlation functions and form factors The XXXXXXXXX limit q→−1q\rightarrow-1q→−1 Discussions List of formulas.