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arXiv:q-alg/9706015

Bosonization of quantum affine groups and its application to the higher spin Heisenberg model

A. Hamid Bougourzi

Published Jun 14, 1997• 9 citations
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Abstract

In this paper, we present a detailed analysis of the diagonalization of the higher spin Heisenberg model using its quantum affine symmetry $U_q(\hat{sl(2)})$. In particular, we describe the bosonizations of the latter algebra, its highest weight representations, vertex operators and screening operators. Finally, we use this bosonization method to compute the vacuum-to-vacuum expectation values and the form factors of any local operator.

DOI
10.48550/arxiv.q-alg/9706015

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In this paper, we present a detailed analysis of the diagonalization of the higher spin Heisenberg model using its quantum affine symmetry Uq(sl(2)^)U_q(\hat{sl(2)})Uq​(sl(2)^​). In particular, we describe the bosonizations of the latter algebra, its highest weight representations, vertex operators and screening operators. Finally, we use this bosonization method to compute the vacuum-to-vacuum expectation values and the form factors of any local operator.
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.
These notes correspond rather accurately to the translation of the lectures given at the Fifth Mexican School of Particles and Fields, held in Guanajuato, Gto., in December 1992. They constitute a brief and elementary introduction to quantum symmetries from a physical point of view, along the lines of the forthcoming book by C. Gómez, G. Sierra, and myself: 1) Introduction. 2) Factorizable S‐matrices. 3) Bethe’s diagonalization of spin chain hamiltonians. 4) Integrable vertex models: the six‐vertex model. 5) The Yang‐Baxter algebra. 6) Physical spectrum of the Heisenberg spin chain. 7) Yang‐Baxter algebras and braid groups. 8) Yang‐Baxter algebras and quantum groups. 9) Affine quantum groups. 10) Hopf algebras. 11) The quantum group Uq(G). 12) Comments. 13) References.
We extend the recent approach of M. Jimbo, K. Miki, T. Miwa, and A. Nakayashiki to derive an integral formula for the N-point correlation functions of arbitrary local operators of the antiferromagnetic spin-1 XXZ model. For this, we realize the quantum affine symmetry algebra Uq(su(2)2)U_q(su(2)_2)Uq​(su(2)2​) of level 2 and its corresponding type I vertex operators in terms of a deformed bosonic field free of a background charge, and a deformed fermionic field. Up to GSO type projections, the Fock space is already irreducible and therefore no BRST projections are involved. This means that no screening charges with their Jackson integrals are required. Consequently, our N-point correlation functions are given in terms of usual classical integrals only, just as those derived by Jimbo et al in the case of the spin-1/2 XXZ model through the Frenkel-Jing bosonization of Uq(su(2)1)U_q(su(2)_1)Uq​(su(2)1​).