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International Journal of Modern Physics A

CALCULATION OF EXCITATION SPECTRA OF THE SPIN MODEL RELATED WITH THE VECTOR REPRESENTATION OF THE QUANTIZED AFFINE ALGEBRA OF TYPE $A_n^{(1)} $

Etsurō Date, Masato Okado

Published Jan 30, 1994• 58 citations
Publisher

Abstract

The energy and momentum of the spin model related to the vector representation of the quantized affine algebra of type [Formula: see text] are computed in the framework of Davies et al. Commutation relations among creation and annihilation operators are also derived.
DOI
10.1142/s0217751x94000194

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EXCITATION SPECTRA OF SPIN MODELS CONSTRUCTED FROM QUANTIZED AFFINE ALGEBRAS OF TYPES Bn(1){\rm B}_n^{\left( 1 \right)}Bn(1)​ and Dn(1)D_n^{\left( 1 \right)}Dn(1)​
The energy and momentum spectra of the spin models constructed from the vector representation of the quantized affine algebras of types [Formula: see text] and [Formula: see text] are computed using the approach of Davies et al.1 The results are for the antiferromagnetic (massive) regime, and they agree with the mass spectrum found from the factorized S matrix theory by Ogievetsky et al.2 The other new result is the explicit realization of the fusion construction for the quantized affine algebras of types [Formula: see text] and [Formula: see text].
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.
Commutation relations of vertex operators related with the spin representation of Uq(Dn(1))U_q(D_n^{(1)})Uq​(Dn(1)​)
We calculate commutation relations of vertex operators for the spin representation of Uq(Dn(1))U_q(D_n^{(1)})Uq​(Dn(1)​) by using recursive formulae of R-matrices. In quantum symmetry approach, we obtain the energy and momentum spectrum of the quantum spin chain model related with the spin representation from these commutation relations.
A new scheme to study the XXZ spin chain in massive regime is introduced by Davies et al in [1]. In this scheme, which is called symmetry approach, the space of states is identified with some Uq(A\_ ) -module. Under this identification, the XXZ Hamiltonian can be constructed on this module as an operator which commutes with the action of Uq(A '). Furthermore, the row transfer matrix and creation (annihilation) operators can be constructed by using vertex operators. In a similar way to [1], the higher spin chain [2] and the models related to the vector representation ofUq(A^) [3], Uq(B^ U q ( D ( n } ) [5] are studied. In this paper, we consider the model related with the spin representation of Uq(Dn ). In physics the model that we consider here is explained as follows. It is a one dimensional quantum spin chain model constructed from the spin representation of Uq(Dn )• The Hamiltonian acts on the space of the infinite tensor product
The formalism of the second quantization is developed entirely within the framework of the Dirac-Jordan representation theory. The equivalence of this formalism and the methods of ordinary coordinate-spin space is shown, and the equations of "connection" are fully developed. It is shown that the "creation" and "annihilation" operators, and indeed all operators which do not commute with the quantized Hamiltonian and hence whose representatives cannot be explicitly given in a configuration space of a fixed number of dimensions are easily treated in terms of the general representation theory and arise quite naturally. Both types of statistics are considered, and the Fock-Dirac density matrix and self-consistent field are treated as a simple illustration.