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arXiv:q-alg/9712014

Commutation relations of vertex operators related with the spin representation of $U_q(D_n^{(1)})$

Yoshiyuki Koga

Published Dec 3, 1997• 0 citations
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Abstract

We calculate commutation relations of vertex operators for the spin representation of $U_q(D_n^{(1)})$ 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.

DOI
10.48550/arxiv.q-alg/9712014

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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
Staggered Polarization of Vertex Models with Uq(widehatsl(n))U_q(\\widehat{sl}(n))Uq​(widehatsl(n))-Symmetry
In this paper we give an explicit formula for level 1 vertex operators related to Uq(widehatsl(n))U_q(\\widehat{sl}(n))Uq​(widehatsl(n)) as operators on the Fock spaces. We derive also their commutation relations. As an applications we culculate the one point functions of the one-dimensional spin chain associated with the vector representation of Uq(widehatsl(n))U_q(\\widehat{sl}(n))Uq​(widehatsl(n)), thereby extending the recent work on the staggered polarization of the XXZXXZXXZ-model.
We systematise and develop a graphical approach to the investigations of quantum integrable vertex statistical models and the corresponding quantum spin chains.The graphical forms of the unitarity and various crossing relations are introduced.Their explicit analytical forms for the case of integrable systems associated with the quantum loop algebra U q (L(sl l+1 )) are given.The commutativity conditions for the transfer operators of lattices with a boundary are derived by the graphical method.Our consideration reveals useful advantages of the graphical approach for certain problems in the theory of quantum integrable systems.
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Some exactly solvable q-state vertex models are investigated. The author employs inversion relations to determine directly the spectra of the transfer matrices in the thermodynamic limit by avoiding the more cumbersome Bethe Ansatz. The results are applied to related quantum spin chains of which two families are SU(q) and SO(q) invariant, respectively. Various quantities are calculated, e.g. energy-momentum excitations and the correlation length. For q=3 the SU(q) invariant chain is the pure biquadratic spin-1 Hamiltonian which turns out to be non-critical. The ground-state energy, the gap, and the correlation length are given.