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arXiv:hep-th/9706086 · Journal of Physics A Mathematical and General

The Monodromy matrices of theXXZmodel in the infinite volume limit

Tetsuji Miwa, Robert Weston

Published Nov 7, 1997• 4 citations
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Abstract

We consider the XXZ model in the infinite volume limit with spin- quantum space and higher spin auxiliary space. Using perturbation theory arguments, we relate the half infinite transfer matrices of this class of model to certain intertwiners introduced by Nakayashiki. We construct the monodromy matrices, and show that the one with spin-1 auxiliary space gives rise to the L-operator.

DOI
10.1088/0305-4470/30/21/022

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We consider the XXZ model in the infinite volume limit with spin- quantum space and higher spin auxiliary space. Using perturbation theory arguments, we relate the half infinite transfer matrices of this class of model to certain intertwiners introduced by Nakayashiki. We construct the monodromy matrices, and show that the one with spin-1 auxiliary space gives rise to the L-operator.
Using notation inherited from the six-vertex model, we construct diagrams that represent the action of the factorizing F-matrices associated to the finite length XXZ spin-12 chain. We provide a diagrammatic proof that these F-matrices factorize the tensor R1…nσ corresponding with elements of the permutation group. We consider in particular the diagram for the tensor R1…nσc, which cyclically permutes the spin chain. This leads us to a diagrammatic construction of the local spin operators Si± and Siz in terms of the monodromy matrix operators.
In the framework of the quantum inverse scattering method, we consider a problem of constructing local operators for one-dimensional quantum integrable models, especially for the lattice versions of the nonlinear Schrödinger and sine-Gordon models. We show that a certain class of local operators can be constructed from the matrix elements of the monodromy matrix in a simple way. They are closely related to the quantum projectors and have nice commutation relations with half of the matrix elements of the elementary monodromy matrix. The form factors of these operators can be calculated by using the standard algebraic Bethe ansatz techniques.
In the present paper we describe the procedure of the Q-operators construction for the q-deformed model, described by the Lax operator, which is important to formulate the Bethe ansatz for the Sin-Gordon model. This Lax operator can also be considered as some massless limit of the Lax operator of SG model. We constructed two R-operators which are the universal intertwiners for the Lax operators. The traces of its monodromies over the auxiliary space are Baxter operators i.e. the operator solutions of T-Q equation. We also found the intertwining relations which imply the mutual commutativity of the corresponding Q-operators.
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