Auxiliary transfer matrices for three-dimensional integrable models

Hector J. de Vega

Christian Korff

The construction of auxiliary matrices for the six-vertex model at a root of unity is investigated from a quantum group theoretic point of view. Employing the concept of intertwiners associated with the quantum loop algebra U q ( 2 ) at q N = 1, a three-parameter family of auxiliary matrices is constructed. The elements of this family satisfy a functional relation with the transfer matrix allowing one to solve the eigenvalue problem of the model and to derive the Bethe ansatz equations. This functional relation is obtained from the decomposition of a tensor product of evaluation representations and involves auxiliary matrices with different parameters. Because of this dependence on additional parameters, the auxiliary matrices break in general the finite symmetries of the six-vertex model, such as spin-reversal or spin-conservation. More importantly, they also lift the extra degeneracies of the transfer matrix due to the loop symmetry present at rational coupling values. The extra parameters in the auxiliary matrices are shown to be directly related to the elements in the enlarged centre Z of the algebra U q ( 2 ) at q N = 1. This connection provides a geometric interpretation of the enhanced symmetry of the six-vertex model at rational coupling. The parameters labelling the auxiliary matrices can be interpreted as coordinates on a hypersurface Spec Z ⊂ ℂ 4 which remains invariant under the action of an infinite-dimensional group G of analytic transformations, called the quantum coadjoint action.

H. Meyer, J C Anglès d'Auriac

We present a statistical analysis of spectra of transfer matrices of classical lattice spin models; this continues the work on the eight-vertex model of the preceding paper [H. Meyer, J.-C. Angl`es d'Auriac, and J.-M. Maillard, Phys. Rev. E 55, 5261 (1997)]. We show that the statistical properties of these spectra can serve as a criterion of integrability. It also provides an operational numerical method to locate integrable varieties. In particular, we distinguish the notions of integrability and criticality, considering the two examples of the three-dimensional Ising critical point and the two-dimensional three-state Potts critical point. For complex spectra, which appear frequently in the context of transfer matrices, we show that the notion of independence of eigenvalues for integrable models still holds.

JUSSI TIMONEN, Yu Chen, R. Bullough

А. А. Овчинников

We obtain the Baxter Q-operators in the Uq(slˆ2) invariant integrable models as a special limits of the quantum transfer matrices corresponding to different spins in the auxiliary space. We derive the Baxter equation from the well-known fusion relations for the transfer matrices. Our method is valid for an arbitrary integrable model corresponding to the quantum group Uq(slˆ2), for example for the XXZ-spin chain.

Costas John Efthimiou