A new Q-matrix in the Eight-Vertex Model

We observe that the exactly solved eight-vertex solid-on-solid model contains an hitherto unnoticed arbitrary field parameter, similar to the horizontal field in the six-vertex model. The parameter is required to describe a continuous spectrum of the unrestricted solid-on-solid model, which has an infinite-dimensional space of states even for a finite lattice. The introduction of the continuous field parameter allows us to completely review the theory of functional relations ...

In this paper we investigate the integrability properties of a two-state vertex model on the square lattice whose microstates at a vertex has always an odd number of incoming or outcoming arrows. This model was named odd eight-vertex model by Wu and Kunz \cite{WK} to distinguish it from the well known eight-vertex model possessing an even number of arrows orientations at each vertex. When the energy weights are invariant under arrows inversion we show that the integrable mani...

The transfer matrix of the square-lattice eight-vertex model on a strip with $L\geqslant 1$ vertical lines and open boundary conditions is investigated. It is shown that for vertex weights $a,b,c,d$ that obey the relation $(a^2+ab)(b^2+ab)=(c^2+ab)(d^2+ab)$ and appropriately chosen $K$-matrices $K^\pm$ this transfer matrix possesses the remarkably simple, non-degenerate eigenvalue $\Lambda_L = (a+b)^{2L}\,\text{tr}(K^+K^-)$. For positive vertex weights, $\Lambda_L$ is shown t...

We construct the Q-operator for generalised eight vertex models associated to higher spin representations of the Sklyanin algebra, following Baxter's 1973 paper. As an application, we prove the sum rule for the Bethe roots.

We study statistical models, specifically transfer matrices corresponding to a multiparameter hierarchy of braid matrices of $(2n)^2\times(2n)^2$ dimensions with $2n^2$ free parameters $(n=1,2,3,...)$. The simplest, $4\times 4$ case is treated in detail. Powerful recursion relations are constructed giving the dependence on the spectral parameter $\theta$ of the eigenvalues of the transfer matrix explicitly at each level of coproduct sequence. A brief study of higher dimension...

We derive sets of functional equations for the eight vertex model by exploiting an analogy with the functional equations of the chiral Potts model. From these equations we show that the fusion matrices have special reductions at certain roots of unity. We explicitly exhibit these reductions for the 3,4 and 5 order fusion matrices and compare our formulation with the algebra of Sklyanin.

Baxter's TQ-equation is solved for the six-vertex model using the representation theory of quantum groups at roots of unity. A novel simplified construction of the Q-operator is given depending on a new free parameter. Specializing this general construction to even roots of unity and lattices with an odd number of columns two linearly independent operator solutions of Baxter's TQ equation are obtained as special limits.

We construct the explicit $Q$-operator incorporated with the $sl_2$-loop-algebra symmetry of the six-vertex model at roots of unity. The functional relations involving the $Q$-operator, the six-vertex transfer matrix and fusion matrices are derived from the Bethe equation, parallel to the Onsager-algebra-symmetry discussion in the superintegrable $N$-state chiral Potts model. We show that the whole set of functional equations is valid for the $Q$-operator. Direct calculations...

In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix of $U_q(\hat{sl}_2)$ over an infinite-dimensional auxiliary space. This auxiliary space is a four-parameter generalization of the q-oscillator representations used previously. We derive generalized T-Q relations in which 3 of these parameters shift. After a suitable restriction of parameters, we give an explicit expression for the Q-operator of the 6-vertex model and show the con...

A summary of the construction procedure of generalized versions of Baxter's Q-operator is given. Illustrated by several figures and diagrams the use of representation theory is explained step-by-step avoiding technical details. The relation with the infinite-dimensional non-abelian symmetries of the six-vertex model at roots of unity is discussed and parallels with the eight-vertex case outlined.