A new Q-matrix in the Eight-Vertex Model

Following Baxter's method of producing Q_{72}-operator, we construct the Q-operator of the root-of-unity eight-vertex model for the crossing parameter $\eta = \frac{2m K}{N}$ with odd $N$ where Q_{72} does not exist. We use this new Q-operator to study the functional relations in the Fabricius-McCoy comparison between the root-of-unity eight-vertex model and the superintegrable N-state chiral Potts model. By the compatibility of the constructed Q-operator with the structure o...

The Q matrix invented by Baxter in 1972 to solve the eight vertex model at roots of unity exists for all values of N, the number of sites in the chain, but only for a subset of roots of unity. We show in this paper that a new Q matrix, which has recently been introduced and is non zero only for N even, exists for all roots of unity. In addition we consider the relations between all of the known Q matrices of the eight vertex model and conjecture functional equations for them.

We study the transfer matrix of the 8 vertex model with an odd number of lattice sites $N.$ For systems at the root of unity points $\eta=mK/L$ with $m$ odd the transfer matrix is known to satisfy the famous ``$TQ$'' equation where ${\bf Q}(v)$ is a specifically known matrix. We demonstrate that the location of the zeroes of this ${\bf Q}(v)$ matrix is qualitatively different from the case of even $N$ and in particular they satisfy a previously unknown equation which is more ...

We demonstrate that the Q matrix introduced in Baxter's 1972 solution of the eight vertex model has some eigenvectors which are not eigenvectors of the spin reflection operator and conjecture a new functional equation for Q(v) which both contains the Bethe equation that gives the eigenvalues of the transfer matrix and computes the degeneracies of these eigenvalues.

The construction of creation operators of exact strings in eigenvectors of the eight vertex model at elliptic roots of unity of the crossing parameter which allow the generation of the complete set of degenerate eigenstates is based on the conjecture that the 'naive' string operator vanishes. In this note we present a proof of this conjecture. Furthermore we show that for chains of odd length the string operator is either proportional to the symmetry operator $S$ or vanishes ...

Whereas the tools to determine the eigenvalues of the eight-vertex transfer matrix T are well known there has been until recently incomplete knowledge about the eigenvectors of T. We describe the construction of eigenvectors of T corresponding to degenerate eigenvalues and discuss the related hidden elliptic symmetry.

We discuss an algebraic method for constructing eigenvectors of the transfer matrix of the eight vertex model at the discrete coupling parameters. We consider the algebraic Bethe ansatz of the elliptic quantum group $E_{\tau, \eta}(sl_2)$ for the case where the parameter $\eta$ satisfies $2 N \eta = m_1 + m_2 \tau $ for arbitrary integers $N$, $m_1$ and $m_2$. When $m_1$ or $m_2$ is odd, the eigenvectors thus obtained have not been discussed previously. Furthermore, we constr...

We extend our studies of the TQ equation introduced by Baxter in his 1972 solution of the 8 vertex model with parameter $\eta$ given by $2L\eta=2m_1K+im_2K'$ from $m_2=0$ to the more general case of complex $\eta.$ We find that there are several different cases depending on the parity of $m_1$ and $m_2$.

In this paper we continue the study of $Q$-operators in the six-vertex model and its higher spin generalizations. In [1] we derived a new expression for the higher spin $R$-matrix associated with the affine quantum algebra $U_q(\widehat{sl(2)})$. Taking a special limit in this $R$-matrix we obtained new formulas for the $Q$-operators acting in the tensor product of representation spaces with arbitrary complex spin. Here we use a different strategy and construct $Q$-operator...

We study the ground state eigenvalues of Baxter's Q-operator for the eight-vertex model in a special case when it describes the off-critical deformation of the $\Delta=-1/2$ six-vertex model. We show that these eigenvalues satisfy a non-stationary Schrodinger equation with the time-dependent potential given by the Weierstrass elliptic P-function where the modular parameter $\tau$ plays the role of (imaginary) time. In the scaling limit the equation transforms into a ``non-sta...