Traces on the Sklyanin algebra and correlation functions of the eight-vertex model

TL;DR

This paper proposes a conjectural formula for correlation functions in the eight-vertex model, linking algebraic structures to physical observables and providing new explicit formulas for certain correlations.

Contribution

It introduces an Ansatz connecting correlation functions to Sklyanin algebra functionals and offers a new formula for next-nearest neighbor correlations.

Findings

Existence of a linear functional on the Sklyanin algebra is established.

The Ansatz relates correlation functions to classical geometric structures.

A new explicit formula for next-nearest neighbor correlations is derived.

Abstract

We propose a conjectural formula for correlation functions of the Z-invariant (inhomogeneous) eight-vertex model. We refer to this conjecture as Ansatz. It states that correlation functions are linear combinations of products of three transcendental functions, with theta functions and derivatives as coefficients. The transcendental functions are essentially logarithmic derivatives of the partition function per site. The coefficients are given in terms of a linear functional on the Sklyanin algebra, which interpolates the usual trace on finite dimensional representations. We establish the existence of the functional and discuss the connection to the geometry of the classical limit. We also conjecture that the Ansatz satisfies the reduced qKZ equation. As a non-trivial example of the Ansatz, we present a new formula for the next-nearest neighbor correlation functions.