Eight-vertex model and non-stationary Lame equation

TL;DR

This paper investigates the eigenvalues of Baxter's Q-operator for the eight-vertex model, revealing they satisfy a non-stationary Schrödinger equation with elliptic potential, connecting to sine-Gordon and Painleve III equations.

Contribution

It establishes a novel link between the eigenvalues of the Q-operator in the eight-vertex model and non-stationary Schrödinger equations with elliptic potentials, extending understanding of integrable models.

Findings

Eigenvalues satisfy a non-stationary Schrödinger equation with Weierstrass elliptic potential.

In the scaling limit, the equation reduces to a non-stationary Mathieu equation.

Connections to sine-Gordon model, dilute polymers, and Painleve III are demonstrated.

Abstract

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-stationary Mathieu equation'' for the vacuum eigenvalues of the Q-operators in the finite-volume massive sine-Gordon model at the super-symmetric point, which is closely related to the theory of dilute polymers on a cylinder and the Painleve III equation.