Eight-vertex model and non-stationary Lamé equation

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 ∆ = − 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 ℘ -function where the modular parameter τ 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 Painlevé III equation. 1

The eight-vertex model and Painlevé VI

“…The conjecture has been verified by an explicit calculation [7] of P n (x, z) for all n ≤ 50. The polynomials P n (x, z) can be effectively calculated using the algebraic form (see [7]) of the equation (7).…”

“…The lattice analog of this continuous quantum field theory corresponds to a special case of Baxter' famous solvable eight-vertex lattice model [6]. In this paper we continue our study [7] of this special model on a finite lattice and unravel its deep connections with Painlevé VI theory.…”

“…are the meromorphic functions of the variable u for any fixed values of q and n. As shown in [7] these functions satisfy the non-stationary Lamé equation…”

“…With a suitable t-dependent normalization of Q ± (θ), Eq. (7) then reduces to the "non-stationary Mathieu equation" [7] t ∂ ∂t…”

Functional relations in the eight - vertex model

“…For an odd N the exponents (2.71) fall into this set only for certain rational values of η/π. A notable example is the case η = π/3, considered in [26][27][28]. The imaginary period relations in (2.67) and (2.68) certainly deserve a detailed consideration.…”