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Eight-vertex model and non-stationary Lame equation

Vladimir V. Bazhanov·Vladimir V. Mangazeev
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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 $Δ=-1/2$ 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 $τ$ 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 Painleve III equation.

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Vladimir V. Bazhanov, Vladimir V. Mangazeev. 2004-11-26. Eight-vertex model and non-stationary Lame equation. https://doi.org/10.1088/0305-4470%2F38%2F8%2Fl01

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