Eight-vertex model and non-stationary Lamé equation

Vladimir V. Bazhanov*, Vladimir V. Mangazeev

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    42 Citations (Scopus)


    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 Schrödinger 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.

    Original languageEnglish
    Pages (from-to)L145-L153
    JournalJournal of Physics A: Mathematical and General
    Issue number8
    Publication statusPublished - 25 Feb 2005


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