A master solution of the quantum Yang-Baxter equation and classical discrete integrable equations

Vladimir V. Bazhanov, Sergey Sergeev

    Research output: Contribution to journalArticle

    32 Citations (Scopus)

    Abstract

    We obtain a new solution of the star-triangle relation with positive Boltzmann weights, which contains as special cases all continuous and discrete spin solutions of this relation, that were previously known. This new master solution defines an exactly solvable two lattice model of statistical mechanics, which involves continuous spin variables, living on a circle, and contains two temperature-like parameters. If one of the these parameters approaches a root of unity (corresponds to zero temperature), the spin variables freezes into discrete positions, equidistantly spaced on the circle. An absolute orientation of these positions on the circle slowly changes between lattice sites by overall rotations. Allowed configurations of these rotations are described by classical discrete integrable equations, closely related to the famous Q4-equations by Adler, Bobenko and Suris. Fluctuations between degenerate ground states in the vicinity of zero temperature are described by a rather general integrable lattice model with discrete spin variables. In some simple special cases, the latter reduces to the Kashiwara-Miwa and chiral Potts models. © 2012 International Press.
    Original languageEnglish
    Pages (from-to)65-96
    Number of pages32
    JournalAdvances in Theoretical and Mathematical Physics
    Volume16
    Issue number1
    DOIs
    Publication statusPublished - Jan 2012

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    Yang-Baxter Equation
    Integrable Equation
    Discrete Equations
    Circle
    Lattice Model
    Chiral Potts Model
    Integrable Models
    Zero
    Roots of Unity
    statistical mechanics
    Ludwig Boltzmann
    Statistical Mechanics
    triangles
    Ground State
    temperature
    unity
    Triangle
    Star
    Fluctuations
    stars

    Cite this

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    abstract = "We obtain a new solution of the star-triangle relation with positive Boltzmann weights, which contains as special cases all continuous and discrete spin solutions of this relation, that were previously known. This new master solution defines an exactly solvable two lattice model of statistical mechanics, which involves continuous spin variables, living on a circle, and contains two temperature-like parameters. If one of the these parameters approaches a root of unity (corresponds to zero temperature), the spin variables freezes into discrete positions, equidistantly spaced on the circle. An absolute orientation of these positions on the circle slowly changes between lattice sites by overall rotations. Allowed configurations of these rotations are described by classical discrete integrable equations, closely related to the famous Q4-equations by Adler, Bobenko and Suris. Fluctuations between degenerate ground states in the vicinity of zero temperature are described by a rather general integrable lattice model with discrete spin variables. In some simple special cases, the latter reduces to the Kashiwara-Miwa and chiral Potts models. {\circledC} 2012 International Press.",
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    A master solution of the quantum Yang-Baxter equation and classical discrete integrable equations. / Bazhanov, Vladimir V.; Sergeev, Sergey.

    In: Advances in Theoretical and Mathematical Physics, Vol. 16, No. 1, 01.2012, p. 65-96.

    Research output: Contribution to journalArticle

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    AB - We obtain a new solution of the star-triangle relation with positive Boltzmann weights, which contains as special cases all continuous and discrete spin solutions of this relation, that were previously known. This new master solution defines an exactly solvable two lattice model of statistical mechanics, which involves continuous spin variables, living on a circle, and contains two temperature-like parameters. If one of the these parameters approaches a root of unity (corresponds to zero temperature), the spin variables freezes into discrete positions, equidistantly spaced on the circle. An absolute orientation of these positions on the circle slowly changes between lattice sites by overall rotations. Allowed configurations of these rotations are described by classical discrete integrable equations, closely related to the famous Q4-equations by Adler, Bobenko and Suris. Fluctuations between degenerate ground states in the vicinity of zero temperature are described by a rather general integrable lattice model with discrete spin variables. In some simple special cases, the latter reduces to the Kashiwara-Miwa and chiral Potts models. © 2012 International Press.

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