Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs

Vladimir V. Bazhanov, Andrew Kels, Sergey Sergeev

    Research output: Contribution to journalArticle

    12 Citations (Scopus)

    Abstract

    In this paper we give an overview of exactly solved edge-interaction models, where the spins are placed on sites of a planar lattice and interact through edges connecting the sites. We only consider the case of a single spin degree of freedom at each site of the lattice. The Yang-Baxter equation for such models takes a particular simple form called the star-triangle relation. Interestigly all known solutions of this relation can be obtained as particular cases of a single 'master solution', which is expressed through the elliptic gamma function and have continuous spins taking values on the circle. We show that in the low-temperature (or quasi-classical) limit these lattice models reproduce classical discrete integrable systems on planar graphs previously obtained and classified by Adler, Bobenko and Suris through the consistency-around-a-cube approach. We also discuss inversion relations, the physicical meaning of Baxter's rapidity-independent parameter in the star-triangle relations and the invariance of the action of the classical systems under the star-triangle (or cube-flip) transformation of the lattice, which is a direct consequence of Baxter's Z-invariance in the associated lattice models.
    Original languageEnglish
    Article number464001
    JournalJournal of Physics A: Mathematical and Theoretical
    Volume49
    Issue number46
    DOIs
    Publication statusPublished - 21 Oct 2016

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    Integrable Systems
    triangles
    Stars
    Triangle
    Star
    stars
    expansion
    Graph in graph theory
    Invariance
    Lattice Model
    Regular hexahedron
    invariance
    Yang-Baxter Equation
    Gamma function
    Classical Limit
    Elliptic function
    gamma function
    Flip
    Discrete Systems
    Planar graph

    Cite this

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    title = "Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs",
    abstract = "In this paper we give an overview of exactly solved edge-interaction models, where the spins are placed on sites of a planar lattice and interact through edges connecting the sites. We only consider the case of a single spin degree of freedom at each site of the lattice. The Yang-Baxter equation for such models takes a particular simple form called the star-triangle relation. Interestigly all known solutions of this relation can be obtained as particular cases of a single 'master solution', which is expressed through the elliptic gamma function and have continuous spins taking values on the circle. We show that in the low-temperature (or quasi-classical) limit these lattice models reproduce classical discrete integrable systems on planar graphs previously obtained and classified by Adler, Bobenko and Suris through the consistency-around-a-cube approach. We also discuss inversion relations, the physicical meaning of Baxter's rapidity-independent parameter in the star-triangle relations and the invariance of the action of the classical systems under the star-triangle (or cube-flip) transformation of the lattice, which is a direct consequence of Baxter's Z-invariance in the associated lattice models.",
    keywords = "3D consistency, classical discrete evolution equations, exactly solved lattice models, star-triangle relation, Yang-Baxter equation",
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    Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs. / Bazhanov, Vladimir V.; Kels, Andrew; Sergeev, Sergey.

    In: Journal of Physics A: Mathematical and Theoretical, Vol. 49, No. 46, 464001, 21.10.2016.

    Research output: Contribution to journalArticle

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    AU - Kels, Andrew

    AU - Sergeev, Sergey

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