Quantum geometry of three-dimensional lattices

Bazhanov Vladimir, Vladimir Mangazeev, Sergey Sergeev

    Research output: Contribution to journalArticle

    33 Citations (Scopus)

    Abstract

    We study geometric consistency relations between angles on three-dimensional (3D) circular quadrilateral lattices—lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable 'ultra-local' Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure leads to new solutions of the tetrahedron equation (the 3D analog of the Yang–Baxter equation). These solutions generate an infinite number of non-trivial solutions of the Yang–Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry. The classical geometry of the 3D circular lattices arises as a stationary configuration giving the leading contribution to the partition function in the quasi-classical limit
    Original languageEnglish
    Pages (from-to)1-27
    Number of pages27
    JournalJournal of Statistical Mechanics: Theory and Experiment
    Volume7
    DOIs
    Publication statusPublished - 2 Jul 2008

    Fingerprint

    Yang-Baxter Equation
    Three-dimensional
    geometry
    Integrable Models
    Canonical Transformation
    Quantum Fluctuations
    Classical Limit
    Poisson Bracket
    Triangular pyramid
    brackets
    Nontrivial Solution
    Quantum Field Theory
    statistical mechanics
    Statistical Mechanics
    3D Model
    Partition Function
    tetrahedrons
    partitions
    Quantization
    algebra

    Cite this

    Vladimir, Bazhanov ; Mangazeev, Vladimir ; Sergeev, Sergey. / Quantum geometry of three-dimensional lattices. In: Journal of Statistical Mechanics: Theory and Experiment. 2008 ; Vol. 7. pp. 1-27.
    @article{29e92b82d41e48c48da553e218bbb263,
    title = "Quantum geometry of three-dimensional lattices",
    abstract = "We study geometric consistency relations between angles on three-dimensional (3D) circular quadrilateral lattices—lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable 'ultra-local' Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure leads to new solutions of the tetrahedron equation (the 3D analog of the Yang–Baxter equation). These solutions generate an infinite number of non-trivial solutions of the Yang–Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry. The classical geometry of the 3D circular lattices arises as a stationary configuration giving the leading contribution to the partition function in the quasi-classical limit",
    keywords = "classical integrability, integrable quantum field theory, quantum integrability (Bethe ansatz), solvable lattice models",
    author = "Bazhanov Vladimir and Vladimir Mangazeev and Sergey Sergeev",
    year = "2008",
    month = "7",
    day = "2",
    doi = "10.1088/1742-5468/2008/07/P07004",
    language = "English",
    volume = "7",
    pages = "1--27",
    journal = "Journal of Statistical Mechanics: Theory and Experiment",
    issn = "1742-5468",
    publisher = "IOP Publishing Ltd.",

    }

    Quantum geometry of three-dimensional lattices. / Vladimir, Bazhanov; Mangazeev, Vladimir; Sergeev, Sergey.

    In: Journal of Statistical Mechanics: Theory and Experiment, Vol. 7, 02.07.2008, p. 1-27.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - Quantum geometry of three-dimensional lattices

    AU - Vladimir, Bazhanov

    AU - Mangazeev, Vladimir

    AU - Sergeev, Sergey

    PY - 2008/7/2

    Y1 - 2008/7/2

    N2 - We study geometric consistency relations between angles on three-dimensional (3D) circular quadrilateral lattices—lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable 'ultra-local' Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure leads to new solutions of the tetrahedron equation (the 3D analog of the Yang–Baxter equation). These solutions generate an infinite number of non-trivial solutions of the Yang–Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry. The classical geometry of the 3D circular lattices arises as a stationary configuration giving the leading contribution to the partition function in the quasi-classical limit

    AB - We study geometric consistency relations between angles on three-dimensional (3D) circular quadrilateral lattices—lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable 'ultra-local' Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure leads to new solutions of the tetrahedron equation (the 3D analog of the Yang–Baxter equation). These solutions generate an infinite number of non-trivial solutions of the Yang–Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry. The classical geometry of the 3D circular lattices arises as a stationary configuration giving the leading contribution to the partition function in the quasi-classical limit

    KW - classical integrability

    KW - integrable quantum field theory

    KW - quantum integrability (Bethe ansatz)

    KW - solvable lattice models

    U2 - 10.1088/1742-5468/2008/07/P07004

    DO - 10.1088/1742-5468/2008/07/P07004

    M3 - Article

    VL - 7

    SP - 1

    EP - 27

    JO - Journal of Statistical Mechanics: Theory and Experiment

    JF - Journal of Statistical Mechanics: Theory and Experiment

    SN - 1742-5468

    ER -