### Abstract

Original language | English |
---|---|

Pages (from-to) | 1-27 |

Number of pages | 27 |

Journal | Journal of Statistical Mechanics: Theory and Experiment |

Volume | 7 |

DOIs | |

Publication status | Published - 2 Jul 2008 |

### Fingerprint

### Cite this

*Journal of Statistical Mechanics: Theory and Experiment*,

*7*, 1-27. https://doi.org/10.1088/1742-5468/2008/07/P07004

}

*Journal of Statistical Mechanics: Theory and Experiment*, vol. 7, pp. 1-27. https://doi.org/10.1088/1742-5468/2008/07/P07004

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

Research output: Contribution to journal › Article

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 -