Weak Universality and Birational Geometry in the Eight-Vertex Model

2014 Fiscal Year Final Research Report

• Project/Area Number: 25610108
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Mathematical physics/Fundamental condensed matter physics
• Research Institution: Nara Medical University
• Principal Investigator: FUJIMOTO Masafumi 奈良県立医科大学, 医学部, 准教授 (30261176)
• Co-Investigator(Renkei-kenkyūsha): OTSUKA Hiromi 首都大学東京, 大学院理工学研究科, 助教 (10254145)
• Research Collaborator: FUJIMOTO Yoshio 奈良県立医科大学, 医学部, 教授 (90192731)
• Project Period (FY): 2013-04-01 – 2015-03-31
• Keywords: 統計力学 / 数理物理 / 格子模型 / 相関関数 / 平衡形 / 代数曲線 / シミュレーション

Outline of Final Research Achievements

In previous studies it was shown that asymptotic behavior of the (two-point) correlation function is expressed in terms of simple algebraic curves of genus 1 for two-dimensional　solvable models.
Reexamining the eight-vertex model, we performed Monte Carlo simulations to investigate the correlation function of the Potts model; note that at the phase transition point the model is equivalent to the six-vertex model. Above the phase transition point we found essentially the same algebraic curves as the eight-vertex model. It was strongly suggested that the algebraic curves are quite general ones which represent the correlation functions of a wide class of lattice models, including unsolvable ones.
It was shown that the algebraic curves can be derived from the point group C_4v or C_6v. We pointed out a relation between the birational geometry and the (weak) universality hypothesis of critical phenomena.

Free Research Field

数理物理・物性基礎