[Paper Review] The Elliptic Function in Statistical Integrable Models II

Kazuyasu Shigemoto|arXiv (Cornell University)|Feb 27, 2013

Algebraic structures and combinatorial models4 citations

TL;DR

This paper establishes a deep connection between the SU(2) group structure and the elliptic function parameterization in the 2D Ising model, showing that the Yang-Baxter equation's integrability condition arises from SU(2) symmetry and that the addition formula of elliptic functions underlies the model's exact solvability. The parameterization via elliptic functions (sn, cn, dn) naturally yields the difference property essential for exact solvability.

ABSTRACT

Two dimensional statistical integrable models, such as the Ising model, the chiral Potts model and the Belavin model, becomes integrable. Because of the SU(2) symmetry of these models, these models become integrable. The integral models are often parameterized by the elliptic function or the elliptic theta function. In this paper, we study the Ising model, and we show that the Yang-Baxter equation of the Ising model can be written as the integrability condition of SU(2), and we gives the natural explanation why the Ising model can be parameterized by the elliptic function by connecting the spherical trigonometry relation and the elliptic function. Addition formula of the elliptic function is the secret of the exact solvability of the Ising model.

Motivation & Objective

• To explain why the Ising model's Boltzmann weights are parameterized by elliptic functions using group-theoretic principles.
• To establish a geometric and algebraic link between spherical trigonometry and the addition formulas of elliptic functions.
• To demonstrate that the difference property in the Yang-Baxter equation—key to exact solvability—arises from SU(2) symmetry and elliptic function identities.
• To derive conserved quantities for hyperelliptic differential equations, generalizing Abel’s addition theorem.
• To provide a natural explanation for the appearance of elliptic functions in integrable models by connecting them to SU(2) group structure.

Proposed method

• Derives the Yang-Baxter equation for the Ising model in terms of SU(2) group generators (σz, σx), showing it satisfies the integrability condition.
• Uses the parameterization: cosh(2Ki) = 1/cn(ui), sinh(2Ki) = sn(ui)/cn(ui), and analogous expressions for L* using elliptic functions.
• Establishes the difference property in the Yang-Baxter equation by expressing parameters as u1 + u3, enabling exact solvability.
• Connects spherical trigonometry (via cosine laws on the 2-sphere) to elliptic function addition theorems using differential equations.
• Applies variable transformation (xi → 1/xi) to hyperelliptic integrals to derive two conserved quantities, generalizing Abel’s addition theorem.
• Validates the n=3 case numerically using symbolic computation (Maxima), confirming the addition formulas for sn, cn, dn with u1+u2+u3=0.

Experimental results

Research questions

• RQ1 Why does the Ising model admit a parameterization in terms of elliptic functions?
• RQ2 How is the difference property in the Yang-Baxter equation—essential for exact solvability—related to SU(2) symmetry?
• RQ3 What is the geometric origin of the addition formula of elliptic functions in the context of statistical mechanics?
• RQ4 Can conserved quantities for hyperelliptic integrals be derived from the structure of the Yang-Baxter equation?
• RQ5 How does the spherical trigonometry of the 2-sphere relate to the algebraic structure of elliptic functions in integrable models?

Key findings

• The Yang-Baxter equation for the Ising model is shown to be equivalent to the integrability condition of the SU(2) group, providing a group-theoretic foundation for its solvability.
• The parameterization of the Ising model’s Boltzmann weights using elliptic functions (sn, cn, dn) naturally leads to the difference property, enabling exact solvability.
• The addition formula of the elliptic function is identified as the fundamental reason behind the exact solvability of the Ising model.
• For the n=3 case, the paper confirms the identity ∑(xi√f4(xi)/F′(xi)) + k∑xi = 0 with xi = sn(ui), u1+u2+u3=0, validating the addition theorem.
• Two conserved quantities are derived for hyperelliptic differential equations: one in terms of xi and another in terms of ξi = 1/xi, generalizing Abel’s addition theorem.
• The transformation from xi to ξi yields a new conserved quantity involving ∑(1/xi) and ∑(1/xi)², with coefficients tied to the polynomial structure of f4(x).