Correlations and Spontaneous Magnetization of the Two-Dimensional Ising Model

1963/02/01 by Elliott W. Montroll, Renfrey B. Potts, John Ward · 5 citations
Physics and Astronomy · Economics, Econometrics and Finance · Computer Science · Mathematics · #Theoretical and Computational Physics #Complex Systems and Time Series Analysis #Topological and Geometric Data Analysis #Pfaffian #Toeplitz matrix #Ising model #Spontaneous magnetization #Magnetization #Mathematics #Generalization #Limiting #Lattice (music) #Mathematical physics #Statistical physics #Condensed matter physics #Physics #Quantum mechanics #Pure mathematics #Mathematical analysis #Magnetic field

paper · doi:10.1063/1.1703955

openalex publication_date 1963/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

In this paper we rederive, by simpler methods, the Onsager-Kaufman formulas for the correlations and the Onsager formula for the spontaneous magnetization of the rectangular two-dimensional Ising lattice. The Pfaffian approach is used to derive the correlations in terms of Pfaffians, and for the correlations in a row a single Toeplitz determinant is obtained which is proved equivalent to the Onsager-Kaufman result. The spontaneous magnetization is obtained as the limiting value of an infinite Toeplitz determinant and its evaluation is facilitated by use of the generalization of a result first published by Szego.

Citations

• Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition
• The Spontaneous Magnetization of a Two-Dimensional Ising Model
• Statistical hydrodynamics

Cited by

• The bulk, surface and corner free energies of the square lattice Ising model
• Local statistics of lattice dimers
• History of the Lenz-Ising Model
• Ising model [wikipedia]
• Square lattice Ising model [wikipedia]