The bulk, surface and corner free energies of the square lattice Ising model

2016/06/30 by R. J. Baxter · 1 citation
Mathematics · Physics and Astronomy · #Anisotropy #Combinatorics #Condensed matter physics #Conjecture #Geometry #Ising model #Isotropy #Lattice (music) #Markov Chains and Monte Carlo Methods #Mathematical physics #Mathematics #Physics #Quantum mechanics #Square (algebra) #Square lattice #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech #math-ph #math.MP #msc:82B20

paper · pdf · doi:10.1088/1751-8113/50/1/014001

published as J. Phys. A:math. Theor. 50 (2017) 014001 (40pp) · 43 pages, 2 figures, paper amended to acknowledge previous work

arxiv created 2016/07/18 · openalex publication_date 2016/11/29 · arxiv updated 2020/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We use Kaufman's spinor method to calculate the bulk, surface and corner free energies f_b, f_s, f_s', f_c of the anisotropic square lattice zero-field Ising model for the ordered ferromagnetic case. For f_b, f_s, f'_s our results of course agree with the early work of Onsager, McCoy and Wu. We also find agreement with the conjectures made by Vernier and Jacobsen (VJ) for the isotropic case. We note that the corner free energy f_c depends only on the elliptic modulus k that enters the working, and not on the argument v, which means that VJ's conjecture applies for the full anisotropic model. The only aspect of this paper that is new is the actual derivation of f_c, but by reporting all four free energies together we can see interesting structures linking them.

Citations

• A unified treatment of Ising model magnetizations
• Surface Critical Phenomena and Scaling in the Eight-Vertex Model
• Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition
• Correlations and Spontaneous Magnetization of the Two-Dimensional Ising Model
• The Spontaneous Magnetization of a Two-Dimensional Ising Model
• Tables of Integrals, Series, and Products
• Surface and corner free energies of the self-dual Potts model
• Corner spontaneous magnetization

Cited by

• Square lattice Ising model [wikipedia]

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