What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution

Viswanathan, Gandhimohan M.; Portillo, Marco Aurelio G.; Raposo, Ernesto P.; da Luz, Marcos G. E.

doi:10.3390/e24111665

Open AccessArticle

What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution

by

Gandhimohan M. Viswanathan

^1,2,*

,

Marco Aurelio G. Portillo

^1,3,

Ernesto P. Raposo

⁴

and

Marcos G. E. da Luz

³

¹

Department of Physics, Federal University of Rio Grande do Norte, Natal 59078-970, RN, Brazil

²

National Institute of Science and Technology of Complex Systems, Federal University of Rio Grande do Norte, Natal 59078-970, RN, Brazil

³

Departmento de Física, Universidade Federal do Paraná, Curitiba 81531-980, PR, Brazil

⁴

Laboratório de Física Teórica e Computacional, Departamento de Física, Universidade Federal de Pernambuco, Recife 50670-901, PE, Brazil

^*

Author to whom correspondence should be addressed.

Entropy 2022, 24(11), 1665; https://doi.org/10.3390/e24111665

Submission received: 10 October 2022 / Revised: 3 November 2022 / Accepted: 8 November 2022 / Published: 15 November 2022

(This article belongs to the Special Issue Ising Model: Recent Developments and Exotic Applications II)

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Abstract

An exact solution of the Ising model on the simple cubic lattice is one of the long-standing open problems in rigorous statistical mechanics. Indeed, it is generally believed that settling it would constitute a methodological breakthrough, fomenting great prospects for further application, similarly to what happened when Lars Onsager solved the two-dimensional model eighty years ago. Hence, there have been many attempts to find analytic expressions for the exact partition function Z, but all such attempts have failed due to unavoidable conceptual or mathematical obstructions. Given the importance of this simple yet paradigmatic model, here we set out clear-cut criteria for any claimed exact expression for Z to be minimally plausible. Specifically, we present six necessary—but not sufficient—conditions that Z must satisfy. These criteria will allow very quick plausibility checks of future claims. As illustrative examples, we discuss previous mistaken “solutions”, unveiling their shortcomings.

Keywords: 3D Ising model; exactly solvable models

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MDPI and ACS Style

Viswanathan, G.M.; Portillo, M.A.G.; Raposo, E.P.; da Luz, M.G.E. What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution. Entropy 2022, 24, 1665. https://doi.org/10.3390/e24111665

AMA Style

Viswanathan GM, Portillo MAG, Raposo EP, da Luz MGE. What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution. Entropy. 2022; 24(11):1665. https://doi.org/10.3390/e24111665

Chicago/Turabian Style

Viswanathan, Gandhimohan M., Marco Aurelio G. Portillo, Ernesto P. Raposo, and Marcos G. E. da Luz. 2022. "What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution" Entropy 24, no. 11: 1665. https://doi.org/10.3390/e24111665

APA Style

Viswanathan, G. M., Portillo, M. A. G., Raposo, E. P., & da Luz, M. G. E. (2022). What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution. Entropy, 24(11), 1665. https://doi.org/10.3390/e24111665

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution

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