{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T09:26:33Z","timestamp":1762075593794,"version":"build-2065373602"},"reference-count":62,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,15]],"date-time":"2022-11-15T00:00:00Z","timestamp":1668470400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CNPq","award":["304532\/2019-3","305062\/2017-4","302051\/2018-0"],"award-info":[{"award-number":["304532\/2019-3","305062\/2017-4","302051\/2018-0"]}]},{"name":"CAPES","award":["304532\/2019-3","305062\/2017-4","302051\/2018-0"],"award-info":[{"award-number":["304532\/2019-3","305062\/2017-4","302051\/2018-0"]}]},{"name":"FACEPE","award":["304532\/2019-3","305062\/2017-4","302051\/2018-0"],"award-info":[{"award-number":["304532\/2019-3","305062\/2017-4","302051\/2018-0"]}]},{"name":"Brazilian National Institute of Science and Technology of Complex Systems","award":["304532\/2019-3","305062\/2017-4","302051\/2018-0"],"award-info":[{"award-number":["304532\/2019-3","305062\/2017-4","302051\/2018-0"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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\u2014but not sufficient\u2014conditions that Z must satisfy. These criteria will allow very quick plausibility checks of future claims. As illustrative examples, we discuss previous mistaken \u201csolutions\u201d, unveiling their shortcomings.<\/jats:p>","DOI":"10.3390\/e24111665","type":"journal-article","created":{"date-parts":[[2022,11,16]],"date-time":"2022-11-16T02:27:34Z","timestamp":1668565654000},"page":"1665","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2301-5593","authenticated-orcid":false,"given":"Gandhimohan M.","family":"Viswanathan","sequence":"first","affiliation":[{"name":"Department of Physics, Federal University of Rio Grande do Norte, Natal 59078-970, RN, Brazil"},{"name":"National Institute of Science and Technology of Complex Systems, Federal University of Rio Grande do Norte, Natal 59078-970, RN, Brazil"}]},{"given":"Marco Aurelio G.","family":"Portillo","sequence":"additional","affiliation":[{"name":"Department of Physics, Federal University of Rio Grande do Norte, Natal 59078-970, RN, Brazil"},{"name":"Departmento de F\u00edsica, Universidade Federal do Paran\u00e1, Curitiba 81531-980, PR, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4178-8266","authenticated-orcid":false,"given":"Ernesto P.","family":"Raposo","sequence":"additional","affiliation":[{"name":"Laborat\u00f3rio de F\u00edsica Te\u00f3rica e Computacional, Departamento de F\u00edsica, Universidade Federal de Pernambuco, Recife 50670-901, PE, Brazil"}]},{"given":"Marcos G. E.","family":"da Luz","sequence":"additional","affiliation":[{"name":"Departmento de F\u00edsica, Universidade Federal do Paran\u00e1, Curitiba 81531-980, PR, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,15]]},"reference":[{"key":"ref_1","unstructured":"Gershenfeld, N. (1998). The Nature of Mathematical Modeling, Cambridge University Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1090\/noti936","article-title":"Closed forms: What they are and why we care","volume":"60","author":"Borwein","year":"2013","journal-title":"Not. Am. Math. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1007\/BF02980577","article-title":"Beitrag zur Theorie des Ferromagnetismus","volume":"31","author":"Ising","year":"1925","journal-title":"Z. Physik"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Tranquillo, J. (2019). An Introduction to Complex Systems: Making Sense of a Changing World, Springer.","DOI":"10.1007\/978-3-030-02589-2"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1103\/PhysRev.65.117","article-title":"Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition","volume":"65","author":"Onsager","year":"1944","journal-title":"Phys. Rev."},{"key":"ref_6","unstructured":"Kotecky, R. (2009). Methods of Contemporary Mathematical Statistical Physics, Springer."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Thompson, C.J. (2015). Mathematical Statistical Mechanics, Princeton University Press.","DOI":"10.1515\/9781400868681"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Sutherland, B. (2005). Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems, World Scientific.","DOI":"10.1142\/5552"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Mattis, D.C. (1993). The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension, World Scientific.","DOI":"10.1142\/1666"},{"key":"ref_10","unstructured":"Lieb, E.H., and Mattis, D.C. (1966). Mathematical Physics in One Dimension, Academic Press Inc."},{"key":"ref_11","unstructured":"Sivasankaran, S., Nayak, P.K., and G\u00fcnay, E. (2020). The Ising Model: Brief Introduction and Its Application. Solid State Physics\u2014Metastable, Spintronics Materials and Mechanics of Deformable Bodies\u2014Recent Progress, IntechOpen."},{"key":"ref_12","unstructured":"Rowlinson, J.S. (2004). On the Continuity of the Gaseous and Liquid States, Dover."},{"key":"ref_13","unstructured":"Maddox, J.R. (1952). Talk Presented at StatPhys 2, Paris, France."},{"key":"ref_14","unstructured":"Maddox, J.R. (1952). Changement de Phases Societ\u00e9 de Chimie Physique, Presses Universitaires de France."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1080\/14786430902776970","article-title":"Comment on \u2018Conjectures on exact solution of three-dimensional (3D) simple orthorhombic Ising lattices\u2019","volume":"89","author":"Perk","year":"2009","journal-title":"Philos. Mag."},{"key":"ref_16","first-page":"244","article-title":"Exact Partition Function of Ising Model in Magnetism in One, Two and Three Dimensions in Nonzero Field","volume":"44","author":"Das","year":"1970","journal-title":"Indian J. Phys."},{"key":"ref_17","first-page":"841","article-title":"Three-dimensional Ising model and transfer matrices","volume":"38","author":"Lou","year":"2000","journal-title":"Chin. J. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"5309","DOI":"10.1080\/14786430701646325","article-title":"Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices","volume":"87","author":"Zhang","year":"2007","journal-title":"Philos. Mag."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Zhang, D. (2021). Exact solution for three-Dimensional Ising model. Symmetry, 13.","DOI":"10.3390\/sym13101837"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"853","DOI":"10.1086\/323419","article-title":"The Ising model in physics and statistical genetics","volume":"69","author":"Majewski","year":"2001","journal-title":"Am. J. Hum. Gen."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Binek, C. (2003). Ising-Type Antiferromagnets, Springer.","DOI":"10.1007\/b10726"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Suzuki, S., Inoue, J.-I., and Chakrabarti, B.K. (2013). Quantum Ising Phases and Transitions in Transverse Ising Models, Springer. [2nd ed.].","DOI":"10.1007\/978-3-642-33039-1"},{"key":"ref_23","unstructured":"McCoy, B.M., and Wu, T.T. (2014). The Two-Dimensional Ising Model, Dover. [2nd ed.]."},{"key":"ref_24","unstructured":"Adler, M. (2016). Monte Carlo Simulations of the Ising Model, Anchor Academic Publishing."},{"key":"ref_25","unstructured":"Fadil, Z. (2020). Semi-Infinite Ising Model by the Renormalization Group: Applicable in Nanotechnology and Spintronics, Lap Lambert Academic Publishing."},{"key":"ref_26","first-page":"2229","article-title":"The applications of the Ising model in statistical thermodynamics and quantum mechanics","volume":"8","author":"Farah","year":"2020","journal-title":"Eur. Acad. Res."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Lipowski, A. (2022). Special Issue \u201cIsing Model: Recent Developments and Exotic Applications\u201d. Entropy, Available online: https:\/\/www.mdpi.com\/journal\/entropy\/special_issues\/Ising_Model.","DOI":"10.3390\/e24121834"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1180","DOI":"10.1126\/science.aab3326","article-title":"Simple universal models capture all classical spin physics","volume":"351","author":"Cubitt","year":"2016","journal-title":"Science"},{"key":"ref_29","unstructured":"Polyakov, A.M. (1987). Gauge Fields and Strings, Harwood Academic Publishers."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"025022","DOI":"10.1103\/PhysRevD.86.025022","article-title":"Solving the 3D Ising model with the conformal bootstrap","volume":"86","author":"Showk","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"185","DOI":"10.5802\/crphys.23","article-title":"3D Ising model: A view from the conformal bootstrap island","volume":"21","author":"Rychkov","year":"2020","journal-title":"Comptes Rendus. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"5189","DOI":"10.1088\/0305-4470\/35\/25\/304","article-title":"Exact partition functions of the Ising model on M \u00d7 N planar lattices with periodic\u2014Aperiodic boundary conditions","volume":"35","author":"Wu","year":"2002","journal-title":"J. Phys. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"041118","DOI":"10.1103\/PhysRevE.76.041118","article-title":"Finite-size effects for the Ising model on helical tori","volume":"76","author":"Izmailian","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1561","DOI":"10.1126\/science.288.5471.1561a","article-title":"Statistical physicists phase out a dream","volume":"288","author":"Cipra","year":"2000","journal-title":"Science"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"883","DOI":"10.1103\/RevModPhys.39.883","article-title":"History of the Lenz-Ising model","volume":"39","author":"Brush","year":"1967","journal-title":"Rev. Mod. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1098\/rsbm.1978.0014","article-title":"Lars Onsager. 27 November 1903\u20135 October 1976","volume":"24","author":"Fisher","year":"1978","journal-title":"Biogr. Mem. Fellows R. Soc."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Ruelle, D. (1999). Statistical Mechanics: Rigorous Results, World Scientific.","DOI":"10.1142\/4090"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Friedli, S., and Velenik, Y. (2017). Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge University Press.","DOI":"10.1017\/9781316882603"},{"key":"ref_39","unstructured":"Feynman, R.P. (1998). Statistical Mechanics: A Set of Lectures, CRC Press."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"4549","DOI":"10.1142\/S0217751X9900213X","article-title":"Loops, surfaces and Grassmann representation in two- and three-dimensional Ising models","volume":"14","author":"Gattringer","year":"1999","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"675","DOI":"10.1007\/s00220-019-03606-1","article-title":"Weak mixing and analyticity of the pressure in the Ising model","volume":"377","author":"Ott","year":"2020","journal-title":"Commun. Math. Phys."},{"key":"ref_42","unstructured":"Ott, S. (2020). Weak mixing and analyticity in Random Cluster and low temperature Ising models. arXiv."},{"key":"ref_43","unstructured":"Marsden, J.E., and Hoffman, M.J. (1998). Basic Complex Analysis, W. H. Freeman. [3rd ed.]."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Krantz, S.G., and Parks, H.R. (2002). A Primer of Real Analytic Functions, Birkh\u00e4user. [2nd ed.].","DOI":"10.1007\/978-0-8176-8134-0"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"5727","DOI":"10.1088\/0305-4470\/29\/17\/042","article-title":"The magnetization of the 3D Ising model","volume":"29","author":"Talapov","year":"1996","journal-title":"J. Phys. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2613","DOI":"10.1103\/PhysRevLett.76.2613","article-title":"Monte Carlo renormalization of the 3D Ising model: Analyticity and convergence","volume":"76","author":"Heringa","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"043301","DOI":"10.1103\/PhysRevE.97.043301","article-title":"Pushing the limits of Monte Carlo simulations for the three-dimensional Ising model","volume":"97","author":"Ferrenberg","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_48","unstructured":"Perk, J.H.H. (2021). Comment on \u201cExact Solution for Three-Dimensional Ising model\u201d by Degang Zhang. arXiv."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/BF01010400","article-title":"Analytical properties of the anisotropic cubic Ising model","volume":"48","author":"Hansel","year":"1987","journal-title":"J. Stat. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1088\/0305-4470\/26\/4\/010","article-title":"Series studies of the Potts model. I: The simple cubic Ising model","volume":"26","author":"Guttmann","year":"1993","journal-title":"J. Phys. A"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1088\/0305-4470\/10\/5\/013","article-title":"Series expansions from the finite lattice method","volume":"10","author":"Enting","year":"1977","journal-title":"J. Phys. A"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"3533","DOI":"10.1088\/0305-4470\/33\/18\/301","article-title":"Size and area of square lattice polygons","volume":"33","author":"Jensen","year":"2000","journal-title":"J. Phys. A"},{"key":"ref_53","first-page":"504","article-title":"Enumerative combinatorics of lattice polymers","volume":"68","author":"Clisby","year":"2021","journal-title":"Not. Am. Math. Soc."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Niculescu, C.P., and Persson, L.-E. (2018). Convex Functions and Their Applications, Springer. [2nd ed.].","DOI":"10.1007\/978-3-319-78337-6"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1103\/PhysRevLett.76.344","article-title":"Solvability of some statistical mechanical systems","volume":"76","author":"Guttmann","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1006\/jsco.2000.0474","article-title":"Fast evaluation of holonomic functions near and in regular singularities","volume":"31","year":"2001","journal-title":"J. Symb. Comput."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"3093","DOI":"10.1080\/14786430802537738","article-title":"Comment on a recent conjectured solution of the three-dimensional Ising model","volume":"88","author":"Wu","year":"2008","journal-title":"Phil. Mag."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1709","DOI":"10.1088\/0305-4470\/19\/9\/041","article-title":"On the combinatorial solution of the Ising model","volume":"19","author":"Rosengren","year":"1986","journal-title":"J. Phys. A."},{"key":"ref_59","unstructured":"Yao, F., and Luks, E. (2000). Statistical mechanics, three-dimensionality and NP-completeness: I. Universality of intractability for the partition function of the Ising model across non-planar lattices. STOC \u201900: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, ACM Press."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"741","DOI":"10.1088\/0305-4470\/33\/4\/308","article-title":"Combinatorial and topological approach to the 3D Ising model","volume":"33","author":"Regge","year":"2000","journal-title":"J. Phys. A"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"P07023","DOI":"10.1088\/1742-5468\/2010\/07\/P07023","article-title":"A generalized Kac-Ward formula","volume":"2010","author":"Cimasoni","year":"2010","journal-title":"J. Stat. Mech."},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Guttman, A.J. (2009). Polygons, Polyominoes and Polycubes, Springer.","DOI":"10.1007\/978-1-4020-9927-4"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/11\/1665\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:18:42Z","timestamp":1760145522000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/11\/1665"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,15]]},"references-count":62,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["e24111665"],"URL":"https:\/\/doi.org\/10.3390\/e24111665","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,11,15]]}}}