{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:24:40Z","timestamp":1760235880591,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,10,1]],"date-time":"2021-10-01T00:00:00Z","timestamp":1633046400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100016349","name":"Sichuan Normal University","doi-asserted-by":"publisher","award":["N\/A"],"award-info":[{"award-number":["N\/A"]}],"id":[{"id":"10.13039\/501100016349","id-type":"DOI","asserted-by":"publisher"}]},{"name":"&quot;Thousand Talents Program&quot; of Sichuan Province, China","award":["N\/A"],"award-info":[{"award-number":["N\/A"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The three-dimensional Ising model in a zero external field is exactly solved by operator algebras, similar to the Onsager\u2019s approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along two directions and the screw boundary condition along the third direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. The critical temperatures, at which order\u2013disorder transitions in the infinite crystal occur along three axis directions, are determined. The analytical expressions for the internal energy and the specific heat are also presented.<\/jats:p>","DOI":"10.3390\/sym13101837","type":"journal-article","created":{"date-parts":[[2021,10,11]],"date-time":"2021-10-11T01:59:47Z","timestamp":1633917587000},"page":"1837","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Exact Solution for Three-Dimensional Ising Model"],"prefix":"10.3390","volume":"13","author":[{"given":"Degang","family":"Zhang","sequence":"first","affiliation":[{"name":"College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610101, China"},{"name":"Institute of Solid State Physics, Sichuan Normal University, Chengdu 610101, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,1]]},"reference":[{"key":"ref_1","first-page":"253","article-title":"Beitrag zur theorie des ferromagnetismus","volume":"31","author":"Ising","year":"1925","journal-title":"Z. 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Phase Transitions and Critical Phenomena, Academic Press."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/10\/1837\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:08:32Z","timestamp":1760166512000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/10\/1837"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,1]]},"references-count":20,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2021,10]]}},"alternative-id":["sym13101837"],"URL":"https:\/\/doi.org\/10.3390\/sym13101837","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,10,1]]}}}