{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T04:50:33Z","timestamp":1764996633059,"version":"build-2065373602"},"reference-count":92,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,7,27]],"date-time":"2025-07-27T00:00:00Z","timestamp":1753574400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Guangdong Provincial Quantum Science Strategic Initiative","award":["GDZX2203001","GDZX2403001","12474489","12474228","SZRCXM2401006","JCYJ20240813153139050","2021ZD0302300"],"award-info":[{"award-number":["GDZX2203001","GDZX2403001","12474489","12474228","SZRCXM2401006","JCYJ20240813153139050","2021ZD0302300"]}]},{"name":"National Natural Science Foundation of China","award":["GDZX2203001","GDZX2403001","12474489","12474228","SZRCXM2401006","JCYJ20240813153139050","2021ZD0302300"],"award-info":[{"award-number":["GDZX2203001","GDZX2403001","12474489","12474228","SZRCXM2401006","JCYJ20240813153139050","2021ZD0302300"]}]},{"name":"Research Funding for Outbound Postdoctoral Fellows in Shenzhen","award":["GDZX2203001","GDZX2403001","12474489","12474228","SZRCXM2401006","JCYJ20240813153139050","2021ZD0302300"],"award-info":[{"award-number":["GDZX2203001","GDZX2403001","12474489","12474228","SZRCXM2401006","JCYJ20240813153139050","2021ZD0302300"]}]},{"DOI":"10.13039\/501100017607","name":"Shenzhen Fundamental Research Program","doi-asserted-by":"publisher","award":["GDZX2203001","GDZX2403001","12474489","12474228","SZRCXM2401006","JCYJ20240813153139050","2021ZD0302300"],"award-info":[{"award-number":["GDZX2203001","GDZX2403001","12474489","12474228","SZRCXM2401006","JCYJ20240813153139050","2021ZD0302300"]}],"id":[{"id":"10.13039\/501100017607","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Innovation Program for Quantum Science and Technology","award":["GDZX2203001","GDZX2403001","12474489","12474228","SZRCXM2401006","JCYJ20240813153139050","2021ZD0302300"],"award-info":[{"award-number":["GDZX2203001","GDZX2403001","12474489","12474228","SZRCXM2401006","JCYJ20240813153139050","2021ZD0302300"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The Ising model is famous in condensed matter and statistical physics. In this work we present a free-fermion formulation of the two-dimensional classical Ising models on honeycomb, triangular and Kagom\u00e9 lattices. Each Ising model is studied in the cases of a zero field and of an imaginary field i(\u03c0\/2)kBT. We employ the decorated lattice technique, star-triangle transformation, and weak-graph expansion method to exactly map each Ising model in both cases into an eight-vertex model on the square lattice. The resulting vertex weights are shown to satisfy the free-fermion condition. In the zero-field case, each Ising model is an even free-fermion model. In the case of the imaginary field, the Ising model on the honeycomb lattice is an even free-fermion model, while the models on the triangular and Kagom\u00e9 lattices are odd free-fermion models. We obtain the exact solution of the Kagom\u00e9 lattice Ising model under the imaginary field i(\u03c0\/2)kBT, a result not previously reported in the literature. We also show that the frustrated Ising models on the triangular and Kagom\u00e9 lattices in the imaginary field still exhibit a non-zero residual entropy.<\/jats:p>","DOI":"10.3390\/e27080799","type":"journal-article","created":{"date-parts":[[2025,7,28]],"date-time":"2025-07-28T10:51:14Z","timestamp":1753699874000},"page":"799","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Free-Fermion Models and Two-Dimensional Ising Models Under Zero Field and Imaginary Field i(\u03c0\/2)kBT"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8039-449X","authenticated-orcid":false,"given":"De-Zhang","family":"Li","sequence":"first","affiliation":[{"name":"Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area, Shenzhen 518045, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2971-5088","authenticated-orcid":false,"given":"Xin","family":"Wang","sequence":"additional","affiliation":[{"name":"Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area, Shenzhen 518045, China"},{"name":"Department of Physics, City University of Hong Kong, Hong Kong SAR, China"},{"name":"City University of Hong Kong Shenzhen Research Institute, Shenzhen 518057, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiao-Bao","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Physics, South China University of Technology, Guangzhou 510640, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1007\/s00407-004-0088-3","article-title":"History of the Lenz-Ising Model 1920\u20131950: From Ferromagnetic to Cooperative Phenomena","volume":"59","author":"Niss","year":"2005","journal-title":"Arch. 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