{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,10]],"date-time":"2026-05-10T10:55:27Z","timestamp":1778410527960,"version":"3.51.4"},"reference-count":64,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,10,29]],"date-time":"2023-10-29T00:00:00Z","timestamp":1698537600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (NSFC)","doi-asserted-by":"publisher","award":["12305032"],"award-info":[{"award-number":["12305032"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (NSFC)","doi-asserted-by":"publisher","award":["LQ23A050003"],"award-info":[{"award-number":["LQ23A050003"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (NSFC)","doi-asserted-by":"publisher","award":["LQ21F050007"],"award-info":[{"award-number":["LQ21F050007"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["12305032"],"award-info":[{"award-number":["12305032"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["LQ23A050003"],"award-info":[{"award-number":["LQ23A050003"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["LQ21F050007"],"award-info":[{"award-number":["LQ21F050007"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["12305032"],"award-info":[{"award-number":["12305032"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["LQ23A050003"],"award-info":[{"award-number":["LQ23A050003"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["LQ21F050007"],"award-info":[{"award-number":["LQ21F050007"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We investigate a generalized Dicke model by introducing two interacting spin ensembles coupled with a single-mode bosonic field. Apart from the normal to superradiant phase transition induced by the strong spin\u2013boson coupling, interactions between the two spin ensembles enrich the phase diagram by introducing ferromagnetic, antiferromagnetic and paramagnetic phases. The mean-field approach reveals a phase diagram comprising three phases: paramagnetic\u2013normal phase, ferromagnetic\u2013superradiant phase, and antiferromagnetic\u2013normal phase. Ferromagnetic spin\u2013spin interaction can significantly reduce the required spin\u2013boson coupling strength to observe the superradiant phase, where the macroscopic excitation of the bosonic field occurs. Conversely, antiferromagnetic spin\u2013spin interaction can strongly suppress the superradiant phase. To examine higher-order quantum effects beyond the mean-field contribution, we utilize the Holstein\u2013Primakoff transformation, which converts the generalized Dicke model into three coupled harmonic oscillators in the thermodynamic limit. Near the critical point, we observe the close of the energy gap between the ground and the first excited states, the divergence of entanglement entropy and quantum fluctuation in certain quadrature. These observations further confirm the quantum phase transition and offer additional insights into critical behaviors.<\/jats:p>","DOI":"10.3390\/e25111492","type":"journal-article","created":{"date-parts":[[2023,10,30]],"date-time":"2023-10-30T12:07:45Z","timestamp":1698667665000},"page":"1492","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Quantum Phase Transitions in a Generalized Dicke Model"],"prefix":"10.3390","volume":"25","author":[{"given":"Wen","family":"Liu","sequence":"first","affiliation":[{"name":"Key Laboratory of Optical Information Detecting and Display Technology of Zhejiang, Zhejiang Normal University, Jinhua 321004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5469-3741","authenticated-orcid":false,"given":"Liwei","family":"Duan","sequence":"additional","affiliation":[{"name":"Department of Physics, Zhejiang Normal University, Jinhua 321004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1103\/PhysRev.93.99","article-title":"Coherence in Spontaneous Radiation Processes","volume":"93","author":"Dicke","year":"1954","journal-title":"Phys. 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