{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:56:39Z","timestamp":1760241399529,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,1,22]],"date-time":"2018-01-22T00:00:00Z","timestamp":1516579200000},"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","doi-asserted-by":"publisher","award":["11372122;11274393;11574404;11275279"],"award-info":[{"award-number":["11372122;11274393;11574404;11275279"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Open Project Program of State Key Laboratory of Theoretical Physics","award":["Y4KF201CJI;"],"award-info":[{"award-number":["Y4KF201CJI;"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>For spin-1 condensates, the spatial degrees of freedom can be considered as being frozen at temperature zero, while the spin-degrees of freedom remain free. Under this condition, the entanglement entropy has been derived exactly with an analytical form. The entanglement entropy is found to decrease monotonically with the increase of the magnetic polarization as expected. However, for the ground state in polar phase, an extremely steep fall of the entropy is found when the polarization emerges from zero. Then the fall becomes a gentle descent after the polarization exceeds a turning point.<\/jats:p>","DOI":"10.3390\/e20010080","type":"journal-article","created":{"date-parts":[[2018,1,22]],"date-time":"2018-01-22T13:40:39Z","timestamp":1516628439000},"page":"80","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Entanglement Entropy of the Spin-1 Condensates at Zero Temperature"],"prefix":"10.3390","volume":"20","author":[{"given":"Zhibing","family":"Li","sequence":"first","affiliation":[{"name":"School of Physics, Sun Yat-Sen University, Guangzhou 510275, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yimin","family":"Liu","sequence":"additional","affiliation":[{"name":"Department of Physics, Shaoguan University, Shaoguan 510205, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Zheng","sequence":"additional","affiliation":[{"name":"Theory of Condensed Matter (T.C.M.) Group, Cavendish Laboratory, J.J.Thomson Avenue, Cambridge CB3 0HE, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chengguang","family":"Bao","sequence":"additional","affiliation":[{"name":"School of Physics, Sun Yat-Sen University, Guangzhou 510275, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"865","DOI":"10.1103\/RevModPhys.81.865","article-title":"Quantum entanglement","volume":"81","author":"Horodecki","year":"2009","journal-title":"Rev. Mod. Phys."},{"key":"ref_2","first-page":"1","article-title":"An introduction to entanglement measures","volume":"7","author":"Plenio","year":"2007","journal-title":"Quantum Inf. Comput."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2027","DOI":"10.1103\/PhysRevLett.80.2027","article-title":"Optical confinement of a Bose-Einstein condensate","volume":"80","author":"Andrews","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"742","DOI":"10.1103\/PhysRevLett.81.742","article-title":"Spinor Bose condensates in optical traps","volume":"81","author":"Ho","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5257","DOI":"10.1103\/PhysRevLett.81.5257","article-title":"Quantum spins mixing in spinor Bose-Einstein condensates","volume":"81","author":"Law","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3896","DOI":"10.1103\/PhysRevA.59.3896","article-title":"Quantum theory of atomic four-wave mixing in Bose-Einstein condensates","volume":"59","author":"Goldstein","year":"1998","journal-title":"Phys. Rev. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4031","DOI":"10.1103\/PhysRevLett.84.4031","article-title":"Fragmented and single condensate ground states of spin-1 Bose gas","volume":"84","author":"Ho","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1066","DOI":"10.1103\/PhysRevLett.84.1066","article-title":"Exact eigenstates and magnetic response of spin-1 and spin-2 Bose-Einstein condensates","volume":"84","author":"Koashi","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"043614","DOI":"10.1103\/PhysRevA.72.043614","article-title":"First excited band of a spinor Bose-Einstein condensate","volume":"72","author":"Bao","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1007\/s11467-005-0001-9","article-title":"One-body and two-body fractional percentage coefficients for spinor Bose-Einstein condensates","volume":"1","author":"Bao","year":"2006","journal-title":"Front. Phys. China"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0166-1280(01)00454-7","article-title":"Weights of the total spins for for systems of permutational symmetry adapted spin-1 particles","volume":"547","author":"Katriel","year":"2001","journal-title":"J. Mol. Struct."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"235002","DOI":"10.1088\/1751-8113\/45\/23\/235002","article-title":"Sensitivity of spin structures of small spin-1 condensates toward a magnetic field studied beyond the mean field theory","volume":"45","author":"Bao","year":"2012","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1007\/s10909-015-1305-y","article-title":"Spin-thermodynamics of ultra-cold spin-1 atoms","volume":"180","author":"Li","year":"2015","journal-title":"J. Low Temp. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1103\/RevModPhys.80.517","article-title":"Entanglement in many-body systems","volume":"80","author":"Amico","year":"2008","journal-title":"Rev. Mod. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1103\/RevModPhys.82.277","article-title":"Colloquium: Area laws for the entanglement entropy","volume":"82","author":"Eisert","year":"2010","journal-title":"Rev. Mod. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/80\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:52:06Z","timestamp":1760194326000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/80"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,22]]},"references-count":15,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1]]}},"alternative-id":["e20010080"],"URL":"https:\/\/doi.org\/10.3390\/e20010080","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2018,1,22]]}}}