{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,8]],"date-time":"2026-02-08T12:30:58Z","timestamp":1770553858637,"version":"3.49.0"},"reference-count":39,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2018,1,25]],"date-time":"2018-01-25T00:00:00Z","timestamp":1516838400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Motivated by the importance of entanglement and correlation indicators in the analysis of quantum systems, we study the equilibrium and the bipartite residual entropy in a two-species Bose\u2013Hubbard dimer when the spatial phase separation of the two species takes place. We consider both the zero and non-zero-temperature regime. We present different kinds of residual entropies (each one associated with a different way of partitioning the system), and we show that they strictly depend on the specific quantum phase characterizing the two species (supermixed, mixed or demixed) even at finite temperature. To provide a deeper physical insight into the zero-temperature scenario, we apply the fully-analytical variational approach based on su(2) coherent states and provide a considerably good approximation of the entanglement entropy. Finally, we show that the effectiveness of bipartite residual entropy as a critical indicator at non-zero temperature is unchanged when considering a restricted combination of energy eigenstates.<\/jats:p>","DOI":"10.3390\/e20020084","type":"journal-article","created":{"date-parts":[[2018,1,25]],"date-time":"2018-01-25T12:25:49Z","timestamp":1516883149000},"page":"84","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Residual Entropy and Critical Behavior of Two Interacting Boson Species in a Double Well"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6395-1258","authenticated-orcid":false,"given":"Fabio","family":"Lingua","sequence":"first","affiliation":[{"name":"Department of Physics, Clark University, Worcester, MA 01610, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8940-6936","authenticated-orcid":false,"given":"Andrea","family":"Richaud","sequence":"additional","affiliation":[{"name":"Department of Applied Science and Technology and u.d.r. CNISM, Politecnico di Torino, I-10129 Torino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vittorio","family":"Penna","sequence":"additional","affiliation":[{"name":"Department of Applied Science and Technology and u.d.r. CNISM, Politecnico di Torino, I-10129 Torino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.aop.2004.09.010","article-title":"The cold atom Hubbard toolbox","volume":"315","author":"Jaksch","year":"2005","journal-title":"Ann. 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