{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T00:45:51Z","timestamp":1780447551109,"version":"3.54.1"},"reference-count":47,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,12]],"date-time":"2025-08-12T00:00:00Z","timestamp":1754956800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Initiative for Excellence\u2013Research University for the AGH University of Krakow"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Silicene is a two-dimensional silicon monolayer with a band gap caused by relatively strong spin\u2013orbit coupling. This band gap can be steered using a vertical electric field. In turn, the change in this electric field value leads to a transition from a topological insulator to a bulk insulator regime. This study aims to develop a phase-space approach to detecting the topological phase transitions in silicene induced by the presence of parallel magnetic and electric fields with the aid of the concept of topological quantum number based on the Wigner\u2013R\u00e9nyi entropy. A reinterpreted definition of the Wigner distribution function is employed to determine this indicator. The topological phase transition in silicene as a function of the electric field in the presence of the magnetic field is confirmed through the use of the topological quantum number determined for the one-half, Shannon and collision entropies.<\/jats:p>","DOI":"10.3390\/e27080857","type":"journal-article","created":{"date-parts":[[2025,8,12]],"date-time":"2025-08-12T07:53:44Z","timestamp":1754985224000},"page":"857","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Phase-Space Approach for Topological Phase Transitions in Silicene"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4020-1582","authenticated-orcid":false,"given":"Maciej","family":"Kalka","sequence":"first","affiliation":[{"name":"Faculty of Physics and Applied Computer Science, AGH University of Krakow, al. A. Mickiewicza 30, 30-059 Krak\u00f3w, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9460-5198","authenticated-orcid":false,"given":"Piotr","family":"Pigo\u0144","sequence":"additional","affiliation":[{"name":"Faculty of Physics and Applied Computer Science, AGH University of Krakow, al. A. Mickiewicza 30, 30-059 Krak\u00f3w, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6716-5182","authenticated-orcid":false,"given":"Bart\u0142omiej J.","family":"Spisak","sequence":"additional","affiliation":[{"name":"Faculty of Physics and Applied Computer Science, AGH University of Krakow, al. A. Mickiewicza 30, 30-059 Krak\u00f3w, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Palma-Chilla, L., and Lazz\u00fas, J.A. (2024). 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