{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:50:16Z","timestamp":1760244616870,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,16]],"date-time":"2022-12-16T00:00:00Z","timestamp":1671148800000},"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>Self-intersecting energy band structures in momentum space can be induced by nonlinearity at the mean-field level, with the so-called nonlinear Dirac cones as one intriguing consequence. Using the Qi-Wu-Zhang model plus power law nonlinearity, we systematically study in this paper the Aharonov\u2013Bohm (AB) phase associated with an adiabatic process in the momentum space, with two adiabatic paths circling around one nonlinear Dirac cone. Interestingly, for and only for Kerr nonlinearity, the AB phase experiences a jump of \u03c0 at the critical nonlinearity at which the Dirac cone appears and disappears (thus yielding \u03c0-quantization of the AB phase so long as the nonlinear Dirac cone exists), whereas for all other powers of nonlinearity, the AB phase always changes continuously with the nonlinear strength. Our results may be useful for experimental measurement of power-law nonlinearity and shall motivate further fundamental interest in aspects of geometric phase and adiabatic following in nonlinear systems.<\/jats:p>","DOI":"10.3390\/e24121835","type":"journal-article","created":{"date-parts":[[2022,12,16]],"date-time":"2022-12-16T04:30:00Z","timestamp":1671165000000},"page":"1835","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Quantization of AB Phase in Nonlinear Systems"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5518-0246","authenticated-orcid":false,"given":"Xi","family":"Liu","sequence":"first","affiliation":[{"name":"NUS Graduate School\u2014Integrative Sciences and Engineering Programme (ISEP), National University of Singapore, Singapore 119077, Singapore"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7952-7101","authenticated-orcid":false,"given":"Qing-Hai","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Physics, National University of Singapore, Singapore 117551, Singapore"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1280-6493","authenticated-orcid":false,"given":"Jiangbin","family":"Gong","sequence":"additional","affiliation":[{"name":"Department of Physics, National University of Singapore, Singapore 117551, Singapore"},{"name":"Center for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"121406","DOI":"10.1103\/PhysRevB.96.121406","article-title":"Nonlinear Dirac cones","volume":"96","author":"Bomantara","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"New, G. (2011). Introduction to Nonlinear Optics, Cambridge University Press.","DOI":"10.1017\/CBO9780511975851"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"115411","DOI":"10.1103\/PhysRevB.102.115411","article-title":"Nonlinearity induced topological physics in momentum space and real space","volume":"102","author":"Tuloup","year":"2020","journal-title":"Phys. Rev. B"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"023402","DOI":"10.1103\/PhysRevA.61.023402","article-title":"Nonlinear Landau-Zener tunneling","volume":"61","author":"Wu","year":"2000","journal-title":"Phys. Rev. A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"023404","DOI":"10.1103\/PhysRevA.66.023404","article-title":"Theory of nonlinear Landau-Zener tunneling","volume":"66","author":"Liu","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"063609","DOI":"10.1103\/PhysRevA.73.063609","article-title":"Towards a generalized Landau-Zener formula for an interacting Bose-Einstein condensate in a two-level system","volume":"73","author":"Witthaut","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"053607","DOI":"10.1103\/PhysRevA.77.053607","article-title":"Two-mode Bose-Einstein condensate in a high-frequency driving field that directly couples the two modes","volume":"77","author":"Zhang","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"073008","DOI":"10.1088\/1367-2630\/10\/7\/073008","article-title":"Nonlinear Landau\u2013Zener processes in a periodic driving field","volume":"10","author":"Zhang","year":"2008","journal-title":"New J. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"085308","DOI":"10.1103\/PhysRevB.74.085308","article-title":"Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors","volume":"74","author":"Qi","year":"2006","journal-title":"Phys. Rev. B"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1038\/nature04233","article-title":"Two-dimensional gas of massless Dirac fermions in graphene","volume":"438","author":"Novoselov","year":"2005","journal-title":"Nature"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1038\/nature04235","article-title":"Experimental observation of the quantum Hall effect and Berry\u2019s phase in graphene","volume":"438","author":"Zhang","year":"2005","journal-title":"Nature"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2857","DOI":"10.1143\/JPSJ.67.2857","article-title":"Berry\u2019s Phase and Absence of Back Scattering in Carbon Nanotubes","volume":"67","author":"Ando","year":"1998","journal-title":"J. Phys. Soc. Jpn."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2147","DOI":"10.1103\/PhysRevLett.82.2147","article-title":"Manifestation of Berry\u2019s Phase in Metal Physics","volume":"82","author":"Mikitik","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"195411","DOI":"10.1103\/PhysRevB.106.195411","article-title":"Topological characteristics of gap closing points in nonlinear Weyl semimetals","volume":"106","author":"Tuloup","year":"2022","journal-title":"Phys. Rev. B"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"052218","DOI":"10.1103\/PhysRevE.103.052218","article-title":"Self-consistent model of the plasma staircase and nonlinear Schr\u00f6dinger equation with subquadratic power nonlinearity","volume":"103","author":"Milovanov","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"802","DOI":"10.1016\/j.mcm.2005.08.010","article-title":"Exact solutions for the fourth order nonlinear Schrodinger equations with cubic and power law nonlinearities","volume":"43","author":"Wazwaz","year":"2006","journal-title":"Math. Comput. Model."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1016\/j.ijleo.2017.03.017","article-title":"Optical solitons for the Schr\u00f6dinger\u2013Hirota equation with power law nonlinearity by the B\u00e4cklund transformation","volume":"138","author":"Kilic","year":"2017","journal-title":"Optik"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Sulem, C., and Sulem, P. (2004). The Nonlinear Schr\u00f6dinger Equation: Self-Focusing and Wave Collapse, Springer Science & Business Media.","DOI":"10.1007\/b98958"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"102157","DOI":"10.1016\/j.rinp.2019.102157","article-title":"A study of optical wave propagation in the nonautonomous Schr\u00f6dinger-Hirota equation with power-law nonlinearity","volume":"13","author":"Osman","year":"2019","journal-title":"Results Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/j.ijleo.2016.11.036","article-title":"Exact solitons to generalized resonant dispersive nonlinear Schr\u00f6dinger\u2019s equation with power law nonlinearity","volume":"130","author":"Mirzazadeh","year":"2017","journal-title":"Optik"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/j.cnsns.2016.07.002","article-title":"Localized modes of the (n + 1)-dimensional Schr\u00f6dinger equation with power-law nonlinearities in PT-symmetric potentials","volume":"43","author":"Dai","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"4246","DOI":"10.1016\/j.ijleo.2014.04.014","article-title":"Optical solitons and optical rogons of generalized resonant dispersive nonlinear Schr\u00f6dinger\u2019s equation with power law nonlinearity","volume":"125","author":"Mirzazadeh","year":"2014","journal-title":"Optik"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Biswas, A., and Konar, S. (2006). Introduction to Non-Kerr Law Optical Solitons, Chapman and Hall\/CRC.","DOI":"10.1201\/9781420011401"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/12\/1835\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:42:34Z","timestamp":1760146954000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/12\/1835"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,16]]},"references-count":23,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["e24121835"],"URL":"https:\/\/doi.org\/10.3390\/e24121835","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,12,16]]}}}