{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T15:56:35Z","timestamp":1781625395603,"version":"3.54.5"},"reference-count":30,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,13]],"date-time":"2022-05-13T00:00:00Z","timestamp":1652400000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Dipartimento di Eccellenza MIUR 2018\u20132022"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Phase transitions\u2014both classical and quantum types\u2014are the perfect playground for appreciating universality at work. Indeed, the fine details become unimportant and a classification in very few universality classes is possible. Very recently, a striking deviation from this picture has been discovered: some antiferromagnetic spin chains with competing interactions show a different set of phase transitions depending on the parity of number of spins in the chain. The aim of this article is to demonstrate that the same behavior also characterizes the most simple quantum spin chain: the Ising model in a transverse field. By means of an exact solution based on a Wigner\u2013Jordan transformation, we show that a first-order quantum phase transition appears at the zero applied field in the odd spin case, while it is not present in the even case. A hint of a possible physical interpretation is given by the combination of two facts: at the point of the phase transition, the degeneracy of the ground state in the even and the odd case substantially differs, being respectively 2 and 2N, with N being the number of spins; the spin of the most favorable kink shows changes at that point.<\/jats:p>","DOI":"10.3390\/sym14050996","type":"journal-article","created":{"date-parts":[[2022,5,15]],"date-time":"2022-05-15T09:48:22Z","timestamp":1652608102000},"page":"996","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Parity-Dependent Quantum Phase Transition in the Quantum Ising Chain in a Transverse Field"],"prefix":"10.3390","volume":"14","author":[{"given":"Daniel","family":"Sacco Shaikh","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e0 degli Studi di Genova, Via Dodecaneso 33, 16146 Genova, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maura","family":"Sassetti","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e0 degli Studi di Genova, Via Dodecaneso 33, 16146 Genova, Italy"},{"name":"CNR SPIN, Via Dodecaneso 33, 16146 Genova, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Niccol\u00f2","family":"Traverso Ziani","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e0 degli Studi di Genova, Via Dodecaneso 33, 16146 Genova, Italy"},{"name":"CNR SPIN, Via Dodecaneso 33, 16146 Genova, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kardar, M. (2007). Statistical Physics of Particles, Cambridge University Press.","DOI":"10.1017\/CBO9780511815898"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kardar, M. (2007). Statistical Physics of Fields, Cambridge University Press.","DOI":"10.1017\/CBO9780511815881"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1757","DOI":"10.1126\/science.1133734","article-title":"Quantum spin Hall effect and topological phase transition in HgTe quantum wells","volume":"314","author":"Bernevig","year":"2006","journal-title":"Science"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1103\/RevModPhys.83.1057","article-title":"Topological insulators and superconductors","volume":"83","author":"Qi","year":"2011","journal-title":"Rev. Mod. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"766","DOI":"10.1126\/science.1148047","article-title":"Quantum Spin Hall Insulator State in HgTe Quantum Wells","volume":"318","author":"Wiedmann","year":"2007","journal-title":"Science"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Sachdev, S. (2011). Quantum Phase Transitions, University Press.","DOI":"10.1017\/CBO9780511973765"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"081001","DOI":"10.1088\/2399-6528\/ab3ab3","article-title":"The frustration of being odd: Universal area law violation in local systems","volume":"3","author":"Giampaolo","year":"2019","journal-title":"J. Phys. Commun."},{"key":"ref_8","first-page":"08302","article-title":"The frustration of being odd: How boundary conditions can destroy local order","volume":"22","author":"Giampaolo","year":"2020","journal-title":"N. J. Phys."},{"key":"ref_9","first-page":"1","article-title":"Quantum phase transition induced by topological frustration","volume":"3","author":"Giampaolo","year":"2020","journal-title":"Commun. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"014429","DOI":"10.1103\/PhysRevB.103.014429","article-title":"Effects of defects in the XY chain with frustrated boundary conditions","volume":"103","author":"Torre","year":"2021","journal-title":"Phys. Rev. B"},{"key":"ref_11","first-page":"1","article-title":"Resilience of the topological phases to frustration","volume":"11","author":"Franchini","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"064408","DOI":"10.1103\/PhysRevB.105.064408","article-title":"Fate of local order in topologically frustrated spin chains","volume":"105","author":"Giampaolo","year":"2022","journal-title":"Phys. Rev. B"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"075","DOI":"10.21468\/SciPostPhys.12.2.075","article-title":"Topological Frustration can modify the nature of a Quantum Phase Transition","volume":"12","author":"Torre","year":"2022","journal-title":"SciPost Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"900","DOI":"10.1038\/nature04693","article-title":"A quantum Newton\u2019s cradle","volume":"440","author":"Kinoshita","year":"2006","journal-title":"Nature"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"136801","DOI":"10.1103\/PhysRevLett.96.136801","article-title":"Time Dependence of Correlation Functions Following a Quantum Quench","volume":"96","author":"Calabrese","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"156403","DOI":"10.1103\/PhysRevLett.97.156403","article-title":"Effect of Suddenly Turning on Interactions in the Luttinger Model","volume":"97","author":"Cazalilla","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"063619","DOI":"10.1103\/PhysRevA.80.063619","article-title":"Quantum quench dynamics of the Luttinger model","volume":"80","author":"Iucci","year":"2009","journal-title":"Phys. Rev. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1103\/RevModPhys.83.863","article-title":"Nonequilibrium dynamics of closed interacting quantum systems","volume":"83","author":"Polkovnikov","year":"2011","journal-title":"Rev. Mod. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"227203","DOI":"10.1103\/PhysRevLett.106.227203","article-title":"Quantum Quench in the Transverse-Field Ising Chain","volume":"106","author":"Calabrese","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"150602","DOI":"10.1103\/PhysRevLett.107.150602","article-title":"Mode-Coupling-Induced Dissipative and Thermal Effects at Long Times after a Quantum Quench","volume":"107","author":"Mitra","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"126406","DOI":"10.1103\/PhysRevLett.109.126406","article-title":"Luttinger-liquid universality in the time evolution after an interaction quench","volume":"109","author":"Karrasch","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"135704","DOI":"10.1103\/PhysRevLett.110.135704","article-title":"Dynamical Quantum Phase Transitions in the Transverse-Field Ising Model","volume":"110","author":"Heyl","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"165131","DOI":"10.1103\/PhysRevB.88.165131","article-title":"Luttinger liquid properties of the steady state after a quantum quench","volume":"88","author":"Kennes","year":"2013","journal-title":"Phys. Rev. B"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"125131","DOI":"10.1103\/PhysRevB.92.125131","article-title":"Quantum quench within the gapless phase of the spin 1\/2 Heisenberg XXZ spin chain","volume":"92","author":"Collura","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1038\/nphys3215","article-title":"Quantum many-body systems out of equilibrium","volume":"11","author":"Eisert","year":"2015","journal-title":"Nat. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"085122","DOI":"10.1103\/PhysRevB.94.085122","article-title":"Out-of-equilibrium density dynamics of a quenched fermionic system","volume":"94","author":"Porta","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"085423","DOI":"10.1103\/PhysRevB.96.085423","article-title":"Quench-induced entanglement and relaxation dynamics in Luttinger liquids","volume":"96","author":"Calzona","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1146\/annurev-conmatphys-031016-025451","article-title":"Quantum Quench Dynamics","volume":"9","author":"Mitra","year":"2018","journal-title":"Ann. Rev. Cond. Mat. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Franchini, F. (2017). An Introduction to Integrable Techniques for One-Dimensional Quantum Systems, Springer International Publishing.","DOI":"10.1007\/978-3-319-48487-7"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"12766","DOI":"10.1038\/s41598-020-69621-8","article-title":"Topological classification of dynamical quantum phase transitions in the xy chain","volume":"10","author":"Porta","year":"2020","journal-title":"Sci. Rep."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/996\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:10:08Z","timestamp":1760137808000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/996"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,13]]},"references-count":30,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["sym14050996"],"URL":"https:\/\/doi.org\/10.3390\/sym14050996","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints202204.0108.v1","asserted-by":"object"}]},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,13]]}}}