{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:10:16Z","timestamp":1760058616323,"version":"build-2065373602"},"reference-count":56,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,9]],"date-time":"2025-04-09T00:00:00Z","timestamp":1744156800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001807","name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo","doi-asserted-by":"publisher","award":["2024\/13426-0","2018\/25225-9","309262\/2019-4"],"award-info":[{"award-number":["2024\/13426-0","2018\/25225-9","309262\/2019-4"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico (CNPq)","doi-asserted-by":"publisher","award":["2024\/13426-0","2018\/25225-9","309262\/2019-4"],"award-info":[{"award-number":["2024\/13426-0","2018\/25225-9","309262\/2019-4"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We investigate the time evolution of the quark condensate toward a chiral symmetry broken phase in hot and dense quark matter using a field-theoretic quark model with nonlocal chiral-invariant four-fermion coupling. By purposely selecting a parameter set in which inhomogeneous phases are energetically disfavored, we nonetheless observe the emergence of metastable patterned configurations that appear to persist for remarkably long timescales. These findings suggest that even when not fully stable, inhomogeneous phases may play a significant role in the dynamics of chiral symmetry breaking and restoration. To gain deeper insight into these phenomena, we also analyze the impact of the dimensionality of coordinate space on both the formation and stability of inhomogeneous chiral condensates.<\/jats:p>","DOI":"10.3390\/sym17040568","type":"journal-article","created":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T07:41:51Z","timestamp":1744270911000},"page":"568","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Inhomogeneous Road to Chiral Symmetry Breaking: A Ginzburg\u2013Landau\u2013Langevin Analysis"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3443-2496","authenticated-orcid":false,"given":"Theo F.","family":"Motta","sequence":"first","affiliation":[{"name":"Instituto de F\u00edsica Te\u00f3rica, Universidade Estadual Paulista, S\u00e3o Paulo 01140-070, SP, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1713-8578","authenticated-orcid":false,"given":"Gast\u00e3o","family":"Krein","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Te\u00f3rica, Universidade Estadual Paulista, S\u00e3o Paulo 01140-070, SP, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1103\/PhysRevLett.4.415","article-title":"Structure of Nuclear Matter","volume":"4","author":"Overhauser","year":"1960","journal-title":"Phys. Rev. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Motta, T.F., Bernhardt, J., Buballa, M., and Fischer, C.S. (2024). Inhomogenuous instabilities at large chemical potential in a rainbow-ladder QCD model. arXiv.","DOI":"10.1103\/PhysRevD.110.074014"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"114019","DOI":"10.1103\/PhysRevD.108.114019","article-title":"Toward a stability analysis of inhomogeneous phases in QCD","volume":"108","author":"Motta","year":"2023","journal-title":"Phys. Rev. D"},{"key":"ref_4","unstructured":"Motta, T.F., Buballa, M., and Fischer, C.S. (2024). New Tool to Detect Inhomogeneous Chiral Symmetry Breaking. arXiv."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"12707","DOI":"10.1088\/0305-4470\/39\/41\/S04","article-title":"From relativistic quantum fields to condensed matter and back again: Updating the Gross-Neveu phase diagram","volume":"39","author":"Thies","year":"2006","journal-title":"J. Phys. A"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"375402","DOI":"10.1088\/1751-8121\/ac820a","article-title":"Detecting inhomogeneous chiral condensation from the bosonic two-point function in the (1 + 1)-dimensional Gross\u2013Neveu model in the mean-field approximation","volume":"55","author":"Koenigstein","year":"2022","journal-title":"J. Phys. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"071603","DOI":"10.1103\/PhysRevLett.129.071603","article-title":"Inhomogeneous Phase of the Chiral Gross-Neveu Model","volume":"129","author":"Ciccone","year":"2022","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"335401","DOI":"10.1088\/1751-8121\/ad6721","article-title":"Revisiting the spatially inhomogeneous condensates in the (1 + 1)-dimensional chiral Gross\u2013Neveu model via the bosonic two-point function in the infinite-N limit","volume":"57","author":"Koenigstein","year":"2024","journal-title":"J. Phys. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1016\/j.physletb.2019.02.045","article-title":"Inhomogeneous chiral phases away from the chiral limit","volume":"791","author":"Buballa","year":"2019","journal-title":"Phys. Lett. B"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.ppnp.2014.11.001","article-title":"Inhomogeneous chiral condensates","volume":"81","author":"Buballa","year":"2015","journal-title":"Prog. Part. Nucl. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3371","DOI":"10.1140\/epjst\/e2020-000101-x","article-title":"Inhomogeneous phases in the quark-meson model with explicit chiral-symmetry breaking","volume":"229","author":"Buballa","year":"2020","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"034022","DOI":"10.1103\/PhysRevD.97.034022","article-title":"Low-temperature behavior of the quark-meson model","volume":"97","author":"Tripolt","year":"2018","journal-title":"Phys. Rev. D"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"074025","DOI":"10.1103\/PhysRevD.80.074025","article-title":"Inhomogeneous phases in the Nambu-Jona-Lasino and quark-meson model","volume":"80","author":"Nickel","year":"2009","journal-title":"Phys. Rev. D"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"096012","DOI":"10.1103\/PhysRevD.110.096012","article-title":"Nonperturbative phase boundaries in the Gross-Neveu model from a stability analysis","volume":"110","author":"Thies","year":"2024","journal-title":"Phys. Rev. D"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Pannullo, L., Wagner, M., and Winstel, M. (2022). Inhomogeneous Phases in the Chirally Imbalanced 2 + 1-Dimensional Gross-Neveu Model and Their Absence in the Continuum Limit. Symmetry, 14.","DOI":"10.3390\/sym14020265"},{"key":"ref_16","first-page":"195","article-title":"Stability of homogeneous chiral phases against inhomogeneous perturbations in 2 + 1 dimensions","volume":"LATTICE2022","author":"Winstel","year":"2023","journal-title":"PoS"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"036022","DOI":"10.1103\/PhysRevD.108.036022","article-title":"Inhomogeneous condensation in the Gross-Neveu model in noninteger spatial dimensions 1 \u2264 d < 3","volume":"108","author":"Pannullo","year":"2023","journal-title":"Phys. Rev. D"},{"key":"ref_18","unstructured":"Koenigstein, A., and Pannullo, L. (2023). Inhomogeneous condensation in the Gross-Neveu model in noninteger spatial dimensions 1 \u2264 d < 3. II. Nonzero temperature and chemical potential. arXiv."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"034503","DOI":"10.1103\/PhysRevD.103.034503","article-title":"Regulator dependence of inhomogeneous phases in the (2 + 1)-dimensional Gross-Neveu model","volume":"103","author":"Buballa","year":"2021","journal-title":"Phys. Rev. D"},{"key":"ref_20","first-page":"156","article-title":"Inhomogeneous phases in the 3 + 1-dimensional Nambu-Jona-Lasinio model and their dependence on the regularization scheme","volume":"LATTICE2022","author":"Pannullo","year":"2023","journal-title":"PoS"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Pannullo, L., Wagner, M., and Winstel, M. (2024). Regularization effects in the Nambu-Jona-Lasinio model: Strong scheme dependence of inhomogeneous phases and persistence of the moat regime. arXiv.","DOI":"10.1103\/PhysRevD.110.076006"},{"key":"ref_22","unstructured":"Pannullo, L. (2024). Inhomogeneous Phases and the Moat Regime in Nambu-Jona-Lasinio-Type Models. [Ph.D. Thesis, Frankfurt Uiversity]."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"016015","DOI":"10.1103\/PhysRevD.102.016015","article-title":"How transverse thermal fluctuations disorder a condensate of chiral spirals into a quantum spin liquid","volume":"102","author":"Pisarski","year":"2020","journal-title":"Phys. Rev. D"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"074025","DOI":"10.1103\/PhysRevD.99.074025","article-title":"Fluctuations in cool quark matter and the phase diagram of Quantum Chromodynamics","volume":"99","author":"Pisarski","year":"2019","journal-title":"Phys. Rev. D"},{"key":"ref_25","unstructured":"Goldenfeld, N. (1992). Lectures on Phase Transitions and the Renormalization Group, Perseus Books."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Onuki, A. (2002). Phase Transition Dynamics, Cambridge University Press.","DOI":"10.1017\/CBO9780511534874"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physletb.2015.04.023","article-title":"Generalized Ginzburg-Landau approach to inhomogeneous phases in nonlocal chiral quark models","volume":"745","author":"Carlomagno","year":"2015","journal-title":"Phys. Lett. B"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"056007","DOI":"10.1103\/PhysRevD.92.056007","article-title":"Inhomogeneous phases in nonlocal chiral quark models","volume":"92","author":"Carlomagno","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"014015","DOI":"10.1103\/PhysRevD.98.014015","article-title":"Time-dependent Ginzburg-Landau approach to the dynamics of inhomogeneous chiral condensates in a nonlocal Nambu\u2013Jona-Lasinio model","volume":"98","author":"Carlomagno","year":"2018","journal-title":"Phys. Rev. D"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"03005","DOI":"10.1051\/epjconf\/201817203005","article-title":"Dynamics of inhomogeneous chiral condensates","volume":"172","author":"Carlomagno","year":"2018","journal-title":"EPJ Web Conf."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2331","DOI":"10.1103\/PhysRevC.58.2331","article-title":"Forming disoriented chiral condensates through fluctuations","volume":"58","author":"Rischke","year":"1998","journal-title":"Phys. Rev. C"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"024912","DOI":"10.1103\/PhysRevC.84.024912","article-title":"Nonequilibrium chiral fluid dynamics including dissipation and noise","volume":"84","author":"Nahrgang","year":"2011","journal-title":"Phys. Rev. C"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Krein, G., and Miller, C. (2021). Nonequilibrium Dynamics of the Chiral Quark Condensate under a Strong Magnetic Field. Symmetry, 13.","DOI":"10.3390\/sym13040551"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Ferrer, E.J., and de la Incera, V. (2021). Magnetic Dual Chiral Density Wave: A Candidate Quark Matter Phase for the Interior of Neutron Stars. Universe, 7.","DOI":"10.3390\/universe7120458"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"114006","DOI":"10.1103\/PhysRevD.71.114006","article-title":"Chiral symmetry and density wave in quark matter","volume":"71","author":"Nakano","year":"2005","journal-title":"Phys. Rev. D"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1016\/j.physletb.2013.10.050","article-title":"Dyson-Schwinger study of chiral density waves in QCD","volume":"727","author":"Buballa","year":"2013","journal-title":"Phys. Lett. B"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.nuclphysa.2010.05.053","article-title":"Quarkyonic Chiral Spirals","volume":"843","author":"Kojo","year":"2009","journal-title":"Nucl. Phys. A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"036008","DOI":"10.1103\/PhysRevD.92.036008","article-title":"Hybrid chiral condensate in the external magnetic field","volume":"92","author":"Nishiyama","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"065022","DOI":"10.1103\/PhysRevD.78.065022","article-title":"Twisted kink crystal in the chiral Gross-Neveu model","volume":"78","author":"Dunne","year":"2008","journal-title":"Physical Review D"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"105012","DOI":"10.1103\/PhysRevD.79.105012","article-title":"Inhomogeneous Condensates in the Thermodynamics of the Chiral NJL(2) model","volume":"79","author":"Basar","year":"2009","journal-title":"Phys. Rev. D"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1140\/epjc\/s2004-01593-y","article-title":"The Pion mass in finite volume","volume":"33","author":"Colangelo","year":"2004","journal-title":"Eur. Phys. J. C"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"FLAG Working Group, Aoki, S., Bernard, C., Blum, T., Colangelo, G., Della Morte, M., D\u00fcrr, S., El-Khadra, A.X., Fukaya, H., and Horsley, R. (2014). Review of Lattice Results Concerning Low-Energy Particle Physics. Eur. Phys. J. C, 74, 2890.","DOI":"10.1140\/epjc\/s10052-014-2890-7"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1016\/0375-9474(94)00481-2","article-title":"A Nonlocal, covariant generalization of the NJL model","volume":"582","author":"Bowler","year":"1995","journal-title":"Nucl. Phys. A"},{"key":"ref_44","first-page":"431","article-title":"Meson Current Correlation Functions in Instanton Vacuum","volume":"62","author":"Diakonov","year":"1985","journal-title":"Sov. Phys. JETP"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1016\/0550-3213(88)90443-9","article-title":"A Chiral Theory of Nucleons","volume":"306","author":"Diakonov","year":"1988","journal-title":"Nucl. Phys. B"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Calzetta, E.A., and Hu, B.L.B. (2008). Nonequilibrium Quantum Field Theory, Cambridge University Press. Cambridge Monographs on Mathematical Physics.","DOI":"10.1017\/CBO9780511535123"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1140\/epja\/i2016-16267-x","article-title":"Topical issue on Exploring Strongly Interacting Matter at High Densities\u2014NICA White Paper","volume":"52","author":"Blaschke","year":"2016","journal-title":"Eur. Phys. J. A"},{"key":"ref_48","first-page":"034006","article-title":"The compressed baryonic matter (CBM) experiment at FAIR\u2014physics, status and prospects","volume":"98","author":"Agarwal","year":"2023","journal-title":"Phys. Scr."},{"key":"ref_49","unstructured":"Aoki, K., Fujioka, H., Gogami, T., Hidaka, Y., Hiyama, E., Honda, R., Hosaka, A., Ichikawa, Y., Ieiri, M., and Isaka, M. (2021). Extension of the J-PARC Hadron Experimental Facility: Third White Paper. arXiv."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"054009","DOI":"10.1103\/PhysRevD.82.054009","article-title":"Influence of vector interaction and Polyakov loop dynamics on inhomogeneous chiral symmetry breaking phases","volume":"82","author":"Carignano","year":"2010","journal-title":"Phys. Rev. D"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"072301","DOI":"10.1103\/PhysRevLett.103.072301","article-title":"How many phases meet at the chiral critical point?","volume":"103","author":"Nickel","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"152302","DOI":"10.1103\/PhysRevLett.127.152302","article-title":"Signatures of Moat Regimes in Heavy-Ion Collisions","volume":"127","author":"Pisarski","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_53","first-page":"016","article-title":"Moat Regimes in QCD and their Signatures in Heavy-Ion Collisions","volume":"CPOD2021","author":"Rennecke","year":"2022","journal-title":"PoS"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Winstel, M., and Valgushev, S. (2024, January 14\u201320). Lattice study of disordering of inhomogeneous condensates and the Quantum Pion Liquid in effective O(N) model. Proceedings of the Excited QCD 2024 Workshop, Benasque, Spain.","DOI":"10.5506\/APhysPolBSupp.17.6-A22"},{"key":"ref_55","unstructured":"Nussinov, Z., Ogilvie, M.C., Pannullo, L., Pisarski, R.D., Rennecke, F., Schindler, S.T., and Winstel, M. (2024). Dilepton production from moaton quasiparticles. arXiv."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"116011","DOI":"10.1103\/PhysRevD.107.116011","article-title":"Particle interferometry in a moat regime","volume":"107","author":"Rennecke","year":"2023","journal-title":"Phys. Rev. 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