{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,22]],"date-time":"2025-12-22T12:45:09Z","timestamp":1766407509111,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,4,1]],"date-time":"2024-04-01T00:00:00Z","timestamp":1711929600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12375119","12141501","2023A1515010936","DE-SC0023516","DE-SC0023175"],"award-info":[{"award-number":["12375119","12141501","2023A1515010936","DE-SC0023516","DE-SC0023175"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Guangdong Basic and Applied Basic Research Foundation","award":["12375119","12141501","2023A1515010936","DE-SC0023516","DE-SC0023175"],"award-info":[{"award-number":["12375119","12141501","2023A1515010936","DE-SC0023516","DE-SC0023175"]}]},{"name":"Fundamental Research Funds for the Central Universities, Sun Yat-sen University","award":["12375119","12141501","2023A1515010936","DE-SC0023516","DE-SC0023175"],"award-info":[{"award-number":["12375119","12141501","2023A1515010936","DE-SC0023516","DE-SC0023175"]}]},{"name":"U.S. Department of Energy, Office of Science, Office of Nuclear Physics","award":["12375119","12141501","2023A1515010936","DE-SC0023516","DE-SC0023175"],"award-info":[{"award-number":["12375119","12141501","2023A1515010936","DE-SC0023516","DE-SC0023175"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we report a study of the low-lying states of deformed 21Ne within the framework of the quantum-number projected generator coordinate method (PGCM), starting from a chiral two-nucleon-plus-three-nucleon (NN+3N) interaction. The wave functions of states are constructed as a linear combination of a set of axially deformed Hartree\u2013Fock\u2013Bogliubov (HFB) wave functions with different quadrupole deformations. These HFB wave functions are projected onto different angular momenta and the correct neutron and proton numbers for 21Ne. The results of the calculations based on the effective Hamiltonians derived by normal-ordering the 3N interaction with respect to three different reference states, including the quantum-number projected HFB wave functions for 20Ne, 22Ne, and an ensemble of them with equal weights, are compared. This study serves as a key step towards ab initio calculations of odd-mass deformed nuclei with the in-medium GCM.<\/jats:p>","DOI":"10.3390\/sym16040409","type":"journal-article","created":{"date-parts":[[2024,4,1]],"date-time":"2024-04-01T05:44:36Z","timestamp":1711950276000},"page":"409","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Quantum-Number Projected Generator Coordinate Method for 21Ne with a Chiral Two-Nucleon-Plus-Three-Nucleon Interaction"],"prefix":"10.3390","volume":"16","author":[{"given":"Wei","family":"Lin","sequence":"first","affiliation":[{"name":"School of Physics and Astronomy, Sun Yat-sen University, Zhuhai 519082, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Enfu","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Physics and Astronomy, Sun Yat-sen University, Zhuhai 519082, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9505-1852","authenticated-orcid":false,"given":"Jiangming","family":"Yao","sequence":"additional","affiliation":[{"name":"School of Physics and Astronomy, Sun Yat-sen University, Zhuhai 519082, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0520-0856","authenticated-orcid":false,"given":"Heiko","family":"Hergert","sequence":"additional","affiliation":[{"name":"Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI 48824-1321, USA"},{"name":"Department of Physics & Astronomy, Michigan State University, East Lansing, MI 48824-1321, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bohr, A., and Mottelson, B.R. (1998). Nuclear Structure. Volume II. Nuclear Deformations, Word Scientific.","DOI":"10.1142\/3530-vol2"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Ring, P., and Schuck, P. (1980). The Nuclear Many-Body Problem, Springer.","DOI":"10.1007\/978-3-642-61852-9"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.ppnp.2013.03.003","article-title":"Electric Dipole Moments of Nucleons, Nuclei, and Atoms: The Standard Model and Beyond","volume":"71","author":"Engel","year":"2013","journal-title":"Prog. Part. Nucl. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Arrowsmith-Kron, G., Athanasakis-Kaklamanakis, M., Au, M., Ballof, J., Berger, R., Borschevsky, A., Breier, A.A., Buchinger, F., Budker, D., and Caldwell, L. (2023). Opportunities for Fundamental Physics Research with Radioactive Molecules. arXiv.","DOI":"10.1088\/1361-6633\/ad1e39"},{"key":"ref_5","unstructured":"Brodeur, M., Buzinsky, N., Caprio, M.A., Cirigliano, V., Clark, J.A., Fasano, P.J., Formaggio, J.A., Gallant, A.T., Garcia, A., and Gandolfi, S. (2023). Nuclear \u03b2 decay as a probe for physics beyond the Standard Model. arXiv."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"103965","DOI":"10.1016\/j.ppnp.2022.103965","article-title":"Beyond-mean-field approaches for nuclear neutrinoless double beta decay in the standard mechanism","volume":"126","author":"Yao","year":"2022","journal-title":"Prog. Part. Nucl. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1102","DOI":"10.1103\/PhysRev.89.1102","article-title":"Nuclear constitution and the interpretation of fission phenomena","volume":"89","author":"Hill","year":"1953","journal-title":"Phys. Rev."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1103\/PhysRev.108.311","article-title":"Collective Motions in Nuclei by the Method of Generator Coordinates","volume":"108","author":"Griffin","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1103\/RevModPhys.75.121","article-title":"Self-consistent mean-field models for nuclear structure","volume":"75","author":"Bender","year":"2003","journal-title":"Rev. Mod. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1016\/j.ppnp.2011.01.055","article-title":"Relativistic Nuclear Energy Density Functionals: Mean-Field and Beyond","volume":"66","author":"Niksic","year":"2011","journal-title":"Prog. Part. Nucl. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"123001","DOI":"10.1088\/1361-6471\/ac288a","article-title":"Symmetry restoration in mean-field approaches","volume":"48","author":"Sheikh","year":"2021","journal-title":"J. Phys. G"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"014302","DOI":"10.1103\/PhysRevC.86.014302","article-title":"No-core Monte Carlo shell-model calculation for 10Be and 12Be low-lying spectra","volume":"86","author":"Liu","year":"2012","journal-title":"Phys. Rev. C"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"054311","DOI":"10.1103\/PhysRevC.98.054311","article-title":"Generator-coordinate reference states for spectra and 0\u03bd\u03b2\u03b2 decay in the in-medium similarity renormalization group","volume":"98","author":"Yao","year":"2018","journal-title":"Phys. Rev. C"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"232501","DOI":"10.1103\/PhysRevLett.124.232501","article-title":"Ab Initio Treatment of Collective Correlations and the Neutrinoless Double Beta Decay of 48Ca","volume":"124","author":"Yao","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1140\/epja\/s10050-022-00692-z","article-title":"Multi-reference many-body perturbation theory for nuclei: I. Novel PGCM-PT formalism","volume":"58","author":"Frosini","year":"2022","journal-title":"Eur. Phys. J. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1140\/epja\/s10050-022-00693-y","article-title":"Multi-reference many-body perturbation theory for nuclei: II. Ab initio study of neon isotopes via PGCM and IM-NCSM calculations","volume":"58","author":"Frosini","year":"2022","journal-title":"Eur. Phys. J. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1140\/epja\/s10050-022-00694-x","article-title":"Multi-reference many-body perturbation theory for nuclei: III. Ab initio calculations at second order in PGCM-PT","volume":"58","author":"Frosini","year":"2022","journal-title":"Eur. Phys. J. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"041302","DOI":"10.1103\/PhysRevC.75.041302","article-title":"The Intruder feature of Mg-31 and the coexistence of many particle and many hole states","volume":"75","author":"Kimura","year":"2007","journal-title":"Phys. Rev. C"},{"key":"ref_19","unstructured":"Kimura, M. (2011). Spectroscopy and intruder configurations of 33Mg and 31Ne studied with antisymmetrized molecular dynamics. arXiv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"162501","DOI":"10.1103\/PhysRevLett.113.162501","article-title":"Beyond Mean-Field Calculations for Odd-Mass Nuclei","volume":"113","author":"Bally","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1140\/epja\/s10050-022-00833-4","article-title":"Structure of 128,129,130Xe through multi-reference energy density functional calculations","volume":"58","author":"Bally","year":"2022","journal-title":"Eur. Phys. J. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1140\/epja\/s10050-023-00955-3","article-title":"The shape of gold","volume":"59","author":"Bally","year":"2023","journal-title":"Eur. Phys. J. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1140\/epja\/i2016-16277-8","article-title":"A symmetry-conserving description of odd nuclei with the Gogny force","volume":"52","author":"Borrajo","year":"2016","journal-title":"Eur. Phys. J. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1016\/j.physletb.2016.11.037","article-title":"Ground-state properties of even and odd Magnesium isotopes in a symmetry-conserving approach","volume":"764","author":"Borrajo","year":"2017","journal-title":"Phys. Lett. B"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"044317","DOI":"10.1103\/PhysRevC.98.044317","article-title":"Symmetry Conserving Configuration Mixing description of odd mass nuclei","volume":"98","author":"Borrajo","year":"2018","journal-title":"Phys. Rev. C"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"034305","DOI":"10.1103\/PhysRevC.109.034305","article-title":"Multireference covariant density-functional theory for the low-lying states of odd-mass nuclei","volume":"109","author":"Zhou","year":"2024","journal-title":"Phys. Rev. C"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1016\/S0375-9474(01)01219-2","article-title":"Particle number projection with effective forces","volume":"696","author":"Anguiano","year":"2001","journal-title":"Nucl. Phys. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"044319","DOI":"10.1103\/PhysRevC.79.044319","article-title":"Particle-number restoration within the energy density functional formalism","volume":"79","author":"Bender","year":"2009","journal-title":"Phys. Rev. C"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"044320","DOI":"10.1103\/PhysRevC.79.044320","article-title":"Particle-number restoration within the energy density functional formalism: Nonviability of terms depending on noninteger powers of the density matrices","volume":"79","author":"Duguet","year":"2009","journal-title":"Phys. Rev. C"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"023002","DOI":"10.1088\/1402-4896\/92\/2\/023002","article-title":"In-Medium Similarity Renormalization Group for Closed and Open-Shell Nuclei","volume":"92","author":"Hergert","year":"2017","journal-title":"Phys. Scr."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/0375-9474(95)00370-G","article-title":"On the solution of the Hartree-Fock-Bogoliubov equations by the conjugate gradient method","volume":"594","author":"Egido","year":"1995","journal-title":"Nucl. Phys. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1140\/epja\/s10050-021-00369-z","article-title":"Symmetry-projected variational calculations with the numerical suite TAURUS: I. Variation after particle-number projection","volume":"57","author":"Bally","year":"2021","journal-title":"Eur. Phys. J. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"044311","DOI":"10.1103\/PhysRevC.81.044311","article-title":"Configuration mixing of angular-momentum projected triaxial relativistic mean-field wave functions","volume":"81","author":"Yao","year":"2010","journal-title":"Phys. Rev. C"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/BF02710281","article-title":"Nonunitary bogoliubov transformations and extension of wick\u2019s theorem","volume":"64","author":"Balian","year":"1969","journal-title":"Nuovo Cim. B"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"956","DOI":"10.1103\/PhysRev.103.956","article-title":"Nucleon-Hole Interaction in jj Coupling","volume":"103","author":"Pandya","year":"1956","journal-title":"Phys. Rev."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"042505","DOI":"10.1103\/PhysRevLett.108.042505","article-title":"Symmetry restoration in Hartree-Fock-Bogoliubov based theories","volume":"108","author":"Bertsch","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"034325","DOI":"10.1103\/PhysRevC.85.034325","article-title":"Evaluation of overlaps between arbitrary fermionic quasiparticle vacua","volume":"85","author":"Avez","year":"2012","journal-title":"Phys. Rev. C"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"041001","DOI":"10.1103\/PhysRevC.68.041001","article-title":"Accurate charge-dependent nucleon-nucleon potential at fourth order of chiral perturbation theory","volume":"68","author":"Entem","year":"2003","journal-title":"Phys. Rev. C"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.ppnp.2010.03.001","article-title":"From low-momentum interactions to nuclear structure","volume":"65","author":"Bogner","year":"2010","journal-title":"Prog. Part. Nucl. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"031301","DOI":"10.1103\/PhysRevC.83.031301","article-title":"Improved nuclear matter calculations from chiral low-momentum interactions","volume":"83","author":"Hebeler","year":"2011","journal-title":"Phys. Rev. C"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1088\/0305-4470\/3\/1\/002","article-title":"Projection in the occupation-number space and the canonical transformation","volume":"3","author":"Fomenko","year":"1970","journal-title":"J. Phys. A Gen. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.physrep.2015.12.007","article-title":"The In-Medium Similarity Renormalization Group: A Novel Ab Initio Method for Nuclei","volume":"621","author":"Hergert","year":"2016","journal-title":"Phys. Rept."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.nds.2015.07.001","article-title":"Nuclear Data Sheets for A = 21","volume":"127","author":"Firestone","year":"2015","journal-title":"Nucl. Data Sheets"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/4\/409\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:22:08Z","timestamp":1760106128000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/4\/409"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,1]]},"references-count":43,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,4]]}},"alternative-id":["sym16040409"],"URL":"https:\/\/doi.org\/10.3390\/sym16040409","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,4,1]]}}}