{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T04:28:18Z","timestamp":1774499298602,"version":"3.50.1"},"reference-count":75,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2024,9,11]],"date-time":"2024-09-11T00:00:00Z","timestamp":1726012800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"Here, we investigate the fractal-lattice Hubbard model using various numerical methods: exact diagonalization, the self-consistent diagonalization of a (mean-field) Hartree-Fock Hamiltonian and state-of-the-art Auxiliary-Field Quantum Monte Carlo. We focus on the Sierpinski triangle with Hausdorff dimension 1.58<\/mml:mn><\/mml:math> and consider several generations. In the tight-binding limit, we find compact localised states, which are also explained in terms of symmetry and linked to the formation of a ferrimagnetic phase at weak interaction. Simulations at half-filling revealed the persistence of this type of magnetic order for every value of interaction strength and a Mott transition for U\/t &#x223C;<\/mml:mo><\/mml:math> 4.5. In addition, we found a remarkable dependence on the Hausdorff dimension regarding i<\/mml:mi>)<\/mml:mo><\/mml:math> the number of compact localised states in different generations, i<\/mml:mi>i<\/mml:mi>)<\/mml:mo><\/mml:math> the scaling of the total many-body ground-state energy in the tight-binding limit, and i<\/mml:mi>i<\/mml:mi>i<\/mml:mi>)<\/mml:mo><\/mml:math> the density of the states at the corners of the lattice for specific values of electronic filling. Moreover, in the presence of an intrinsic spin-orbit coupling, the zero-energy compact localized states become entangled and give rise to inner and outer corner modes.<\/jats:p>","DOI":"10.22331\/q-2024-09-11-1469","type":"journal-article","created":{"date-parts":[[2024,9,11]],"date-time":"2024-09-11T14:10:15Z","timestamp":1726063815000},"page":"1469","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":4,"title":["The Fractal-Lattice Hubbard Model"],"prefix":"10.22331","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-2632-6306","authenticated-orcid":false,"given":"Monica","family":"Conte","sequence":"first","affiliation":[{"name":"Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584CC Utrecht, The Netherlands,"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8236-9052","authenticated-orcid":false,"given":"Vinicius","family":"Zampronio","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica Te\u00f3rica e Experimental - UFRN, Av. Sen. Salgado Filho 3000, 59078-970, Natal - RN, Brazil,"},{"name":"Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584CC Utrecht, The Netherlands,"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7784-8104","authenticated-orcid":false,"given":"Malte","family":"R\u00f6ntgen","sequence":"additional","affiliation":[{"name":"Laboratoire d\u2019Acoustique de l\u2019Universit\u00e9 du Mans, Unite Mixte de Recherche 6613, Centre National de la Recherche Scientifique, Avenue O. Messiaen, F-72085 Le Mans Cedex 9, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4190-3893","authenticated-orcid":false,"given":"Cristiane Morais","family":"Smith","sequence":"additional","affiliation":[{"name":"Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584CC Utrecht, The Netherlands,"}]}],"member":"9598","published-online":{"date-parts":[[2024,9,11]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Daniel P. Arovas, Erez Berg, Steven A. Kivelson, and Srinivas Raghu. ``The Hubbard Model''. Annu. Rev. Condens. Matter Phys. 13, 239\u2013274 (2022).","DOI":"10.1146\/annurev-conmatphys-031620-102024"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Elliott H. Lieb. ``The Hubbard model: Some rigorous results and open problems''. Pages 59\u201377. Springer Berlin Heidelberg. (2004).","DOI":"10.1007\/978-3-662-06390-3_4"},{"key":"2","doi-asserted-by":"crossref","unstructured":"D.C. Mattis. ``The many-body problem: An encyclopedia of exactly solved models in one dimension''. World Scientific. (1993). url: https:\/\/books.google.nl\/books?id=BGdHpCAMiLgC.","DOI":"10.1142\/9789812796523"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Elliott H. Lieb and Werner Liniger. ``Exact analysis of an interacting bose gas. i. the general solution and the ground state''. Phys. Rev. 130, 1605\u20131616 (1963).","DOI":"10.1103\/PhysRev.130.1605"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Christie S. Chiu, Geoffrey Ji, Anton Mazurenko, Daniel Greif, and Markus Greiner. ``Quantum State Engineering of a Hubbard System with Ultracold Fermions''. Phys. Rev. Lett. 120, 243201 (2018).","DOI":"10.1103\/PhysRevLett.120.243201"},{"key":"5","doi-asserted-by":"publisher","unstructured":"D. Tusi, L. Franchi, L. F. Livi, K. Baumann, D. Benedicto Orenes, L. Del Re, R. E. Barfknecht, T.-W. Zhou, M. Inguscio, G. Cappellini, M. Capone, J. Catani, and L. Fallani. ``Flavour-selective localization in interacting lattice fermions''. Nat. Phys. 18, 1201\u20131205 (2022).","DOI":"10.1038\/s41567-022-01726-5"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Jin Yang, Liyu Liu, Jirayu Mongkolkiattichai, and Peter Schauss. ``Site-Resolved Imaging of Ultracold Fermions in a Triangular-Lattice Quantum Gas Microscope''. PRX Quantum 2, 020344 (2021).","DOI":"10.1103\/PRXQuantum.2.020344"},{"key":"7","doi-asserted-by":"publisher","unstructured":"Christie S. Chiu, Geoffrey Ji, Annabelle Bohrdt, Muqing Xu, Michael Knap, Eugene Demler, Fabian Grusdt, Markus Greiner, and Daniel Greif. ``String patterns in the doped Hubbard model''. Science 365, 251\u2013256 (2019).","DOI":"10.1126\/science.aav3587"},{"key":"8","doi-asserted-by":"publisher","unstructured":"Aaron Szasz, Johannes Motruk, Michael P. Zaletel, and Joel E. Moore. ``Chiral spin liquid phase of the triangular lattice hubbard model: A density matrix renormalization group study''. Phys. Rev. X 10, 021042 (2020).","DOI":"10.1103\/PhysRevX.10.021042"},{"key":"9","doi-asserted-by":"publisher","unstructured":"Tomonori Shirakawa, Takami Tohyama, Jure Kokalj, Sigetoshi Sota, and Seiji Yunoki. ``Ground-state phase diagram of the triangular lattice Hubbard model by the density-matrix renormalization group method''. Phys. Rev. B 96, 205130 (2017).","DOI":"10.1103\/PhysRevB.96.205130"},{"key":"10","doi-asserted-by":"publisher","unstructured":"Aaron Szasz and Johannes Motruk. ``Phase diagram of the anisotropic triangular lattice Hubbard model''. Phys. Rev. B 103, 235132 (2021).","DOI":"10.1103\/PhysRevB.103.235132"},{"key":"11","doi-asserted-by":"publisher","unstructured":"Zheng Zhu, D. N. Sheng, and Ashvin Vishwanath. ``Doped Mott insulators in the triangular-lattice Hubbard model''. Phys. Rev. B 105, 205110 (2022).","DOI":"10.1103\/PhysRevB.105.205110"},{"key":"12","doi-asserted-by":"publisher","unstructured":"Davis Garwood, Jirayu Mongkolkiattichai, Liyu Liu, Jin Yang, and Peter Schauss. ``Site-resolved observables in the doped spin-imbalanced triangular Hubbard model''. Phys. Rev. A 106, 013310 (2022).","DOI":"10.1103\/PhysRevA.106.013310"},{"key":"13","doi-asserted-by":"publisher","unstructured":"Vito Marino, Federico Becca, and Luca F. Tocchio. ``Stripes in the extended $t-t^\\prime$ Hubbard model: A Variational Monte Carlo analysis''. SciPost Phys. 12, 180 (2022).","DOI":"10.21468\/SciPostPhys.12.6.180"},{"key":"14","doi-asserted-by":"publisher","unstructured":"Luca F. Tocchio, Arianna Montorsi, and Federico Becca. ``Magnetic and spin-liquid phases in the frustrated $t{-}{t}^{{'}}$ hubbard model on the triangular lattice''. Phys. Rev. B 102, 115150 (2020).","DOI":"10.1103\/PhysRevB.102.115150"},{"key":"15","doi-asserted-by":"publisher","unstructured":"Luca F. Tocchio, Arianna Montorsi, and Federico Becca. ``Hubbard model on triangular $n$-leg cylinders: Chiral and nonchiral spin liquids''. Phys. Rev. Research 3, 043082 (2021).","DOI":"10.1103\/PhysRevResearch.3.043082"},{"key":"16","doi-asserted-by":"publisher","unstructured":"Yukitoshi Motome and Masatoshi Imada. ``A Quantum Monte Carlo method and its applications to multi-orbital Hubbard models''. J. Phys. Soc. Jpn. 66, 1872\u20131875 (1997).","DOI":"10.1143\/jpsj.66.1872"},{"key":"17","doi-asserted-by":"publisher","unstructured":"Vinicius Zampronio and Tommaso Macr\u00ec . ``Chiral superconductivity in the doped triangular-lattice Fermi-Hubbard model in two dimensions''. Quantum 7, 1061 (2023).","DOI":"10.22331\/q-2023-07-20-1061"},{"key":"18","doi-asserted-by":"publisher","unstructured":"Alexander Wietek, Riccardo Rossi, Fedor \u0160imkovic, Marcel Klett, Philipp Hansmann, Michel Ferrero, E. Miles Stoudenmire, Thomas Sch\u00e4fer, and Antoine Georges. ``Mott Insulating States with Competing Orders in the Triangular Lattice Hubbard Model''. Phys. Rev. X 11, 041013 (2021).","DOI":"10.1103\/PhysRevX.11.041013"},{"key":"19","doi-asserted-by":"publisher","unstructured":"Jie Xu, Chia-Chen Chang, Eric J Walter, and Shiwei Zhang. ``Spin-and charge-density waves in the Hartree\u2013Fock ground state of the two-dimensional Hubbard model''. J. Phys.: Condens. Matter 23, 505601 (2011).","DOI":"10.1088\/0953-8984\/23\/50\/505601"},{"key":"20","doi-asserted-by":"publisher","unstructured":"Tomonori Shirakawa, Takami Tohyama, Jure Kokalj, Sigetoshi Sota, and Seiji Yunoki. ``Ground-state phase diagram of the triangular lattice Hubbard model by the density-matrix renormalization group method''. Phys. Rev. B 96, 205130 (2017).","DOI":"10.1103\/PhysRevB.96.205130"},{"key":"21","doi-asserted-by":"publisher","unstructured":"Bin-Bin Chen, Ziyu Chen, Shou-Shu Gong, D. N. Sheng, Wei Li, and Andreas Weichselbaum. ``Quantum spin liquid with emergent chiral order in the triangular-lattice Hubbard model''. Phys. Rev. B 106, 094420 (2022).","DOI":"10.1103\/PhysRevB.106.094420"},{"key":"22","doi-asserted-by":"publisher","unstructured":"Elbio Dagotto. ``Correlated electrons in high-temperature superconductors''. Rev. Mod. Phys. 66, 763\u2013840 (1994).","DOI":"10.1103\/RevModPhys.66.763"},{"key":"23","doi-asserted-by":"publisher","unstructured":"S. N. Kempkes, M. R. Slot, S. E. Freeney, S. J. M. Zevenhuizen, D. Vanmaekelbergh, I. Swart, and C. Morais Smith. ``Design and characterization of electrons in a fractal geometry''. Nat. Phys. 15, 127\u2013131 (2019).","DOI":"10.1038\/s41567-018-0328-0"},{"key":"24","doi-asserted-by":"publisher","unstructured":"Xiao-Yun Xu, Xiao-Wei Wang, Dan-Yang Chen, Cristiane Morais Smith, and Xian-Min Jin. ``Quantum transport in fractal networks''. Nat. Phot. 15, 1\u20138 (2021).","DOI":"10.1038\/s41566-021-00845-4"},{"key":"25","doi-asserted-by":"publisher","unstructured":"Shriya Pai and Abhinav Prem. ``Topological states on fractal lattices''. Phys. Rev. B 100, 155135 (2019).","DOI":"10.1103\/PhysRevB.100.155135"},{"key":"26","doi-asserted-by":"publisher","unstructured":"Sourav Manna, Snehasish Nandy, and Bitan Roy. ``Higher-order topological phases on fractal lattices''. Phys. Rev. B 105, L201301 (2022).","DOI":"10.1103\/PhysRevB.105.L201301"},{"key":"27","unstructured":"Junkai Li, Qingyang Mo, Jian-Hua Jiang, and Zhaoju Yang. ``Higher-order topological phase in an acoustic fractal lattice'' (2022). arXiv:2205.05298."},{"key":"28","doi-asserted-by":"crossref","unstructured":"R. Canyellas, Chen Liu, R. Arouca, L. Eek, Guanyong Wang, Yin Yin, Dandan Guan, Yaoyi Li, Shiyong Wang, Hao Zheng, Canhua Liu, Jinfeng Jia, and C. Morais Smith. ``Topological edge and corner states in bi fractals on insb'' (2023). arXiv:2309.09860.","DOI":"10.1038\/s41567-024-02551-8"},{"key":"29","doi-asserted-by":"publisher","unstructured":"E. Y. Loh, J. E. Gubernatis, R. T. Scalettar, S. R. White, D. J. Scalapino, and R. L. Sugar. ``Sign problem in the numerical simulation of many-electron systems''. Phys. Rev. B 41, 9301\u20139307 (1990).","DOI":"10.1103\/PhysRevB.41.9301"},{"key":"30","doi-asserted-by":"publisher","unstructured":"Matthias Troyer and Uwe-Jens Wiese. ``Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations''. Phys. Rev. Lett. 94, 170201 (2005).","DOI":"10.1103\/PhysRevLett.94.170201"},{"key":"31","doi-asserted-by":"publisher","unstructured":"Daniel Leykam, Alexei Andreanov, and Sergej Flach. ``Artificial flat band systems: From lattice models to experiments''. Adv. Phys. 3, 1473052 (2018).","DOI":"10.1080\/23746149.2018.1473052"},{"key":"32","doi-asserted-by":"publisher","unstructured":"Huy Nguyen, Hao Shi, Jie Xu, and Shiwei Zhang. ``CPMC-lab: A matlab package for constrained path Monte Carlo calculations''. Comp. Phys. Commu. 185, 3344\u20133357 (2014).","DOI":"10.1016\/j.cpc.2014.08.003"},{"key":"33","doi-asserted-by":"publisher","unstructured":"Malte R\u00f6ntgen, C. V. Morfonios, I. Brouzos, F. K. Diakonos, and P. Schmelcher. ``Quantum network transfer and storage with compact localized states induced by local symmetries''. Phys. Rev. Lett. 123, 080504 (2019).","DOI":"10.1103\/PhysRevLett.123.080504"},{"key":"34","doi-asserted-by":"publisher","unstructured":"S. N. Kempkes, P. Capiod, S. Ismaili, J. Mulkens, L. Eek, I. Swart, and C. Morais Smith. ``Compact localized boundary states in a quasi-1D electronic diamond-necklace chain''. Quantum Front 2, 1 (2023).","DOI":"10.1007\/s44214-023-00026-0"},{"key":"35","doi-asserted-by":"publisher","unstructured":"Daniel Leykam and Sergej Flach. ``Perspective: Photonic flatbands''. APL Phot. 3, 070901 (2018).","DOI":"10.1063\/1.5034365"},{"key":"36","doi-asserted-by":"publisher","unstructured":"Bill Sutherland. ``Localization of electronic wave functions due to local topology''. Phys. Rev. B 34, 5208\u20135211 (1986).","DOI":"10.1103\/PhysRevB.34.5208"},{"key":"37","doi-asserted-by":"publisher","unstructured":"Elliott H. Lieb. ``Two theorems on the Hubbard model''. Phys. Rev. Lett. 62, 1201\u20131204 (1989).","DOI":"10.1103\/PhysRevLett.62.1201"},{"key":"38","doi-asserted-by":"publisher","unstructured":"F. Albert Cotton. ``Chemical applications of group theory''. Wiley-Interscience. New York (1990). 3 edition.","DOI":"10.1021\/ed041p113.2"},{"key":"39","unstructured":"L. D. Landau and L. M. Lifshitz. ``Quantum mechanics non-relativistic theory, third edition: Volume 3''. Butterworth-Heinemann. (1981). 3 edition. url: http:\/\/www.worldcat.org\/isbn\/0750635398."},{"key":"40","doi-asserted-by":"publisher","unstructured":"C. L. Kane and E. J. Mele. ``Z2 topological order and the quantum spin Hall effect''. Phys. Rev. Lett. 95, 146802 (2005).","DOI":"10.1103\/PhysRevLett.95.146802"},{"key":"41","doi-asserted-by":"publisher","unstructured":"Ralph van Gelderen and C. Morais Smith. ``Rashba and intrinsic spin-orbit interactions in biased bilayer graphene''. Phys. Rev. B 81, 125435 (2010).","DOI":"10.1103\/PhysRevB.81.125435"},{"key":"42","doi-asserted-by":"publisher","unstructured":"Alexander Wei\u00dfe and H. Fehske. ``Exact diagonalization techniques''. In Computational Many Body Physics. Pages 529\u2013544. Springer International Publishing, Berlin\/Heidelberg (2008).","DOI":"10.1007\/978-3-540-74686-7_18"},{"key":"43","doi-asserted-by":"publisher","unstructured":"Patrick Fazekas. ``Lecture notes on electron correlation and magnetism''. World Scientific, 1999. (1999).","DOI":"10.1142\/2945"},{"key":"44","doi-asserted-by":"publisher","unstructured":"Shiwei Zhang. ``Ab initio electronic structure calculations by auxiliary-field quantum Monte Carlo''. In Handbook of Materials Modeling. Pages 1\u201327. Springer International Publishing (2018).","DOI":"10.1007\/978-3-319-42913-7_47-1"},{"key":"45","doi-asserted-by":"publisher","unstructured":"Wirawan Purwanto and Shiwei Zhang. ``Quantum monte carlo method for the ground state of many-boson systems''. Phys. Rev. E 70, 056702 (2004).","DOI":"10.1103\/PhysRevE.70.056702"},{"key":"46","doi-asserted-by":"publisher","unstructured":"Hao Shi and Shiwei Zhang. ``Symmetry in auxiliary-field quantum monte carlo calculations''. Phys. Rev. B 88, 125132 (2013).","DOI":"10.1103\/PhysRevB.88.125132"},{"key":"47","doi-asserted-by":"publisher","unstructured":"Thomas Garm Pedersen. ``Graphene fractals: Energy gap and spin polarization''. Phys. Rev. B 101, 235427 (2020).","DOI":"10.1103\/PhysRevB.101.235427"},{"key":"48","doi-asserted-by":"publisher","unstructured":"Marcin Raczkowski, Robert Peters, Th\u1ecb Thu Ph\u00f9ng, Nayuta Takemori, Fakher F. Assaad, Andreas Honecker, and Javad Vahedi. ``Hubbard model on the honeycomb lattice: From static and dynamical mean-field theories to lattice quantum monte carlo simulations''. Phys. Rev. B 101, 125103 (2020).","DOI":"10.1103\/PhysRevB.101.125103"},{"key":"49","doi-asserted-by":"publisher","unstructured":"W. F. Brinkman and T. M. Rice. ``Application of gutzwiller's variational method to the metal-insulator transition''. Phys. Rev. B 2, 4302\u20134304 (1970).","DOI":"10.1103\/PhysRevB.2.4302"},{"key":"50","doi-asserted-by":"publisher","unstructured":"S. Sorella and E. Tosatti. ``Semi-metal-insulator transition of the hubbard model in the honeycomb lattice''. EPL 19, 699 (1992).","DOI":"10.1209\/0295-5075\/19\/8\/007"},{"key":"51","doi-asserted-by":"publisher","unstructured":"Marcin Raczkowski, Robert Peters, Th\u1ecb Thu Ph\u00f9ng, Nayuta Takemori, Fakher F. Assaad, Andreas Honecker, and Javad Vahedi. ``Hubbard model on the honeycomb lattice: From static and dynamical mean-field theories to lattice quantum Monte Carlo simulations''. Phys. Rev. B 101, 125103 (2020).","DOI":"10.1103\/PhysRevB.101.125103"},{"key":"52","doi-asserted-by":"publisher","unstructured":"Biplab Pal and Kush Saha. ``Flat bands in fractal-like geometry''. Phys. Rev. B 97, 195101 (2018).","DOI":"10.1103\/PhysRevB.97.195101"},{"key":"53","doi-asserted-by":"publisher","unstructured":"Sougata Biswas and Arunava Chakrabarti. ``Designer quantum states on a fractal substrate: Compact localization, flat bands and the edge modes''. Physica E 153, 115762 (2023).","DOI":"10.1016\/j.physe.2023.115762"},{"key":"54","doi-asserted-by":"publisher","unstructured":"Atanu Nandy. ``Controlled imprisonment of wave packet and flat bands in a fractal geometry''. Phys. Scr. 96, 045802 (2021).","DOI":"10.1088\/1402-4896\/abdcf6"},{"key":"55","doi-asserted-by":"publisher","unstructured":"Yuqing Xie, Limin Song, Wenchao Yan, Shiqi Xia, Liqin Tang, Daohong Song, Jun-Won Rhim, and Zhigang Chen. ``Fractal-like photonic lattices and localized states arising from singular and nonsingular flatbands''. APL Phot. 6, 116104 (2021).","DOI":"10.1063\/5.0068032"},{"key":"56","doi-asserted-by":"publisher","unstructured":"Haissam Hanafi, Philip Menz, and Cornelia Denz. ``Localized States Emerging from Singular and Nonsingular Flat Bands in a Frustrated Fractal-Like Photonic Lattice''. Adv. Opt. Mater. 10, 2102523 (2022).","DOI":"10.1002\/adom.202102523"},{"key":"57","doi-asserted-by":"publisher","unstructured":"Tom Westerhout, Edo van Veen, Mikhail I. Katsnelson, and Shengjun Yuan. ``Plasmon confinement in fractal quantum systems''. Phys. Rev. B 97, 205434 (2018).","DOI":"10.1103\/PhysRevB.97.205434"},{"key":"58","unstructured":"Askar A. Iliasov, Mikhail I. Katsnelson, and Andrey A. Bagrov. ``Strong enhancement of superconductivity on finitely ramified fractal lattices'' (2024). arXiv:2310.11497."},{"key":"59","doi-asserted-by":"publisher","unstructured":"Sourav Manna, Biplab Pal, Wei Wang, and Anne E. B. Nielsen. ``Anyons and fractional quantum hall effect in fractal dimensions''. Phys. Rev. Res. 2, 023401 (2020).","DOI":"10.1103\/PhysRevResearch.2.023401"},{"key":"60","doi-asserted-by":"publisher","unstructured":"Blazej Jaworowski, Michael Iversen, and Anne E. B. Nielsen. ``Approximate Hofstadter- and Kapit-Mueller-like parent Hamiltonians for Laughlin states on fractals''. Phys. Rev. A 107, 063315 (2023).","DOI":"10.1103\/PhysRevA.107.063315"},{"key":"61","doi-asserted-by":"publisher","unstructured":"Akihisa Koga and Sam Coates. ``Ferrimagnetically ordered states in the Hubbard model on the hexagonal golden-mean tiling''. Phys. Rev. B 105, 104410 (2022).","DOI":"10.1103\/PhysRevB.105.104410"},{"key":"62","doi-asserted-by":"publisher","unstructured":"Elliott H. Lieb and F. Y. Wu. ``Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension''. Phys. Rev. Lett. 20, 1445\u20131448 (1968).","DOI":"10.1103\/PhysRevLett.20.1445"},{"key":"63","doi-asserted-by":"publisher","unstructured":"F. D. M. Haldane. ``General Relation of Correlation Exponents and Spectral Properties of One-Dimensional Fermi Systems: Application to the Anisotropic $s=\\frac{1}{2}$ Heisenberg Chain''. Phys. Rev. Lett. 45, 1358\u20131362 (1980).","DOI":"10.1103\/PhysRevLett.45.1358"},{"key":"64","doi-asserted-by":"publisher","unstructured":"F D M Haldane. ``'Luttinger liquid theory' of one-dimensional quantum fluids. I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas''. J. Phys. C: Solid State Phys. 14, 2585\u20132609 (1981).","DOI":"10.1088\/0022-3719\/14\/19\/010"},{"key":"65","doi-asserted-by":"publisher","unstructured":"F.D.M. Haldane. ``Demonstration of the \"Luttinger liquid\" character of Bethe-ansatz-soluble models of 1-D quantum fluids''. Phys. Lett. A 81, 153\u2013155 (1981).","DOI":"10.1016\/0375-9601(81)90049-9"},{"key":"66","unstructured":"V. Zampronio. ``Cp-afqmc'' (2022)."},{"key":"67","doi-asserted-by":"publisher","unstructured":"Shiwei Zhang, J. Carlson, and J. E. Gubernatis. ``Constrained Path Quantum Monte Carlo Method for Fermion Ground States''. Phys. Rev. Lett. 74, 3652\u20133655 (1995).","DOI":"10.1103\/PhysRevLett.74.3652"},{"key":"68","doi-asserted-by":"publisher","unstructured":"Shiwei Zhang, J. Carlson, and J. E. Gubernatis. ``Constrained path Monte Carlo method for fermion ground states''. Phys. Rev. B 55, 7464\u20137477 (1997).","DOI":"10.1103\/PhysRevB.55.7464"},{"key":"69","doi-asserted-by":"publisher","unstructured":"H. F. Trotter. ``On the product of semi-groups of operators''. Proc. Amer. Math. Soc. 10, 545\u2013551 (1959).","DOI":"10.1090\/S0002-9939-1959-0108732-6"},{"key":"70","doi-asserted-by":"publisher","unstructured":"J. E. Hirsch. ``Two-dimensional Hubbard model: Numerical simulation study''. Phys. Rev. B 31, 4403\u20134419 (1985).","DOI":"10.1103\/PhysRevB.31.4403"},{"key":"71","doi-asserted-by":"publisher","unstructured":"Peter J. Reynolds, David M. Ceperley, Berni J. Alder, and William A. Lester. ``Fixed-node quantum Monte Carlo for molecules a)-b)''. J. Chem. Phys. 77, 5593\u20135603 (1982). arXiv:https:\/\/doi.org\/10.1063\/1.443766.","DOI":"10.1063\/1.443766"},{"key":"72","doi-asserted-by":"publisher","unstructured":"Mingpu Qin, Hao Shi, and Shiwei Zhang. ``Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method''. Phys. Rev. B 94, 085103 (2016).","DOI":"10.1103\/PhysRevB.94.085103"},{"key":"73","doi-asserted-by":"publisher","unstructured":"X. Y. Zhang, Elihu Abrahams, and G. Kotliar. ``Quantum Monte Carlo algorithm for constrained fermions: Application to the infinite-${U}$ Hubbard model''. Phys. Rev. Lett. 66, 1236\u20131239 (1991).","DOI":"10.1103\/PhysRevLett.66.1236"},{"key":"74","doi-asserted-by":"publisher","unstructured":"Wirawan Purwanto and Shiwei Zhang. ``Quantum Monte Carlo method for the ground state of many-boson systems''. Phys. Rev. E 70, 056702 (2004).","DOI":"10.1103\/PhysRevE.70.056702"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2024-09-11-1469\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2024,9,11]],"date-time":"2024-09-11T14:10:24Z","timestamp":1726063824000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2024-09-11-1469\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,11]]},"references-count":75,"URL":"https:\/\/doi.org\/10.22331\/q-2024-09-11-1469","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,9,11]]},"article-number":"1469"}}