Chiral superconductivity in the doped triangular-lattice Fermi-Hubbard model in two dimensions

Vinicius Zampronio1,2 and Tommaso Macrì3,2

1Institute for Theoretical Physics, Utrecht University, 3584CS Utrecht, Netherlands
2Departamento de Física Teórica e Experimental, Federal University of Rio Grande do Norte 59078-950 Natal-RN, Brazil
3ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


The triangular-lattice Fermi-Hubbard model has been extensively investigated in the literature due to its connection to chiral spin states and unconventional superconductivity. Previous simulations of the ground state of the doped system rely on quasi-one-dimensional lattices where true long-range order is forbidden. Here we simulate two-dimensional and quasi-one-dimensional triangular lattices using state-of-the-art Auxiliary-Field Quantum Monte Carlo. Upon doping a non-magnetic chiral spin state, we observe evidence of chiral superconductivity supported by long-range order in Cooper-pair correlation and a finite value of the chiral order parameter. With this aim, we first locate the transition from the metallic to the non-magnetic insulating phase and the onset of magnetic order. Our results pave the way towards a better understanding of strongly correlated lattice systems with magnetic frustration.

Chiral superconductivity is a fascinating new phenomenon where Cooper pairs exhibit motion in a direction determined by their spin orientation, resulting in an intriguing interplay between spin and motion. One remarkable characteristic of chiral superconductors is the presence of robust chiral edge currents that remain unaffected by impurities, making them highly useful for applications in quantum computing. Chirality commonly arises in magnetic-frustrated lattice systems, characterized by interactions between magnetic moments that cannot be mutually satisfied, leading to complex and disordered magnetic states such as quantum spin liquids. The behavior of chiral spin states in frustrated lattices can be effectively described by the Fermi-Hubbard model, a fundamental concept in condensed-matter physics. The Hubbard Hamiltonian, which accounts for on-site interactions, extends beyond conventional band theory and successfully captures the intricate physics of Mott insulators, quantum spin liquids, and unconventional superconductors, although our understanding of these systems remains incomplete. Despite its simplicity, the Hubbard model is analytically tractable only in a few scenarios, and numerical methods are generally preferred. In our study, we utilized Quantum Monte Carlo simulations to investigate the Hubbard model on the triangular lattice, the simplest form of a frustrated lattice. Our results demonstrate the existence of a non-magnetic chiral spin state at intermediate interactions and at half filling. Moreover, below half filling, we observe the emergence of chiral superconductivity. Significantly, our Quantum Monte Carlo method enables simulations of two-dimensional systems and, for the first time, provides evidence of true long-range order in the ground state of this particular system.

► BibTeX data

► References

[1] Daniel P. Arovas, Erez Berg, Steven A. Kivelson, and Srinivas Raghu. ``The Hubbard model''. Annual Review of Condensed Matter Physics 13, 239–274 (2022).

[2] Masatoshi Imada, Atsushi Fujimori, and Yoshinori Tokura. ``Metal-insulator transitions''. Rev. Mod. Phys. 70, 1039–1263 (1998).

[3] J. E. Hirsch. ``Two-dimensional Hubbard model: Numerical simulation study''. Phys. Rev. B 31, 4403–4419 (1985).

[4] Leon Balents. ``Spin liquids in frustrated magnets''. Nature 464, 199–208 (2010).

[5] Lucile Savary and Leon Balents. ``Quantum spin liquids: a review''. Reports on Progress in Physics 80, 016502 (2016).

[6] Yi Zhou, Kazushi Kanoda, and Tai-Kai Ng. ``Quantum spin liquid states''. Rev. Mod. Phys. 89, 025003 (2017).

[7] Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma. ``Non-Abelian anyons and topological quantum computation''. Rev. Mod. Phys. 80, 1083–1159 (2008).

[8] P.W. Anderson. ``Resonating valence bonds: A new kind of insulator?''. Materials Research Bulletin 8, 153–160 (1973).

[9] P. W. Anderson. ``The Resonating Valence Bond State in La$_2$CuO$_4$ and Superconductivity''. Science 235, 1196–1198 (1987).

[10] Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, and G. Saito. ``Spin Liquid State in an Organic Mott Insulator with a Triangular Lattice''. Phys. Rev. Lett. 91, 107001 (2003).

[11] Y. Kurosaki, Y. Shimizu, K. Miyagawa, K. Kanoda, and G. Saito. ``Mott Transition from a Spin Liquid to a Fermi Liquid in the Spin-Frustrated Organic Conductor ${\kappa}$-(ET)$_{2}$Cu$_{2}$(CN)$_{3}$''. Phys. Rev. Lett. 95, 177001 (2005).

[12] Satoshi Yamashita, Yasuhiro Nakazawa, Masaharu Oguni, Yugo Oshima, Hiroyuki Nojiri, Yasuhiro Shimizu, Kazuya Miyagawa, and Kazushi Kanoda. ``Thermodynamic properties of a spin-1/​2 spin-liquid state in a $\kappa$-type organic salt''. Nature Physics 4, 459–462 (2008).

[13] Takayuki Isono, Hiromichi Kamo, Akira Ueda, Kazuyuki Takahashi, Motoi Kimata, Hiroyuki Tajima, Satoshi Tsuchiya, Taichi Terashima, Shinya Uji, and Hatsumi Mori. ``Gapless Quantum Spin Liquid in an Organic Spin-1/​2 Triangular-Lattice ${\kappa}{-}$H$_{3}$(Cat-EDT-TTF)$_{2}$''. Phys. Rev. Lett. 112, 177201 (2014).

[14] Björn Miksch, Andrej Pustogow, Mojtaba Javaheri Rahim, Andrey A. Bardin, Kazushi Kanoda, John A. Schlueter, Ralph Hübner, Marc Scheffler, and Martin Dressel. ``Gapped magnetic ground state in quantum spin liquid candidate $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$''. Science 372, 276–279 (2021).

[15] Olexei I. Motrunich. ``Variational study of triangular lattice spin-$1/​2$ model with ring exchanges and spin liquid state in ${\kappa}$-(ET)$_{2}$Cu$_{2}$(CN)$_{3}$''. Phys. Rev. B 72, 045105 (2005).

[16] Sung-Sik Lee and Patrick A. Lee. ``U(1) Gauge Theory of the Hubbard Model: Spin Liquid States and Possible Application to ${\kappa}$-(BEDT-TTF)$_{2}$Cu$_{2}$(CN)$_{3}$''. Phys. Rev. Lett. 95, 036403 (2005).

[17] Darrell F. Schroeter, Eliot Kapit, Ronny Thomale, and Martin Greiter. ``Spin Hamiltonian for which the Chiral Spin Liquid is the Exact Ground State''. Phys. Rev. Lett. 99, 097202 (2007).

[18] D. N. Sheng, Olexei I. Motrunich, and Matthew P. A. Fisher. ``Spin Bose-metal phase in a spin-$\frac{1}{2}$ model with ring exchange on a two-leg triangular strip''. Phys. Rev. B 79, 205112 (2009).

[19] Hong-Yu Yang, Andreas M. Läuchli, Frédéric Mila, and Kai Phillip Schmidt. ``Effective Spin Model for the Spin-Liquid Phase of the Hubbard Model on the Triangular Lattice''. Phys. Rev. Lett. 105, 267204 (2010).

[20] Tessa Cookmeyer, Johannes Motruk, and Joel E. Moore. ``Four-Spin Terms and the Origin of the Chiral Spin Liquid in Mott Insulators on the Triangular Lattice''. Phys. Rev. Lett. 127, 087201 (2021).

[21] Fengcheng Wu, Timothy Lovorn, Emanuel Tutuc, and A. H. MacDonald. ``Hubbard Model Physics in Transition Metal Dichalcogenide Moiré Bands''. Phys. Rev. Lett. 121, 026402 (2018).

[22] Yanhao Tang, Lizhong Li, Tingxin Li, Yang Xu, Song Liu, Katayun Barmak, Kenji Watanabe, Takashi Taniguchi, Allan H. MacDonald, Jie Shan, and Kin Fai Mak. ``Simulation of Hubbard model physics in WSe$_2$/​WS$_2$ moiré superlattices''. Nature 579, 353–358 (2020).

[23] 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).

[24] Jirayu Mongkolkiattichai, Liyu Liu, Davis Garwood, Jin Yang, and Peter Schauss. ``Quantum gas microscopy of a geometrically frustrated Hubbard system'' (2022).

[25] Steven R. White and A. L. Chernyshev. ``Neél Order in Square and Triangular Lattice Heisenberg Models''. Phys. Rev. Lett. 99, 127004 (2007).

[26] S. Raghu, S. A. Kivelson, and D. J. Scalapino. ``Superconductivity in the repulsive Hubbard model: An asymptotically exact weak-coupling solution''. Phys. Rev. B 81, 224505 (2010).

[27] Rahul Nandkishore, Ronny Thomale, and Andrey V. Chubukov. ``Superconductivity from weak repulsion in hexagonal lattice systems''. Phys. Rev. B 89, 144501 (2014).

[28] Yuval Gannot, Yi-Fan Jiang, and Steven A. Kivelson. ``Hubbard ladders at small ${U}$ revisited''. Phys. Rev. B 102, 115136 (2020).

[29] Peyman Sahebsara and David Sénéchal. ``Hubbard Model on the Triangular Lattice: Spiral Order and Spin Liquid''. Phys. Rev. Lett. 100, 136402 (2008).

[30] A. Yamada. ``Magnetic properties and Mott transition in the Hubbard model on the anisotropic triangular lattice''. Phys. Rev. B 89, 195108 (2014).

[31] Manuel Laubach, Ronny Thomale, Christian Platt, Werner Hanke, and Gang Li. ``Phase diagram of the Hubbard model on the anisotropic triangular lattice''. Phys. Rev. B 91, 245125 (2015).

[32] Hidekazu Morita, Shinji Watanabe, and Masatoshi Imada. ``Nonmagnetic Insulating States near the Mott Transitions on Lattices with Geometrical Frustration and Implications for $\kappa$-(ET)$_2$Cu$_2$(CN)$_3$''. Journal of the Physical Society of Japan 71, 2109–2112 (2002).

[33] Takuya Yoshioka, Akihisa Koga, and Norio Kawakami. ``Quantum Phase Transitions in the Hubbard Model on a Triangular Lattice''. Phys. Rev. Lett. 103, 036401 (2009).

[34] A. E. Antipov, A. N. Rubtsov, M. I. Katsnelson, and A. I. Lichtenstein. ``Electron energy spectrum of the spin-liquid state in a frustrated Hubbard model''. Phys. Rev. B 83, 115126 (2011).

[35] Takashi Koretsune, Yukitoshi Motome, and Akira Furusaki. ``Exact Diagonalization Study of Mott Transition in the Hubbard Model on an Anisotropic Triangular Lattice''. Journal of the Physical Society of Japan 76, 074719 (2007).

[36] 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).

[37] 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).

[38] Aaron Szasz and Johannes Motruk. ``Phase diagram of the anisotropic triangular lattice Hubbard model''. Phys. Rev. B 103, 235132 (2021).

[39] 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).

[40] 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).

[41] 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).

[42] Hunpyo Lee, Gang Li, and Hartmut Monien. ``Hubbard model on the triangular lattice using dynamical cluster approximation and dual fermion methods''. Phys. Rev. B 78, 205117 (2008).

[43] T. Watanabe, H. Yokoyama, Y. Tanaka, and J. Inoue. ``Predominant magnetic states in the Hubbard model on anisotropic triangular lattices''. Phys. Rev. B 77, 214505 (2008).

[44] Luca F. Tocchio, Hélène Feldner, Federico Becca, Roser Valentí, and Claudius Gros. ``Spin-liquid versus spiral-order phases in the anisotropic triangular lattice''. Phys. Rev. B 87, 035143 (2013).

[45] Alexander Wietek, Riccardo Rossi, Fedor Šimkovic, Marcel Klett, Philipp Hansmann, Michel Ferrero, E. Miles Stoudenmire, Thomas Schäfer, and Antoine Georges. ``Mott Insulating States with Competing Orders in the Triangular Lattice Hubbard Model''. Phys. Rev. X 11, 041013 (2021).

[46] Patrick A. Lee, Naoto Nagaosa, and Xiao-Gang Wen. ``Doping a Mott insulator: Physics of high-temperature superconductivity''. Rev. Mod. Phys. 78, 17–85 (2006).

[47] B J Powell and Ross H McKenzie. ``Quantum frustration in organic Mott insulators: from spin liquids to unconventional superconductors''. Reports on Progress in Physics 74, 056501 (2011).

[48] Kazushi Kanoda and Reizo Kato. ``Mott Physics in Organic Conductors with Triangular Lattices''. Annual Review of Condensed Matter Physics 2, 167–188 (2011).

[49] Annabelle Bohrdt, Lukas Homeier, Christian Reinmoser, Eugene Demler, and Fabian Grusdt. ``Exploration of doped quantum magnets with ultracold atoms''. Annals of Physics 435, 168651 (2021).

[50] Zheng Zhu, D. N. Sheng, and Ashvin Vishwanath. ``Doped Mott insulators in the triangular-lattice Hubbard model''. Phys. Rev. B 105, 205110 (2022).

[51] Wilhelm Kadow, Laurens Vanderstraeten, and Michael Knap. ``Hole spectral function of a chiral spin liquid in the triangular lattice Hubbard model''. Phys. Rev. B 106, 094417 (2022).

[52] Yixuan Huang and D. N. Sheng. ``Topological Chiral and Nematic Superconductivity by Doping Mott Insulators on Triangular Lattice''. Phys. Rev. X 12, 031009 (2022).

[53] Yixuan Huang, Shou-Shu Gong, and D. N. Sheng. ``Quantum Phase Diagram and Spontaneously Emergent Topological Chiral Superconductivity in Doped Triangular-Lattice Mott Insulators''. Phys. Rev. Lett. 130, 136003 (2023).

[54] 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).

[55] Matthias Troyer and Uwe-Jens Wiese. ``Computational Complexity and Fundamental Limitations to Fermionic Quantum Monte Carlo Simulations''. Phys. Rev. Lett. 94, 170201 (2005).

[56] Shiwei Zhang, J. Carlson, and J. E. Gubernatis. ``Constrained Path Quantum Monte Carlo Method for Fermion Ground States''. Phys. Rev. Lett. 74, 3652–3655 (1995).

[57] Shiwei Zhang, J. Carlson, and J. E. Gubernatis. ``Constrained path Monte Carlo method for fermion ground states''. Phys. Rev. B 55, 7464–7477 (1997).

[58] Huy Nguyen, Hao Shi, Jie Xu, and Shiwei Zhang. ``Cpmc-lab: A Matlab package for Constrained Path Monte Carlo calculations''. Computer Physics Communications 185, 3344–3357 (2014).

[59] J. P. F. LeBlanc, Andrey E. Antipov, Federico Becca, Ireneusz W. Bulik, Garnet Kin-Lic Chan, Chia-Min Chung, Youjin Deng, Michel Ferrero, Thomas M. Henderson, Carlos A. Jiménez-Hoyos, E. Kozik, Xuan-Wen Liu, Andrew J. Millis, N. V. Prokof'ev, Mingpu Qin, Gustavo E. Scuseria, Hao Shi, B. V. Svistunov, Luca F. Tocchio, I. S. Tupitsyn, Steven R. White, Shiwei Zhang, Bo-Xiao Zheng, Zhenyue Zhu, and Emanuel Gull. ``Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms''. Phys. Rev. X 5, 041041 (2015).

[60] 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).

[61] R. P. Feynman. ``Atomic Theory of the Two-Fluid Model of Liquid Helium''. Phys. Rev. 94, 262–277 (1954).

[62] Manuela Capello, Federico Becca, Michele Fabrizio, Sandro Sorella, and Erio Tosatti. ``Variational Description of Mott Insulators''. Phys. Rev. Lett. 94, 026406 (2005).

[63] J. Kokalj and Ross H. McKenzie. ``Thermodynamics of a Bad Metal–Mott Insulator Transition in the Presence of Frustration''. Phys. Rev. Lett. 110, 206402 (2013).

[64] W. F. Brinkman and T. M. Rice. ``Application of Gutzwiller's Variational Method to the Metal-Insulator Transition''. Phys. Rev. B 2, 4302–4304 (1970).

[65] Henk Eskes, Andrzej M. Oleś, Marcel B. J. Meinders, and Walter Stephan. ``Spectral properties of the Hubbard bands''. Phys. Rev. B 50, 17980–18002 (1994).

[66] P. H. Y. Li, R. F. Bishop, and C. E. Campbell. ``Quasiclassical magnetic order and its loss in a spin-$\frac{1}{2}$ Heisenberg antiferromagnet on a triangular lattice with competing bonds''. Phys. Rev. B 91, 014426 (2015).

[67] A. L. Chernyshev and M. E. Zhitomirsky. ``Spin waves in a triangular lattice antiferromagnet: Decays, spectrum renormalization, and singularities''. Phys. Rev. B 79, 144416 (2009).

[68] 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, 251–256 (2019).

[69] Konrad Viebahn, Matteo Sbroscia, Edward Carter, Jr-Chiun Yu, and Ulrich Schneider. ``Matter-Wave Diffraction from a Quasicrystalline Optical Lattice''. Phys. Rev. Lett. 122, 110404 (2019).

[70] Matteo Sbroscia, Konrad Viebahn, Edward Carter, Jr-Chiun Yu, Alexander Gaunt, and Ulrich Schneider. ``Observing Localization in a 2D Quasicrystalline Optical Lattice''. Phys. Rev. Lett. 125, 200604 (2020).

[71] A. Mendoza-Coto, R. Turcati, V. Zampronio, R. Díaz-Méndez, T. Macrì, and F. Cinti. ``Exploring quantum quasicrystal patterns: A variational study''. Phys. Rev. B 105, 134521 (2022).

[72] Ronan Gautier, Hepeng Yao, and Laurent Sanchez-Palencia. ``Strongly Interacting Bosons in a Two-Dimensional Quasicrystal Lattice''. Phys. Rev. Lett. 126, 110401 (2021).

[73] Matteo Ciardi, Tommaso Macrì, and Fabio Cinti. ``Finite-temperature phases of trapped bosons in a two-dimensional quasiperiodic potential''. Phys. Rev. A 105, L011301 (2022).

[74] T. Macrì and T. Pohl. ``Rydberg dressing of atoms in optical lattices''. Phys. Rev. A 89, 011402 (2014).

[75] Peter Schauss. ``Quantum simulation of transverse Ising models with Rydberg atoms''. Quantum Science and Technology 3, 023001 (2018).

[76] Nicolò Defenu, Tobias Donner, Tommaso Macrì, Guido Pagano, Stefano Ruffo, and Andrea Trombettoni. ``Long-range interacting quantum systems'' (2021).

[77] Elmer Guardado-Sanchez, Benjamin M. Spar, Peter Schauss, Ron Belyansky, Jeremy T. Young, Przemyslaw Bienias, Alexey V. Gorshkov, Thomas Iadecola, and Waseem S. Bakr. ``Quench Dynamics of a Fermi Gas with Strong Nonlocal Interactions''. Phys. Rev. X 11, 021036 (2021).

[78] V. Zampronio. ``CP-AFQMC'' (2022).

[79] H. F. Trotter. ``On the product of semi-groups of operators''. Proc. Amer. Math. Soc. 10, 545–551 (1959).

[80] Peter J. Reynolds, David M. Ceperley, Berni J. Alder, and William A. Lester. ``Fixed-node quantum Monte Carlo for molecules a)-b)''. The Journal of Chemical Physics 77, 5593–5603 (1982).

[81] 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, 1236–1239 (1991).

[82] Wirawan Purwanto and Shiwei Zhang. ``Quantum Monte Carlo method for the ground state of many-boson systems''. Phys. Rev. E 70, 056702 (2004).

[83] Natanael C Costa, José P de Lima, Thereza Paiva, Mohammed El Massalami, and Raimundo R dos Santos. ``A mean-field approach to Kondo-attractive-Hubbard model''. Journal of Physics: Condensed Matter 30, 045602 (2018).

Cited by

[1] Ji-Si Xu, Zheng Zhu, Kai Wu, and Zheng-Yu Weng, "Hubbard Model on Triangular Lattice: Role of Charge Fluctuations", arXiv:2306.11096, (2023).

[2] Yang Yu, Shaozhi Li, Sergei Iskakov, and Emanuel Gull, "Magnetic phases of the anisotropic triangular lattice Hubbard model", Physical Review B 107 7, 075106 (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2023-09-22 11:33:57). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2023-09-22 11:33:55).