Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method

Qin, Mingpu; Shi, Hao; Zhang, Shiwei

doi:10.1103/physrevb.94.085103

Title: Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method

Journal Article · Thu Aug 04 00:00:00 EDT 2016 · Physical Review B

DOI:https://doi.org/10.1103/physrevb.94.085103· OSTI ID:1565498

Qin, Mingpu ^[1]; Shi, Hao ^[1]; Zhang, Shiwei ^[1]

College of William and Mary, Williamsburg, VA (United States)

Ground-state properties of the Hubbard model on a two-dimensional square lattice are studied by the auxiliary-field quantum Monte Carlo method. Accurate results for energy, double occupancy, effective hopping, magnetization, and momentum distribution are calculated for interaction strengths of U/t from 2 to 8, for a range of densities including half-filling and n = 0.3,0.5,0.6, 0.75, and 0.875. At half-filling, the results are numerically exact. Away from half-filling, the constrained path Monte Carlo method is employed to control the sign problem. Our results are obtained with several advances in the computational algorithm, which are described in this paper in detail. We discuss the advantages of generalized Hartree-Fock trial wave functions and its connection to pairing wave functions, as well as the interplay with different forms of Hubbard-Stratonovich decompositions. We study the use of different twist angle sets when applying the twist averaged boundary conditions. We propose the use of quasirandom sequences, which improves the convergence to the thermodynamic limit over pseudorandom and other sequences. With it and a careful finite size scaling analysis, we are able to obtain accurate values of ground-state properties in the thermodynamic limit. Detailed results for finite-sized systems up to 16 × 16 are also provided for benchmark purposes.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States). Oak Ridge Leadership Computing Facility (OLCF)

Sponsoring Organization:: USDOE Office of Science (SC); Simons Foundation; National Science Foundation (NSF)

Grant/Contract Number:: SC0008627; ACI-1053575

OSTI ID:: 1565498

Alternate ID(s):: OSTI ID: 1282432

Journal Information:: Physical Review B, Vol. 94, Issue 8; ISSN 2469-9950

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 47 works

Citation information provided by
Web of Science

References (63)

Quantum Monte Carlo study of the two-dimensional fermion Hubbard model Varney, C. N.; Lee, C. -R.; Bai, Z. J. Physical Review B, Vol. 80, Issue 7 https://doi.org/10.1103/PhysRevB.80.075116	journal	August 2009
Pairing Correlations in the Two-Dimensional Hubbard Model Zhang, Shiwei; Carlson, J.; Gubernatis, J. E. Physical Review Letters, Vol. 78, Issue 23 https://doi.org/10.1103/PhysRevLett.78.4486	journal	June 1997
Backflow correlations in the Hubbard model: An efficient tool for the study of the metal-insulator transition and the large- $U$ limit Tocchio, Luca F.; Becca, Federico; Gros, Claudius Physical Review B, Vol. 83, Issue 19 https://doi.org/10.1103/PhysRevB.83.195138	journal	May 2011
Ground-state properties of strongly interacting Fermi gases in two dimensions Shi, Hao; Chiesa, Simone; Zhang, Shiwei Physical Review A, Vol. 92, Issue 3 https://doi.org/10.1103/PhysRevA.92.033603	journal	September 2015
Sign problem in the numerical simulation of many-electron systems Loh, E. Y.; Gubernatis, J. E.; Scalettar, R. T. Physical Review B, Vol. 41, Issue 13 https://doi.org/10.1103/PhysRevB.41.9301	journal	May 1990
Spin and Charge Order in the Doped Hubbard Model: Long-Wavelength Collective Modes Chang, Chia-Chen; Zhang, Shiwei Physical Review Letters, Vol. 104, Issue 11 https://doi.org/10.1103/PhysRevLett.104.116402	journal	March 2010
Quantum cluster theories Maier, Thomas; Jarrell, Mark; Pruschke, Thomas Reviews of Modern Physics, Vol. 77, Issue 3 https://doi.org/10.1103/RevModPhys.77.1027	journal	October 2005
Linearized auxiliary fields Monte Carlo technique: Efficient sampling of the fermion sign Sorella, Sandro Physical Review B, Vol. 84, Issue 24 https://doi.org/10.1103/PhysRevB.84.241110	journal	December 2011
Density Matrix Embedding: A Simple Alternative to Dynamical Mean-Field Theory Knizia, Gerald; Chan, Garnet Kin-Lic Physical Review Letters, Vol. 109, Issue 18 https://doi.org/10.1103/PhysRevLett.109.186404	journal	November 2012
Numerical study of the two-dimensional Hubbard model White, S. R.; Scalapino, D. J.; Sugar, R. L. Physical Review B, Vol. 40, Issue 1 https://doi.org/10.1103/PhysRevB.40.506	journal	July 1989
Auxiliary field Monte-Carlo for quantum many-body ground states Sugiyama, G.; Koonin, S. E. Annals of Physics, Vol. 168, Issue 1 https://doi.org/10.1016/0003-4916(86)90107-7	journal	April 1986
Ground-state staggered magnetization of two-dimensional quantum Heisenberg antiferromagnets Huse, David A. Physical Review B, Vol. 37, Issue 4 https://doi.org/10.1103/PhysRevB.37.2380	journal	February 1988
Density matrix formulation for quantum renormalization groups White, Steven R. Physical Review Letters, Vol. 69, Issue 19 https://doi.org/10.1103/PhysRevLett.69.2863	journal	November 1992
Two-dimensional Hubbard model: Numerical simulation study Hirsch, J. E. Physical Review B, Vol. 31, Issue 7 https://doi.org/10.1103/PhysRevB.31.4403	journal	April 1985
Monte Carlo calculations of coupled boson-fermion systems. I Blankenbecler, R.; Scalapino, D. J.; Sugar, R. L. Physical Review D, Vol. 24, Issue 8 https://doi.org/10.1103/PhysRevD.24.2278	journal	October 1981
Ground-state correlations of quantum antiferromagnets: A Green-function Monte Carlo study Trivedi, Nandini; Ceperley, D. M. Physical Review B, Vol. 41, Issue 7 https://doi.org/10.1103/PhysRevB.41.4552	journal	March 1990
Coupled-cluster theory in quantum chemistry Bartlett, Rodney J.; Musiał, Monika Reviews of Modern Physics, Vol. 79, Issue 1 https://doi.org/10.1103/RevModPhys.79.291	journal	February 2007
The Resonating Valence Bond State in La2CuO4 and Superconductivity Anderson, P. W. Science, Vol. 235, Issue 4793 https://doi.org/10.1126/science.235.4793.1196	journal	March 1987
Phase diagram of the two-dimensional negative- U Hubbard model Scalettar, R. T.; Loh, E. Y.; Gubernatis, J. E. Physical Review Letters, Vol. 62, Issue 12 https://doi.org/10.1103/PhysRevLett.62.1407	journal	March 1989
The advantages of the general Hartree–Fock method for future computer simulation of materials Hammes‐Schiffer, Sharon; Andersen, Hans C. The Journal of Chemical Physics, Vol. 99, Issue 3 https://doi.org/10.1063/1.465305	journal	August 1993
Finite-Size Error in Many-Body Simulations with Long-Range Interactions Chiesa, Simone; Ceperley, David M.; Martin, Richard M. Physical Review Letters, Vol. 97, Issue 7 https://doi.org/10.1103/PhysRevLett.97.076404	journal	August 2006
Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms LeBlanc, J. P. F.; Antipov, Andrey E.; Becca, Federico Physical Review X, Vol. 5, Issue 4 https://doi.org/10.1103/PhysRevX.5.041041	journal	December 2015
Discrete Hubbard-Stratonovich transformation for fermion lattice models Hirsch, J. E. Physical Review B, Vol. 28, Issue 7 https://doi.org/10.1103/PhysRevB.28.4059	journal	October 1983
Finite-size scaling with modified boundary conditions Sorella, Sandro Physical Review B, Vol. 91, Issue 24 https://doi.org/10.1103/PhysRevB.91.241116	journal	June 2015
Spin- and charge-density waves in the Hartree–Fock ground state of the two-dimensional Hubbard model Xu, Jie; Chang, Chia-Chen; Walter, Eric J. Journal of Physics: Condensed Matter, Vol. 23, Issue 50 https://doi.org/10.1088/0953-8984/23/50/505601	journal	November 2011
Colloquium : Theory of intertwined orders in high temperature superconductors Fradkin, Eduardo; Kivelson, Steven A.; Tranquada, John M. Reviews of Modern Physics, Vol. 87, Issue 2 https://doi.org/10.1103/RevModPhys.87.457	journal	May 2015
Eliminating spin contamination in auxiliary-field quantum Monte Carlo: Realistic potential energy curve of F2 Purwanto, Wirawan; Al-Saidi, W. A.; Krakauer, Henry The Journal of Chemical Physics, Vol. 128, Issue 11 https://doi.org/10.1063/1.2838983	journal	March 2008
Computational Complexity and Fundamental Limitations to Fermionic Quantum Monte Carlo Simulations Troyer, Matthias; Wiese, Uwe-Jens Physical Review Letters, Vol. 94, Issue 17 https://doi.org/10.1103/PhysRevLett.94.170201	journal	May 2005
Role of backflow correlations for the nonmagnetic phase of the $t – t^{'}$ Hubbard model Tocchio, Luca F.; Becca, Federico; Parola, Alberto Physical Review B, Vol. 78, Issue 4 https://doi.org/10.1103/PhysRevB.78.041101	journal	July 2008
Auxiliary-field quantum Monte Carlo method for strongly paired fermions Carlson, J.; Gandolfi, Stefano; Schmidt, Kevin E. Physical Review A, Vol. 84, Issue 6 https://doi.org/10.1103/PhysRevA.84.061602	journal	December 2011
Constrained path Monte Carlo method for fermion ground states Zhang, Shiwei; Carlson, J.; Gubernatis, J. E. Physical Review B, Vol. 55, Issue 12 https://doi.org/10.1103/PhysRevB.55.7464	journal	March 1997
Universality, low-temperature properties, and finite-size scaling in quantum antiferromagnets Fisher, Daniel S. Physical Review B, Vol. 39, Issue 16 https://doi.org/10.1103/PhysRevB.39.11783	journal	June 1989
Metal-insulator transitions Imada, Masatoshi; Fujimori, Atsushi; Tokura, Yoshinori Reviews of Modern Physics, Vol. 70, Issue 4, p. 1039-1263 https://doi.org/10.1103/RevModPhys.70.1039	journal	October 1998
$\frac{t}{U}$ expansion for the Hubbard model MacDonald, A. H.; Girvin, S. M.; Yoshioka, D. Physical Review B, Vol. 37, Issue 16 https://doi.org/10.1103/PhysRevB.37.9753	journal	June 1988
Symmetry in auxiliary-field quantum Monte Carlo calculations Shi, Hao; Zhang, Shiwei Physical Review B, Vol. 88, Issue 12 https://doi.org/10.1103/PhysRevB.88.125132	journal	September 2013
Spin density waves in the Hubbard model: A DMFT approach Peters, Robert; Kawakami, Norio Physical Review B, Vol. 89, Issue 15 https://doi.org/10.1103/PhysRevB.89.155134	journal	April 2014
Finite-size effects in Heisenberg antiferromagnets Neuberger, Herbert; Ziman, Timothy Physical Review B, Vol. 39, Issue 4 https://doi.org/10.1103/PhysRevB.39.2608	journal	February 1989
Correlated electrons in high-temperature superconductors Dagotto, Elbio Reviews of Modern Physics, Vol. 66, Issue 3 https://doi.org/10.1103/RevModPhys.66.763	journal	July 1994
CPMC-Lab: A Matlab package for Constrained Path Monte Carlo calculations Nguyen, Huy; Shi, Hao; Xu, Jie Computer Physics Communications, Vol. 185, Issue 12 https://doi.org/10.1016/j.cpc.2014.08.003	journal	December 2014
Finite-size scaling of the ground-state parameters of the two-dimensional Heisenberg model Sandvik, Anders W. Physical Review B, Vol. 56, Issue 18 https://doi.org/10.1103/PhysRevB.56.11678	journal	November 1997
Twist-averaged boundary conditions in continuum quantum Monte Carlo algorithms Lin, C.; Zong, F. H.; Ceperley, D. M. Physical Review E, Vol. 64, Issue 1 https://doi.org/10.1103/PhysRevE.64.016702	journal	June 2001
Electron correlations in narrow energy bands Hubbard, J. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Vol. 276, Issue 1365, p. 238-257 https://doi.org/10.1098/rspa.1963.0204	journal	November 1963
Algorithm 247: Radical-inverse quasi-random point sequence Halton, J. H. Communications of the ACM, Vol. 7, Issue 12 https://doi.org/10.1145/355588.365104	journal	December 1964
Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension Lieb, Elliott H.; Wu, F. Y. Physical Review Letters, Vol. 20, Issue 25 https://doi.org/10.1103/PhysRevLett.20.1445	journal	June 1968
Numerical study of the two-dimensional Hubbard model for various band fillings Moreo, A.; Scalapino, D. J.; Sugar, R. L. Physical Review B, Vol. 41, Issue 4 https://doi.org/10.1103/PhysRevB.41.2313	journal	February 1990
Bold Diagrammatic Monte Carlo Technique: When the Sign Problem Is Welcome Prokof’ev, Nikolay; Svistunov, Boris Physical Review Letters, Vol. 99, Issue 25 https://doi.org/10.1103/PhysRevLett.99.250201	journal	December 2007
Infinite variance in fermion quantum Monte Carlo calculations Shi, Hao; Zhang, Shiwei Physical Review E, Vol. 93, Issue 3 https://doi.org/10.1103/PhysRevE.93.033303	journal	March 2016
Dual fermion approach to nonlocal correlations in the Hubbard model Rubtsov, A. N.; Katsnelson, M. I.; Lichtenstein, A. I. Physical Review B, Vol. 77, Issue 3 https://doi.org/10.1103/PhysRevB.77.033101	journal	January 2008
Spatially inhomogeneous phase in the two-dimensional repulsive Hubbard model Chang, Chia-Chen; Zhang, Shiwei Physical Review B, Vol. 78, Issue 16 https://doi.org/10.1103/PhysRevB.78.165101	journal	October 2008
Metal-insulator transitions Paesler, Michael Journal of Non-Crystalline Solids, Vol. 130, Issue 1 https://doi.org/10.1016/0022-3093(91)90162-y	journal	June 1991
Phase Diagram of the Two-Dimensional Negative- $U$ Hubbard Model Scalettar, R. T.; Loh, E. Y.; Gubernatis, J. E. Physical Review Letters, Vol. 63, Issue 2 https://doi.org/10.1103/physrevlett.63.218	journal	July 1989
Quantum cluster theories Maier, Thomas; Jarrell, Mark; Pruschke, Thomas Karlsruhe https://doi.org/10.5445/ir/1000012189	text	January 2005
Finite-size correction in many-body electronic structure calculations Kwee, Hendra; Zhang, Shiwei; Krakauer, Henry arXiv https://doi.org/10.48550/arxiv.0711.0921	text	January 2007
Spatially inhomogeneous phase in the two-dimensional repulsive Hubbard model Chang, Chia-Chen; Zhang, Shiwei arXiv https://doi.org/10.48550/arxiv.0805.4831	text	January 2008
Spin- and charge-density waves in the Hartree-Fock ground state of the two-dimensional Hubbard model Xu, Jie; Chang, Chia-Chen; Walter, Eric J. arXiv https://doi.org/10.48550/arxiv.1107.0976	text	January 2011
Multi-reference symmetry-projected variational approaches for ground and excited states of the one-dimensional Hubbard model Rodríguez-Guzmán, R.; Jiménez-Hoyos, Carlos A.; Schutski, R. arXiv https://doi.org/10.48550/arxiv.1304.4192	text	January 2013
Symmetry-projected Wave Functions in Quantum Monte Carlo Calculations Shi, Hao; Jiménez-Hoyos, Carlos A.; Rodríguez-Guzmán, R. arXiv https://doi.org/10.48550/arxiv.1402.0018	text	January 2014
Infinite Variance in Fermion Quantum Monte Carlo Calculations Shi, Hao; Zhang, Shiwei arXiv https://doi.org/10.48550/arxiv.1511.04084	text	January 2015
Quantum Monte Carlo method using phase-free random walks with Slater determinants Zhang, Shiwei; Krakauer, Henry arXiv https://doi.org/10.48550/arxiv.cond-mat/0208340	text	January 2002
The Finite Size Error in Many-body Simulations with long-Ranged Interactions Chiesa, Simone; Ceperley, David M.; Martin, Richard M. arXiv https://doi.org/10.48550/arxiv.cond-mat/0605004	text	January 2006
Correlated Electrons in High Temperature Superconductors Dagotto, Elbio arXiv https://doi.org/10.48550/arxiv.cond-mat/9311013	text	January 1993
A Constrained Path Monte Carlo Method for Fermion Ground States Zhang, Shiwei; Carlson, J.; Gubernatis, J. E. arXiv https://doi.org/10.48550/arxiv.cond-mat/9607062	text	January 1996
Pairing Correlations in the Two-Dimensional Hubbard Model Zhang, Shiwei; Carlson, J.; Gubernatis, J. E. arXiv https://doi.org/10.48550/arxiv.cond-mat/9703017	text	January 1997

Cited By (14)

Auxiliary-field quantum Monte Carlo calculations of the structural properties of nickel oxide Zhang, Shuai; Malone, Fionn D.; Morales, Miguel A. The Journal of Chemical Physics, Vol. 149, Issue 16 https://doi.org/10.1063/1.5040900	journal	October 2018
An auxiliary-Field quantum Monte Carlo perspective on the ground state of the dense uniform electron gas: An investigation with Hartree-Fock trial wavefunctions Lee, Joonho; Malone, Fionn D.; Morales, Miguel A. The Journal of Chemical Physics, Vol. 151, Issue 6 https://doi.org/10.1063/1.5109572	journal	August 2019
An efficient method for strongly correlated electrons in one dimension Mitxelena, Ion; Piris, Mario Journal of Physics: Condensed Matter, Vol. 32, Issue 17 https://doi.org/10.1088/1361-648x/ab6d11	journal	January 2020
Calculating ground-state properties of correlated fermionic systems with BCS trial wave functions in Slater determinant path-integral approaches Vitali, Ettore; Rosenberg, Peter; Zhang, Shiwei Physical Review A, Vol. 100, Issue 2 https://doi.org/10.1103/physreva.100.023621	journal	August 2019
Visualizing the BEC-BCS crossover in a two-dimensional Fermi gas: Pairing gaps and dynamical response functions from ab initio computations Vitali, Ettore; Shi, Hao; Qin, Mingpu Physical Review A, Vol. 96, Issue 6 https://doi.org/10.1103/physreva.96.061601	journal	December 2017
Numerically exact study of Weyl superconductivity Rosenberg, Peter; Aryal, Niraj; Manousakis, Efstratios Physical Review B, Vol. 100, Issue 10 https://doi.org/10.1103/physrevb.100.104522	journal	September 2019
Variational cluster approach to thermodynamic properties of interacting fermions at finite temperatures: A case study of the two-dimensional single-band Hubbard model at half filling Seki, Kazuhiro; Shirakawa, Tomonori; Yunoki, Seiji Physical Review B, Vol. 98, Issue 20 https://doi.org/10.1103/physrevb.98.205114	journal	November 2018
Metal-insulator transition in the ground state of the three-band Hubbard model at half filling Vitali, Ettore; Shi, Hao; Chiciak, Adam Physical Review B, Vol. 99, Issue 16 https://doi.org/10.1103/physrevb.99.165116	journal	April 2019
Stripe order in the underdoped region of the two-dimensional Hubbard model Zheng, Bo-Xiao; Chung, Chia-Min; Corboz, Philippe Science, Vol. 358, Issue 6367 https://doi.org/10.1126/science.aam7127	journal	November 2017
Computation of dynamical correlation functions for many fermion systems with auxiliary-field quantum Monte Carlo Vitali, Ettore; Shi, Hao; Qin, Mingpu arXiv https://doi.org/10.48550/arxiv.1606.04785	text	January 2016
Coupling quantum Monte Carlo and independent-particle calculations: self-consistent constraint for the sign problem based on density or density matrix Qin, Mingpu; Shi, Hao; Zhang, Shiwei arXiv https://doi.org/10.48550/arxiv.1608.07154	text	January 2016
Stripe order in the underdoped region of the two-dimensional Hubbard model Zheng, Bo-Xiao; Chung, Chia-Min; Corboz, Philippe arXiv https://doi.org/10.48550/arxiv.1701.00054	text	January 2017
Visualizing the BEC-BCS crossover in the two-dimensional Fermi gas: pairing gaps and dynamical response functions from ab initio computations Vitali, Ettore; Shi, Hao; Qin, Mingpu arXiv https://doi.org/10.48550/arxiv.1705.07929	text	January 2017
Calculating ground state properties of correlated fermionic systems with BCS trial wave functions in Slater determinant path-integral approaches Vitali, Ettore; Rosenberg, Peter; Zhang, Shiwei arXiv https://doi.org/10.48550/arxiv.1905.05012	text	January 2019

Similar Records

Numerical results on the short-range spin correlation functions in the ground state of the two-dimensional Hubbard model

Journal Article · Fri Aug 25 00:00:00 EDT 2017 · Physical Review B · OSTI ID:1565498

Qin, Mingpu; Shi, Hao; Zhang, Shiwei

Symmetry in auxiliary-field quantum Monte Carlo calculations

Journal Article · Mon Sep 23 00:00:00 EDT 2013 · Physical Review. B, Condensed Matter and Materials Physics · OSTI ID:1565498

Shi, Hao; Zhang, Shiwei

Auxiliary field quantum Monte Carlo for multiband Hubbard models: Controlling the sign and phase problems to capture Hund's physics

Journal Article · Thu Jun 20 00:00:00 EDT 2019 · Physical Review. B · OSTI ID:1565498

Hao, Hongxia; Rubenstein, Brenda M.; Shi, Hao

Related Subjects

75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
Materials Science
Physics

Title: Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method

Citation Formats

References (63)

Cited By (14)

Similar Records

Related Subjects