Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9810371 · Physical review. B, Condensed matter

Incremental expansions for the ground-state energy of the two-dimensional Hubbard model

Jiri Malek, Sergej Flach A, K. Kladko

Published Feb 15, 1999• 4 citations
arXivHTMLPDF

Abstract

A generalization of Faddeev's approach of the three-body problem to the many-body problem leads to the method of increments. This method was recently applied to account for the ground-state properties of Hubbard-Peierls chains [J. Malek, K. Kladko, and S. Flach, JETP Lett. 67, 1052 (1998)]. Here we generalize this approach to two-dimensional square lattices and explicitly treat the incremental expansion up to third order. Comparing our numerical results with various other approaches (Monte Carlo, cumulant approaches) we show that incremental expansions are very efficient because good accuracy with these approaches is achieved treating lattice segments composed of eight sites only.

DOI
10.1103/physrevb.59.r5273

Related papers

Large-Uexpansions for the Hubbard model atT=0

J. Oitmaa, Jon Henderson

We report results for the ground-state energy of the Hubbard model in the form of perturbation expansions in t/U, i.e., expansions about the strong-coupling or ``atomic'' limit. Expansions to order (t/U${)}^{40}$ are obtained for finite cells with number of sites \ensuremath{\le}10, for the half-filled case, for both the linear chain and two-dimensional (2D) square lattice. Analysis of the series by Pad\'e approximant methods yields, in the 1D case, results consistent with the known exact results. For the largest cell, N=10, on the square lattice there is an indication of a singularity on the real t axis, which may indicate a phase transition at U/t\ensuremath{\simeq}4.

Physical review. B, Condensed matter•Oct 1, 1991•9 citations

Theoretical study of interacting low dimensional electron systems

Jianzhong Zheng

This thesis addresses the question of a strongly interacting many-body problem: the Hubbard model on one- and two-dimensional lattices. The technique employed is the small-cluster method wherein the full many-body Hamiltonian for finite clusters with periodic boundary conditions (PBC's) is solved exactly. We first apply this technique to an extended one-dimensional chain and a two-dimensional square-lattice Hubbard model. We then construct a two-dimensional overlayer/substrate Hubbard model with first- and second-nearest-neighbor hopping, on-site and nearest-neighbor interactions, and overlayer-site orbital energy. The ground-state properties are investigated and discussed in a many-body picture.

•May 29, 2019•0 citations

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

Mingpu Qin, Hao Shi, Shiwei Zhang

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 $n=0.3,0.5,0.6,\phantom{\rule{0.16em}{0ex}}0.75$, 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 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 $16\ifmmode\times\else\texttimes\fi{}16$ are also provided for benchmark purposes.

arXiv:1605.09421•Physical review. B./Physical review. B•Aug 4, 2016•100 citations

Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms

Leblanc Jpf, Andrey E. Antipov, Federico Becca +22 more

Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.

•Jan 1, 2015•599 citations

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 +23 more

Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.

•Jan 1, 2015•456 citations

Ground-state properties of the two-dimensional Hubbard model

H. Q. Lin

Using all appropriate symmetries for a two-dimensional square system, we have been able to diagonalize exactly the Hubbard Hamiltonian on the 4\ifmmode\times\else\texttimes\fi{}4 and 18-site lattices. We calculate the ground-state energies, hole-hole interactions, and various correlation functions as functions of the Coulomb repulsion U and band filling, and compare the Hubbard-model results with those obtained from the t-J and strong-coupling models.

Physical review. B, Condensed matter•Oct 1, 1991•28 citations
A generalization of Faddeev's approach of the three-body problem to the many-body problem leads to the method of increments. This method was recently applied to account for the ground-state properties of Hubbard-Peierls chains [J. Malek, K. Kladko, and S. Flach, JETP Lett. 67, 1052 (1998)]. Here we generalize this approach to two-dimensional square lattices and explicitly treat the incremental expansion up to third order. Comparing our numerical results with various other approaches (Monte Carlo, cumulant approaches) we show that incremental expansions are very efficient because good accuracy with these approaches is achieved treating lattice segments composed of eight sites only.
We report results for the ground-state energy of the Hubbard model in the form of perturbation expansions in t/U, i.e., expansions about the strong-coupling or ``atomic'' limit. Expansions to order (t/U)40{)}^{40})40 are obtained for finite cells with number of sites \ensuremath{\le}10, for the half-filled case, for both the linear chain and two-dimensional (2D) square lattice. Analysis of the series by Pad\'e approximant methods yields, in the 1D case, results consistent with the known exact results. For the largest cell, N=10, on the square lattice there is an indication of a singularity on the real t axis, which may indicate a phase transition at U/t\ensuremath{\simeq}4.
This thesis addresses the question of a strongly interacting many-body problem: the Hubbard model on one- and two-dimensional lattices. The technique employed is the small-cluster method wherein the full many-body Hamiltonian for finite clusters with periodic boundary conditions (PBC's) is solved exactly. We first apply this technique to an extended one-dimensional chain and a two-dimensional square-lattice Hubbard model. We then construct a two-dimensional overlayer/substrate Hubbard model with first- and second-nearest-neighbor hopping, on-site and nearest-neighbor interactions, and overlayer-site orbital energy. The ground-state properties are investigated and discussed in a many-body picture.
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/tU/tU/t from 2 to 8, for a range of densities including half-filling and n=0.3,0.5,0.6,0.75n=0.3,0.5,0.6,\phantom{\rule{0.16em}{0ex}}0.75n=0.3,0.5,0.6,0.75, and 0.8750.8750.875. 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 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 $16\ifmmode\times\else\texttimes\fi{}16$ are also provided for benchmark purposes.
Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.
Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.