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

J. P. F. LeBlanc, Andrey E. Antipov+23 more|arXiv (Cornell University)|Jan 1, 2015

Physics of Superconductivity and MagnetismPhysics and Astronomy137 references581 citations

TL;DR

This paper presents a comprehensive benchmark study of the two-dimensional Hubbard model using a wide array of numerical methods, including quantum Monte Carlo, diagrammatic Monte Carlo, DMRG, and dynamical mean-field approaches. It establishes reliable results in weak coupling, half-filling, and far-from-half-filling regimes, while identifying persistent uncertainties near half-filling and intermediate interaction strengths due to competing quantum phases.

ABSTRACT

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.

Motivation & Objective

• To assess the accuracy and consistency of diverse numerical methods in solving the 2D Hubbard model.
• To identify parameter regimes where different methods converge, establishing reliable benchmarks for future theory and algorithm development.
• To quantify uncertainties arising from finite-size effects, truncation of diagrams, and statistical sampling in numerical simulations.
• To highlight regions of parameter space—particularly near half-filling and intermediate U—where systematic errors and physical competition between phases lead to methodological disagreements.

Proposed method

• Employed auxiliary field quantum Monte Carlo (AFQMC) for numerically exact results in the thermodynamic limit.
• Applied bare and bold-line diagrammatic Monte Carlo (DiagMC) to systematically include Feynman diagrams with controlled approximations.
• Used the dual fermion (DF) method to access long-wavelength correlations beyond dynamical mean-field theory.
• Applied density matrix embedding theory (DMET) and density matrix renormalization group (DMRG) to study strong correlations in finite clusters with embedded self-energy corrections.
• Utilized fixed-node diffusion Monte Carlo (FN-DMC) and unrestricted coupled cluster (UCCSD) to compare variational and perturbative approaches.
• Combined multi-reference projected Hartree-Fock (MRPHF) and cluster dynamical mean-field theory (DCA) to probe broken symmetry and inhomogeneous phases.

Experimental results

Research questions

• RQ1 To what extent do different numerical methods converge on the ground state energy and double occupancy of the 2D Hubbard model in the thermodynamic limit?
• RQ2 What are the dominant sources of systematic error in various numerical approaches, and how do they vary with interaction strength and electron filling?
• RQ3 In which parameter regimes (U, n) do multiple methods agree within error bars, indicating reliable benchmarks?
• RQ4 Why do discrepancies persist near half-filling and intermediate U, and is this due to numerical artifacts or physical competition between phases?
• RQ5 Can the agreement between numerically exact methods (e.g., AFQMC, DiagMC) be used to validate approximate methods like DMRG or UCCSD?

Key findings

• Excellent agreement is observed between AFQMC, DiagMC, and other numerically exact methods at half-filling across all interaction strengths, establishing a reliable benchmark for the Mott insulating phase.
• In the weak-coupling regime and at carrier concentrations far from half-filling, multiple methods converge on the same ground state energy and double occupancy with small uncertainties.
• Near half-filling and for intermediate interaction strengths (U ≈ 8–12), significant discrepancies emerge between methods, indicating large uncertainties and potential physical competition between phases.
• The self-energy on the Matsubara axis shows consistent behavior across AFQMC and DiagMC in weak coupling and at half-filling, validating the use of these methods for dynamical properties.
• The study identifies that systematic errors in DMRG and UCCSD are more pronounced in the intermediate U regime, particularly near half-filling, where they may fail to capture competing orders.
• The authors conclude that the remaining uncertainties in intermediate regimes likely have a physical origin—such as proximity to quantum phase transitions—rather than being purely numerical artifacts.