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

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. 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. arXiv:1505.02290v2 [cond-mat.str-el] 15

Dynamical mean-field theory study of stripe order and d -wave superconductivity in the two-dimensional Hubbard model

“…The uniform solution is similar to those in earlier work [53,54], although we did not study exactly the same parameters. The plaquette-DMFT energy is in reasonable agreement with recent numerical results [78], although the differences between methods are generally larger than, e.g., differences in the energies of different wavelength stripe states. Some systematic errors are naturally expected because of the mean-field nature of the method.…”

“…These results have been extrapolated to the thermodynamic limit in embedded cluster size (DMET) or system size (FN), while our calculations are performed for the 2 × 2 cluster. The DMET error estimates include all sources of error, while the (very small) FN error bars do not include errors from the fixed node approximation [78]. In absolute terms, the energies are in reasonably good agreement with the other methods, which gives confidence that the energy calculation procedure is technically correctly implemented and reliable.…”

“…(1), which does not include the chemical potential terms. For comparison, we have included some results from density matrix embedding theory (DMET) and fixednode diffusion Monte Carlo (FN) from LeBlanc et al [78]. These results have been extrapolated to the thermodynamic limit in embedded cluster size (DMET) or system size (FN), while our calculations are performed for the 2 × 2 cluster.…”

“…Some systematic errors are naturally expected because of the mean-field nature of the method. At half-filling, where high-accuracy reference data is available [78], plaquette-DMFT overshoots the ground-state energy by about 0.01 hopping units. Away from half-filling, agreement between the different reference methods is not sufficient to judge the magnitude of the error.…”

Coupling quantum Monte Carlo and independent-particle calculations: Self-consistent constraint for the sign problem based on the density or the density matrix

“…A constraint is applied in some space to restrict the Monte Carlo sampling, which introduces a systematic bias but in turn removes the exponential growth in variance and restores the algebraic complexity of the algorithm. The majority of QMC calculations have employed this approach, including many on spin and fermion models 9,10 , and almost all on realistic systems in condensed matter physics [11][12][13] , nuclear physics 14 , and quantum chemistry [15][16][17] .…”