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

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.

Popular Summary

The two-dimensional Hubbard model is one of the simplest models for interacting fermions: Its Hamiltonian consists of a single band of electrons. The potential energy contribution is modeled by an on-site interaction, and the kinetic energy contribution is modeled by nearest-neighbor and next-nearest-neighbor hopping terms. This Hamiltonian is believed to model the low-energy behavior of cuprate superconductors, and it can be emulated using ultracold fermionic gases. In two dimensions, no analytical solutions to this Hamiltonian are known beyond the perturbative limit, despite the simplicity of the Hamiltonian, and solutions derived from numerical methods have proven to be challenging. Here, we present numerical solutions obtained from a wide range of numerical methods, for widely different parts of phase space of the model, in the thermodynamic limit.

We consider a Hubbard model defined on a two-dimensional square lattice, and we calculate the properties of systems with many interacting electrons using different numerical methods and considering strong, intermediate, and weak on-site repulsion. We pay particular attention to the estimation of the errors introduced by approximations employed by the different methods, and we systematically control these errors wherever possible. We find that errors arise from extrapolating to a thermodynamic limit and that some models diverge in the regime of intermediate coupling. The resulting data yield a summary of the state of the art of the field and delineate areas of parameter space of the model that are known very well and areas where additional improvement is necessary.

Our results constitute a stringent set of reference (i.e., benchmark) data with which future results from numerical methods can be compared and tested.

Figure 1

Extrapolations of the ground-state energy at $U/t=8$, $n=1$. Main panel: AFQMC and FN extrapolated as a function of the inverse cube of the system’s linear dimension $L$ along with DMRG extrapolated in the cube of the inverse cylinder circumference (also denoted $L$). DMRG data are presented both for rotated (with $\sqrt{2}$) and unrotated wrapping of cylinders. Inset: DMET data for clusters of size and geometry indicated, plotted against the reciprocal of the square root of the total number of sites in the cluster $N_c$.

Figure 2

Energies obtained at $U/t=4$, $T/t=0.25$ and $n=0.8$ from $G^2\mathrm{\Gamma }$ (magenta) and $[G^{(0)}{]}^2{\mathrm{\Gamma }}^{(0)}$ (turquoise) as a function of the inverse square of the diagram order parameter $\alpha$ (upper axis label) along with DCA results obtained from finite clusters plotted against the inverse of the cluster size $N_c$ (lower axis labels).

Figure 3

Temperature dependence of the energy for $n=1$ for $U/t=8$ obtained by DCA (black circles) and DF (red cross) and compared to zero-temperature results compiled from various techniques. Solid symbols represent finite systems; open symbols represent extrapolations to the TL.

Figure 4

Thermodynamic-limit ground-state energy for $n=1$ for $U/t=8$ as obtained by various algorithms (open symbols). Also shown are the finite-size systems (filled symbols with adjacent labels) from which the TL ground-state energy was obtained. Data are from AFQMC (red crosses), DMET (blue triangles), UCCSD (maroon diamonds), MRPHF (purple triangles), DMRG (orange squares), and FN (green triangles). Horizontal thin dotted lines show the best estimates for the ground-state energy.

Figure 5

Double occupancy data for $U/t=8$ and $n=1$. Main panel: Temperature dependence of double occupancy, obtained from DCA (finite $T$, black circles) and DF (finite $T$, red plus sign), and the $T=0$ techniques AFQMC (red crosses), DMET (blue triangles), UCCSD (maroon diamonds), MRPHF (purple triangles), DMRG (orange squares), and FN (green triangles). Solid symbols represent finite systems; open symbols represent extrapolations to the thermodynamic limit. Inset: Data at $T=0$ reproduced with an arbitrary $x$-axis offset, from MRPHF, UCCSD, FN, DMET, DMRG, and AFQMC.

Figure 6

Data for $n=0.875$ for $U/t=8$. Main panel: Temperature dependence of $E/t$ compiled from various techniques. Solid symbols represent finite systems; open symbols represent extrapolations to the thermodynamic limit. Finite-$T$ results are shown for DCA (black circles), and zero-$T$ data from AFQMC (red crosses), DMET (blue triangles), UCCSD (maroon diamonds), DMRG (orange squares), and FN (green triangles). Top left inset: Zoom-in of the zero-$T$ data from DMET, AFQMC, FN, and DMRG including finite-system-size data (as labeled).

Figure 7

Data for $n=0.875$ for $U/t=8$. Main panel: Temperature dependence of double occupancy $D$, compiled from various techniques. Solid symbols represent finite systems; open symbols represent extrapolations to the thermodynamic limit. Finite-$T$ results are shown for DCA (black circles), and zero-$T$ data from DMET (blue triangles), MRPHF (purple triangles), UCCSD (maroon diamonds), DMRG (orange squares), and FN (green triangles). Inset: Zoom-in of the zero-$T$ data from FN, AFQMC, DMET, and DMRG.

Figure 8

Data for $n=1$ for $U/t=8$ with $t^{\prime }/t=-0.2$. Main panel: Temperature dependence of $E/t$ compiled from various techniques. Solid symbols represent finite systems; open symbols represent extrapolations to the thermodynamic limit. Finite-$T$ results are shown for DCA (black circles), and zero-$T$ data from AFQMC (red crosses), DMET (blue triangles), UCCSD (maroon diamonds), MRPHF (purple triangles), DMRG (orange squares), and FN (green triangles). Top left inset: Zoom-in of the zero-$T$ data from MRPHF, UCCSD, DMET, FN, DMRG, and AFQMC, including finite-system-size data (as labeled) for MRPHF, FN, DMET, and DMRG. Bottom right panel: Enlarged region of the top left inset showing DMET, DMRG, FN, and AFQMC data at $T=0$, including error bars, extrapolated to the infinite system size.

Figure 9

Data for $n=1$ for $U/t=4$. Main panel: Temperature dependence of $E/t$ compiled from various techniques. Solid symbols represent finite systems; open symbols represent extrapolations to the thermodynamic limit. Finite-$T$ results are shown for DCA (black circles) and DiagMC (turquoise stars), and zero-$T$ data from AFQMC (red crosses), DMET (blue triangles), UCCSD (maroon diamonds), MRPHF (purple triangles), DMRG (orange squares), and FN (green triangles). Top left inset: Zoom-in of the zero-$T$ data from MRPHF, DMET, FN, DMRG, and AFQMC, including finite-system-size data (as labeled) for MRPHF, FN, DMRG, and DMET.

Figure 10

Data for $n=1$ for $U/t=4$. Main panel: Temperature dependence of double occupancy $D$, compiled from various techniques. Solid symbols represent finite systems; open symbols represent extrapolations to the thermodynamic limit. Finite-$T$ results are shown for DCA and DiagMC (turquoise stars), and zero-$T$ data from DMET (blue triangles), UCCSD (maroon diamonds), MRPHF (purple triangles), DMRG (orange squares), and FN (green triangles). Inset: Zoom-in of the zero-$T$ data from DMRG, FN, UCCSD (only finite-system data), DMET, and AFQMC.

Figure 11

Data for $n=0.8$ for $U/t=4$. Main panel: Temperature dependence of $E/t$ compiled from various techniques. Solid symbols represent finite systems; open symbols represent extrapolations to the thermodynamic limit. Finite-$T$ results are shown for DCA (black circles) and DiagMC (pink and turquoise asterisks), and zero-$T$ data from AFQMC (red crosses), DMRG (orange squares), FN (green triangles), and DMET (blue triangles). Top left inset: Zoom-in of the $T/t=0.25$ data from DCA and two types of DiagMC for different orders $\alpha =3,4,\dots$. Bottom right inset: Plot of the imaginary part of the local self-energy $\mathrm{Im}\mathrm{\Sigma }(i{\omega }_n)$ from DiagMC for different expansion orders $\alpha$ and from DCA (black circles covered by magenta stars).

Figure 12

Thermodynamic-limit ground-state energy for $n=0.8$ for $U/t=4$ as obtained by various algorithms (open symbols). Also shown are the finite-size systems (filled symbols with adjacent labels) from which the thermodynamic-limit ground-state energy was obtained. Data are from AFQMC (red crosses), DMET (blue triangles), UCCSD (maroon diamonds), UCC on a $10\times 4$ lattice (shaded diamonds), MRPHF (purple triangles), DMRG (orange squares), and FN (green triangles). Horizontal thin dotted lines show the best estimates for the ground-state energy.

Figure 13

Data for $n=1$ for $U/t=2$. Main panel: Temperature dependence of $E/t$ compiled from various techniques. Solid symbols represent finite systems; open symbols represent extrapolations to the thermodynamic limit. Finite-$T$ results are shown for DCA (black circles) and DiagMC (blue asterisks), and zero-$T$ data from AFQMC (red crosses), DMET (blue triangles), UCCSD (maroon diamonds), MRPHF (purple triangles), DMRG (orange squares), and FN (green triangles). Top left inset: Zoom-in of the zero-$T$ data from UCCSD, DMET, FN, DMRG, AFQMC, and DMET. Bottom right panel: Enlarged region of the top left inset showing thermodynamic-limit data for DMRG, DMET, UCCSD, FN, and AFQMC at $T=0$, including error bars, extrapolated to the infinite system size.

Figure 14

Data for $n=1$ for $U/t=2$. Main panel: Temperature dependence of double occupancy $D$, compiled from various techniques. Solid symbols represent finite systems; open symbols represent extrapolations to the thermodynamic limit. Finite-$T$ results are shown for DCA (black circles) and DiagMC (turquoise asterisks), and zero-$T$ data from DMET (blue triangles), UCCSD (maroon diamonds), MRPHF (purple triangles), DMRG (orange squares), and FN (green triangles). Inset: Zoom-in of the zero-$T$ data from FN, UCCSD, MRPHF, DRMG, DMET, and AFQMC.

Figure 15

Imaginary part of the local self-energy, $\mathrm{Im}{\mathrm{\Sigma }}_{\mathrm{loc}}(\mathrm{i}{\omega }_{\mathrm{n}})$ at $U/t=2$, $T/t=0.5$ and $n=0.3$ from DCA and DiagMC. In the case of DiagMC-$G^2\mathrm{\Gamma }$, we label $\alpha$, the series order from Eq. (7).

Figure 16

Comparison of the real part of the self-energy, $\mathrm{Re}\mathrm{\Sigma }(k,i{\omega }_0)$, at the lowest Matsubara frequency $i{\omega }_0$, obtained from DF (red) compared to 72-site DCA calculations (black) plotted as a function of momentum $k$, throughout the Brillouin zone for $n=1.0$, $U/t=8$, $T/t=0.5$. The dual fermion and DCA self-energies are plotted as step functions. Also included are interpolated results obtained by diagrammatic determinantal Monte Carlo (DDMC) [120, 140, 142, 143] (dashed black) with a gray shading to indicate the level of uncertainty.

Figure 17

Comparison of the frequency dependence of the imaginary part of the self-energy, $\mathrm{Im}\mathrm{\Sigma }(k,i{\omega }_n)$, at fixed $k=(\pi ,0)$ obtained from DF (red) compared to 72-site DCA calculations (black) plotted for $n=1.0$, $U/t=8$, $T/t=0.5$. Also shown are results from the dynamical vertex approximation $D\mathrm{\Gamma }A$ (blue) [124, 144].