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arXiv:1904.10529 · Journal of Physics Condensed Matter

Approximate expression for the ground-state energy of the two- and three-dimensional Hubbard model at arbitrary filling obtained from dimensional scaling

LUCAS NASCIMENTO VILELA, Klaus Capelle, L. Nunes de Oliveira, V. L. Campo

Published Jul 17, 2019• 3 citations
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Abstract

We generalize the linear discrete dimensional scaling approach for the repulsive Hubbard model to obtain a nonlinear scaling relation that yields accurate approximations to the ground-state energy in both two and three dimensions, as judged by comparison to auxiliary-field quantum Monte Carlo (QMC) data. Predictions are made for the per-site ground-state energies in two and three dimensions for n (filling factor) and U (Coulomb interaction) values for which QMC data are currently unavailable.

DOI
10.1088/1361-648x/ab32ac

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We generalize the linear discrete dimensional scaling approach for the repulsive Hubbard model to obtain a nonlinear scaling relation that yields accurate approximations to the ground-state energy in both two and three dimensions, as judged by comparison to auxiliary-field quantum Monte Carlo (QMC) data. Predictions are made for the per-site ground-state energies in two and three dimensions for n (filling factor) and U (Coulomb interaction) values for which QMC data are currently unavailable.
A simple, physically motivated, scaling hypothesis, which becomes exact in important limits, yields estimates for the ground-state energy of large, composed, systems in terms of the ground-state energy of its building blocks. The concept is illustrated for the electron liquid, and the Hubbard model. By means of this scaling argument the energy of the one-dimensional half-filled Hubbard model is estimated from that of a two-site Hubbard dimer, obtaining quantitative agreement with the exact one-dimensional Bethe-Ansatz solution, and the energies of the two- and three-dimensional half-filled Hubbard models are estimated from the one-dimensional energy, recovering exact results for $U\ensuremath{\rightarrow}0$ and $U\ensuremath{\rightarrow}\ensuremath{\infty}$ and coming close to quantum Monte Carlo data for intermediate UUU.
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.
Abstract We use the Auxiliary Field Quantum Monte Carlo technique to perform a simple study of the two dimensional attractive Hubbard model and calculate the total ground state energy to compare with another technique in the literature for U = –4t. Our finite size extrapolated result corroborate the approximated lattice density-functional theory method for this value of U at half-filling. To stabilize the Auxiliary Field Quantum Monte Carlo algorithm one must resort to one of the two decomposition methods: the modified Grand-Schmidt decomposition and the singular value decomposition. Two characteristics like accuracy and total execution time were considered to compare the decomposition algorithms, and the results are presented bellow.
A Quantum Monte Carlo method of calculating operator expectation values for the ground state of the nearest-neighbor Heisenberg model on large square lattices is presented, along with some comparisons of the results obtained from this method and alternative methods for 2D lattices. In the simulations the ground state is projected out from an arbitrary state and sampled in a valence\nbond basis spanning the spin singlet subspace. The projection method is then extended to be used on a simplified Hubbard model at half-filling, which is\naptly called the T0-less Hubbard model, for arbitrary relative strengths of the electron hopping and electron-electron repulsion energy parameters. Results for calculated ground state energies in 1D are presented. These calculations are performed using a spin-charge separated basis, in which the valence bonds\nare generalized to also carry charge.
We develop a systematic self-consistent perturbative expansion for the self-energy of Hubbard-like models. The interaction lines in the Feynman diagrams are dynamically screened by the charge fluctuations in the system. Although the formal expansion is exact---assuming that the model under the study is perturbative---only if diagrams to all orders are included, it is shown that for large-on-site-Coulomb-repulsion-UUU systems weak-coupling expansions to a few orders may already converge. In order to test the approximation at intermediate-to-high temperatures, we use the exact charge-fluctuation susceptibility from quantum Monte Carlo (QMC) simulation studies as input, which determines the exact screened interaction, and compare our results for the self-energy to the QMC results. We also make comparisons with fluctuation-exchange approximation. We show that the screened interaction for the large-UUU system can be vanishingly small at a certain intermediate electron filling, and it is found that our approximation for the imaginary part of the one-particle self-energy agrees well with the QMC results in the low-energy scales at this particular filling. But the usefulness of the approximation is hindered by the fact that it has the incorrect filling dependence when the filling deviates from this value. We also calculate the exact QMC Fermi surfaces for the two-dimensional (2D) Hubbard model for several fillings. Our results near half filling show extreme violation of the concepts of the band theory; in fact, instead of growing, the Fermi surface vanishes when doped toward the half-filled Mott-Hubbard insulator. Sufficiently away from half filling, noninteractinglike Fermi surfaces are recovered. These results combined with the Luttinger theorem might show that diagrammatic expansions for the nearly-half-filled Hubbard model are unlikely to be possible; however, the nonperturbative part of the solution seems to be less important as the filling gradually moves away from one half. Results for the 2D one-band Hubbard model for several hole dopings are presented. Implications of this study for the high-temperature superconductors are also discussed.