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Physical review. B, Condensed matter

Numerical study of the two-dimensional Hubbard model for various band fillings

Adriana Moreo, Douglas J. Scalapino, Robert L. Sugar, Steven R. White, N. E. Bickers

Published Feb 1, 1990• 187 citations
Publisher

Abstract

Quantum Monte Carlo methods are used to study the physical properties of the two-dimensional Hubbard model for various band fillings. Results for the energy, magnetic moment, magnetic structure factor, and integrated optical spectral weight are presented. A comparison of the half-filled (〈n〉=1) and quarter-filled (〈n〉=0.5) bands is made using results for the momentum distribution 〈${n}_{k}$〉, the compressibility K, and the one-electron self-energy \ensuremath{\Sigma}(k,i${\ensuremath{\omega}}_{n}$), which shows that a gap exists in the one-electron excitations at half-filling but is absent for the one-quarter-filled case. At a filling of 〈n〉=0.87, the locus of k values for which 〈${n}_{k}$〉=0.5 is found to be close to the free Fermi surface for this filling.

DOI
10.1103/physrevb.41.2313

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Quantum Monte Carlo methods are used to study the physical properties of the two-dimensional Hubbard model for various band fillings. Results for the energy, magnetic moment, magnetic structure factor, and integrated optical spectral weight are presented. A comparison of the half-filled (〈n〉=1) and quarter-filled (〈n〉=0.5) bands is made using results for the momentum distribution 〈nk{n}_{k}nk​〉, the compressibility K, and the one-electron self-energy \ensuremath{\Sigma}(k,i\ensuremathωn{\ensuremath{\omega}}_{n}\ensuremathωn​), which shows that a gap exists in the one-electron excitations at half-filling but is absent for the one-quarter-filled case. At a filling of 〈n〉=0.87, the locus of k values for which 〈nk{n}_{k}nk​〉=0.5 is found to be close to the free Fermi surface for this filling.
We investigate the dynamical properties of the one-dimensional Hubbard model using the quantum Monte Carlo method and the numerical analytic continuation method based on the maximum entropy principle. Results for the spin, charge and current excitation spectrum are presented for various values of the coupling and the band filling for the system of 24 sites with the periodic boundary condition. We find that the total spectral density of the spin excitation exhibits a gapless structure irrespective of the coupling or the band filling. The charge excitation spectrum, on the other hand, clearly shows the gap formation at the half filled band and shifts its weight toward the higher energy region with increase of the coupling. This charge excitation gap, however, disappears with the hole doping. We also discuss the momentum dependence of the spectral functions.
Electronic states of a two-dimensional Hubbard model near half filling is investigated by use of the composite-operator method, which is improved by including two-site correlations in combined use with the resolvent method. The densities of states obtained for the upper and lower Hubbard bands have three-peek structures at half filling and give shapes above a certain temperature similar to the ones obtained in the numerical simulation by Dagotto, Ortolani, and Scalapino [Phys. Rev. B 46, 3183 (1992)], but show narrowing of the bands at low temperature due to the formation of spin-singlet states among the nearest-neighbor electrons. A doped case shows a coherent peak at the Fermi level and the feature of the density of states is very similar to the one from numerical simulation. The one-particle spectra in the (k,k) direction is also presented. The results show that, at half filling, the spectral weight is not evenly distributed between two bands; most of the weight of the lower Hubbard band appears between (0,0) and (\ensuremath{\pi}/2a,\ensuremath{\pi}/2a), while that of the upper Hubbard band lies in the region between (\ensuremath{\pi}/2a,\ensuremath{\pi}/2a) and (\ensuremath{\pi}/a,\ensuremath{\pi}/a), keeping roughly the original band distribution. With hole doping, a flat band develops at the Fermi level with a large Fermi surface and the lower Hubbard band extends through the whole Brillouin zone.
We report on a numerical study of the two-dimensional Hubbard model and describe two new algorithms for the simulation of many-electron systems. These algorithms allow one to carry out simulations within the grand canonical ensemble at significantly lower temperatures than had previously been obtained and to calculate ground-state properties with fixed numbers of electrons. We present results for the two-dimensional Hubbard model with half- and quarter-filled bands. Our results support the existence of long-range antiferromagnetic order in the ground state at half-filling and its absence at quarter-filling. Results for the magnetic susceptibility and the momentum occupation along with an upper bound to the spin-wave spectrum are given. We find evidence for an attractive effective d-wave pairing interaction near half-filling but have not found evidence for a phase transition to a superconducting state.
Abstract The path‐integral formulation along with the quantum Monte Carlo method is used to study the physical properties of the one‐dimensional Hubbard model for various band fillings and values of interaction strength U. Results for singlet and triplet pairing correlations, local square magnetic moment, total energy, probability of double occupancy, and nearest‐neighbour charge‐density correlations are presented. These results are explored at β (inverse temperature) = 2.0, 1.0, and 0.5 for the linear chain N = 16. These results show that the effect of correlation on the physical quantities is small for low band‐filling and it becomes most evident for the half‐filled band at finite temperatures.
Recent computer simulations performed on the one-dimensional Hubbard model with a new algorithm developed by Sorella et al. confirm the absence of a Fermi surface in this model at half filling. However, these same simulations suggest the existence of a Fermi surface at zero temperature away from half filling, contrary to expectations based on renormalization-group analysis. The results found here, using a more standard algorithm and a simple low-temperature extrapolation, appear consistent with the above-mentioned numerical results in the half-filled and quarter-filled band cases. However, it is argued that the discontinuity at the Fermi level found in all these simulations is likely to come from the zero-temperature finite-size dependence of the quasiparticle weight Z, which is also discussed here.
The Hubbard model has been widely used to describe magnetism due to itinerant electrons. However, except for the one-dimensional case, no exact solution of the model exists. We have studied the two-dimensional Hubbard model on a square lattice using a Monte Carlo simulation technique. Simulations were performed on lattices of up to 8×8 in spatial size. We will discuss results for thermodynamic properties and correlation functions as a function of temperature and interaction strength. For the half-filled band case, our results indicate that the system exhibits long-range antiferromagnetic order in the ground state for any nonzero interaction, although substantially reduced from the classical Néel state due to both charge and spin fluctuations. For the non-half-filled band case, our results suggest that no long-range magnetic order exists. We also don’t find any indication of short-ranged ferromagnetic correlations in the parameter range studied. We have also studied the predictions of various approximate treatments for this model, particularly the static approximation and the coherent potential approximation (CPA). By comparison with the Monte Carlo results, we discuss the validity of these schemes. Finally, we discuss the effect of changing the band structure on the magnetic properties of the system.