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arXiv:cond-mat/9402097 · Physical review. B, Condensed matter

Systematic 1/Sstudy of the two-dimensional Hubbard model at half-filling

Andrey V. Chubukov, Karen A. Musaelian

Published Sep 1, 1994• 17 citations
arXivHTMLPDF

Abstract

The two-dimensional Hubbard model is extended by placing 2S orbitals at each lattice site and studied in a systematic 1/S expansion. The 1/S results for the magnetic susceptibility and the spectra of spin-wave excitations at half-filling are consistent with the large S calculations for the Heisenberg antiferromagnet. The 1/S corrections to the fermionic spectrum lift the degeneracy along the edge of the magnetic Brillouin zone yielding minima at (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2). Relation to previous papers on the subject is discussed.

DOI
10.1103/physrevb.50.6238

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The two-dimensional Hubbard model is extended by placing 2S orbitals at each lattice site and studied in a systematic 1/S expansion. The 1/S results for the magnetic susceptibility and the spectra of spin-wave excitations at half-filling are consistent with the large S calculations for the Heisenberg antiferromagnet. The 1/S corrections to the fermionic spectrum lift the degeneracy along the edge of the magnetic Brillouin zone yielding minima at (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2). Relation to previous papers on the subject is discussed.
We develop series expansions for the ground state properties of the Hubbard model by introducing an Ising anisotropy into the Hamiltonian. For the two-dimensional square lattice half-filled Hubbard model, the ground state energy, local moment, sublattice magnetization, uniform magnetic susceptibility, and spin stiffness are calculated as a function of U/t, where U is the Coulomb constant and t is the hopping parameter. Magnetic susceptibility data indicate a crossover around U\ensuremath{\approxeq}4 between spin density wave antiferromagnetism and Heisenberg antiferromagnetism. Comparisons with Monte Carlo simulations, random phase approximation result, and mean-field solutions are also made.
We report large scale determinant quantum Monte Carlo calculations of the effective bandwidth, momentum distribution, and magnetic correlations of the square lattice fermion Hubbard Hamiltonian at half-filling. The sharp Fermi surface of the noninteracting limit is significantly broadened by the electronic correlations but retains signatures of the approach to the edges of the first Brillouin zone as the density increases. Finite-size scaling of simulations on large lattices allows us to extract the interaction dependence of the antiferromagnetic order parameter, exhibiting its evolution from weak-coupling to the strong-coupling Heisenberg limit. Our lattices provide improved resolution of the Green's function in momentum space, allowing a more quantitative comparison with time-of-flight optical lattice experiments.
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.
The T = 0 spin stiffness ρ s and the transverse susceptibility χ ⊥ of the square lattice half-filled Hubbard model are calculated as a function of the Hubbard parameter ratio U / t by series expansions around the Ising limit. We find that the calculated spin stiffness, transverse susceptibility, and sublattice magnetization for the Hubbard model smoothly approach the Heisenberg values for large U / t . The results are compared for different U / t with RPA and other numerical studies. The uniform susceptibility data indicate a crossover around U / t ≈ 4 between weak-coupling (spin density wave) behavior at small U and strong-coupling (Heisenberg) behavior at large U .
We have extended calculations of the zero-temperature excitation spectrum of the one-dimensional Hubbard model to the case where the number of electrons is less than the number of sites in the chain. The results are computed as a function of the ratio Ut\frac{U}{t}tU​, where UUU represents the on-site Coulomb repulsion and ttt is the transfer integral, assumed to be nonzero only for nearest neighbors. Exact calculations are made for the energy and momentum of excitations having single-particle character. Unlike the situation for the half-filled band, we find no gap in the excitation spectrum. We have also considered excitations of the spin-wave type. These are shown to vary linearly with momentum for small momentum. The group velocity for small momentum is found to be inversely proportional to the magnetic susceptibility.