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Physical review. B, Condensed matter

Ground-state energy of the Hubbard model at half filling

G. Polatsek, K. W. Becker

Published Jul 15, 1996• 15 citations
Publisher

Abstract

A recently proposed projection operator technique based on the introduction of cumulants is used to investigate the ground-state energy of the Hubbard model at half filling on a square lattice in two dimensions, in the antiferromagnetic phase. Our results in the intermediate and large regime of the ratio U/t are below those of variational Monte Carlo calculations, which set an upper bound for the energy. By including the most dominant spin-fluctuation processes, the present calculation also obeys the correct infinite U/t limit, where the Hubbard model at half filling reduces to the Heisenberg antiferromagnet. The relation of our approach to recent coupled-cluster calculations for the Hubbard model is also discussed. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.1637

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A recently proposed projection operator technique based on the introduction of cumulants is used to investigate the ground-state energy of the Hubbard model at half filling on a square lattice in two dimensions, in the antiferromagnetic phase. Our results in the intermediate and large regime of the ratio U/t are below those of variational Monte Carlo calculations, which set an upper bound for the energy. By including the most dominant spin-fluctuation processes, the present calculation also obeys the correct infinite U/t limit, where the Hubbard model at half filling reduces to the Heisenberg antiferromagnet. The relation of our approach to recent coupled-cluster calculations for the Hubbard model is also discussed. \textcopyright{} 1996 The American Physical Society.
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 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 have calculated the ground-state energy and the fluctuation contribution to the free-energy for the Hubbard model at half filling. We have used a gauge that leads to the Hartree-Fock solution in the saddle-point approximation. The mean-field ground-state energy calculated here agrees with the results obtained by Dagotto et al. [Phys. Rev. B 45, 10 741 (1992)] with use of the exact-diagonalization techniques on a 4\ifmmode\times\else\texttimes\fi{}4 site cluster. In the large-U limit, the ground-state energy per site is obtained as Eg{\mathit{E}}_{\mathit{g}}Eg​/N=-U/4-J+O(J2{\mathit{J}}^{2}J2/U). The fluctuation contribution to the free-energy was found to have a stronger temperature and U dependence than the mean-field contribution, and to play a dominant role for strongly correlated systems.
Recent refinements of analytical and numerical methods have improved our understanding of the ground-state phase diagram of the two-dimensional (2D) Hubbard model. Here, we focus on variational approaches, but comparisons with both quantum cluster and Gaussian Monte Carlo methods are also made. Our own ansatz leads to an antiferromagnetic ground state at half filling with a slightly reduced staggered order parameter (as compared to simple mean-field theory). Away from half filling, we find d-wave superconductivity, but confined to densities where the Fermi surface passes through the antiferromagnetic zone boundary (if hopping between both nearest-neighbour and next-nearest-neighbour sites is considered). Our results agree surprisingly well with recent numerical studies using the quantum cluster method. An interesting trend is found by comparing gap parameters Δ (antiferromagnetic or superconducting) obtained with different variational wave functions. Δ varies by an order of magnitude and thus cannot be taken as a characteristic energy scale. In contrast, the order parameter is much less sensitive to the degree of sophistication of the variational schemes, at least at and near half filling.
A variational ground state of the repulsive Hubbard model on a square lattice is investigated numerically for an intermediate coupling strength (U=8t)(U=8t)(U=8t) and for moderate sizes (from $6\ifmmode\times\else\texttimes\fi{}6$ to $10\ifmmode\times\else\texttimes\fi{}10$). Our ansatz is superior to other widely used variational wave functions. The results for order parameters and correlation functions provide insight into the antiferromagnetic state at half-filling as well as strong evidence for a superconducting phase away from half-filling.