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Europhysics Letters (EPL)

Spin Waves in the Half-Filled Hubbard Model beyond the Random Phase Approximation

P. J. H. Denteneer, J. M. J. van Leeuwen

Published May 20, 1993• 9 citations
Publisher

Abstract

The velocity of long-wavelength spin waves for the two-dimensional Hubbard model on a square lattice at half-filling is calculated using the variational Monte Carlo method with a Gutzwiller-type wave function. We show that the result from the Random Phase Approximation (RPA) for the spin wave velocity can also be found as the square root of the ratio of the helicity modulus and the perpendicular susceptibility as calculated in the Hartree-Fock approximation (HFA). Since the Gutzwiller wave function improves upon the HFA, our result for the spin wave velocity improves upon the RPA result. For the on-site repulsion U between 3 and 12 times the hopping integral t the spin wave velocity increases by 34 to 15% compared to the RPA result.

DOI
10.1209/0295-5075/22/6/003

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The velocity of long-wavelength spin waves for the two-dimensional Hubbard model on a square lattice at half-filling is calculated using the variational Monte Carlo method with a Gutzwiller-type wave function. We show that the result from the Random Phase Approximation (RPA) for the spin wave velocity can also be found as the square root of the ratio of the helicity modulus and the perpendicular susceptibility as calculated in the Hartree-Fock approximation (HFA). Since the Gutzwiller wave function improves upon the HFA, our result for the spin wave velocity improves upon the RPA result. For the on-site repulsion U between 3 and 12 times the hopping integral t the spin wave velocity increases by 34 to 15% compared to the RPA result.
The slave-boson mean-field method is used to study the two-dimensional Hubbard model. A magnetic phase diagram allowing for paramagnetism, weak and strong ferromagnetism and antiferromagnetism is constructed and compared to the corresponding phase diagram using the Hartree-Fock approximation (HFA). Magnetically ordered regions are reduced by a factor of about three along both the t/U and density axes compared to the HFA. Using the spin-rotation-invariant formulation of the slave-boson method the helicity modulus is computed and for half filling is found practically to coincide with that found using variational Monte Carlo calculations with the Gutzwiller wave function. Off half filling the results can be used to compare with quantum Monte Carlo calculations of the effective hopping parameter. Contrary to the case of half filling, the slave-boson approach is seen to greatly improve the results of the HFA when off half filling.
As a continuation of a previous paper [J. Phys. Soc. Jpn. 56 (1987) 1490], the variational Monte-Carlo method is extended to include the antiferromagnetic long-range order. The theory is based on the Gutzwiller-type correlation factor and its effect is exactly taken into account by the Monte-Carlo procedure. An application is made to the half-filled-band case of one-dimensional lattice (50 sites), two-dimensional square lattice (up to 20×20 sites) and three-dimensional simple cubic lattice (6×6×6 sites). The result is qualitatively different from previous studies relying on the random-phase-type "Gutzwiller approximation." The variational energy for two and three dimensions is favorably compared with Hirsch's quantum Monte-Carlo data.
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.
A variational estimate for the spin-wave velocity of the one-band Hubbard model on a square lattice in the spin-density-wave regime is studied. The estimate is given by the ratio of the f-sum rule to the static structure factor of the transverse-spin response function. The known results for the Heisenberg model are used to obtain results for those quantities in the large-U limit of the Hubbard model. The f-sum rule and static structure factor are calculated using the random-phase approximation (RPA). The spin-wave velocity calculated in the RPA, in the spin-density wave ground state, violates the variational bound.
The Gutzwiller wave function for the three-band Hubbard model on a two-dimensional CuO2{\mathrm{CuO}}_{2}CuO2​ plane is studied by using the variational Monte Carlo (VMC) method in the limit Ud{\mathit{U}}_{\mathit{d}}Ud​\ensuremath{\rightarrow}\ensuremath{\infty}, where Ud{\mathit{U}}_{\mathit{d}}Ud​ is the Coulomb repulsion between holes on a Cu site. The VMC results for the energy and the fraction of d holes are compared with those of the Rice-Ueda type Gutzwiller approximation (GA). The difference between the VMC and the GA results are most pronounced at half-filling, and away from half-filling the two results agree well in both the hole-doped and the electron-doped cases. The doping dependence of the momentum distribution function is also studied.
We use a variational Monte Carlo method (VMC) and a quantum Monte Carlo method to study the two-dimensional half-filled Hubbard model. We use the Gutzwiller wave function as a trial wave function in VMC and measure energy, correlation functions, and momentum distribution for lattices up to 12\ifmmode\times\else\texttimes\fi{}12 sites. We then use a Monte Carlo algorithm to study the Hubbard model with the same boundary conditions as in VMC at very low temperatures. By comparing the results from these two methods, we assess the accuracy of the Gutzwiller wave function as a representation of the ground state of the Hubbard model. Major differences are found, implying that the Gutzwiller wave function is not a useful approximation to the ground state of the Hubbard model in two dimensions.