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AIP conference proceedings

Comparison of the magnetic properties of the Hubbard Hamiltonian in the weak correlation limmit to the HFA

L. C. Bartel, Howard S. Jarrett

Published Jan 1, 1975• 0 citations
Publisher

Abstract

Resonance broadening is included along with spin‐disorder scattering to study the magnetic properties of the Hubbard Hamiltonian in the weak correlation limit. The calculated susceptibility is compared to the exchange enhanced Hartree–Fock approximation (HFA) susceptibility, and limits are set on the applicability of the HFA. For a semicircular density of states, HFA appears to be applicable for I/Δ<0.5 and a half‐filled band, but for low electron (or hole) densities the range of applicability in I/Δ decreases.

DOI
10.1063/1.30120

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Resonance broadening is included along with spin‐disorder scattering to study the magnetic properties of the Hubbard Hamiltonian in the weak correlation limit. The calculated susceptibility is compared to the exchange enhanced Hartree–Fock approximation (HFA) susceptibility, and limits are set on the applicability of the HFA. For a semicircular density of states, HFA appears to be applicable for I/Δ<0.5 and a half‐filled band, but for low electron (or hole) densities the range of applicability in I/Δ decreases.
The applicability of the Hartree-Fock and random phase approximations to models of strongly correlated electrons is discussed. The 2D Hubbard model is analyzed. An antiferromagnetic phase (at half filling) and Fermi liquid behavior (at low electron densities) are found. In addition, the system shows an unconventional regime near half filling. It is characterized by the formation of spin and charge textures. Local magnetic moments and spin waves are also present, and the quasiparticle pole at the Fermi energy vanishes. The charge excitation spectrum is gapless. The overall features resemble those of the 1D Luttinger liquid.
We present an approach to the magnetic properties of the half-filled Hubbard model, based on an approximate mapping of its low-energy transverse spin excitations on to those of an effective underlying Heisenberg model, but with effective spin interactions which are self-consistently determined and not confined solely to nearest-neighbor couplings. The mapping is exact in strong-coupling and is found to be accurate over a very wide range of interaction strengths, down to weak coupling. At zero temperature, it permits ready evaluation at finite U of the one-loop effects of zero-point spin fluctuations on, e.g., the sublattice magnetization. At finite temperatures, thermodynamic properties of the system in the thermal paramagnet are studied via a physically transparent Onsager reaction field approach, which amounts to a self-consistent treatment of paramagnetic spin correlations. This is central not only in recovering the correct dimensionality dependence of antiferromagnetic long-ranged order, but also for the d=3 case of primary interest here yields a N\'eel temperature in close agreement with known strong- and weak-coupling limits. Spin correlation functions and magnetic susceptibilities also show very good agreement with quantum Monte Carlo calculations over an appreciable temperature range in which the low-lying transverse spin excitations are thermally dominant. \textcopyright{} 1996 The American Physical Society.
Common density-matrix functionals, the M\"uller and the power functional, have been benchmarked for the half-filled Hubbard dimer, which allows us to model the bond dissociation problem and the transition from the weakly to the strongly correlated limit. Unbiased numerical calculations are combined with analytical results. Despite the well known successes of the M\"uller functional, the ground state is degenerate with a one-dimensional manifold of ferromagnetic solutions. The resulting infinite magnetic susceptibility indicates another qualitative flaw of the M\"uller functional. The derivative discontinuity with respect to particle number is not present indicating an incorrect metal-like behavior. The power functional actually favors the ferromagnetic state for weak interaction. Analogous to the Hartree-Fock approximation, the power functional undergoes a transition beyond a critical interaction strength, in this case, however, to a noncollinear antiferromagnetic state.
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.