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arXiv:cond-mat/0107050 · Journal of Physics Condensed Matter

Random-phase approximation for multi-band Hubbard models

Jörg Bünemann, Florian Gebhard

Published Oct 19, 2001• 5 citations
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Abstract

We derive the random-phase approximation for spin excitations in general multi-band Hubbard models, starting from a collinear ferromagnetic Hartree-Fock ground state. The results are compared with those from a recently introduced variational many-body approach to spin waves in itinerant ferromagnets. As we exemplify for Hubbard models with one and two bands, the two approaches lead to qualitatively different results. The discrepancies can be traced back to the fact that the Hartree-Fock theory fails to describe properly the local moments which naturally arise in a correlated-electron theory.

DOI
10.1088/0953-8984/13/44/312

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We derive the random-phase approximation for spin excitations in general multi-band Hubbard models, starting from a collinear ferromagnetic Hartree-Fock ground state. The results are compared with those from a recently introduced variational many-body approach to spin waves in itinerant ferromagnets. As we exemplify for Hubbard models with one and two bands, the two approaches lead to qualitatively different results. The discrepancies can be traced back to the fact that the Hartree-Fock theory fails to describe properly the local moments which naturally arise in a correlated-electron theory.
The subject of the present paper is the theoretical description of collective electronic excitations, i.e., spin waves, in the Hubbard model. Starting with the widely used random-phase approximation, which combines Hartree-Fock theory with the summation of the two-particle ladder, we extend the theory to a more sophisticated single particle approximation, namely, the spectral-density ansatz. Doing so we have to introduce a ``screened'' Coulomb interaction rather than the bare Hubbard interaction in order to obtain physically reasonable spinwave dispersions. The discussion following the technical procedure shows that comparison of standard RPA with our new approximation reduces the occurrence of a ferromagnetic phase further with respect to the phase diagrams delivered by the single particle theories.
All the linear excitations are calculated in the random-phase approximation around a doped inhomogeneous Hartree-Fock state as well as the undoped uniform antiferromagnetic state in the one-dimensional Hubbard model. Upon doping, shape modes appear, localized around an introduced hole, as well as some ``shake-off'' branches related to combinations of local and extended Hartree-Fock orbitals. These modes are in addition to spin wave and ultragap branches in the pure antiferromagnetic state. Some characteristics of the shape modes are similar to bare processes in the t-t'-J model derived from the Hubbard model at strong coupling. Spectral weights in different susceptibility channels are calculated and shown to have peaks in low-energy regions, arising from specific shape modes.
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.
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.
We formulate a multi-band generalization of the time-dependent Gutzwiller theory. This approach allows for the calculation of general two-particle response functions, which are crucial for an understanding of various experiments in solid-state physics. As a first application, we study the momentum- and frequency-resolved magnetic susceptibility in a two-band Hubbard model. As in the case of underlying ground-state approaches, we find significant differences between the results of our method and those from a time-dependent Hartree–Fock approximation.
Based on the variational Gutzwiller theory, we present a method for the computation of response functions for multiband Hubbard models with general local Coulomb interactions. The improvement over the conventional random-phase approximation is exemplified for an infinite-dimensional two-band Hubbard model where the incorporation of the local multiplet structure leads to a much larger sensitivity of ferromagnetism on the Hund coupling. Our method can be implemented into local-density approximation and Gutzwiller schemes and will therefore be an important tool for the computation of response functions for strongly correlated materials.