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Journal of the Nigerian Association of Mathematical Physics

The derivation of the analytic expressions of the dynamical susceptibility based on the Hubbard-Hirsch Hamiltonian

B E Iyorsor, Robinson O. Okanigbuan, Joa Idiodi

Published May 22, 2008• 0 citations
Publisher

Abstract

Investigation have been carried out on the spin wave energy of a ferromagnet using the random-phase approximation (RPA) in the Stoner model. Based on the Hubbard-Hirsch Hamiltonian, in this paper we study the spin excitation in an itinerant-electron system with the Hartree-Fock Approximation (HFA) to derive the analytic expressions of the dynamical susceptibility of the spin system, by setting up the equation of motion and solving it in a generalized HFA of the susceptibility tensor, Χ-+, and quoting the result for the first derivative of the Green's function. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 579-582

DOI
10.4314/jonamp.v10i1.40181

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Investigation have been carried out on the spin wave energy of a ferromagnet using the random-phase approximation (RPA) in the Stoner model. Based on the Hubbard-Hirsch Hamiltonian, in this paper we study the spin excitation in an itinerant-electron system with the Hartree-Fock Approximation (HFA) to derive the analytic expressions of the dynamical susceptibility of the spin system, by setting up the equation of motion and solving it in a generalized HFA of the susceptibility tensor, Χ-+, and quoting the result for the first derivative of the Green's function. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 579-582
Based on the Hubbard-Hirsch model, we studied the dynamical susceptibility and spin excitation of an itinerant-electron system within the random-phase approximation by using a Green's-function technique. It is shown that the two-center matrix elements in the Hubbard-Hirsch model not only change the dynamic property of the susceptibility, but also widen the gap between two spin subbands, which suppresses the electron-hole single excitations. In contrast, the region of collective excitations of the spin becomes larger. \textcopyright{} 1996 The American Physical Society.
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 transverse dynamical susceptibility and the spin-wave stiffness constant of spin-density-wave (SDW) states in itinerant-electron systems are studied. The Hamiltonian is transformed by using a transformation which rotates the spin coordinate system locally, so that the new zzz direction lies in the direction of the local spin. For this transformed Hamiltonian, a self-consistent many-body theory developed by Fedro and Wilson and extended by Kishore is exploited to derive an exact expression for the transverse dynamical susceptibility. From the long-wavelength limit of the spin-wave spectrum, given by the poles of the transverse dynamical susceptibility, an exact formula for the spin-wave stiffness constant is obtained. Within the Hartree-Fock approximation, it reduces to the Fisher's formula for antiferromagnets.
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.
A simple calculation of the wave-vector- and frequency-dependent paramagnetic spin susceptibility of an interacting electron gas is presented. The equation of motion of the double-time Green's function is solved using a moment-conserving decoupling approximation. For the short-range interaction, our expression for the dynamic spin susceptibility reduces to a previous result of Wolff. In the static and long-wavelength limit, our result for the Stonerenhanced spin susceptibility coincides with that obtained from consideration of the groundstate energy in the Hartree-Fock approximation.
A self-consistent many-body theory for the many-particle Green's function is developed and applied to derive an exact microscopic formula for the transverse dynamical susceptibility. As a concrete application of the above general results we consider the Hubbard Hamiltonian and obtain the spin-wave stiffness constant for an itinerant ferromagnet. It is found that earlier expressions for the stiffness constant obtained by Edwards and Hertz and Edwards are special cases of our general expression.