Hubbard Model

Abstract

In order to describe the magnetic metals like the classical ferromagnets Fe, Co and Ni (“band ferromagnets”), the Heisenberg model is conceptually not suitable. In the theory of band magnetism, the most popularly employed model today is the Hubbard model, which will be the focal point of this chapter. First this model will be “derived” and its region of application fixed. Inspite of its very simple structure, the many-body problem defined by this model is not exactly solvable.

The molecular field approximation (“Stoner model”) seems to approximate the ground state properties quite well and predicts a para/ferromagnetic phase transition, though with an unrealistically high Curie temperature T _C. The magnetism results from a rigid shift of the ↑- and ↓-sub-bands, the shift itself being proportional to the magnetization. The Stoner model neglects the important electronic correlations, which cannot be neglected if one really wants to understand magnetism.

For the Hubbard model, it is possible to make only a few exact statements out of which, without claiming any completeness, we will discuss in this chapter a few of them (infinitely narrow band, exactly half-filled band, Mermin–Wagner theorem, spectral moments and high-energy expansions, regions of weak and strong coupling, etc). They will provide the first glimpse of general physics of the model.

To understand the collective magnetism in the Hubbard model, it is necessary to understand and investigate electronic correlations. We present an interpolation method which “interpolates” between two limiting cases and at the end is identical to the so-called “Hubbard-I solution”, which was obtained by Hubbard in his original work by using a decoupling procedure for the Greens functions. This, however, leads to a criterion for ferromagnetism which is difficult to fulfil. A spectral density ansatz, with exactly fitted spectral moments, provides plausible criteria for the model variables such as temperature and band filling. In contrast to the Hubbard-I solution, in this case, in the quasiparticle energies, there appears a spin-dependent band shift which proves to be decisive for the possibility of collective ferromagnetic order. With a modified alloy analogy fitted with exact high-energy expansions and evaluated in the coherent potential approximation (CPA), one can observe the influence of quasiparticle damping on magnetic stability. The most reliable statements about collective magnetism available today are obtained from the “dynamical mean field theory”, which will be discussed at the end of this chapter.

We will limit the discussion exclusively to ferromagnetism. Antiferro- and ferrimagnetism will not be considered.