Since the discovery of the High-T c superconductors, [1], the Hubbard model has been the subject of intense investigations following Anderson’s proposal [2] that the model should capture the essential physics of the cuprate superconductors. From the earlier attempts to obtain the magnetic phase diagram on the square lattice (for an overview see the book by Mattis [3]) one can deduce that antiferromagnetic order exists in the vicinity of the half-filled band whereas ferromagnetic ordering might take place in the phase diagram for strong repulsive interaction strength and moderate hole doping of the half-filled band. Obviously antiferromagnetic and ferromagnetic orders compete in this part of the phase diagram. More recent calculations [4] established that the ground state of the Hubbard model on the square lattice shows long-ranged antiferromagnetic ordering with a charge transfer gap. However, the problem of mobile holes in an antiferromagnetic background remains mostly unsolved. Suggestions for a very wide ferromagnetic domain in the phase diagram based on the restricted Hartree-Fock Approximation have been made by several authors [5] on the cubic lattice, and on the square lattice [6–8]. This domain appears for large interaction and moderate hole doping in which case the Hartree-Fock Approximation ceases to be controlled. Within this framework one expects to obtain reliable results for moderate U where the paramagnetic phase is indeed unstable towards an incommensurate spin structure at a critical density n c (U) [9]. The Gutzwiller Approximation (GA) [10–12] has been applied [13], even for large U, yielding results similar to the Hartree-Fock Approximation. However, for large U, a ferromagnetic domain appears only if the density is larger than some critical value. In the Kotliar and Ruckenstein slave boson technique [14] the GA appears as a saddle-point approximation of this field theoretical representation of the Hubbard model. In the latter a metal-insulator transition occurs at half-filling as recently discussed by Lavagna [15]. The contribution of the thermal fluctuations has been calculated [16] and turned out to be incomplete as this representation, even though exact, is not manifestly spin-rotation invariant. Spin-rotation invariant [17] and spin and charge-rotation invariant [18] formulations have been proposed and the first one was used to calculate correlation functions [19] and spin fluctuation contributions to the specific heat [20]. Comparisons of ground state energy with Quantum Monte-Carlo simulations, including antiferromagnetic ordering [21] and spiral states [22], or with exact diagonalisation data [23] have been done and yield excellent agreement, and a magnetic phase diagram has been proposed [24].
