Phase diagrams of the two-dimensional Hubbard model

Abstract

The magnetism of strongly correlated electrons in narrow energy bands is a relevant phenomenon for several technologically important materials. In this paper, we investigate the conditions for ferromagnetism of the two-dimensional Hubbard model by the functional-integral method. Using the static and uniform approximation for the partition function of the system we have determined the functional free energy for a squared lattice. Thus we have obtained magnetic phase diagrams equivalent to the Hartree-Fock approximation ones. We have observed the existence of a critical temperature (TC) and of a critical onsite Coulombian electrons interaction (UC) for occurrence of spontaneous magnetization in the context of our approximations. We have obtained several phase diagrams relating T, U and n (average number of electrons per site). We graphically verified the dependence of TC and UCwith respect to n, besides the dependence of TC with respect to n and U simultaneously. In the half-filledband case (n=1) we have obtained an analytical expression for UC, for some given temperatures. The determination of TC and UC for the spontaneous magnetization occurrence contributes to show the functional characteristics of this method. The shape of the curves relative to temperature agrees qualitatively with that expected for itinerant electrons magnetic systems. We have shown that besides the existence of a minimum value for the Coulombian interaction energy for occurrence of spontaneous magnetization, there is a saturation point, that is, a limiting value for the magnetization increase, as expected.