Quantum field theories on a lattice are generally not exactly soluble. This holds in particular for the Hubbard model as a theoretical description of correlation effects in solids. As a consequence, accurate investigations of physical properties of the solutions of such models are quite complicated. These difficulties are particularly severe at low temperatures, where all sorts of types of symmetry breaking are a priori possible. Interestingly, it is nevertheless possible to make exact statements concerning the behavior of solutions of the Hubbard model, at least in certain limits. These exact statements can be derived with the use of an appropriate formulation of perturbation theory.
E.g., in the articles, cited below, it is shown that the predictions of Hartree-Fock theory are modified drastically by quantum fluctuations, even in the extreme weak coupling limit and in the limit of high dimensions. It is also shown that, in sufficiently high spatial dimensions, the doped Hubbard model describes electronic phase separation, that the renormalization due to quantum fluctuations also occurs in the three-dimensional system, and that the saddle point solution is renormalized infinitely strongly in two spatial dimensions.
Thermodynamics of the Extended Hubbard Model in high dimensions
P. G. J. van Dongen
Phys. Rev. Lett. 67 757-760 (1991)
Phase separation in the extended Hubbard model at weak coupling
P. G. J. van Dongen
Phys. Rev. Lett. 74 182-185 (1995)
Symmetry breaking in the Hubbard model at weak coupling
T. Schauerte and P. G. J. van Dongen
Phys. Rev. B 65, 081105 (2002)
Orbital-selective Mott transitions in a doped two-band Hubbard model with crystal field splitting
E. Jakobi, N. Blümer, and P. G. J. van Dongen
Phys. Rev. B 87, 205135 (2013)