Strongly Interacting Spinless Fermions in D=1 and 2 Dimensions: A Perturbative-Variational Approach

We propose a perturbative-variational approach to interacting fermion systems on 1D and 2D lattices at half-filling. We address relevant issues such as the existence of Long Range Order, quantum phase transitions and the evaluation of ground state energy. In 1D our method is capable of unveiling the existence of a critical point in the coupling constant at $(t/U)_c=0.7483$ as in fact occurs in the exact solution at a value of 0.5. In our approach this phase transition is described as an example of Bifurcation Phenomena in the variational computation of the ground state energy. In 2D the van Hove singularity plays an essential role in changing the asymptotic behaviour of the system for large values of $t/U$. In particular, the staggered magnetization for large $t/U$ does not display the Hartree-Fock law $(t/U) e^{-2 \pi \sqrt{t/U}}$ but instead we find the law $(t/U) e^{- \frac{\pi ^2}{3} t/U}$. Moreover, the system does not exhibit bifurcation phenomena and thus we do not find a critical point separating a CDW state from a fermion "liquid" state.