The one-dimensional optical polaron is treated on the basis of perturbation theory in the weak-coupling limit. A special matrix diagrammatic technique is developed. It is shown how to evaluate all terms of the perturbation theory for the ground-state energy of a polaron to any order by means of this technique. The ground-state energy is calculated up to eighth order of perturbation theory. The effective mass of an electron is obtained up to sixth order of perturbation theory. The radius of convergence of the series obtained is estimated.
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−2πt/U but instead we find the law (t/U)e−3π2t/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.