A Novel Technique for the Simulation of Interacting Fermion Systems

In the first part of our thesis we analyze, as rigorously as possible, this "sign problem" in the HST formulation. We show in fact that in many non trivial cases the method is stable for arbitrary large imaginary time. Although this property has. not been proved to hold in general, the HST has opened new possibilities 24,26,32 for the simulation of interacting fermions. In the second part we describe an algorithm that enables to use the LD for the statistical sampling of weights with negative sign. In the last part we show some of the results obtained with this new method. We consider a simple, but very interesting model; the Hubbard model in 1 and 2D. In 1D we found, quite surprisingly a very sharp jump, at the Fermi momentum kF, in the momentum distribution function. This is especially evident at low density. Then with a careful size scaling analysis we show that this jump is only a finite size effect, and it is instead replaced, for the infinite system, by a power low singularity close to kp, in agreement with previous argument based on Renormalization Group Methods. This behaviour suggest that the lD Hubbard model is a marginal conductor away from half filling because the quasi particle are not Fermi-liquid like. In 2D at half filling we studied the properties of the ground state in order to understand if a possible Mott transition takes place by increasing the onsite interaction U. We found that our numerical results for cluster up to 242 sites strongly suggest the existence of long range antiferromagnetic AF order even for relatively weak interaction. We performed also a size scaling analysis of the momentum distribution and we found no evidence of a Fermi surface at half filing. Then our numerical simulation suggests that the ground state of the half filled Hubbard model is an antiferromagnetic insulator even in the small coupling regime. Away from half filling in 2D the main difficulty is that the statistical weight obtained with the HST is non positive definite and its average sign can reach extremely small values for low temperatures. In this case we performed calculations by using a different but related statistical weight which gives the same ground state properties of the exact non positive weight provided the average sign of the weight does not vanishes exponentially as the temperature is decreased. It is not possible, at the moment, to verify numerically such a convergence condition for such large size, because the statistical evaluation of the average sign is prohibitive whenever this average sign is an extremely small number. In the present numerical calculation, as in the calculation performed by other people, it is not possible to distinguish whether the average sign is vanishing or converging to some small constant. Tests on small size clusters, which can be exactly diagonalized, show that the calculation performed in this way, i.e. neglecting the sign, can be considered at least as a good approximation. Within this approximation we found that away from half filling the 2D antiferromagnetic order is initially destroyed albeit without any clear Fermi liquid behaviour. Then it is possible that the 2D Hubbard model away from half filling has a special kind of ground state which may provide the basis for the understanding of High-Tc superconductivity. We are at the moment systematically improving our calculation away from half-filling and for larger coupling constant.