A Rigorous Proof of Fermi Liquid Behavior for Jellium Two-Dimensional Interacting Fermions

Using the method of continuous constructive renormalization group around the Fermi surface, it is proved that a jellium two-dimensional interacting system of Fermions at low temperature $T$ remains analytic in the coupling constant $\lambda$ for $|\lambda| |\log T| \le K$ where $K$ is some numerical constant and $T$ is the temperature. Furthermore in that range of parameters, the first and second derivatives of the self-energy remain bounded, a behavior which is that of Fermi liquids and in particular excludes Luttinger liquid behavior. Our results prove also that in dimension two any transition temperature must be non-perturbative in the coupling constant, a result expected on physical grounds. The proof exploits the specific momentum conservation rules in two dimensions.Comment: 4 pages, no figure

Short-range spin correlations and induced local spin-singlet amplitude in the Hubbard model

In this paper, from the microscopic Hubbard Hamiltonian we extract the local spin-singlet amplitude due to short-range spin correlations, and quantify its strength near half-filling. As a first application of the present approach, we study a problem of the energy dispersion and its d-wave modulation in the insulating cuprates, Sr$_{2}$CuO$_{2}$Cl$_{2}$ and Ca$_{2}$CuO$_{2}$Cl$_{2}$. Without any adjustable parameters, most puzzling issues are naturally and quantitatively explained within the present approach.Comment: 6 pages, 3 figure

Two-Particle-Self-Consistent Approach for the Hubbard Model

Even at weak to intermediate coupling, the Hubbard model poses a formidable challenge. In two dimensions in particular, standard methods such as the Random Phase Approximation are no longer valid since they predict a finite temperature antiferromagnetic phase transition prohibited by the Mermin-Wagner theorem. The Two-Particle-Self-Consistent (TPSC) approach satisfies that theorem as well as particle conservation, the Pauli principle, the local moment and local charge sum rules. The self-energy formula does not assume a Migdal theorem. There is consistency between one- and two-particle quantities. Internal accuracy checks allow one to test the limits of validity of TPSC. Here I present a pedagogical review of TPSC along with a short summary of existing results and two case studies: a) the opening of a pseudogap in two dimensions when the correlation length is larger than the thermal de Broglie wavelength, and b) the conditions for the appearance of d-wave superconductivity in the two-dimensional Hubbard model.Comment: Chapter in "Theoretical methods for Strongly Correlated Systems", Edited by A. Avella and F. Mancini, Springer Verlag, (2011) 55 pages. Misprint in Eq.(23) corrected (thanks D. Bergeron

The Hubbard model within the equations of motion approach

The Hubbard model has a special role in Condensed Matter Theory as it is considered as the simplest Hamiltonian model one can write in order to describe anomalous physical properties of some class of real materials. Unfortunately, this model is not exactly solved except for some limits and therefore one should resort to analytical methods, like the Equations of Motion Approach, or to numerical techniques in order to attain a description of its relevant features in the whole range of physical parameters (interaction, filling and temperature). In this manuscript, the Composite Operator Method, which exploits the above mentioned analytical technique, is presented and systematically applied in order to get information about the behavior of all relevant properties of the model (local, thermodynamic, single- and two- particle ones) in comparison with many other analytical techniques, the above cited known limits and numerical simulations. Within this approach, the Hubbard model is shown to be also capable to describe some anomalous behaviors of the cuprate superconductors.Comment: 232 pages, more than 300 figures, more than 500 reference