Beyond simple variational approach for strongly correlated electron systems

For weakly interacting many-body quantum systems one can apply the extensive framework of perturbative quantum field theory. On the other hand, strong interactions can not be treated practically in perturbation theory because of the absence of a small control parameter. In this context variational methods are a powerful tool to obtain reliable informations regarding the properties of the ground state of the strongly correlated electron system. In this thesis we present novel formulations of two widely used methods: the Gutzwiller wave function and the slave-spin theory. These techniques share a similar philosophy and the extensions are based on the common attitude towards a more detailed description of the high-energy incoherent excitations characterizing interacting electron systems together with low-energy quasiparticles excitations, captured by standard approaches. Our task is not only crucial for a faithful description of the insulating phase but also improves the variational characterization of the low-energy quasiparticle excitations. We apply the novel ghost-Gutzwiller wave function technique to tackle an intriguing phenomenon, namely the exciton Mott transition in photoexcited semiconductors. Despite being a quite old topic that goes back to the 1970's, the nature of the exciton Mott transition still defies a complete understanding. By considering an idealised model of photoexcited semiconductors we unveil the important role of the exciton binding energy in determining the nature of the transition. Moreover, our results uncover rather anomalous electron-hole liquid phase next to the transition, which still sustains excitonic excitations although being a degenerate Fermi liquid of heavy mass quasiparticles. By means of the Dirac-Frenkel variational principle, we generalize the ghost-Gutzwiller wave function to study the out of equilibrium dynamics of strongly correlated electron systems. Numerical results on the single-band Hubbard model show a remarkable agreement with those obtained with time-dependent dynamical mean field theory. We believe that the method opens the way to several promising developments and future applications. Concerning the slave-particle approach, we consider Anderson impurity models and we show that, within our formulation, the constraint on the slave-spin variable can be removed, leaving thermal average free by any projection on the "physical" subspace of the enlarged Hilbert space. To the best of our knowledge, this represents an exception and shows that our formulation is more convenient than other slave-particle approaches, where the constraint have to be implemented explicitly. The method, suited to deal with impurity models, finds direct application in studying time-dependent transport across an interacting quantum dot. To this aim we formulate, by means of the Keldysh Green's function, the slave-spin mean-field theory in the out of equilibrium framework and we apply the method to the dynamics of a driven quantum dot. Finally we study the current-voltage characteristic of an interacting quantum dot tunnel-coupled to the edge of a superconductive nanowire. The appearance of the topological Majorana edge mode is signalized by distinctive features in the charge transport properties of the junction. Apart from the well known half-integer zero bias conductance we find that the topological region is characterized by a vanishing Fano factor, that can be measured in experiments to detect the presence of a Majorana zero mode hybridized with the quantum dot.