Mott physics and spin fluctuations: a functional viewpoint

We present a formalism for strongly correlated systems with fermions coupled to bosonic modes. We construct the three-particle irreducible functional $\mathcal{K}$ by successive Legendre transformations of the free energy of the system. We derive a closed set of equations for the fermionic and bosonic self-energies for a given $\mathcal{K}$. We then introduce a local approximation for $\mathcal{K}$, which extends the idea of dynamical mean field theory (DMFT) approaches from two- to three-particle irreducibility. This approximation entails the locality of the three-leg electron-boson vertex $\Lambda(i\omega,i\Omega)$, which is self-consistently computed using a quantum impurity model with dynamical charge and spin interactions. This local vertex is used to construct frequency- and momentum-dependent electronic self-energies and polarizations. By construction, the method interpolates between the spin-fluctuation or GW approximations at weak coupling and the atomic limit at strong coupling. We apply it to the Hubbard model on two-dimensional square and triangular lattices. We complement the results of Phys.Rev. B 92, 115109 by (i) showing that, at half-filling, as DMFT, the method describes the Fermi-liquid metallic state and the Mott insulator, separated by a first-order interacting-driven Mott transition at low temperatures, (ii) investigating the influence of frustration and (iii) discussing the influence of the bosonic decoupling channel.