The primary subject of this thesis is the two-dimensional Hubbard model. Using the functional renormalization Group in static approximation we will present the phase diagram of the 2D-Hubbard model and refine it by including frequency dependencies and full self-energy effects. The focus will fall on the vicinity of the quantum critical point, where the system shows a non-Fermi-liquid-like behavior. In mean-field calculations, we show that the frequency dependence of the self-energy has a substantial impact on the ground state. Furthermore, we investigate the relation between the density-density interaction and the quasiparticle lifetime. During the course of our study, we will develop a saddle-point formulation of Schwinger-Dyson equations and present a new renormalization group scheme which is valid for all fermionic systems.
We analyze the one-dimensional extended Hubbard model with a single static impurity by using a computational technique based on the functional renormalization group. This extends previous work for spinless fermions to spin-
1∕2 fermions. The underlying approximations are devised for weak interactions and arbitrary impurity strengths, and have been checked by comparing with density-matrix renormalization-group data. We present results for the density of states, the density profile, and the linear conductance. Two-particle backscattering leads to striking effects, which are not captured if the bulk system is approximated by its low-energy fixed point, the Luttinger model. In particular, the expected decrease of spectral weight near the impurity and of the conductance at low energy scales is often preceded by a pronounced increase, and the asymptotic power laws are modified by logarithmic corrections.
We analyze the interplay of antiferromagnetism and pairing in the two-dimensional Hubbard model with a moderate repulsive interaction. Coupled charge, magnetic, and pairing fluctuations above the energy scale of spontaneous symmetry breaking are treated by a functional renormalization group flow, while the formation of gaps and order below that scale is treated in mean-field theory. The full frequency dependences of the interaction vertices and gap functions are taken into account. We compute the magnetic and pairing gap functions as a function of doping
p and compare with results from a static approximation. In spite of the strong frequency dependences of the effective interactions and of the pairing gap, important physical results from previous static functional renormalization group calculations are confirmed. In particular, there is a sizable doping regime with robust pairing coexisting with N\'eel or incommensurate antiferromagnetism. The critical temperature for magnetic order is interpreted as the pseudogap crossover temperature. Computing the Kosterlitz-Thouless temperature from the superfluid phase stiffness, we obtain a superconducting dome in the
(p,T) phase diagram centered around 15% hole doping.