Magnetic correlation length at quantum criticality: The role of Fermi surface geometries

Strongly correlated electron systems exhibit some of the most fascinating phenomena of condensed matter physics. Beyond the famous example of the Mott-Hubbard metal-to-insulator transition and the occurrence of classical phase transitions to magnetic and charge ordered states as well as superconductivity, several quantum phase transitions can be found in the phase diagrams of strongly correlated systems. These transitions are quite intriguing, because they occur at zero temperature, where quantum fluctuations dominate the physics in contrast to their classical, thermal counterparts. The study of quantum phase transitions does not represent, in any case, a mere academic exercise, as their occurrence affect broad finite-temperature sectors of the phase diagrams of correlated materials. Their theoretical description for correlated electron systems faces big challenges, both analytical and numerical, so that a comprehensive theory could not be established hitherto. This master thesis aims at improving our fundamental theoretical understanding of quantum phase transitions in correlated bulk metals by performing thorough numerical and analytical investigations. In particular, for the former we resort to the dynamical mean-field theory(DMFT), which includes purely local temporal correlations, while analytical derivations will mostly exploit the random phase approximation (RPA). These quantum many-body methods are applied to one of the most fundamental model systems in condensed matter physics, the Hubbard model on three dimensional lattices. By means of DMFT and RPA, the magnetic susceptibility and the corresponding correlation length in all the relevant regimes around a magnetic quantum critical point in the Hubbard model are determined, clarifying the previously not understood behavior of the latter quantity. This provides further evidence for a significant violation of the predictions of the conventional Hertz-Millis-Moriya theory, triggered by specific geometrical properties of the underlying Fermi surfaces, and for their effects on the whole phase-diagram around the corresponding magnetic quantum critical points.