The substantial effect of quantum fluctuations in strongly correlated electron materials often results in a rich phase diagram with many interesting states of matter. This thesis aims to study some of the mechanisms that give rise to such phenomena.

In Part I, we focus on iron-based superconductors, and use bilinear-biquadratic spin models to study the different magnetic orderings present in both iron pnictides and chalcogenides, specifically FeSe and FeTe_(1-x)Se_x. After benchmarking different methods for the theoretical representation of electron spins, we find our models give good qualitative descriptions of the phases observed in the aforementioned materials. We also find that the dynamical spin structure factors are in agreement with experimental inelastic neutron scattering (INS) results.

In Part II, we switch our focus to so-called heavy fermion materials, where the strong interactions between electrons endow the charge carriers with a very heavy mass. In particular, we investigate the multichannel Kondo model, which is appropriate to describe real materials with multiple conduction electron bands. When just enough of these bands are present to exactly screen the impurity spin, the heavy fermion Fermi liquid state is formed. We present a novel technique for solving this problem in both one- and two-impurity systems. In the latter case, we quantify the transition from the phase with strong magnetic correlations between the two local moments to the heavy fermion Fermi liquid regime. Extending this model further, we aim to capture the superconducting correlations between the conduction electrons on the two impurities. Using the language of auxiliary particles, called bosons and holons, we find that superconductivity does indeed arise in the region where correlations between bosons and between holons are both present, provided that an attractive interaction between the latter exceeds a certain minimum value.

The phenomena described in this thesis provide just a few examples that showcase the richness in the quantum world of strongly interacting particles. We hope that the theoretical methods developed in this work will help shed more light on these quantum phenomena.