Rabambi, Teflon Phumudzo2015-09-082015-09-082015-02http://hdl.handle.net/10539/18572A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, 2015.Recently there has been signi cant interest in new types of metals called non-Fermi liquids, which cannot be described by Landau Fermi liquid theory. Landau Fermi liquid theory is a theoretical model used to describe low energy interacting fermions or quasiparticles. There is a growing interest in constructing an e ective eld theory for these types of metals. One of the paradigms to understand these metals is by the use of Wilsonian renormalization group (RG) to study a theoretical toy model consisting of fermions coupled to a gapless order parameter eld. Here we will study fermions coupled to gapless bosons (order parameter) below the upper critical dimension (d = 3). We will treat both fermions and bosons on equal footing and construct an e ective eld theory which only integrates out high momentum modes. Then we compute the one-loop RG ows for the Yukawa coupling and four-Fermi interaction. We will discuss log2 and log3 subleties associated with the one loop RG ows for the four-Fermi interaction and how they can be circumvented.enFermi liquid theory.Renormalization group.Quantum liquids.Thesis