UBC Theses and Dissertations
2+1d quantum field theories in large N limit Omid, Hamid
In Chapter 1, we present a brief introduction to the tight-binding model of graphene and show that in the low-energy continuum limit, it can be modeled by reduced QED₂₊₁ . We then review renormalization group technique which is used in the next chapters. In Chapter 2, we consider a quantum field theory in 3+1d with the defect of a large number of fermion flavors, N. We study the next-to-leading order contributions to the fermions current-current correlation function by performing a large N expansion. We find that the next-to-leading order contributions 1/N to the current-current correlation function is significantly suppressed. The suppression is a consequence of a surprising cancellation between the two contributing Feynman diagrams. We calculate the model's conductivity via the Kubo formula and compare our results with the observed conductivity for graphene. In Chapter 3, we study graphene's beta function in large N. We use the large N expansion to explore the renormalization of the Fermi velocity in the screening dominated regime of charge neutral graphene with a Coulomb interaction. We show that inclusion of the fluctuations of the magnetic field lead to a cancellation of the beta function to the leading order in 1/N. The first non-zero contribution to the beta function turns out to be of order 1/N². We perform a careful analysis of possible infrared divergences and show that the superficial infrared divergences do not contribute to the beta function. In Chapter 4, we study the phase structure of a Φ⁶ theory in large N. The leading order of the large N limit of the O(N) symmetric phi-six theory in three dimensions has a phase which exhibits spontaneous breaking of scale symmetry accompanied by a massless dilaton. In this chapter, we show that this “light dilaton” is actually a tachyon. This indicates an instability of the phase of the theory with spontaneously broken approximate scale invariance. We rule out the existence of Bardeen-Moshe-Bander phase. In this thesis, we show that Large N expansion is a powerful tool which in regimes that the system is interacting strongly could be used as an alternative to coupling expansion scheme.
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