Exact Renormalization Group and the O(N) Model[HBNI Th213]

Abstract

Studying finite cutoff CFT holds much importance in both Condensed Matter Physics and AdS/CFT conjecture. As lattice systems always have an inherent scale, it is important to devise a theory that will be valid for all energy scales. On the other hand, the scheme of Holographic RG requires the study of ERG in boundary CFT.

In this thesis, we apply Exact RG to study a fixed point action and composite operators in finite cutoff O(N) model. In Exact RG the higher energy modes of a theory are integrated out with help of an analytic function resulting in no loss of information as one flows down to the lower energy. This method gives us the most general expression of an action or composite operators which is/are valid at all energy scales.

The main component of our work is Polchinski’s ERG equation. The terms in the action have interpretations in terms of the Feynman diagram which eases the procedure of calculation.

When one has a fixed point action at hand, it is important to find the corresponding irrelevant and relevant operators. Because irrelevant operators define a critical surface, and relevant direction defines directions away from the critical surface. From the AdS/CFT perspective also studying perturbations around the boundary CFT is important, because they give rise to different bulk dual fields. In the second part of our work we have constructed two important composite operators near the WF fixed points. In continuum theory, the composite operators mix with the same dimension operators. Here expressions are more general, composite operators mixes with all operators which are allowed in the theory. As we are nearby the fixed point, we have calculated the anomalous dimensions of these operators too. In continuum limit our result matches with results found from the Dimensional Regularization. Also our method of finding composite operators is independent of choice of cutoff functions. This makes it useful for the Holographic RG purpose. However, to find the anomalous dimensions of these operators we have used a specific cutoff function for ease of calculation.