Abstract

Conformal field theories (CFTs) play a very significant role in modern physics, appearing in such diverse fields as particle physics, condensed matter and statistical physics and in quantum gravity both as the string worldsheet theory and through the AdS/CFT correspondence. In recent years major breakthroughs have been made in solving these CFTs through a method called numerical conformal bootstrap. This method uses consistency conditions on the CFT data in order to find and constrain conformal field theories and obtain precise measurements of physical observables. In this thesis we apply the conformal bootstrap to study among others the O(2)- and the ARP^3- models in 3D. In the first chapter we extend the conventional scalar numerical conformal bootstrap to a mixed system of correlators involving a scalar field charged under a global U(1) symmetry and the associated conserved spin-1 current J. The inclusion of a conserved spinning operator is an important advance in the numerical bootstrap program. Using numerical bootstrap techniques we obtain bounds on new observables not accessible in the usual scalar bootstrap. Concentrating on the O(2) model we extract rigorous bounds on the three-point function coefficient of two currents and the unique relevant scalar singlet, as well as those of two currents and the stress tensor. Using these results, and comparing with a quantum Monte Carlo simulation of the O(2) model conductivity, we give estimates of the thermal one-point function of the relevant singlet and the stress tensor. We also obtain new bounds on operators in various sectors. In the second chapter we investigate the existence of a second-order phase transition in the ARP^3 model. This model has a global O(4) symmetry and a discrete Z_2 gauge symmetry. It was shown by a perturbative renormalization group analysis that its Landau-Ginzburg-Wilson effective description does not have any stable fixed point, thus disallowing a second-order phase transition. However, it was also shown that lattice simulations contradict this, finding strong evidence for the existence of a second-order phase transition. In this chapter we apply conformal bootstrap methods to the correlator of four scalars t transforming in the traceless symmetric representation of O(4) in order to investigate the existence of this second order phase transition. We find various features that stand out in the region predicted by the lattice data. Moreover, under reasonable assumptions a candidate island can be isolated. We also apply a mixed t-s bootstrap setup in which this island persists. In addition we study the kink-landscape for all representations appearing in the t times t OPE for general N. Among others, we find a new family of kinks in the upper-bound on the dimension of the first scalar operator in the "Box" and "Hook" representations.

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