New Developments in Vector, Matrix and Tensor Quantum Field Theories

Abstract

This thesis is devoted to studies of quantum field theories with dynamical fields in the vector, matrix or tensor representations of O(N) symmetry groups. These models provide interesting classes of exactly solvable models that can be examined in detail and give insights into the properties of other, more complicated quantum field theories. The introduction to the thesis reviews the general ideas about why such systems are interesting and exactly solvable. The different classes of Feynman diagrams which dominate the large N limits are exhibited. The introduction is partially based on work with Igor R. Klebanov and Preethi Pallegar. Chapter 2 is based on work with Igor R. Klebanov, Preethi Pallegar, Gabriel Gaitan and Kiryl Pakrouski. It is dedicated to the study of quantum Majorana fermionic models. The refined energy bound is derived for these models. We study these models numerically, analytically and qualitavely. All of these results hint that the vector quantum mechanics undergoes second order phase transition and has a limiting temperature in the large N limit. Chapter 3 is based on papers with Igor R. Klebanov, Simone Giombi, Grigory Tarnopolsky and Shiroman Prakash. It is dedicated to the search of a stable bosonic tensor and SYK-like theories in higher dimensions. We propose two models: prismatic and supersymmetric that have a positive potential and therefore these theories should be stable unlike the bosonic tensor models with quartic tetrahedral interactions. Chapter 4 is based on papers with Igor R. Klebanov and Christian Jepsen. It studies the properties of RG flow and bifurcations that could arise in this case. We consider a theory of symmetric traceless matrices in d=3-e dimensions and analytically continue the rank of these matrices to fractional values. At some values of N we managed to find a fixed point whose stability matrix has a pair of purely imaginary eigenvalues. From the theory of ordinary differential equations it is known that it corresponds to Hopf bifurcation and that at some values of N a stable limit cycle exists, that we also find numerically. Such an approach is extended to the case of O(N) x O(M) group, where it is shown that a homoclinic RG flow exists, which starts and terminates at the same fixed point.