Quantum Critical Phenomena of Relativistic Fermions in 1+1d and 2+1d

In this dissertation, we study the phase structures and the quantum critical phenomena of relativistic lattice fermions with $\O(2N_f)$ symmetry in one and two spatial dimensions, motivated by the ability to perform efficient Monte Carlo simulations. Close to a quantum critical point, physics is universal and can be described by continuum quantum field theories. We perform a perturbative analysis of all independent four-fermion interactions allowed by the $\O(2N_f)$ symmetry near the free-fermion fixed point. We then analyze the resulting continuum field theories using various techniques. In one spatial dimension, we use the powerful tools from conformal field theory and non-abelian bosonization to understand the renormalization group flows, the correlation functions, and the spectra. In the case of $N_f=2$, we find that by tuning a Hubbard coupling, our model undergoes a second-order phase transition, which can be described by an $\SU(2)_1$ Wess-Zumino-Witten model perturbed by a marginal coupling. We confirm these results using the meron-cluster algorithm, and locate the critical point precisely using exact diagonalization based on the spectrum of the Wess-Zumino-Witten model. In two spatial dimensions, we analyze the model using $\varepsilon$ expansion, large $N_f$ expansion and effective potential methods. In the case of $N_f=2$, we find a novel critical point where the anti-ferromagnetic order and superconducting-CDW order become simultaneously quantum critical, which seems to have been missed in literature. We compare these predictions with the numerical results obtained using the fermion-bag algorithm by Emilie Huffman.