We discuss the boundary critical behaviors of two-dimensional (2D) quantum phase transitions with fractionalized degrees of freedom in the bulk, motivated by the fact that usually it is the one-dimensional boundary that is exposed and can be conveniently probed in many experimental platforms. In particular, we mainly discuss boundary criticality of two examples: (i) the quantum phase transition between a 2D
Z2 topological order and an ordered phase with spontaneous symmetry breaking; (ii) the continuous quantum phase transition between metal and a particular type of Mott insulator [
U(1) spin liquid]. In particular, we obtain the critical exponents and scaling laws of these exotic quantum phase transitions when the systems are probed from the boundary in proposed experimental setup. These critical exponents obtained are significantly different from those one would see through bulk measurements. This theoretical study could be relevant to many purely 2D systems, where recent experiments have found correlated insulator, superconductor, and metal in the same phase diagram.
In this thesis I will outline various quantum phase transitions in 2D models of magnets that are amenable to simulation with quantum Monte Carlo techniques. The key player in this work is the theory of deconfined criticality, which generically allows for zero temperature quantum phase transitions between phases that break distinct global symmetries. I will describe models with different symmetries including SU(N), SO(N), and "easy-plane" SU(N) and I will demonstrate how the presence or absence of continuous transitions in these models fits together with the theory of deconfined criticality.