Universal non-local observables at quantum critical points

Abstract

This dissertation is devoted to the study of universal observables in quantum critical systems, most of which have a non-local character. The overarching goal of this work is to compute new universal observables of critical points to aid theoretical understanding of these systems. We first consider the finite-size energy spectrum of quantum critical points on the torus. We compute the spectrum for the Wilson-Fisher conformal field theory in the $\epsilon = 3 - d$ expansion, where the energy spectrum maps onto a strongly-coupled problem in quantum mechanics. We also compute the energy spectrum to leading order in $1/N$, and we compare the two expansions. We then study the torus spectra associated with a class of confinement transitions in states with $\mathbb{Z}_2$ topological order. After introducing these universality classes, we show that the critical torus spectrum can be used to detect nontrivial effects like spontaneous symmetry breaking and emergent gauge degrees of freedom. We compare our analytic results with numerical simulations where available, demonstrating the utility of the torus spectrum as a useful characterization of the universality class of a quantum critical point. We then present a computation of the von Neumann entanglement entropy of the Wilson-Fisher and Gross-Neveu conformal field theories in the large $N$ limit. We obtain an exact mapping to the von Neumann entanglement entropy of a free quantum field theory, allowing an exact determination of the entanglement entropy in a number of cases. We also study a critical point displaying impurity-driven critical behavior in a boson superfluid-insulator phase transition. The presence of an impurity drives the system to a new interacting universality class, which has critical exponents associated with the scaling dimensions of the impurity degree of freedom. We present the universal quantum field theory of this transition and compute the critical exponents and finite-temperature compressibility at the critical point using the $\epsilon$ expansion.