Topological Order and Conformal Quantum Critical Points

TL;DR

This paper explores the nature of topological and conventional order in 2D quantum systems, revealing conformal invariance at critical points and connecting these to classical models, gauge theories, and quantum computation.

Contribution

It introduces a class of 2D quantum critical points with conformal invariance, linking them to classical models, gauge theories, and topological quantum computation.

Findings

Ground-state correlators match classical 2D models

Quantum critical points exhibit conformal invariance

Phase diagram includes topologically ordered phases

Abstract

We discuss a certain class of two-dimensional quantum systems which exhibit conventional order and topological order, as well as two-dimensional quantum critical points separating these phases. All of the ground-state equal-time correlators of these theories are equal to correlation functions of a local two-dimensional classical model. The critical points therefore exhibit a time-independent form of conformal invariance. These theories characterize the universality classes of two-dimensional quantum dimer models and of quantum generalizations of the eight-vertex model, as well as Z_2 and non-abelian gauge theories. The conformal quantum critical points are relatives of the Lifshitz points of three-dimensional anisotropic classical systems such as smectic liquid crystals. In particular, the ground-state wave functional of these quantum Lifshitz points is just the statistical (Gibbs) weight of the ordinary 2D free boson, the 2D Gaussian model. The full phase diagram for the quantum eight-vertex model exhibits quantum critical lines with continuously-varying critical exponents separating phases with long-range order from a Z_2 deconfined topologically-ordered liquid phase. We show how similar ideas also apply to a well-known field theory with non-abelian symmetry, the strong-coupling limit of 2+1-dimensional Yang-Mills gauge theory with a Chern-Simons term. The ground state of this theory is relevant for recent theories of topological quantum computation.