  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.
