  We present the critical theory of a number of zero temperature phase
transitions of quantum antiferromagnets and interacting boson systems in two
dimensions. The most important example is the transition of the S = 1/2 square
lattice antiferromagnet between the Neel state (which breaks spin rotation
invariance) and the paramagnetic valence bond solid (which preserves spin
rotation invariance but breaks lattice symmetries). We show that these two
states are separated by a second order quantum phase transition. The critical
theory is not expressed in terms of the order parameters characterizing either
state (as would be the case in Landau-Ginzburg-Wilson theory) but involves
fractionalized degrees of freedom and an emergent, topological, global
conservation law. A closely related theory describes the superfluid-insulator
transition of bosons at half-filling on a square lattice, in which the
insulator has a bond density wave order. Similar considerations are shown to
apply to transitions of antiferromagnets between the valence bond solid and the
Z_2 spin liquid: the critical theory has deconfined excitations interacting
with an emergent U(1) gauge force. We comment on the broader implications of
our results for the study of quantum criticality in correlated electron
systems.
