  The theory of second order phase transitions is one of the foundations of
modern statistical mechanics and condensed matter theory. A central concept is
the observable `order parameter', whose non-zero average value characterizes
one or more phases and usually breaks a symmetry of the Hamiltonian. At large
distances and long times, fluctuations of the order parameter(s) are described
by a continuum field theory, and these dominate the physics near such phase
transitions. In this paper we show that near second order quantum phase
transitions, subtle quantum interference effects can invalidate this paradigm.
We present a theory of quantum critical points in a variety of experimentally
relevant two-dimensional antiferromagnets. The critical points separate phases
characterized by conventional `confining' order parameters. Nevertheless, the
critical theory contains a new emergent gauge field, and `deconfined' degrees
of freedom associated with fractionalization of the order parameters. We
suggest that this new paradigm for quantum criticality may be the key to
resolving a number of experimental puzzles in correlated electron systems.
