  The entanglement entropy of a pure quantum state of a bipartite system $A
\cup B$ is defined as the von Neumann entropy of the reduced density matrix
obtained by tracing over one of the two parts. Critical ground states of local
Hamiltonians in one dimension have entanglement that diverges logarithmically
in the subsystem size, with a universal coefficient that for conformally
invariant critical points is related to the central charge of the conformal
field theory. We find the entanglement entropy for a standard class of $z=2$
quantum critical points in two spatial dimensions with scale invariant ground
state wave functions: in addition to a nonuniversal ``area law'' contribution
proportional to the size of the $AB$ boundary, there is generically a universal
logarithmically divergent correction. This logarithmic term is completely
determined by the geometry of the partition into subsystems and the central
charge of the field theory that describes the equal-time correlations of the
critical wavefunction.
