  We consider two-dimensional Fermi liquids in the vicinity of a quantum
transition to a phase with commensurate, antiferromagnetic long-range order.
Depending upon the Fermi surface topology, mean-field spin-density-wave theory
predicts two different types of such transitions, with mean-field dynamic
critical exponents $z=1$ (when the Fermi surface does not cross the magnetic
zone boundary, type $A$) and $z=2$ (when the Fermi surface crosses the magnetic
zone boundary, type $B$). The type $A$ system only displays $z=1$ behavior at
all energies and its scaling properties are similar (though not identical) to
those of an insulating Heisenberg antiferromagnet. Under suitable conditions
precisely stated in this paper, the type $B$ system displays a crossover from
relaxational behavior at low energies to type $A$ behavior at high energies. A
scaling hypothesis is proposed to describe this crossover: we postulate a
universal scaling function which determines the entire, temperature-,
wavevector-, and frequency-dependent, dynamic, staggered spin susceptibility in
terms of 4 measurable, $T=0$, parameters (determining the distance, energy, and
order parameter scales, plus one crossover parameter). The scaling function
contains the full scaling behavior in all regimes for both type $A$ and $B$
systems. The crossover behavior of the uniform susceptibility and the specific
heat is somewhat more complicated and is also discussed. Explicit computation
of the crossover functions is carried out in a large $N$ expansion on a
mean-field model. Some new results for the critical properties on the ordered
side of the transition are also obtained in a spin-density wave formalism. The
possible relevance of our results to the doped cuprate compounds is briefly
discussed.

Condensed Matter