  We present some exact results for the effect of disorder on the critical
properties of an anisotropic XY spin chain in a transverse field. The continuum
limit of the corresponding fermion model is taken and in various cases results
in a Dirac equation with a random mass. Exact analytic techniques can then be
used to evaluate the density of states and the localization length. In the
presence of disorder the ferromagnetic-paramagnetic or Ising transition of the
model is in the same universality class as the random transverse field Ising
model solved by Fisher using a real space renormalization group decimation
technique (RSRGDT). If there is only randomness in the anisotropy of the
magnetic exchange then the anisotropy transition (from a ferromagnet in the $x$
direction to a ferromagnet in the $y$ direction) is also in this universality
class. However, if there is randomness in the isotropic part of the exchange or
in the transverse field then in a non-zero transverse field the anisotropy
transition is destroyed by the disorder. We show that in the Griffiths' phase
near the Ising transition that the ground state energy has an essential
singularity. The results obtained for the dynamical critical exponent, the
typical correlation length, and the temperature dependence of the specific heat
near the Ising transition agree with the results of the RSRGDT and numerical
work.
