  This thesis contains two results for the low temperature behavior of quantum
spin systems. First, we present a lower bound for the spin-1 XXZ chain in
finite volumes in terms of the gap of the two-site Hamiltonian. The estimate is
derived by a method developed by Nachtergaele in (cond-mat/9410110) called the
Martingale Method. Our bound relies on an assumption which we have, as yet,
been unable to verify analytically in all cases. We present numerical evidence
that strongly indicates our assumption is valid.
  The second result is a proof that the spin-1/2, d-dimensional XY model in the
presence of an external magnetic field does not undergo a phase transition at
low temperature, provided that the strength of the field is great enough. Using
a contour expansion inspired by Kennedy, we show that the weights of contours
satisfy a condition of Kotecky and Preiss which allows us to express the free
energy of the system as a cluster expansion. As part of the setup we give a
simple proof that the all-spin-up state is the unique ground state when the
external magnetic field has strength at least 2d.
