Long range order in the anisotropic quantum ferromagnetic Heisenberg model

We study the anisotropic quantum mechanical ferromagnetic Heisenberg model. By anisotropic we mean that the x and y exchange constants are equal but smaller than the z exchange constant. We show that for any amount of anisotropy there is long range order in two or more dimensions at low enough temperature. We also develop a convergent low temperature expansion and use it to prove exponential decay of the truncated correlation functions.