Magnetic fluctuations in a finite two-dimensional XY model

We calculate the two-dimensional probability P(M-x, M-y) for the magnetization in a two-dimensional XY model of finite size. We show that, for arbitrarily large N, there is a topological difference between the distributions in the low-temperature spin wave regime and in the high-temperature paramagnetic regime. In the low-temperature phase P(M-x, M-y) is a well-defined ring function and we calculate an upper bound for P(0, 0) less than or equal to 0.0019(T/J)(2). Even so, this is consistent with the susceptibility per spin, chi, being divergent. We show further that the distribution function Q(M) for the scalar magnetization has a universal form, scaling with the single variable y = JT(-1) MLT/4 pi J, where L is the system size and J is the coupling constant. We show that chi has considerable structure, with two terms of order N, one due to the spin waves and the other due to vortices. This leads to a peak in chi at the Kosrerlitz-Thouless-Berezinskii transition.