Universal magnetic fluctuations with a field-induced length scale

We calculate the probability density function for the order-parameter fluctuations in the low-temperature phase of the two-dimensional XY model of magnetism near the line of critical points. A finite correlation length xi, is introduced with a small magnetic field h, and an expression for xi (h) is developed by treating nonlinear contributions to the field energy using a Hartree approximation. We find analytically a series of universal non-Gaussian distributions of the finite-size scaling form P(m,L,xi)similar toL(beta/nu) P-L(mL(beta/nu), xi /L) and present a function of the form P(x)similar to {exp[x-exp(x)]}(a(h)) that gives the probability density functions to an excellent approximation. We propose a(h) as an indirect measure of the length scale of correlations in a wide range of complex systems.