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 ξ, is introduced with a small magnetic field h, and an expression for ξ(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,ξ)~L^β/νPL(mL^β/ν,ξ/L) and present a function of the form P(x)~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.