Relevance of soft modes for order parameter fluctuations in the two-dimensional XY model

We analyse the spin wave approximation for the two-dimensional XY model, directly in reciprocal space. In this limit the model is diagonal and the normal modes are statistically independent. Despite this simplicity non-trivial critical properties are observed and exploited. We confirm that the observed asymmetry for the probability density function for order parameter fluctuations comes from the divergence of the mode amplitudes across the Brillouin zone. We show that the asymmetry is a many-body effect despite the importance played by the zone centre. The precise form of the function is dependent on the details of the microscopic generating function for the modes. We discuss the relevance of this to more complex or non-equilibrium systems where similar distributions are observed.