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Logarithmic Corrections to Scaling in the Two Dimensional $XY$--Model

By expressing thermodynamic functions in terms of the edge and density of Lee--Yang zeroes, we relate the scaling behaviour of the specific heat to that of the zero field magnetic susceptibility in the thermodynamic limit of the $XY$--model in two dimensions. Assuming that finite--size scaling holds, we show that the conventional Kosterlitz--Thouless scaling predictions for these thermodynamic functions are not mutually compatable unless they are modified by multiplicative logarithmic corrections. We identify these logarithmic corrections analytically in the case of the specific heat and numerically in the case of the susceptibility. The techniques presented here are general and can be used to check the compatibility of scaling behaviour of odd and even thermodynamic functions in other models too.