Infinite set of crossover exponents of the XY model and f(α) approach

For transitions described by the XY model, it has been shown that each Fourier component of the angular-structure factor is controlled by a different crossover exponent. Each crossover exponent is associated with a particular symmetry-breaking field. It is shown here that although, as for multifractals, the angular structure itself can be obtained from the crossover exponents, in contrast to multifractals, the asymptotic form of this angular-structure factor is described by the Legendre transform of the (analytically continued) infinite set of crossover exponents only in the special case of two dimensions.