Crossover at a bicritical point: Asymptotic behavior of a field theory with quadratic symmetry breaking

Abstract

We discuss the case of an O(N)-symmetric theory whose symmetry is broken by a quadratic term which produces two different masses (correlation lengths) one associated with M components and one with N − M. This theory describes physical systems having a bicritical point. To this problem we contribute by showing how within renormalized field theory the calculation of the shift in the critical temperature (leading to a crossover exponent) is separated from the calculation of the crossover function. The computation of the latter is then carried out in terms of the natural temperature variable, relative to the anisotropic critical temperature. This feature allows one to bypass the complicated corrections to scaling generated in earlier approaches by the use of a different variable. Within this frame-work which allows many tests of universality the results are explicit and readily generalizable to the ordered state and to higher orders in ϵ. To field theory we contribute by showing how the mass-dependent renormalization procedure allows the interpolation between asymptotic behaviors with internal O(N)- and O(M)-symmetries. The interpolating parameter is the ratio between the momentum scale and the mass difference breaking the symmetry.