Effective exponents near bicritical points

The phase diagram of a system with two order parameters, with n₁ and n₂ components, respectively, contains two phases, in which these order parameters are non-zero. Experimentally and numerically, these phases are often separated by a first-order “flop” line, which ends at a bicritical point. For n=n₁+n₂=3 and d=3 dimensions (relevant, e.g., to the uniaxial antiferromagnet in a uniform magnetic field), this bicritical point is found to exhibit a crossover from the isotropic n-component universal critical behavior to a fluctuation-driven first-order transition, asymptotically turning into a triple point. Using a novel expansion of the renormalization group recursion relations near the isotropic fixed point, combined with a resummation of the sixth-order diagrammatic expansions of the coefficients in this expansion, we show that the above crossover is slow, explaining the apparently observed second-order transition. However, the effective critical exponents near that transition, which are calculated here, vary strongly as the triple point is approached.