Field-induced Bose-Einstein condensation of interacting dilute magnons   in three-dimensional spin systems: A renormalization-group study
  We use the Renormalization Group method to study the magnetic field influence on the Bose-Einstein condensation of interacting dilute magnons in three dimensional spin systems. We first considered a model with SU(2) symmetry (universality class $z=1$) and we obtain for the critical magnetic field a power law dependence on the critical temperature, $[H_c(T)-H_c(0)]\sim T^2$. In the case of U(1) symmetry (universality class $z=2$) the dependence is different, and the magnetic critical field depends linearly on the critical temperature, $[H_c(T)-H_c(0)]\sim T$. By considering a more relevant model, which includes also the system's anisotropy, we obtain for the same symmetry class a $T^{3/2}$ dependence of the magnetic critical field on the critical temperature. We discuss these theoretical predictions of the renormalization group in connection with experimental results reported in the literature. 
M. Crisan, I. Tifrea, D Bodea, and I. Grosu