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, $[{H}_{c}(T)\ensuremath{-}{H}_{c}(0)]\ensuremath{\sim}{T}^{2}$. 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, $[{H}_{c}(T)\ensuremath{-}{H}_{c}(0)]\ensuremath{\sim}T$. 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.