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We use the Renormalization Group method to study the magnetic field influenceon the Bose-Einstein condensation of interacting dilute magnons in threedimensional spin systems. We first considered a model with SU(2) symmetry(universality class $z=1$) and we obtain for the critical magnetic field apower law dependence on the critical temperature, $[H_c(T)-H_c(0)]\sim T^2$. Inthe case of U(1) symmetry (universality class $z=2$) the dependence isdifferent, and the magnetic critical field depends linearly on the criticaltemperature, $[H_c(T)-H_c(0)]\sim T$. By considering a more relevant model,which includes also the system's anisotropy, we obtain for the same symmetryclass a $T^{3/2}$ dependence of the magnetic critical field on the criticaltemperature. We discuss these theoretical predictions of the renormalizationgroup in connection with experimental results reported in the literature.Comment: 6 pages, Revte

Field-induced Bose-Einstein condensation of interacting dilute magnons in three-dimensional spin systems: A renormalization-group study