Bruekner approach to the spin-wave gap critical index for the two-layer Heisenberg antiferromagnet

We consider the two-layer Heisenberg antiferromagnet near a zero temperature quantum phase transition from a disordered dimer phase to a collinear Neel state. At approaching the transition point the spin-wave gap vanishes as $\Delta \propto (J_\perp-J_{\perp c})^{\nu}$. To account for strong correlations between the S=1 elementary excitations we apply the Brueckner diagram approach which gives the critical index $\nu\approx 0.5$. We demonstrate also that the linearized in density Brueckner equations give the mean field result $\nu=1$. Finally an expansion of the Brueckner equations in powers of the density, combined with the scaling hypothesis, give $\nu\approx 0.67$. This value reasonably agrees with that of the nonlinear O(3) $\sigma$-model. Our approach demonstrates that for other quantum spin models the critical index can be different from that in the nonlinear $\sigma$-model. We discuss the conditions for this to occur.