Effect of spatial correlations of the impurity magnetization on the transition temperature to the $A_1$-phase of superfluid $^3$He

The effect of correlations of magnetic impurities on the superfluid transition temperature to the ${{A}_{1}}$-phase of superfluid $^3$He in an isotropic aerogel has been examined. It has been shown that, in contrast to the pure case and the case of nonmagnetic impurities, the presence of correlations in the magnetization of impurities in the system leads to the emergence of a region where the superfluid transition temperature exhibits a nonlinear dependence on the external magnetic field. It has been found that such anomalous behavior is expected to occur in the region of relatively small magnetic fields. The width of the nonlinear region is determined by the relations between the magnetic correlation length, the coherence length of superfluid $^3$He, and the mean free path of quasiparticles in the aerogel. This result significantly differs from the previous result obtained in the mean-field theory, where the nonlinear region is determined by the saturation field of the solid paramagnetic $^3$He film coating the aerogel surface. The derived theoretical dependence has been qualitatively compared to recent experimental results on measuring the superfluid transition temperature to the ${{A}_{1}}$-phase in nematic aerogel.