Longitudinal and transverse static spin fluctuations in layered ferro- and antiferromagnets

We analyse the momentum dependence of static non-uniform susceptibilities of layered local-moment systems below Curie (Neel) temperature within the 1/S-expansion, the renormalization-group approach, and first order of 1/N-expansion. We argue that the previously known results of the spin-wave theory and renormalization-group approach for the transverse spin susceptibility acquire strong corrections already at sufficiently low temperatures, which appear due to the interaction of the incomping magnon having momentum q with magnons with momenta k<q. Such corrections can not be treated in the standard renormalization-group approach, but can be described by both, 1/S and 1/N expansions. The results of these expansions can be successfully extrapolated to the magnetic transition temperature T=T_M yielding the correct weight of static spin fluctuations, determined by the O(3) symmetry. For the longitudinal susceptibility, the summation of leading terms of 1/S expansion within the parquet approach allows to fulfill the sumrule for the weights of transverse and longitudinal fluctuations in a broad temperature region below T_M outside the critical regime. We also discuss the effect of longitudinal spin fluctuations on the (sublattice) magnetization of layered systems.