Critical behavior of RMn$_2$O$_5$ oxides near the magnetic phase transition to a structure incommensurate in two spatial directions

A phase transition from the paramagnetic state to the long-period magnetic structure in RMn$_2$O$_5$ oxides with the star of the wave vector determining the incommensurability of long-range magnetic order in two spatial directions has been investigated. An effective Hamiltonian of the system that allows one to describe this transition in the framework of the renormalization group approach has been constructed. It has been shown that there is a stable critical point of transformations of this group at which there occurs a second-order phase transition. The critical indices have been found. The obtained results have been compared with the results for phase transitions occurring in these oxides in accordance with the star of the wave vector, which provides incommensurability in one of the spatial directions. It has been found that fluctuations of the four-component order parameter due to the low spatial symmetry of these compounds do not change the order of the phase transition, which was found in terms of the Landau theory.