Helicoidal magnetic ordering in crystals: exact periodic solutions of equations of state with fourth-order anisotropy

In this paper we obtain an exact periodic solution for the system of equations of state corresponding to the spiral phase of crystalline ordering. We used method of integral rational basis of invariants to construct an inhomogeneous Landau potential. As a result the inhomogeneous Landau potential takes account of anisotropic invariants consisting of OP components as well as anisotropic invariants that comprise space derivatives of OP components. It is demonstrated that it is taking into account the latter that leads to the exact periodic solution for the system of equations of state that describes the spiral phase. A phase diagram of states is built for active Lifshitz representations at the second-order phase transition point. Analogy between inhomogeneous states of A-phase and second-order superconductors in the magnetic field is discussed. It is suggested to describe the vortex state in the magnetic field as a deformation of the magnetic sublattice with dislocations.