Thermodynamic theory of square skyrmion lattice in tetragonal frustrated antiferromagnets

High-temperature part of the phase diagram of tetragonal frustrated antiferromagnets is discussed within the framework of the mean-field approach. Based on recent experimental and theoretical studies, we consider a model including frustrated Heisenberg exchange, biquadratic exchange, anisotropic exchange, and magnetodipolar interaction. It is analytically demonstrated that a subtle interplay among these interactions results in a variety of phase diagrams in the temperature-magnetic field plane. We argue that the phase diagram obtained for the easy axis along the high-symmetry direction $[001]$ and moderate biquadratic exchange reproduces all crucial features of the phase diagram experimentally observed for~\gdru~compound. The topological properties of the respective double-modulated square skyrmion lattice phase are discussed. The developed analytical approach can also be useful for the refinement of the microscopic model parameters when comparing its predictions with experimental findings.