Thermodynamic theory of square skyrmion lattice in tetragonal frustrated antiferromagnets

The 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 findings, we generalize previous theoretical studies by considering a model that includes frustrated Heisenberg exchange, biquadratic exchange, magnetodipolar interaction, anisotropic exchange, and single-ion anisotropy. 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 one of the proposed diagrams reproduces all crucial features of the phase diagram experimentally observed for ${\mathrm{GdRu}}_{2}{\mathrm{Si}}_{2}$ compound. Besides magnetodipolar interaction and additional easy-axis contribution, it requires moderate biquadratic exchange. We show that, despite the remarkable square skyrmion lattice being stable even if only magnetodipolar interaction or compass anisotropy or biquadratic exchange is included, their ``symbiosis'' allows for greatly enhancing its stability region. It is also demonstrated that higher-order harmonics play an important role in the stabilization of the square skyrmion lattice. The developed analytical approach can be useful for the refinement of the microscopic model parameters when comparing its predictions with experimental findings.