We have studied the finite-temperature magnetic phase transition using a combination of first-principles calculations and the second-order Holstein-Primakoff approximation of the anisotropic Heisenberg model for hexagonal MnX (X = N, P, As, and Sb) monolayers. The MnX monolayers are all half-metal with dynamically and thermally stable atomic structures at T = 300 K. The hexagonal MnN is an out-of-plane easy-axis ferro- magnetically ordered monolayer with the Curie temperature close to the room temperature. However, the other three MnX (X = P, As, and Sb) monolayers have an easy axis inside the plane with Curie temperature close to zero. The ab initio energy difference between spin configurations is mapped into an anisotropic Heisenberg spin Hamiltonian which is solved using the second-order Holstein-Primakoff approximation. The anharmonic (magnon-magnon) interaction softens the magnetic excitation energy and reduces the magnon energy gap at the point, which is crucial for the finite-temperature long-range magnetic order in two dimensions. As a result, the inclusion of magnon-magnon interaction dramatically reduces the Curie temperature of hexagonal MnX monolayers. Although the magnon-magnon interaction is a perturbation in energy dispersion, it has an important effect on the finite-temperature long-range magnetic order in two-dimensional monolayers.

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