  We investigate the effect of magnon-magnon interactions on the dispersion and
polarization of magnons in collinear antiferromagnetic (AF) insulators at
finite temperatures. In two-sublattice AF systems with either uniaxial or
biaxial magnetocrystalline anisotropies, we implement a self-consistent
Hartree-Fock mean-field approximation to explore the nonlinear thermal
interactions. The resulting nonlinear magnon interactions separate into
two-magnon intra- and interband scattering processes. Furthermore, we compute
the temperature dependence of the magnon bandgap and AF resonance modes due to
nonlinear magnon interactions for square and hexagonal lattices. In addition,
we study the effect of magnon interactions on the polarization of magnon modes.
We find that although the noninteracting eigenmodes in the uniaxial case are
circularly polarized, but in the presence of nonlinear thermal interactions the
local U(1) symmetry of the Hamiltonian is broken. The attractive nonlinear
interactions squeeze the low-energy magnon modes and make them elliptical. In
the biaxial case, on the other hand, the bare eigenmodes of low energy magnons
are elliptically polarized but thermal nonlinear interactions squeeze them
further. Direct measurements of the predicted temperature-dependent AF
resonance modes and their polarization can be used as a tool to probe the
nonlinear interactions. Our findings establish a framework for exploring the
effect of thermal magnon interactions in technologically important magnetic
systems, such as magnetic stability of recently discovered two-dimensional
magnetic materials, coherent transport of magnons, Bose-Einstein condensation
of magnons, and magnonic topological insulators.
