Theory of terahertz-driven magnetic switching in rare-earth orthoferrites: The case of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>TmFe</mml:mi><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mn>3</mml:mn></mml:ms

We report a theoretical formalism that describes a dynamic magnetic response of rare-earth orthoferrites, particularly those with non-Kramers rare-earth () ions, when driven by strong terahertz fields. We derive a total thermodynamic potential for the exchange coupled -Fe system by constructing an effective Hamiltonian and employing a mean-field theory approximation. We investigate static properties of the and Fe subsystems across the spin-reorientation phase transitions and obtain resonance frequencies for Fe and magnetic sublattices as a function of temperature. Taking an example of the archetypical orthoferrite <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>TmFe</a:mi><a:msub><a:mi mathvariant="normal">O</a:mi><a:mn>3</a:mn></a:msub></a:mrow></a:math>, we perform numerical modeling to accurately describe the behavior of its anisotropy functions vs temperature. Finally, we analyze switching dynamics of Fe spins and nonlinear effects in the subsystem of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>TmFe</c:mi><c:msub><c:mi mathvariant="normal">O</c:mi><c:mn>3</c:mn></c:msub></c:mrow></c:math> driven by strong terahertz radiation.