Third-harmonic generation (THG) is a key nonlinear optical process for ultrafast imaging, terahertz signal generation, and symmetry-sensitive probes, often dominating in centrosymmetric materials where lower-order responses vanish. However, the role of band geometry, Fermi surface effects, and disorder in enabling large and tunable THG remains poorly understood. Here, we develop a finite-frequency quantum kinetic theory of THG based on the density matrix formalism, deriving the third-harmonic conductivity tensor. Our framework isolates five distinct band-geometric contributions to interband and intraband processes, separates Fermi sea from Fermi surface terms, and incorporates disorder effects phenomenologically. We further provide a complete symmetry classification of THG for all 122 magnetic point groups. Applying the theory to graphene and the spin-split altermagnet ${\mathrm{RuO}}_2$, we trace its THG response to specific geometric terms. These results establish a predictive foundation for designing materials with enhanced and tunable THG in the finite-frequency regime.

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• Magnetic order
• Optical & microwave phenomena
• Optoelectronics
• Third order nonlinear optical processes
• Topological materials
• Altermagnets
• Crystalline systems
• Boltzmann theory
• Discrete symmetries in condensed matter
• Methods in electromagnetism
• Methods in transport
• Symmetries in condensed matter