In recent years, the investigation of nonlinear electromagnetic responses has received significant attention due to its potential for elucidating the quantum properties of matter. Although remarkable progress has been achieved in developing quantum theories of nonlinear responses to electric field, a comprehensive quantum theory framework that systematically addresses nonlinear responses to both electric and magnetic fields has yet to be thoroughly discussed. Here, we present a systematic quantum theory of nonlinear electromagnetic response using the Matsubara Green's function approach, which explicitly incorporates the wave vector dependence of external electromagnetic fields. We reveal the general properties of transport coefficients. We apply our theory to second-order responses, deriving the nonlinear Hall effects and magneto-nonlinear Hall effects in both time-reversal symmetric and time-reversal breaking systems. These effects stem from diverse quantum geometric quantities. Additionally, we analyze the contributions arising from the Zeeman interaction. Our work presents a unified quantum theory of nonlinear electromagnetic response, paving the way for further exploration of novel phenomena in this field.

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• Electrical conductivity
• Quantum Hall effect
• Quantum transport
• Second order nonlinear optical processes
• Green's function methods