(Optical) conductivity is one of the most important characteristics of materials. Recently, nonlinear conductivities are of increasing experimental and theoretical interest, but the subject is largely open. I will discuss our recent approach to the fundamentals of the nonlinear optical conductivities. A system under the periodic boundary condition in one direction can be regarded as a "ring". A static magnetic flux through the ring induces the quantum mechanical Aharonov-Bohm effect. Furthermore, increasing the flux over time induces a uniform electric field in the system. The uniform electric conductivity can be regarded as the response of the system to the increase of the magnetic flux. This not only allows a unified formulation of the "frequency sum rule" (f-sum rule) and "Kohn formula", which have been important in many applications of linear response theory, but their natural generalization to nonlinear conductivities at all orders.

Professor Masaki Oshikawa has made significant contributions to the areas of condensed matter physics, statistical physics and topological order. He is a recipient of the JSPS Prize (2008) and the Award for Science and Technology (2022) from the Minister of Education, Culture, Sports, Science and Technology of Japan.

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