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In his paper Quantized Hall conductivity in two dimensions, Laughlin considers a ribbon of two-dimensional metal bent into a loop of circumference $L$, and pierced everywhere by a magnetic field $H_o$ normal to its surface. He says that the current through the loop is equal to the adiabatic derivative of the electronic energy $U$ of the system wrt the magnetic flux $\phi$ through the loop: $$I=c\frac{\partial U}{\partial \phi}=\frac{c}{L}\frac{\partial U}{\partial A}$$

My question is what is the origin of this equation and why is the current equal to the adiabatic derivative of the electronic energy $U$?

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  • $\begingroup$ I think your question might be already answered in here. See the comments right below the question. Laughlin's own Nobel Lecture also has more details on this equation pp 271-272. $\endgroup$
    – Waterfall
    Aug 28, 2020 at 23:33

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