| Тойм: | We present a simple method that allows us to calculate the electromagnetic response of noninteracting electrons in a strong magnetic field to arbitrary order in the gradients of external electric and magnetic fields. We illustrate the method on both nonrelativistic and massless Dirac electrons filling N Landau levels. First, we derive an exact relation between the electromagnetic response of the nonrelativistic and Dirac electrons in the lowest Landau level. Next, we obtain a closed form expression for the polarization operator in the large- N (or weak magnetic field) limit. We explicitly show that in the large- N limit the random phase approximation (RPA) computation of the polarization tensor agrees—in leading and subleading order in N —with a Fermi liquid computation to all orders in the gradient expansion and for arbitrary value of the g factor. Finally, we show that in the large- N limit the nonrelativistic polarization tensor agrees with Dirac's in the leading and subleading orders in N , provided that the Berry phase of the Dirac cone is taken into account via replacement N ⟶ N + 1 / 2 .
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