Exact Electromagnetic Response of Landau Level Electrons

We present a simple method that allows to calculate the electromagnetic response of non-interacting electrons in strong magnetic field to arbitrary order in the gradients of external electric and magnetic fields. We illustrate the method on both non-relativistic and massless Dirac electrons filling $N$ Landau levels. First, we derive an exact relation between the electromagnetic response of the non-relativistic 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 sub-leading order in $N$ - with a Fermi liquid computation to {\it all} orders in the gradient expansion and for arbitrary value of the $\mathrm{g}$-factor. Finally, we show that in the large $N$ limit the non-relativistic polarization tensor agrees with Dirac's in the leading and sub-leading orders in $N$, provided that Berry phase of the Dirac cone is taken into account via replacement $N\longrightarrow N+\frac{1}{2}$.