Entanglement entropy for integer quantum Hall effect in two and higher dimensions

We analyze the entanglement entropy, in real space, for the higher dimensional integer quantum Hall effect on ${\mathbb{C}\mathbb{P}}^k$ (any even dimension) for abelian and nonabelian magnetic background fields. In the case of $\nu =1$ we perform a semiclassical calculation which gives the entropy as proportional to the phase-space area. This exhibits a certain universality in the sense that the proportionality constant is the same for any dimension and for any background, abelian or nonabelian. We also point out some distinct features in the profiles of the eigenfunctions of the two-point correlator that underline the difference in the value of entropies between $\nu =1$ and higher Landau levels.