Quantum Hall effect on the GrassmanniansGr2(CN)

Abstract

Quantum Hall effects on the complex Grassmann manifolds ${\mathbf{Gr}}_{2}({\mathbb{C}}^{N})$ are formulated. We set up the Landau problem in ${\mathbf{Gr}}_{2}({\mathbb{C}}^{N})$ and solve it using group theoretical techniques and provide the energy spectrum and the eigenstates in terms of the $SU(N)$ Wigner $\mathcal{D}$ functions for charged particles on ${\mathbf{Gr}}_{2}({\mathbb{C}}^{N})$ under the influence of Abelian and non-Abelian background magnetic monopoles or a combination of these. In particular, for the simplest case of ${\mathbf{Gr}}_{2}({\mathbb{C}}^{4})$, we explicitly write down the $U(1)$ background gauge field as well as the single- and many-particle eigenstates by introducing the Pl\"ucker coordinates and show by calculating the two-point correlation function that the lowest Landau level at filling factor $\ensuremath{\nu}=1$ forms an incompressible fluid. Our results are in agreement with the previous results in the literature for the quantum Hall effect on $\mathbb{C}{P}^{N}$ and generalize them to all ${\mathbf{Gr}}_{2}({\mathbb{C}}^{N})$ in a suitable manner. Finally, we heuristically identify a relation between the $U(1)$ Hall effect on ${\mathbf{Gr}}_{2}({\mathbb{C}}^{4})$ and the Hall effect on the odd sphere ${S}^{5}$, which is yet to be investigated in detail, by appealing to the already-known analogous relations between the Hall effects on $\mathbb{C}{P}^{3}$ and $\mathbb{C}{P}^{7}$ and those on the spheres ${S}^{4}$ and ${S}^{8}$, respectively.