SO(5) Landau model and 4D quantum Hall effect in the SO(4) monopole background

We investigate the SO(5) Landau problem in the SO(4) monopole gauge field background by applying the techniques of the non-linear realization of quantum field theory. The SO(4) monopole carries two topological invariants, the second Chern number and a generalized Euler number, specified by the SU(2) monopole and antimonopole indices, I+ and I-. The energy levels of the SO(5) Landau problem are grouped into Min(I+, I-) 1 sectors, each of which holds Landau levels. In the n-sectors, Nth Landau level eigenstates constitute the SO(5) irreducible representation with (p, q)(5) = (N + I+ + I- - n, N + n)(5) whose function form is obtained from the SO(5) nonlinear realization matrix. In the n = 0 sector, the emergent quantum geometry of the lowest Landau level is identified as the fuzzy four-sphere with radius being proportional to the difference between I+ and I-. The Laughlin-like wavefunction is constructed by imposing the SO(5) lowest Landau level projection to the many-body wavefunction made of the Slater determinant. We also analyze the relativistic version of the SO(5) Landau model to demonstrate the Atiyah-Singer index theorem in the SO(4) gauge field configuration.

Automated summary

This study investigates the SO(5) Landau problem in the background of the SO(4) monopole gauge field using techniques of non-linear realization of quantum field theory. It analyzes the energy levels distribution, wave function form, and constructs Laughlin-like wavefunctions.