Incompressible Quantum Hall Liquid on the Four-Dimensional Sphere

The quantum Hall effect (QHE) is a cornerstone of topological physics, inspiring extensive explorations of its high-dimensional generalizations such as experimental realizations in synthetic systems including cold atoms, photonic lattices, and metamaterials. However, the many-body effect in the higher dimensional QHE system remains poorly understood. We explore this problem by formulating the microscopic wave functions on a four-dimensional sphere inspired by Laughlin's seminal work. Employing a generalized pseudopotential framework, we derive an exact microscopic Hamiltonian consisting of two-body projection operators that annihilate the microscopic wave functions. Diagonalizations on finite system sizes show that the quasihole states remain zero energy while the quasiparticle states exhibit a finite gap, in consistency with an incompressible state. Furthermore, the pair distribution is calculated to substantiate the liquidlike nature of the wave function. Our Letter provides a preliminary understanding to the fractional quantum Hall states in high dimensions.