Incompressible quantum liquid on the four-dimensional sphere

TL;DR

This paper investigates the many-body effects in the four-dimensional quantum Hall effect, formulating microscopic wavefunctions and Hamiltonians to reveal incompressible, liquid-like topological states in high-dimensional systems.

Contribution

It introduces a microscopic wavefunction and Hamiltonian framework for 4D QHE, advancing understanding of fractional topological states in higher dimensions.

Findings

Quasi-hole states are zero energy, indicating incompressibility.

Quasi-particle states have a finite energy gap.

Wavefunction exhibits liquid-like pairing distribution.

Abstract

The study of quantum Hall effect (QHE) is a foundation of topological physics, inspiring extensive explorations of its high-dimensional generalizations. Notably, the four dimensional (4D) QHE has been experimentally realized in synthetic quantum systems, including cold atoms, photonic lattices, and metamaterials. However, the many-body effect in the 4D QHE system remains poorly understood. In this study, we explore this problem by formulating the microscopic wavefunctions inspired by Laughlin's seminal work. Employing a generalized pseudo-potential framework, we derive an exact microscopic Hamiltonian consisting of two-body projectors that annihilate the microscopic wavefunctions. Diagonalizations on a small size system show that the quasi-hole states remain zero energy while the quasi-particle states exhibit a finite gap, in consistency with an incompressible state. Furthermore, the pairing distribution is calculated to substantiate the liquid-like nature of the wavefunction. Our work provides a preliminary understanding to the fractional topological states in high dimension.