Incompressible quantum liquid on the four-dimensional sphere

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.

Figures (3)

• Figure 1: The energy spectra are shown in blocks marked by the quantum numbers $(M_J, M_K)$ for the Hamiltonian (\ref{['eq12']}) with $N = 4$ particles occupying the LLL orbitals in the SO(5) IRREP $(3,0)_{\text{\tiny SO(5)}}$. The two datasets with blue and red spots represent two branches of states with integer and half-integer SO(4) quantum numbers. The number of orbitals $N_o$ is 20. The energy unit is taken as $\hbar^2/(2MR^2)$, and the horizontal axis shows the index of states in the ascending order in terms of energy. The zero energy ground state is an SO(5) singlet and unique. The lowest excitation is another SO(5) single state with a gap $\Delta=4.8$.
• Figure 2: The energy spectra of the same system as in Fig. \ref{['fig:spectra']}: ($a$) quasi-hole states with $N=3$, ($b$) quasi-particle states with $N=5$. Only the block of $(\frac{1}{2},0)$ is represented. The ground states of the quasi-hole case remain at zero energy, while those of the quasi-particle case exhibit a finite gap. The ground states for both the quasi-hole and quasi-particle cases belong to the 20-dimensional SO(5) IRREP $(3,0)_{\text{\tiny {SO(5)} }}$.
• Figure 3: The pair distribution function of the 4D determinant-type Laughlin wavefunction lacks a peak at long distance, indicating liquid-like behavior.