Incompressible quantum liquid on the four-dimensional sphere

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.

💡 Research Summary

In this work the authors address the largely unexplored many‑body physics of the four‑dimensional (4D) quantum Hall effect (QHE) by constructing explicit fractional quantum Hall (FQH) wavefunctions on a four‑sphere (S⁴) threaded by a Yang monopole. Starting from the single‑particle Hamiltonian H = (P + A)²/2M with an SU(2) gauge field A that realizes the Yang monopole, they identify the lowest Landau level (LLL) as the SO(5) irreducible representation (2I, 0) whose degeneracy grows as d(2I, 0)=⅟3!(2I+1)(2I+2)(2I+3). The LLL orbitals are expressed as homogeneous polynomials of four-component spinors ψα, related to the coordinates on S⁴ via the second Hopf map, while an internal SU(2) “isospin” is described by a two‑component spinor u obtained from the first Hopf map.

The authors then develop a systematic group‑theoretical decomposition of two‑particle states. The product (2I, 0)⊗(2I, 0) splits into a sum of SO(5) channels (n₁+n₂, n₁−n₂) with 0 ≤ n₂ ≤ n₁ ≤ 2I. In particular, the channels (4I−2n, 0) are generated by the SO(5) invariant bilinear S(x,n;x′,n′)=ψα(x,n)Rαβψβ(x′,n′), where R is the charge‑conjugation matrix. The parity of n determines fermionic (odd n) or bosonic (even n) statistics.

A generalized pseudo‑potential Hamiltonian is introduced:  H = ∑{n₁,n₂} V{n₁,n₂} P_{n₁,n₂}, where P_{n₁,n₂} projects onto the SO(5) irrep (n₁+n₂, n₁−n₂) and V_{n₁,n₂} are interaction strengths. This construction is a direct 4D analogue of Haldane’s pseudo‑potential on the 2‑sphere and guarantees that any many‑body state annihilated by all the selected projectors is a zero‑energy ground state.

Two families of Laughlin‑type wavefunctions are then built. The determinant‑type state is obtained by antisymmetrizing a product of LLL spinors and raising the resulting Slater determinant to the m‑th power (the authors focus on m = 3). For I = ½ (the smallest non‑trivial LLL) and N = 4 particles, the wavefunction reads Ψ_{m=3}^{det}=