Fractional Quantum Hall States on CP2 Space

TL;DR

This paper explores four-dimensional fractional quantum Hall states on CP2, introducing new wavefunctions, analyzing their properties, and extending pseudopotential formalism to higher dimensions, with implications for experimental realization.

Contribution

It defines and analyzes 4D Laughlin wavefunctions, extending pseudopotential formalism, and investigates their ground states and excitations on CP2 geometry.

Findings

Two types of 4D Laughlin states are exactly annihilated by specific short-range Hamiltonians.

The quasi-hole degeneracy shows anomalous counting, indicating multiple wavefunction forms.

The work extends pseudopotential formalism to higher dimensions using coherent state wavefunctions.

Abstract

We study four-dimensional fractional quantum Hall states on CP2 geometry from microscopic approaches. While in 2d the standard Laughlin wave function, given by a power of Vandermonde determinant, admits a product representation in terms of the Jastrow factor, this is no longer true in higher dimensions. In 4d we can define two different types of Laughlin wavefunctions, the Determinant-Laughlin (Det-Laughlin) and Jastrow-Laughlin (Jas-Laughlin) states. We find that they are exactly annihilated by, respectively, two-particle and three-particle short ranged interacting Hamiltonians. We then mainly focus on the ground state, low energy excitations and the quasi-hole degeneracy of Det-Laughlin state. The quasi-hole degeneracy exhibits an anomalous counting, indicating the existence of multiple forms of quasi-hole wavefunctions. We argue that these are captured by the mathematical framework of the "commutative algebra of N-points in the plane". We also generalize the pseudopotential formalism to dimensions higher than two, by considering coherent state wavefunction of bound states. The microscopic wavefunctions and Hamiltonians studied in this work pave the way for systematic study of high dimensional topological phase of matter that is potentially realizable in cold atom and optical experiments.