Quantum Hall Effect on the Flag Manifold F_2

TL;DR

This paper investigates the quantum Hall effect on the complex flag manifold F_2, deriving Landau levels, wavefunctions, and edge dynamics using algebraic and geometric methods, revealing incompressible fluid behavior at strong magnetic fields.

Contribution

It introduces a novel algebraic and geometric framework for analyzing the quantum Hall effect on the flag manifold F_2, including explicit wavefunctions and edge state descriptions.

Findings

Wavefunctions are SU(3) Wigner D-functions satisfying constraints

Landau levels and Hamiltonian derived from algebraic structures

System exhibits incompressible fluid behavior at high magnetic fields

Abstract

The Landau problem on the flag manifold ${\bf F}_2 = SU(3)/U(1)\times U(1)$ is analyzed from an algebraic point of view. The involved magnetic background is induced by two U(1) abelian connections. In quantizing the theory, we show that the wavefunctions, of a non-relativistic particle living on ${\bf F}_2$, are the SU(3) Wigner ${\cal D}$-functions satisfying two constraints. Using the ${\bf F}_2$ algebraic and geometrical structures, we derive the Landau Hamiltonian as well as its energy levels. The Lowest Landau level (LLL) wavefunctions coincide with the coherent states for the mixed SU(3) representations. We discuss the quantum Hall effect for a filling factor $\nu =1$. where the obtained particle density is constant and finite for a strong magnetic field. In this limit, we also show that the system behaves like an incompressible fluid. We study the semi-classical properties of the system confined in LLL. These will be used to discuss the edge excitations and construct the corresponding Wess-Zumino-Witten action.