Microscopic Study of Edge Excitations of Spin-Polarized and Spin-Unpolarized $\nu=2/3$ Fractional Quantum Hall Effect

TL;DR

This paper provides a microscopic analysis of edge excitations in the $ u=2/3$ fractional quantum Hall states, revealing the nature of counter-propagating modes and spin-charge separation.

Contribution

It offers a microscopic perspective confirming the existence of a single backward moving neutral mode and elucidates the spin-charge separation in these states.

Findings

One backward moving neutral mode confirmed.

Spin and charge modes move in opposite directions.

Microscopic model aligns with effective theory predictions.

Abstract

The edge of spin unpolarized or spin polarized $\nu=2/3$ fractional quantum Hall states is predicted by the effective theory to support a backward moving neutral mode in addition to a forward moving charge mode. We study this issue from a microscopic perspective where these states are identified with effective filling factor of 2 of composite fermions, but with an effective magnetic field that is antiparallel to the external field. A simple counting from the composite fermion description suggests that there might be two backward moving edge modes, but explicit calculations show that one of these is projected out of the low energy sector, while the remaining mode provides a good microscopic account of the actual counter propagating edge mode. The forward moving modes are identified as "Schur modes," obtained by multiplying the ground state wave function by the symmetric Schur polynomials. The edge of the 2/3 spin unpolarized state provides a particularly striking realization of "spin charge separation" in a one dimensional Tomonaga Luttinger liquids, with the spin and charge modes moving in opposite directions.