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The edge of spin unpolarized or spin polarized $\nu=2/3$ fractional quantumHall states is predicted by the effective theory to support a backward movingneutral mode in addition to a forward moving charge mode. We study this issuefrom a microscopic perspective where these states are identified with effectivefilling factor of 2 of composite fermions, but with an effective magnetic fieldthat is antiparallel to the external field. A simple counting from thecomposite fermion description suggests that there might be two backward movingedge modes, but explicit calculations show that one of these is projected outof the low energy sector, while the remaining mode provides a good microscopicaccount of the actual counter propagating edge mode. The forward moving modesare identified as "Schur modes," obtained by multiplying the ground state wavefunction by the symmetric Schur polynomials. The edge of the 2/3 spinunpolarized state provides a particularly striking realization of "spin chargeseparation" in a one dimensional Tomonaga Luttinger liquids, with the spin andcharge modes moving in opposite directions.

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