Composite fermions and the half-filled state

Abstract

Under appropriate conditions electron-hole symmetry should apply to a partially filled Landau level of a two-dimensional electron gas. This suggests that the application of Jain's composite fermion (CF) picture to either electrons or holes should lead to equivalent results. Surprisingly, for a system of ${N}_{e}$ electrons on a Haldane sphere, this is not true for three values of the Landau level degeneracy $2S+1$. When ${N}_{e}\ensuremath{-}1l~Sl~{N}_{e},$ the sum of the electron filling factor $\ensuremath{\nu}$ and the hole filling factor $\ensuremath{\mu}$, as determined from Jain's picture, is smaller than unity. Because of this, use of the relation $\ensuremath{\nu}=1\ensuremath{-}\ensuremath{\mu}$ can lead to ``twin'' or ``alias'' states having different values of $\ensuremath{\nu}$ for the same ${N}_{e}$ and $2S+1$. One example is the ``half-filled'' state. It is determined by requiring the effective (mean-field) flux ${2S}^{*}$ ``seen'' by one CF to vanish. Different results are obtained when ${S}_{e}^{*}{=S}_{e}\ensuremath{-}{(N}_{e}\ensuremath{-}1)$ and ${S}_{h}^{*}=S\ensuremath{-}{(N}_{h}\ensuremath{-}1)$ are set equal to zero. The same problem arises in the CF hierarchy picture when the number of quasielectrons ${n}_{\mathrm{QE}}$ is related to the effective flux ${2S}^{*}$ by ${2(n}_{\mathrm{QE}}\ensuremath{-}1)l~{2S}^{*}l~{2n}_{\mathrm{QE}}.$