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 $N_e-1<~S<~N_e,$ the sum of the electron filling factor $\nu$ and the hole filling factor $\mu$, as determined from Jain’s picture, is smaller than unity. Because of this, use of the relation $\nu =1\mathrm{}-\mu$ can lead to “twin” or “alias” states having different values of $\nu$ 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 $S_e^{*}{=S}_e-{(N}_e-1)$ and $S_h^{*}=S-{(N}_h-1)$ 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 ${2(n}_{\mathrm{QE}}-1)<~{2S}^{*}<~{2n}_{\mathrm{QE}}.$

• Received 25 June 1997

DOI:https://doi.org/10.1103/PhysRevB.56.14941