Ground-state degeneracy and fractionally charged excitations in the anomalous quantum Hall effect

The Hamiltonian describing two-dimensional electrons in a high magnetic field is diagonalized exactly for a small number of particles. In addition to the energy spectrum the mean occupation number $\rho \mathrm{}\left(j\right)=〈C_j^{†}{C}_j〉$ of the $j$th Landau state in the lowest Landau level is also calculated. For $\nu =\frac{n}{m}$ with $m$ an odd integer, $\rho \mathrm{}\left(j\right)\mathrm{}$ has a period $m \left[\rho \mathrm{}\right(j)=\rho \mathrm{}(j+m\left)\right]$, and there are $m$ distinct ground states—in striking analogy with a one-dimensional charge-density-wave system. In terms of $\rho \mathrm{}\left(j\right)\mathrm{}$, profiles of the ⅓ kinks are obtained in the ground state for $\nu$ close to ⅓. Creation energy of the kink is obtained from the energy gap. The $\nu =\frac{1}{2}$ case is markedly different.