Incompressible states of the interacting composite fermions in negative effective magnetic fields at $\nu =4/13,5/17$, and 3/10

By developing an algorithm for evaluating the basis states for the composite fermions with negative effective magnetic field, we perform the composite-fermion-diagonalization study for the fully spin-polarized fractional quantum Hall states at the filling factors $\nu =3/10$, 4/13, and 5/17 in the range $2/7<\nu <1/3$. These observed states correspond to a partially filled second effective Landau level, for the composite fermions carrying four vortices, with filling factor $\overline{\nu }=1/2$, 1/3, and 2/3, respectively, analogous to the previously studied states of composite fermions with two attached vortices in the range $1/3<\nu <2/5$. We show that the character of these states in the range $2/7<\nu <1/3$ replicates the same for the states in the range $1/3<\nu <2/5$ having identical $\overline{\nu }$: chiral $p$-wave pairing with an anti-Pfaffian correlation of composite fermions carrying six quantized vortices produces an incompressible state at $\nu =3/10$; an unconventional interaction between composite fermions, resulting from the suppression of fermion pairs with relative angular momentum three and producing the fractional quantum Hall effect of composite fermions in the second effective Landau level with $\overline{\nu }=1/3$ and its particle-hole conjugate filling factor 2/3, reproduces incompressible states at $4/13$ and $5/17$ filling factors. We further estimate the thermodynamic limit of the ground-state energies and calculate the lowest energy gap for the neutral collective excitations of these states.

Figure 1

The low-lying energy spectra obtained by CFD with both the “Pf shift” (right panels) and the “APf shift” (left panels) at fully spin-polarized $\nu =3/10$, for systems with various values of $N$ and $2Q$ shown inside the panels. The dashes represent the energy spectra obtained by exact diagonalization of the Coulomb interaction in the full LLL Hilbert space. The statistical uncertainty, estimated from a Metropolis Monte Carlo evaluation of integrals is less than the diameter of any circle. The energy per particle $E$ includes the interaction with the background.

Figure 2

The low-lying energy spectra obtained by CFD at fully spin-polarized $\nu =4/13$ and 5/17 emerging out of the unconventional FQHE of ${}^4\mathrm{CFs}$ at 1/3 and 2/3 filling in second $\mathrm{\Lambda }\mathrm{L}$. $N$ is the total number of electrons and $2Q$ is the total flux quanta passing through the system. The diameter of any circle is smaller than the statistical uncertainty estimated from a Metropolis Monte Carlo evaluation of integrals. The energy per particle $E$ includes the interaction with the background.

Figure 3

(a) The ground-state interaction energy per particle, ${\mathrm{E}}_{\mathrm{gs}}$, obtained by CFD at fully spin-polarized $\nu =3$/10 (with APf shift), 4/13, and 5/17 are plotted against $1/N$ and extrapolated to the thermodynamic limit. (b) The minimum energy gap, $\mathrm{\Delta }$, of a neutral excitation is plotted as a function of $1/N$. We do not consider the ground-state energy as well as the excitation gap of the smallest systems with $N=6$ and 8 to neglect the large deviation in results due to the strong finite size effect.

Figure 4

The ground-state interaction energy per particle, ${\mathrm{E}}_{\mathrm{gs}}$, obtained from Monte Carlo calculations for the parafermionic state at $\nu =5/17$ for $N=5$, 10, and 15 is plotted against $1/N$ and extrapolated to the thermodynamic limit. Since the energy for $N=5$ may have a large finite size effect, we consider the thermodynamic limit of the energy by linear fitting with $N=10$ and 15 only.