Landau-Level Mixing and the Emergence of Pfaffian Excitations for the $5/2$ Fractional Quantum Hall Effect

We report on exact diagonalization studies for fully spin polarized $5/2$ fractional quantum Hall effect, incorporating Landau-level mixing through the Bishara-Nayak effective interaction. We find that there is an experimentally accessible region in the phase diagram where the Pfaffian model accurately describes not only the ground state but also the neutral and charged excitations. These results are consistent with the observed persistence of the $5/2$ Hall effect down to very low magnetic fields; they are also relevant to the experimental attempts to detect non-Abelian braid statistics.

Figure 1

Panel (a) shows the squared overlaps of the exact ground state of $H_{\mathrm{BN}}$ with both Pf and APf as a function of $\kappa$ for the indicated values of $N$. The top axis shows the corresponding magnetic field for GaAs. The squared overlaps of these states are also shown with the corresponding $L=0$ states of noninteracting composite fermions (lines with same color and style but thinner), labeled CFFS (CF Fermi sea [32]). Panel (b) shows the squared overlaps of the exact ground state of $H_1$ with both Pf and CFFS. Panel (c) displays the squared overlap between the ground state of $H_{\mathrm{BN}}$ at $2Q=2N-3$ with the hole conjugate of its ground state at $2Q=2N+1$; the deviation from unity is a measure of the $p\mathrm{\text{−}}h$ symmetry breaking. The pair correlation function of the $H_{\mathrm{BN}}$ ground state is shown in panel (d) for several values of $\kappa$ (with $\kappa =0$ giving the Coulomb result) along with that of the Pfaffian wave function. Panel (e) depicts comparisons for Pf and APf ($N=16$ and 14) with the $H_{\mathrm{BN}}$ ground state for a quantum well of thickness $w=3\lambda$ (three magnetic lengths).

Figure 2

Spectra of 16 particles at flux $2Q=2N-3=29$ for (a) $V_{\mathrm{Coulomb}}$, (b) $H_{\mathrm{BN}}$, and (c) $H_{\mathrm{Pf}}$ (black as well as colored dots). The energy expectation values of the Pf wave functions are also shown for $V_{\mathrm{Coulomb}}$ and $H_{\mathrm{BN}}$ in panels (a) and (b) (dashes) except when they are so high that they fall outside the frame. The energies are given in units of $e^2/\lambda$ ($\lambda$ is the magnetic length) in panels (a) and (b) and in units of $W_3$ in (c). The numbers near the ground state (red [medium gray]) and the low energy neutral excitations (blue [dark gray]) indicate the squared overlaps with the corresponding Pf eigenstates. Panels (d) and (e) show the evolution of the overlaps of the exact states of $H_{\mathrm{BN}}$ and $H_1$ with the Pf states as a function of $\kappa$ and $\delta V_1$, respectively; results for only alternate $L$ are shown for clarity.

Figure 3

Same as in Fig. 2 but for two Pf quasiholes at $2Q=2N-2=30$.

Figure 4

Same as in Fig. 2 but for two Pf quasiparticles at $2Q=2N-4=28$.

Figure 5

Squared overlaps of the Pf ground and QP/QH states with the exact eigenstates as a function of interaction parameters. Panels (a) and (e) are for the Pf ground state; (c) and (f) for $\mathrm{Pf}+2\mathrm{QHs}$; (d) and (g) for $\mathrm{Pf}+2\mathrm{QPs}$. All of these are for $N=14$ particles, with $L=0$ in (a) and (e) and $L=1$ in (c), (d), (f), and (g) which corresponds to maximally separated QPs or QHs. The comparison with the APf is shown in panel (b) for 12 particles. In (a)–(d), the results are given as a function of the 3-body pseudopotentials $W_3$ and $W_6$, with $W_5=0.367W_3$; the white arrows trace the BN interaction, with the two ends representing $\kappa =0$ and 2. Panels (e)–(g) show the results as a function of $\kappa$ and the quantum well width $w$, quoted in units of the magnetic length $\lambda$. The contours mark squared overlaps of 0.8, 0.85, 0.9 and 0.95.