In-plane electric fields and the $\nu =\frac{5}{2}$ fractional quantum Hall effect in a disk geometry

The $\nu =\frac{5}{2}$ fractional quantum Hall effect is of experimental and theoretical interest due to the possible non-Abelian statistics of the excitations in the electron liquid. A small voltage difference across a sample applied in experiments to probe the system is often ignored in theoretical studies due to the Galilean invariance in the thermodynamic limit. No experimental sample, however, is Galilean invariant. In this Rapid Communication, we explore the effects of the probe electric fields in a disk geometry with finite thickness. We find that weak probe fields enhance the Moore-Read Pfaffian state, but sufficiently strong electric fields destroy the incompressible state. In a disk geometry, the behavior of the system depends on the polarity of the applied radial field, which can potentially be observed in experiments using a Corbino disk configuration. Our simulation also shows that the application of such a field enhances the coherence length of quasiholes propagating through the edge channels.

Figure 1

(a) Matrix elements of the external potential for systems with 25 (red), 50 (blue), and 100 (green) particles. (b) As $\kappa$ is varied, the size of the electric potential for a fixed value for $U$ varies.

Figure 2

The phase diagram as $U$, $d$, and $\kappa$ are varied for $N=8$, 10, and 12 electrons in $S=2N-2$ states. The green region highlights the Pfaffian state in all nine diagrams and a collapsed state with the lowest angular momentum highlighted in black. In the first row, $N=8$ with (a) $U=-0.01$, (b) $U=0$ [24], (c) $U=0.01$. In the second row, $N=10$ with (d) $U=0$ [24], (e) $U=0.0001$, (f) $U=0.01$. In the third row, $N=12$ with (g) $U=0.0001$, (h) $U=0$, and (i) $U=-0.0001$, and all $U$'s are in units of $\frac{e^2}{\varepsilon {\ell }_B^2}$.

Figure 3

The energy spectra when $d=1{\ell }_B$, $\kappa =0.3$ and (a) $U=0.0001\frac{e^2}{\varepsilon {\ell }_B^2}$, (b) $U=0$, (c) $U=-0.0001\frac{e^2}{\varepsilon {\ell }_B^2}$, with the edge modes highlighted in red. The dispersion relations for the Fermi (red +) and Bose (blue *) for (d) $U=0.0001\frac{e^2}{\varepsilon {\ell }_B^2}$, (e) $U=0$, (f) $U=-0.0001\frac{e^2}{\varepsilon {\ell }_B^2}$.