Full Classification of Transport on an Equilibrated $5/2$ Edge via Shot Noise

The nature of the bulk topological order of the $5/2$ non-Abelian fractional quantum Hall state and the steady state of its edge are long-studied questions. The most promising non-Abelian model bulk states are the Pfaffian (Pf), anti-Pffafian (APf), and particle-hole symmetric Pfaffian (PHPf). Here, we propose to employ a set of dc current-current correlations (electrical shot noise) in order to distinguish among the Pf, APf, and PHPf candidate states, as well as to determine their edge thermal equilibration regimes: full vs partial. Using other tools, measurements of GaAs platforms have already indicated consistency with the PHPf state. Our protocol, realizable with available experimental tools, is based on fully electrical measurements.

Figure 1

Current experimental and theoretical status for classification of the non-Abelian $5/2$ bulk states and their edge thermal equilibration regimes. Crosses (red, green, blue) represent the experimental exclusion of corresponding scenarios. Red, green, blue crosses exclude scenarios via measurements of two-terminal thermal conductance [13], interface noise [14], and interface thermal conductance [15], respectively. Theoretical proposals by Park et al. [34] and Yutushui et al. [50] used thermal transport with electrical shot noise at the edge and electrical transport with no equilibration, respectively, to confirm corresponding scenarios. Our work can distinguish among the yellow highlighted scenarios, thus completing the table.

Figure 2

(a) The edge structures of Pf, APf, and PHPf states: integer, charge-$1/2$ boson [4], and charge-neutral Majorana modes are depicted as black, blue, and wiggly red lines, respectively. Arrowheads show the chirality. The lowest Landau level is the same for all three states, but the second Landau level is different. (b) Our proposed device: a middle region of the Hall bar is depleted from filling $\nu$ to filling ${\nu }_i(<\nu )$ creating two interfaces (“line”). Specifically, we take $\{\nu ,{\nu }_i\}$ as $\{5/2,2\}$ or $\{5/2,7/3\}$ or $\{3,5/2\}$. We call the boundary of the vacuum with $\nu$ the “outer” and with ${\nu }_i$ the “upper.” We have four contacts, the source $S$ (biased by a dc voltage $V_0$), a ground contact $G$, and two drains $D_1$, $D_2$. Each circular arrow shows the chirality of charge propagation of the respective filling. The geometric lengths $L_{\mathrm{arm}}$ and $L_{\mathrm{int}}$ are assumed to be of the same order of magnitude.

Figure 3

Noise generation in our proposed device (see Fig. 2) for full (a) and partial (b),(c) thermal equilibration, where in the latter scenario the modes in the lowest Landau level localize in the line segment. Each circular arrow indicates the chirality of charge propagation of the respective filling. Voltage drops occur at the hot spots $H_1$, $H_2$ (red circles), resulting in the noise spots $M$, $N$, $O$, $P$ (green circles) [57]. In (a) we have $\{\nu ,{\nu }_i\}$ as $\{5/2,2\}$ or $\{5/2,7/3\}$ or $\{3,5/2\}$. In (b) we effectively have the filling $\{{\nu }^{\mathrm{eff}},{\nu }_i^{\mathrm{eff}}\}=\{1/2,0\}$ for $\{\nu ,{\nu }_i\}=\{5/2,2\}$. In (c) we effectively have the filling $\{{\nu }^{\mathrm{eff}},{\nu }_i^{\mathrm{eff}}\}=\{1/2,1/3\}$ for $\{\nu ,{\nu }_i\}=\{5/2,7/3\}$ and $\{{\nu }^{\mathrm{eff}},{\nu }_i^{\mathrm{eff}}\}=\{1,1/2\}$ for $\{\nu ,{\nu }_i\}=\{3,5/2\}$.