Shot noise signatures of candidate states for the fractional quantum Hall $ν= 12/5$ state

TL;DR

The paper tackles the problem of identifying the bulk ground state of the ν=$12/5$ fractional quantum Hall plateau, where non-Abelian anyons such as parafermions or Fibonacci anyons may reside. It develops a shot-noise-based protocol in a filter geometry, leveraging a hydrodynamic edge model with hotspots and noise-spots to predict qualitative Fano-factor behavior across full and partial thermal equilibration without requiring precise equilibration lengths. A state-by-state analysis yields distinct Fano-factor patterns for HH, BS with N=1, BS with N=-3, and A-RR, and introduces a practical flowchart that uses measurements across three interface configurations to discriminate among candidates. The approach is experimentally feasible with current techniques, extends shot-noise diagnostics to the ν=$12/5$ regime, and has significant implications for identifying non-Abelian anyons suitable for topological quantum computing, including potential realization of universal quantum gates with Fibonacci anyons.

Abstract

Fractional quantum Hall (FQH) states are highly sought after because of their ability to host non-abelian anyons, whose braiding statistics make them excellent candidates for qubits in topological quantum computing. Multiple theoretical studies on the $ν=\frac{12}{5}$ FQH state predict various quasi-particle states hosted by the $\frac{12}{5}$ plateau, which include $\mathbb Z_3$ parafermions and Majorana modes. In this work, we provide a systematic protocol to distinguish among four possible candidate wavefunctions of the $\frac{12}{5}$ plateau using zero-frequency shot noise experiments on a filter-geometry. Qualitative comparisons of Fano-Factors provide a robust way to predict the candidate state across both the full and partial thermal equilibration regimes without prior knowledge of the experimental information, like thermal equilibration length, to allow for more realistic experiments.

Figures (3)

• Figure 1: A representation of the edge structure for all candidate states considered in this work. The black dotted line separates the Lower Landau Levels (LLL) from the Higher Landau Levels (HLL). The thick black lines represent the Integer modes with $\nu=1$ and thick blue lines represent fractional modes with $\nu=1/5$. The dotted orange line represents $\mathbb Z_3$ parafermion modes and the dashed red lines represent Majorana modes. The BS state has two variants which we consider where the number of Majorana modes can either be $N=1$ or $N=-3$ where the negative means that they states are moving upstream. It is important to note that while the overall composition of the constituent edge modes is accurately represented by the figure, it gives no sense of which modes are facing the bulk.
• Figure 2: (a) Illustration of an interface consisting of two different filling fractions such that the upstream and downstream modes contain different edge structures is modeled as shown. The chiral modes are shown in blue and red in the filling fractions $\nu$ and $\nu_i$, respectively. The grey lines between the two sides show the tunneling channels between the upstream and downstream modes to facilitate equilibration. (b) A schematic of the device used in this work. Here, contacts $S, D_1, D_2$, and $G$ are Source, 2 drains, and ground, respectively. The convention $\nu > \nu_i$ is chosen so that the current always splits by the interface. Points labeled $O,L,P,Q$ represent noise spots, and points $H_1,H_2$ are hot spots.
• Figure 3: The flowchart shows step-by-step our protocol explained in \ref{['sec:method']}. Each decision point helps eliminate or confirm a state entirely based on qualitative comparisons of Fano-Factor values, bypassing the need to know the exact values of these Fanofactors.