Non-Abelian Fibonacci quantum Hall states in 4-layer rhombohedral stacked graphene

Abstract

In 1991, it was proposed that fourfold-degenerate Landau levels formed by a single species of electrons could host a non-Abelian fractional quantum Hall (FQH) state with Fibonacci anyons at filling fraction $\hat{I}½=\frac{2}{3}$. In this work, we investigate how such degenerate Landau levels can be realized in rhombohedral-stacked tetralayer graphene. We identify the following key conditions which may stabilize the Fibonacci state: (1) A magnetic field of around 20 Tesla is required if surface and interior carbons have the same energy level. If substrate hybridization raises the surface carbon energy level by $\hat{I}{”}_2=30$\,meV relative to interior carbon, the required field will have a larger range: 15 -- 20 Tesla. For $\hat{I}{”}_2=45$\,meV, the range reaches a maximum: 7 -- 20 Tesla. (2) The displacement field must be tuned to achieve Landau level degeneracy. $\hat{I}½=\frac{3}{5}$ Fibonacci FQH states may also be realized in pentalayer rhombohedral graphene with a magnetic field of 12 Tesla, and $\hat{I}½=\frac{1}{2}$ states with Ising anyons may occur in trilayer graphene for magnetic fields of 12 -- 20 Tesla at $\hat{I}{”}_2=0$ or 5 -- 20 Tesla at $\hat{I}{”}_2=45$\,meV. We also study a simple interaction model to explore spin/valley polarization effects, and we see that the Fibonacci statemay occur at $\hat{I}½=2/3+$ integer filling fractions, where the integer is 0 and 4 for sufficiently weak interaction, or can shift to 2 and 5 under a stronger interaction. The case $\hat{I}{”}_2=45$\,meV also produces states at negative filling fraction, e.g. $-\frac{2}{3}$, $-4\frac{2}{3}$. Here $\hat{I}½$ is defined with respect to the Hall conductance, $\ddot{I}{ƒ}_{xy}=\hat{I}½\frac{e^2}{h}$.

11 pages, 6 figures