Emergence of Neutral Modes in Laughlin-like Fractional Quantum Hall Phases

Chiral gapless boundary modes are characteristic of quantum Hall (QH) states. For hole-conjugate fractional QH phases counterpropagating edge modes (upstream and downstream) are expected. In the presence of electrostatic interactions and disorder these modes may renormalize into charge and upstream neutral modes. Orthodox models of Laughlin phases anticipate only a downstream charge mode. Here we show that in the latter case, in the presence of a smooth confining potential, edge reconstruction leads to the emergence of pairs of counterpropagating modes, which, by way of mode renormalization, may give rise to nontopological upstream neutral modes, possessing nontrivial statistics. This may explain the experimental observation of ubiquitous neutral modes, and the overwhelming suppression of anyonic interference in Mach-Zehnder interferometry platforms. We also point out other signatures of such edge reconstruction.

Figure 1

Schematic of two a priori possible configurations at the edge of the bulk $\nu =1/3$ phase. For a sharp confining potential, there is a single ($\nu =1/3$) quantum Hall droplet. For smoother edge potentials, an additional side strip, with filling factor (a) $1/3$ or (b) $1/5$, composed of $N_S$ electrons may be nucleated along the edge. The side strip is separated from the bulk (comprising $N_B$ electrons) by $L_S$ guiding centers.

Figure 2

Results of the variational calculations with 50 electrons. (a)–(c) The energy ($⟨H⟩$) of the states in the two variational classes as a function of the total angular momentum at (a) sharp ($d=0.01\ell$), (b) moderately smooth ($d=2.25\ell$), and (c) very smooth ($d=2.50\ell$) confining potentials, where $\ell$ is the magnetic length. In all cases, the energy of the unreconstructed state ($⟨H{⟩}_{\mathrm{ur}}$) has been subtracted. The blue (red) circles show energy of states with a side strip of $\nu =1/3$ ($\nu =1/5$). The black square marks the state with the lowest energy. (d) The lowest possible energy in the two variational classes as a function of the smoothness of the confining potential (parameterized by $d/\ell$). The blue (red) line corresponds to states with a side strip of $\nu =1/3$ ($\nu =1/5$). For sharp edges the ground state is the unreconstructed $\nu =1/3$ state with angular momentum $3675\hslash$. This state supports a single chiral $e/3$ mode. For moderately smooth potentials ($2.17<d/\ell <2.42$), an additional $\nu =1/3$ side strip is nucleated, which defines a pair of counterpropagating $e/3$ modes (in addition to the chiral $e/3$ mode arising from the bulk). For very smooth potentials ($d>2.42\ell$), a $\nu =1/5$ side strip is generated, which supports a pair of $e/5$ edge modes.

Figure 3

For the edge structure with a $\nu =1/5$ strip, interactions (represented by the red curve) and disorder-induced electron tunneling (blue dashed line) between the inner two edge modes may lead to emergence of a renormalized downstream charge (${\varphi }_c$) and an upstream neutral mode (${\varphi }_n$). The outermost mode is assumed to be completely decoupled from the inner two modes. This idealization is justified by the variational analysis, which shows that as the confining potential becomes shallower, the width of the side strip (proportional to $N_S$) increases faster than the separation of the strip ($L_S$).