Edge-mode velocities and thermal coherence of quantum Hall interferometers

We present comprehensive results on the edge-mode velocities in a quantum Hall droplet with realistic interaction and confinement at various filling fractions. We demonstrate that the charge-mode velocity scales roughly with the valence Landau level filling fraction and the Coulomb energy in the corresponding Landau level. At Landau level filling fraction $\nu =5/2$, the stark difference between the bosonic charge-mode velocity and the fermionic neutral-mode velocity can manifest itself in the thermal smearing of the non-Abelian quasiparticle interference. We estimate the dependence of the coherence temperature on the confining potential strength, which may be tunable experimentally to enhance the non-Abelian state.

Figure 1

(Color online) (a) Ground state energy $E\left(M\right)$ in each total angular momentum $M$ subspace for $N=12$ electrons with 22 orbitals in the 1LL $\left(\nu =5/2\right)$. The neutralizing background charge is at a distance $d/l_B=0.4$, 0.6, and 0.8 away from the 2DEG. The total angular momentum of the global ground state $M_{\text{gs}}$ (indicated by arrows) increases from 121, 126, to 136, respectively. The global ground state at $d=0.6l_B$ has the same total angular momentum $M_{\text{MR}}=N\left(2N-3\right)/2=126$ as the corresponding $N=12$ Moore-Read state. We have shifted the curves vertically for $d=0.4l_B$ and $0.8l_B$ for clarity. (b) $M_{\text{gs}}$ as a function of $d$ at $\nu =5/2$. The solid dots indicate the plateau at which the Moore-Read state is groundstatable. (c) $M_{\text{gs}}$ as a function of $d$ for 20 electrons in 30 orbitals $\left(\nu =2/3\right)$. We can understand the three plateaus of $M_{\text{gs}}$ as $\nu =1/3$ Laughlin droplets of 5–7 holes embedded in a $\nu =1$ electron droplet. Note that they are in the same phase, but the corresponding distances between two counterpropagating edges are different (Ref. 8). (d) $M_{\text{gs}}$ as a function of $d$ for 6 electrons in 18 orbitals $\left(\nu =1/3\right)$. The solid dots indicate the plateau at which the Laughlin state is groundstatable. Beyond $d\approx 1.5l_B$, the system undergoes an edge-reconstruction transition (Refs. 10, 12).

Figure 2

(Color online) Edge-mode velocity analysis for $\nu =1/3$. (a) Low-energy excitations in a $N=8$ system with background charge at $d=0.6l_B$, with edge excitations marked by solid bars. (b) Energies of single chiral boson excitations, which are the ground states in individual momentum spaces [marked in red in (a)]. (c) Chiral boson (or charge mode) velocity $v_c$ defined either by the slope of the best quadratic fit in (b) at the origin $\left(v_c^d\right)$ or by the ground-state energy difference at $\Delta M=0$ and 1 $\left(v_c^{\Delta }\right)$ for systems of 5–10 electrons. The finite-size scaling of $v_c^{\Delta }$ shows a weaker nonlinear effect, although the two definitions tend to give the same result in the thermodynamic limit.

Figure 3

(Color online) Summary of the charge-mode velocity $v_c$ (solid red candles) and neutral-mode velocity $v_n$ (empty blue candles), extrapolated to the thermodynamic limit (except for $\nu =2/3$), at various valence LL filling ${\nu }^{\ast }=\nu -\lfloor \nu \rfloor$. The size of the candles reflects the range of velocities in the corresponding groundstatable range of the confining potential strength (or $d$). The candle data for $\nu =2/3$ are based on 20 electrons in 30 orbitals for various confining potential strength, while the error bars reflect our estimate of the uncertainty due to finite-size effects. The solid line is the dimensional analysis result $v_c={\nu }^{\ast }{e}^2/ϵ\hslash$.

Figure 4

(Color online) Coherence temperature $T^{\ast }$ (based on the estimate of $v_c$ and $v_n$) as a function of $d$ for both $e/4$ (upper line) and $e/2$ (lower line) quasiparticles in the Moore-Read state in an $L=1\mu \text{m}$ sample. The broken lines at $d>0.62l_B$ are obtained by the extrapolation of the velocities, as the Moore-Read state is no longer groundstatable in a system of 12 electrons in 26 orbitals. A stripe phase may emerge below $d=l_B$ (Ref. 20).