Dynamic enhancement of conductance in fractional quantum Hall constriction

TL;DR

This work analyzes the AC conductance of a quantum Hall edge junction with a four-quadrant gate geometry, focusing on Andreev-like processes that enable transformer action between filling fractions $\nu_1$ and $\nu_2$. Using a folded 1D bosonized model and a plasmon-scattering (Bogoliubov) approach, the authors derive a frequency-dependent, current-conserving conductance framework that relates the incoming and outgoing edge modes via a scattering matrix $\mathcal{S}(\omega)$ and a 3×3 $G^{T}_{AC}$. They compare point-like and finite-range (screened Coulomb) interedge interactions, showing that the AC gain can exceed the DC bound of $\tfrac{3}{2}$ and can reach $\sqrt{3}$ at resonances, with long-range interactions producing richer resonance/anti-resonance structures and nonperiodic behavior for irrational velocity ratios. Wave-packet dynamics confirm the presence of fractional negative pulses and reveal time-domain signatures of the AC transformer action, with estimated GHz-range resonances indicating experimental feasibility in gate-tuned QH devices.

Abstract

A disparity in the charge of quasi-particle excitations across a tunnel junction can trigger Andreev-like processes, creating an effect similar to that of a step-up transformer. We study such a junction in its strong coupling limit in the context of quantum Hall states. Specifically, for filling fractions $ν=1$ and $1/3$, we show the DC gain in the transformer action is bounded by 3/2, irrespective of the interedge interaction range, while the AC gain is bounded by $\sqrt{3}$ and is sensitive to the range of the interaction. This setup presents a unique possibility of frequency-tunable resonances and anti-resonances across the QPC.

Figures (7)

• Figure 1: (a) Schematic diagram motivated from Ref.[Andrea_Young_transformer] and adopted in our proposed theoretical model. In (b), the corresponding edge state configuration of a QPC has been shown schematically. In (c), $\phi_{i,I/O}$ denotes the bosonic field corresponding to the incoming/outgoing edge state of the $i^{th}$ QH edge. $\alpha,\beta,\gamma$ denote the finite range density-density repulsive interactions between the QH edges. The boundary condition at $x=0$ is denoted through the current splitting matrix $S$.
• Figure 2: $|\Tilde{g}_{21}(\omega)|$ is plotted as a function of frequency $\omega$ for an injection from $\nu_1=1$ side and the value for $\nu_2$ is 1/3. $|\Tilde{g}_{21}(\omega)|$ is plotted (with L = 1) for (a) $\alpha=0.55$, (b) $\alpha=0.55,\beta=0.14 \textrm{ and }\gamma=0.589$, both with point-like interaction and (c) $\alpha=0.55,\beta=0.14,\gamma=0.589$ with screened Coulomb interaction ($\eta=12.34$). In (b) and (c), the anti-resonance points are where $|\Tilde{g}_{21}(\omega)|$ intersects the brown line.
• Figure 3: For a QH junction of $\nu=1$ and $1/3$, $\mathrm{Im}[\Tilde{g}_{21}(\omega)]$ vs $\mathrm{Re}[\Tilde{g}_{21}(\omega)]$ is plotted as a function of frequency ($\omega$) with different $v_{1}/v_{2}$ ratios and interaction parameters for point-like (top row) and screened Coulomb ($\eta=12.34$) (bottom row) interaction. (a) and (c) are plotted for $\alpha=0.55, \beta=\gamma = 0$ ($v_{1}/v_{2}=1$). (b) is plotted for $\alpha=0.55, \beta=\gamma=0.25833$ ($v_{1}/v_{2}=\sqrt{2}$). (d) is plotted for $v_{1}/v_{2}=1.5$ at $\omega=0$ with $\alpha=0.55, \beta=\gamma=0.298077$. The red and black dots indicate the initial and final frequencies of the trajectory, respectively.
• Figure 4: $V(k)$ and $\mathbb{V}(k)$ have been plotted together with the parameters $d_t=30$ nm and $d_b=15$ nm for $V(k)$ and $d=0.01$ and $\eta=12.34$ for $\mathbb{V}(k)$.
• Figure 5: $|\Tilde{g}_{21}(\omega)|$ is plotted as a function of frequency $\omega$ for four values of $\eta$: (a) 0.1, (b) 10, (c) 100, and (d) 1000, with $L = 1$.