Quantum hall transformer in a quantum point contact over the full range of transmission

TL;DR

The paper addresses the observed robust e^2/2h conductance in a quantum Hall transformer realized at a quantum point contact and the unclear microscopic mechanism behind strong coupling. It develops an alternative description based on a sliding charge density wave in a Luttinger liquid with a spatially varying K(x) and a pinning potential, establishing a quantitative link to the boundary sine-Gordon model through perturbative and matrix-product-state (MPS) methods. The authors derive full-range conductance behavior, connect the perturbative backscattering scale to an observable energy E_B, and map lattice-model results to BSG parameters, showing consistency with the experimental high-voltage universal regime and the dissipationless dc transformer limit. They further argue that a wide QPC can effectively realize a single percolating Luttinger liquid channel, providing a natural RG-consistent framework for interpreting QHT physics and offering concrete predictions for ac conductance, shot noise, and microwave responses. The work thus clarifies the microscopic origin of the BSG-like transport and lays groundwork for future experiments to probe the boundary dynamics in QPCs between fractional quantum Hall edges.

Abstract

A recent experiment [Cohen et al., Science 382, 542 (2023)] observed a robustly quantized $e^2/2h$ conductance in a quantum point contact between fractional quantum Hall edges (a quantum Hall transformer), which then vanishes at low temperature. While this behavior can be described by a boundary sine gordon (BSG) model derived from electron tunneling, the microscopic motivation for such strong tunneling is unclear. We use an alternative model based on sliding charge density waves and a density inhomogeneity to clarify the BSG description and calculate the conductance over the full range of transmission. Using a perturbative method and a matrix product state calculation, we draw a quantitative connection between the BSG model and the physical quantum Hall system.

Figures (6)

• Figure 1: Sign-flipping reflection of a charge wavepacket from the interface between $\nu=1/3$ and $\nu=1$ quantum Hall edge states, illustrating the quantum hall transformer behavior. The charge is normalized by a factor $\rho_0$ to illustrate the $2e^*\rightarrow -e^*,\; 3e^*=e$ charge transfer.
• Figure 2: Scaling exponent of the perturbative correction to harmonic conductance for various junctions widths $d$, calculated as the log-log slope of $\Delta G(T)$. The horizontal lines denote the low- and high-temperature limits, given by the effective Luttinger parameter $K'$ and the Luttinger parameter at the impurity $K(x=0)$. (inset) The spatial profiles of the Luttinger parameter $K(x)$---which has a piecewise exponential form--- scaled by the length parameter $d$ and the backscattering impurity denoted by a green star.
• Figure 3: (a) ac conductivity for various impurity strengths $u$ and packet width $15$ demonstrating near $G=e^2/2h$ for $u=0$. The real part of the data is fit to Eq. \ref{['eq:majorana_conductance']} to calculate barrier energy $E_\mathrm{B}$, with fits shown as dashed lines. (b) Fit value of $E_\mathrm{B}$ as a function of impurity strength $u$, demonstrating a quadratic relationship. Data from the narrower packet is fit to $E_\mathrm{B}=A(u+u_0)^2$, which yields $A=0.55$ and $u_0=0.04$, the latter of which represents the intrinsic backs-cattering of the Luttinger parameter crossover.
• Figure S1: Asymptotic behavior of noise $P$ representing $V\rightarrow \infty$ (top) and $V\rightarrow 0$ (bottom) asymptotic behavior. The $T=0$ case matches $P\rightarrow \delta I$ in the large voltage limit and $P\rightarrow 2 I$ in the small voltage limit. The impurity energy is $E_\mathrm{B}=1$.
• Figure S2: Schematic of a QPC between filling $\nu_-=1$ and $\nu_+=1/3$. The arrows represent the direction of movement of the electron at each edge. The dotted line represents a domain wall with effective chiral charge $\nu_--\nu_+=2/3$ with disorder kane1994randomness. The chemical potential of each segment of edge is shown to be $\mu_{\pm}+\mathrm{sgn}(y)\, j/2\nu_{\pm}$ where $j$ is the total current flowing through the QPC and the edge velocity near the ends is chosen to be unity.