Role of scaling dimensions in generalized noises in fractional quantum Hall tunneling due to a temperature bias

TL;DR

This work analyzes temperature-biased transport in a fractional quantum Hall quantum point contact using chiral Luttinger liquid theory to extract the scaling dimensions of tunneling quasiparticles. It derives comprehensive analytic and numerical results for delta-\(T\) noise, heat-current noise, and mixed noise, including cross-correlations, in both small- and large-temperature-bias limits, and introduces an effective density of states to unify the description with scattering concepts. The paper shows that scaling dimensions imprint distinct signatures on delta-\(T\) and heat-noise, enabling determination of exchange statistics and edge structure in Laughlin states; mixed noise provides a diagnostic of particle-hole symmetry breaking and thermoelectric response. Overall, the results extend noise spectroscopy as a practical tool for probing strongly correlated edge physics and offer a framework applicable to other one-dimensional, strongly interacting systems beyond the Laughlin FQH regime.

Abstract

Continued improvement of heat control in mesoscopic conductors brings novel tools for probing strongly correlated electron phenomena. Motivated by these advances, we comprehensively study transport due to a temperature bias in a quantum point contact device in the fractional quantum Hall regime. We compute the charge-current noise (so-called delta-$T$ noise), heat-current noise, and mixed noise and elucidate how these observables can be used to infer strongly correlated properties of the device. Our main focus is the extraction of so-called scaling dimensions of the tunneling anyonic quasiparticles, of critical importance to correctly infer their anyonic exchange statistics.

Figures (8)

• Figure 1: A quantum point contact device in the fractional quantum Hall regime at Laughlin filling $\nu=(2n+1)^{-1}$, with $n$ a positive integer. The source contacts $1$ and $2$ have temperatures $T_1$ and $T_2$, respectively, and inject one right (${\hat{\phi}}_R$) and left (${\hat{\phi}}_L$) moving edge mode at these temperatures, respectively. Tunneling of charge and heat ($I_T$ and $J_T$ respectively) between the edge modes occur at $x=0$. In this work, we analyze the resulting charge and heat currents and their fluctuations in drain contacts $3$ and $4$.
• Figure 2: (a-b) Second- and fourth-order delta-$T$ noise expansion coefficients $\mathcal{C}^{(2)}$, $\mathcal{C}^{(4)}$, $\mathcal{D}^{(2)}$, and $\mathcal{D}^{(4)}$ (Eq. \ref{['eq:C2']}, \ref{['eq:C4']}, \ref{['eq:D2']}, and \ref{['eq:D4']}, respectively) as functions of the scaling dimension $\lambda$. Panels (c-d) show the difference $\mathcal{D}^{(n)}-\mathcal{C}^{(n)}$ that appears in the expansion for the full cross correlation noise \ref{['eq:S34_expanded']}. Triangles and circles mark the values for $\lambda=\nu$ (panels a and c) and $\lambda=1/\nu$ (panels b and d) for fillings $\nu=1,1/3,1/5,1/7$.
• Figure 3: Tunneling delta-$T$ noise \ref{['eq:chargenoise_largebias']} in the large bias regime, normalized to the equilibrium noise $S_0^{II}$, as a function of the scaling dimension $\lambda$. Circles mark the values for $\lambda=\nu$ for $\nu=1,1/3,1/5,1/7$ (left panel) and $\lambda=1/\nu$ for $\nu=1,1/3,1/5$ (right panel). The free-electron value $2\ln 2$, given by Eq. \ref{['eq:STT_large_bias']}, is highlighted.
• Figure 4: Numerically computed backscattering charge-current noise $S^{II}_{TT}$, normalized to $S_0^{II}$ (solid, dark green line) for different scaling dimensions $\lambda$. The values $\lambda=1/3,\,1/5$ correspond to the ideal ones in the weak backscattering regime at fillings $\nu=1/3,\,1/5$, while $\lambda=3,\,5$ are the ideal values in the strong backscattering regime at the same filling. We also plot the small-$\Delta T$ expansions [see Eq. \ref{['eq:chargeNoise_exp']}] at second and fourth order, (light green, dashed and yellow, dashed curves, respectively). The large bias limits \ref{['eq:chargenoise_largebias']} are given as black, dot-dashed lines. The noise is plotted vs $T_1/T_2=[1+\Delta T /(2\bar{T})]/[1-\Delta T /(2\bar{T})]$. Note that the large bias limit $T_1/T_2\gg 1$ is obtained for $\Delta T \to 2\bar{T}$, $T_1\to T_\mathrm{hot}$, whereas in the opposite limit $T_1/T_2\ll 1$, $T_2\to T_\mathrm{hot}$.
• Figure 5: Second- and fourth-order delta-$T$ noise expansion coefficients $\mathcal{C}^{(2)}_{Q}$, $\mathcal{C}^{(4)}_{Q}$, $\mathcal{D}^{(2)}_{Q}$, and $\mathcal{D}^{(4)}_{Q}$ (Eq. \ref{['eq:C2Q']}, \ref{['eq:C4Q']}, \ref{['eq:D2Q']}, and \ref{['eq:D4Q']}, respectively) as functions of the scaling dimension $\lambda$. Triangles and circles mark the values for $\lambda=1,1/3,1/5,1/7$ (panels a and c) and $\lambda=1,3,5$ (panels b and d).