Universal chiral Luttinger liquid behavior in a graphene fractional quantum Hall point contact

TL;DR

The paper demonstrates universal chiral Luttinger liquid behavior at a graphene-based quantum point contact joining ν=1 and ν=1/3 edge states. It reveals and characterizes both weak- and strong-coupling regimes: in the weak regime, the conductance follows a universal $G \propto V^2$ and $G \propto T^2$ scaling, while in the strong regime the conductance saturates near $G \to e^2/(2h)$ via Andreev-like scattering of fractional quasiparticles, enabling a nearly dissipationless DC voltage transformer with gain close to 3/2. The crossover is described by an exact quantum impurity model, with a universal scaling function that collapses data across temperatures and gate-voltages when expressed in terms of the characteristic scale $T_0$. These results, validated across multiple magnetic fields, establish graphene as a clean platform for probing topological order, edge reconstruction phenomena, and quantum statistics in 2D electron systems, with potential applications in interferometry and solid-state topological devices.

Abstract

One dimensional conductors are described by Luttinger liquid theory, which predicts a power-law suppression of the density of states near the Fermi level. The scaling exponent is non-universal in the general case, but is predicted to be quantized for the chiral edge states of the fractional quantum Hall effect. Here, we report conductance measurements across a point contact linking integer and fractional quantum Hall edge states. At weak coupling, we observe the predicted universal quadratic scaling with temperature and voltage. At strong coupling, the conductance saturates to e^2/2h, arising from perfect Andreev reflection of fractionalized quasi-particles at the point contact. We use the strong coupling physics to realize a nearly dissipationless DC voltage step-up transformer, whose gain of 3/2 arises directly from topological fractionalization of electrical charge.

Figures (12)

• Figure 1: Universal conductance scaling at weak coupling.(a) Optical micrograph of the device with schematic depiction of chiral edge states. (b) Device schematic showing the patterned top graphite layer, graphene monolayer, and global bottom graphite gatecohen_tunable_2022. The tunneling conductance across the junction is determined from the transmitted current $I$ and diagonal voltage $V_D$ as $G \equiv I / V_D$. (c)$G$ measured as function of $V$ at $T_{\text{probe}} = 56mK$, with $V_{\text{NS}} = −2.465V$ and $B = 10$ T. The inset shows a parabolic fit to the low-$V$ regime, giving $T_0=9.02\pm 0.007K$ as defined in Eq. \ref{['eq:lutt_cond']}. The main panel shows $G - G_{\text{min}}$, where $G_{\text{min}}=7.5\times 10^{-4} e^2/h$ is the minimum conductance. Fitting a power-law gives an exponent of $2.00 \pm 0.06$cohen_supplementary_2022, where the error represents the standard deviation in the fit parameter. (d)$G$ measured at $V=0$ as a function of temperature at the same gate voltages as panel (c). The dashed line is a plot of the conductance given by the first term of Eq. \ref{['eq:lutt_cond']}, using $T_0=9.02$ K. (e) Nonlinear differential conductance for $T_{\text{probe}}$ =202mK, 245mK, 290mK, 344mK, 450mK, 549mK, 618mK, 666mK at $V_{\text{NS}} = −2.456V$. (f) The same data as in panel (e) after scaling $G$ and $V$ as described in the main text. The black dashed line is $\widetilde{G}$ as predicted by Eq. \ref{['eq:lutt_cond']}.
• Figure 2: Andreev-like quasiparticle scattering.(a)$G$ versus $V_\mathrm{NS}$ taken at $B= 9$T with a finite DC voltage bias of $145 \mu V$. Here, $V_\mathrm{BG}$=2.0V, $V_\mathrm{E}$=-1.460V, and $V_\mathrm{W}$=-1.775V to maintain $\nu = 1$ and $\nu = 1/3$ on the 'East' and 'West' sides of the junction respectively. (b) Schematic representation of the Andreev scattering process for fractionally charged quasiparticles in the strong coupling limitsandler_andreev_1998 of a $\nu =1/3$ to $\nu = 1$ point heterojunction. (c) Ratio of the reflected voltage $V_\mathrm{r}$ to the source voltage $V_\mathrm{i}$ versus $V_\mathrm{NS}$; all other gate voltages are the same as in panel (a).
• Figure 3: Crossover from weak to strong coupling.(a) Schematic of renormalization group flow in the $\Gamma$-$U$, where $U$ represents additional perturbations to Eq. \ref{['eq:lagrangian']}. (b)$G$ as a function of the voltage on the North/South gates, $V_\mathrm{NS}$, and the DC voltage bias $V$ at $B=10T$. (c) Line cuts of panel (b) at the values of $V_{\text{NS}}$ indicated by the colored points. Black dashed lines are plotted using the value of $G$ predicted by Eq. \ref{['eq:integral_conductance']} where the parameter $T_0$ is extracted from the low-bias conductance. (d) The zero-bias conductance also scales with temperature in agreement with Eq. \ref{['eq:integral_conductance']} for low energies. While the data deviates from the model at high energies, $G$ nevertheless exceeds $G = \frac{e^2}{3h}$ for $T \gg T_0$, indicating strong coupling at high $T$. (e) The data from panel d, after scaling $T$ by $T_0$. The curves collapse onto the universal scaling formula Eq. \ref{['eq:integral_conductance']}, shown in black.
• Figure 4: Zero frequency voltage step-up transformer. (a) The differential gain $dV_\mathrm{o}/dV_\mathrm{i}$ and resulting integrated DC gain $\beta = V_\mathrm{o}/V_\mathrm{i}$, measured in the configuration shown in the inset, with B=9T, $T_{\text{probe}}$=48mK, $V_\mathrm{E}$=-1.460V, $V_\mathrm{W}$=-1.775V, $V_\mathrm{NS}$=-3.225V, $V_\mathrm{BG}$=2.0V. The FQH Andreev scattering process yields an enhancement of the output voltage on the FQH side chklovskii_consequences_1998, with the DC gain predicted to reach a value of 1.5 in the dissipationless limit. Experimentally, we find a gain $\beta = 1.46$, despite the nonlinearity at low bias arising from the instability suppression of the Andreev scattering at low energies. (b) The DC power dissipation ratio, calculated from $\beta$ via $P_\mathrm{out}/P_\mathrm{in} = (2 \beta / 3)^2 - (2 \beta / 3) + 1$, is plotted versus V, and reaches a maximum value of 97.6% halperin_dc_2003.
• Figure S1: Signatures of reconstruction in $G$ between $\nu = 1$ edge modes.(a) The conductance measured across the QPC with both the East and West regions in $\nu = 1$ at B = 8T. The East and West gate voltages are swept in the opposition direction of $V_{\text{BG}}$ along the y axis, through the range $V_\mathrm{EW} \in (0.6V, -2.191V)$, to maintain a fixed filling factor while varying the voltage difference and thereby the potential sharpness. (b) The simulated electric potential at the monolayer, corresponding to the operating point I. (c) Same as (b) but at the operating point II, where the potential is much softer. (d) Simulated potential along the contours marked in grey and black in panels B and C, respectively. The softness is quantified by the maximum magnitude of the in-plane confining electric field, $E_{\parallel}$ (i.e. simply the gradient of the potential normal to the boundary between the N(/S) and E(/W) regions).