🔬 mesoscale physics

Ballistic-to-diffusive transition in engineered counter-propagating quantum Hall channels

This paper experimentally demonstrates that charge transport in engineered quantum Hall samples with counter-propagating edge states transitions from a ballistic regime (when channel numbers are unequal) to a critical diffusive regime (when channel numbers are equal) by utilizing tunable Landauer reservoirs to control charge equilibration.

Original authors: Aifei Zhang, Kenji Watanabe, Takashi Taniguchi, Patrice Roche, Carles Altimiras, François D. Parmentier, Olivier Maillet

Published 2026-05-14

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CC BY 4.0

Original authors: Aifei Zhang, Kenji Watanabe, Takashi Taniguchi, Patrice Roche, Carles Altimiras, François D. Parmentier, Olivier Maillet

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a highway where cars (electrons) are forced to drive in a single file line. In most "Quantum Hall" systems, all the lanes go in the same direction. Traffic flows smoothly, perfectly, and without any friction or accidents. This is the "ballistic" regime: the cars get from point A to point B with zero energy lost.

However, some exotic quantum systems have a twist: they have lanes going in opposite directions. Some cars drive clockwise, others counter-clockwise. Usually, when these opposing lanes are close together, the cars crash into each other, swap places, and mix their energy. This causes traffic jams and energy loss, turning the smooth highway into a chaotic, "diffusive" mess where the distance traveled matters a lot.

The Experiment: Building a Custom Highway
The researchers in this paper didn't just observe this chaos; they built a custom "highway" to study exactly how it happens. They used a special material (graphene) to create two separate strips of road.

Strip A has a certain number of lanes going one way.
Strip B has a certain number of lanes going the other way.
They connected these strips together with a series of "rest stops" (called Landauer reservoirs). These rest stops act like mixing bowls where cars from the clockwise lanes and counter-clockwise lanes can stop, swap drivers, and re-equilibrate before getting back on the road.

By changing the "filling factor" (essentially the number of cars in each lane), they could control exactly how many lanes were going up versus how many were going down.

The Discovery: Two Types of Traffic
The team discovered that the behavior of this traffic depends entirely on the balance between the two directions:

The "Unequal Traffic" Scenario (Ballistic):
Imagine you have 6 lanes going one way and only 3 going the other. Even though they mix at the rest stops, the sheer volume of the 6 lanes dominates. The "extra" 3 lanes just keep flowing through without getting stuck. The system behaves like a perfect, frictionless highway again. The cars travel ballistically, and the energy loss is negligible, except for a tiny "hot spot" right near the exit where the mixing finally happens.
The "Equal Traffic" Scenario (Diffusive):
Now, imagine you have exactly 3 lanes going one way and 3 going the other. This is the critical tipping point. Because the traffic is perfectly balanced, every car that leaves the clockwise lane has a car waiting to swap with it in the counter-clockwise lane.
Instead of a smooth flow or a single crash at the exit, the cars are constantly swapping and mixing all along the entire highway. The energy loss isn't concentrated at one spot; it's spread out evenly across the whole road. The system becomes "diffusive." The further the cars have to travel, the more resistance they face, and the conductance drops in a predictable, linear way (like walking through a crowded room where you bump into people the whole time).

The "Magic" Length Scale
The researchers found a way to measure a "mixing length."

If the traffic is unequal, the mixing length is short. The cars settle down quickly, and the rest of the road is smooth.
If the traffic is perfectly equal, this "mixing length" becomes infinite. The cars never stop mixing; the whole road is a zone of constant interaction.

Why This Matters
This experiment is like a simulator. Real exotic quantum systems (like those involving fractional charges or spin) are messy and hard to control. By building this engineered "counter-propagating" highway with simple integer lanes, the scientists created a clean, controllable model. They proved that you can switch a system from a perfect, frictionless flow to a chaotic, diffusive mess just by balancing the number of lanes.

They showed that when the opposing forces are equal, the system enters a "critical" state where the rules of transport change completely, behaving like a standard resistor (Ohmic) rather than a quantum super-highway. This helps scientists understand how energy and charge move in more complex, mysterious quantum materials without needing to build those complex materials first.

Technical Summary: Ballistic-to-Diffusive Transition in Engineered Counter-Propagating Quantum Hall Channels

Problem Statement
Quantum Hall (QH) systems typically exhibit robust, ballistic, and dissipationless edge transport when channels co-propagate. However, systems hosting counter-propagating edge states—such as the Quantum Spin Hall (QSH) effect or hole-conjugate fractional quantum Hall (FQH) states (e.g., $\nu = 2/3$ )—display complex transport regimes. In these systems, Coulomb interactions and inter-channel tunneling can lead to charge equilibration along the edge, causing transport properties to become length-dependent. Theoretical models predict two distinct regimes based on the balance of upstream and downstream conductances:

Ballistic Regime: Occurs when upstream and downstream conductances are unequal ( $\nu_{up} \neq \nu_{down}$ ). Transport remains quantized with negligible dissipation at large scales.
Diffusive Regime: Occurs when upstream and downstream conductances are equal ( $\nu_{up} = \nu_{down}$ ). Transport becomes diffusive, characterized by inexact conductance quantization and dissipation distributed along the entire edge.

Experimentally probing the transition between these regimes is challenging because it requires precise control over the number of equilibrating channels, the strength of the equilibration process, and the ability to measure dissipation along the edge.

Methodology
The authors engineered a tunable experimental platform to simulate counter-propagating edge channels using integer Quantum Hall states. The device consists of two monolayer graphene Hall bars (Device A and Device B), encapsulated in hexagonal boron nitride (hBN) and individually gated by graphite back gates.

Configuration: The two Hall bars are connected in a "top-to-tail" fashion, where the $k$ -th contact of Device A is connected to the $k$ -th contact of Device B. This creates an effective single Hall bar with counter-propagating edge channels determined by the filling factors $\nu_A$ and $\nu_B$ of the respective devices.
Equilibration Mechanism: Intermediate contacts between the two bars act as tunable Landauer reservoirs. These metallic contacts force charge redistribution and energy equilibration between the counter-propagating channels.
Measurement: The system is cooled to 10 mK under a 14 T magnetic field. By fixing $\nu_A$ and sweeping $\nu_B$ , the authors tune the number of counter-propagating channels. They measure voltage drops across floating Ohmic contacts and the two-point conductance of the effective Hall bar as a function of the number of intermediate reservoirs ( $N$ ), effectively varying the sample length.

Key Results

Asymmetric Case ( $\nu_A \neq \nu_B$ ):
- The voltage profile along the edge exhibits an exponential decay (or rise) characterized by a dimensionless equilibration distance $\delta = 1/|\log(\nu_A/\nu_B)|$ .
- Dissipation is localized in a "hot spot" near the downstream contact where the voltage drops.
- The two-point conductance converges exponentially to the equilibrated value $G_\infty = |\nu_A - \nu_B|e^2/h$ as the number of reservoirs $N$ increases.
- This behavior confirms a ballistic transport regime outside the localized dissipation region.
Symmetric/Critical Case ( $\nu_A = \nu_B$ ):
- The voltage profile transitions from exponential to linear, indicating a uniform voltage drop across all intermediate contacts.
- Dissipation becomes delocalized over the entire edge length, characteristic of a diffusive, Ohmic regime.
- The two-point conductance decays algebraically ( $G \propto 1/N$ ) rather than exponentially, remaining significantly above zero even for large $N$ .
- This confirms a transition to a critical, scale-invariant diffusive regime where the effective equilibration length diverges.
Universality and Scaling:
- By rescaling the voltage drops and conductance data by the equilibration length $\delta$ , all data for asymmetric cases collapse onto universal curves (exponential decay and $\coth$ function).
- The discrete system of Landauer reservoirs maps effectively to a continuous edge model, validating theoretical scattering formalisms.

Significance and Claims
The paper claims to experimentally demonstrate and fully explore the transition between ballistic and diffusive transport regimes in counter-propagating edge channels using a system with fully controlled parameters.

Simulation of Exotic States: The approach allows for the simulation of charge equilibration in hole-conjugate states (like $\nu=2/3$ ) and other exotic QH effects using well-understood integer QH states. The authors note that while the microscopic equilibration mechanism in their device (reservoir redistribution) differs from that in FQH states (inter-edge tunneling), the macroscopic scaling laws (exponential vs. algebraic) and the divergence of the equilibration length near the critical point are identical.
Conceptual Simplicity: The experimental design is highlighted for its simplicity, allowing direct measurement of voltage drops and dissipation along the edge, which is difficult in standard FQH or QSH samples.
Future Directions: The authors suggest this platform could be extended to investigate heat flow and charge-heat separation (e.g., observing fractional electrical conductance with integer thermal conductance) via noise measurements, particularly in systems where interactions are added.

The work provides a controlled experimental realization of theoretical predictions regarding the interplay between channel imbalance and equilibration in topological edge states, offering a new tool to study non-universal transport regimes.

Technical Summary: Ballistic-to-Diffusive Transition in Engineered Counter-Propagating Quantum Hall Channels

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