🔬 mesoscale physics

Long-lived Laughlin pairs in a depleted quantum Hall edge channel

This paper presents a microscopic theory and numerical simulations demonstrating that repulsive Coulomb interactions in depleted quantum Hall edge channels can stabilize long-lived Laughlin pairs of electrons, suggesting that their creation and detection are feasible with existing single-electron circuit technology.

Original authors: Girts Barinovs, Agris Buzs, Vyacheslavs Kashcheyevs

Published 2026-08-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Girts Barinovs, Agris Buzs, Vyacheslavs Kashcheyevs

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

In the quiet, ultra-cold world of quantum physics, electrons usually behave like solitary travelers. When they move along the edge of a special material called a quantum Hall system, they are forced into a single-file line by a powerful magnetic field. For decades, scientists have learned to control these individual travelers, sending them one by one through tiny circuits to study the fundamental nature of electricity. This field, known as electron quantum optics, has allowed researchers to split beams of electrons and watch them interfere with one another, much like waves of light. However, a new question has emerged from this controlled environment: if two electrons are sent to collide, can their natural repulsion, the force that usually pushes them apart, actually force them to stick together and travel as a single unit?

This question challenges the intuition that repulsion always leads to separation. In a strong magnetic field, the rules of motion change. Electrons do not move in straight lines but instead drift along the contours of invisible electric landscapes. When two electrons are close to each other in this environment, their mutual push creates a circular dance around a common center. While this motion is well understood in theory, it has been unclear whether such a pair could survive long enough to be observed in a real experiment, or if the surrounding environment would immediately tear them apart. The stability of these pairs depends on the shape of the electric fields guiding them and whether the energy holding them together is strong enough to resist the forces trying to pull them apart.

Researchers at the University of Latvia have now provided a detailed answer to this puzzle. They developed a microscopic theory to describe how two electrons can form a bound state, which they call a Laughlin pair, named after a physicist who first proposed similar ideas in the context of exotic quantum fluids. Using a combination of advanced mathematical modeling and computer simulations, the team investigated whether these pairs could exist in the specific conditions created by current laboratory devices. Their work focuses on the "depleted" edge channels of quantum Hall systems, where the electron density is low enough that individual particles can be manipulated without interference from a sea of other electrons.

The study reveals that under the right conditions, these repulsive pairs are not only possible but remarkably stable. The researchers calculated that the lifetime of such a pair depends heavily on its energy relative to a specific threshold. If the pair has enough energy, it becomes trapped in a state where it can circulate for a very long time before eventually breaking apart. Their calculations show that this lifetime grows exponentially as the pair's energy increases. For the specific materials used in existing experiments, such as gallium arsenide, the theory predicts that the lowest-energy version of this pair could survive for about three orders of magnitude longer than the time it takes to travel through the device. In practical terms, this means the pair would likely reach the end of the experimental setup long before it had a chance to decay.

To confirm that these pairs could actually be created, the team simulated a collision between two electrons moving toward each other in a realistic electric landscape. They tracked the motion of the electrons from the moment they approached, through the collision, and into the aftermath. The simulation showed that after the initial scattering, where most of the energy is released and the electrons fly apart, a small, persistent core remains. This core matches the theoretical description of the Laughlin pair, possessing a specific internal structure and a unique pattern of zeros in its wave function that distinguishes it from ordinary scattering states. The researchers visualized this process using a method that maps the probability of finding the electrons in different locations, showing how the chaotic motion of the collision settles into a stable, circulating pattern.

The findings suggest that the technology already exists to create, transport, and detect these repulsively paired electrons. The devices currently used to manipulate single electrons in laboratories are capable of supporting these pairs, provided the electric fields are shaped correctly. The researchers identified that the stability of the pair is a result of the one-dimensional nature of the edge channel, which restricts the ways the electrons can separate. This kinematic constraint acts as a protective mechanism, allowing the pair to remain intact even if it is not the lowest possible energy state. This discovery opens a new path for exploring how electrons interact in strong magnetic fields and suggests that similar pairing mechanisms might apply to other exotic quantum particles known as anyons, which are predicted to exist in fractional quantum Hall states.

By bridging the gap between abstract theory and experimental reality, this work places the preparation of these long-lived pairs within reach. It demonstrates that the repulsive force between electrons, often seen as a barrier to control, can be harnessed to create a new type of composite particle. The ability to generate and study these pairs could lead to a deeper understanding of quantum correlations and potentially offer new ways to manipulate quantum information. The study does not claim to have observed these pairs in a new experiment, but rather provides the theoretical proof and quantitative guidance that existing devices are already in the right regime to do so. It turns a theoretical possibility into a practical target for future experiments, suggesting that the next step is to build the specific conditions needed to watch these elusive pairs form and travel.

Technical Summary: Long-lived Laughlin pairs in a depleted quantum Hall edge channel

Problem Statement
While electron quantum optics has successfully enabled the manipulation of single-electron excitations in quantum Hall edge channels, a fundamental question remains regarding the role of repulsive interactions: can Coulomb repulsion itself generate a stable, propagating two-electron composite? In strong magnetic fields, the drift of guiding centers is perpendicular to applied forces. For two nearby electrons, mutual Coulomb repulsion generates a tangential force, leading to quantized relative circulation (Laughlin pairs). However, in a realistic experimental setting with non-uniform guiding potentials, the stability of these positive-energy pairs against dissociation is unclear. Specifically, it is unknown whether the proposed collision pathway to pair formation survives full two-dimensional dynamics or if the pairs dissociate too rapidly to be observed in existing depleted-edge devices.

Methodology
The authors develop a microscopic theory based on the physical two-electron Hamiltonian in two dimensions, incorporating a strong magnetic field and a smooth external electric potential.

Hamiltonian and Projection: The system is modeled using a two-electron Hamiltonian where the center-of-mass and relative coordinates separate. The relative motion is governed by a repulsive central Coulomb potential and a quadratic electric-field gradient (characterized by a tensor $\hat{V}$ ). The authors project the relative-motion Hamiltonian onto the Lowest Landau Level (LLL), reducing the problem to a dimensionless form controlled by the ratio of the interaction energy scale to the magnetic confinement energy ( $\chi$ ) and the anisotropy parameter ( $\kappa$ ).
Stationary Theory: To determine the existence and lifetimes of quasibound states, the authors employ the complex-scaling method. This technique allows for the calculation of complex resonance energies ( $E_m - i\gamma_m$ ), where the imaginary part $\gamma_m$ corresponds to the decay rate. They also utilize analytic tunneling theory to derive approximations for these decay rates.
Visualization: The internal structure of the resonances is visualized using Husimi distributions ( $Q(z)$ ) and the stellar representation (zeros of the wave function). These tools map the relative-coordinate probability density and the phase-space structure, linking the deformed quasibound states to ideal Laughlin pairs.
Dynamical Simulation: To test the stationary theory, the authors perform time-dependent Schrödinger equation (TDSE) simulations of a two-electron collision using the full finite-field Hamiltonian (without LLL projection). They initialize two Gaussian wave packets on opposite incoming branches of a saddle potential and track the evolution of the probability density and its phase singularities (zeros).

Key Contributions and Results

Stability Threshold and Branch Counting: The study establishes a dissociation threshold energy $\varepsilon_{th} = (3/2)\chi^{1/3}$ . Above this threshold, a finite number of quasibound states exist. The maximum angular momentum index $\bar{m}_{max}$ of supported states scales as $\chi^{-2/3}$ . For spin-polarized electrons (odd parity), only odd- $m$ branches are supported.
Exponential Lifetime Scaling: A central finding is that the decay rates $\gamma_m$ decrease exponentially as the pair energy increases above the dissociation threshold. The authors derive an analytic tunneling estimate (Eq. 18) showing that lifetimes grow exponentially with the energy difference $(E_m - E_{th})$ . This implies that higher-order branches are significantly more stable than those near the threshold.
Wave Function Structure: Visualization reveals that while the quadrupole field breaks rotational symmetry, the quasibound states retain a recognizable "branch identity" inherited from the ideal Laughlin pairs. The Husimi densities show a pile-up near interaction-induced saddle points, and the wave-function zeros preserve the topological structure (e.g., a central zero for $m=1$ ) while acquiring additional zeros associated with the outgoing scattering channels.
Dynamical Formation: The time-dependent collision simulations demonstrate that a long-lived core survives after the prompt scattering components escape. The density profile and the single central zero of this remnant qualitatively match the independently calculated stationary $m=1$ resonance. This confirms that the collision pathway can successfully populate the quasibound state.
Experimental Relevance: Applying the theory to reported GaAs parameters (specifically for depleted edge channels and collision circuits), the authors find that existing devices likely already support the lowest spin-polarized pair ( $m=1$ ). The estimated lifetime is approximately three orders of magnitude longer than typical propagation times in these devices.

Significance and Claims
The paper claims that the preparation, propagation, and detection of repulsively paired electrons are within the reach of existing single-electron circuit technology. The results suggest that kinematic stabilization under constrained one-dimensional propagation can stabilize topologically allowed internal anyon states, even if they are not energetically favorable relative to separated particles.

The authors modestly frame their contribution as providing a quantitative framework for understanding these states. They do not claim to have experimentally observed these pairs but argue that the theoretical lifetimes are sufficiently long to make such an observation feasible. Furthermore, they suggest that the mechanism of kinematic stabilization may extend to anyonic quantum Hall edge excitations in populated channels, offering a potential pathway to stabilize internal anyon states that would otherwise dissociate in a bulk environment. The work bridges the gap between the theoretical concept of Laughlin pairs and the practical constraints of mesoscopic electron optics.

Technical Summary: Long-lived Laughlin pairs in a depleted quantum Hall edge channel

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