Anyon Exchange Phase from Antidot Interferometry

In the strange landscape of two-dimensional physics, particles do not always behave like familiar bosons or fermions. In certain fractional quantum Hall systems, quasiparticles emerge known as anyons. These particles possess a fractional charge and a unique property. When two anyons are exchanged, the system’s wavefunction acquires a fractional phase. Physicists have measured the fractional charge and the "braiding phase" (the phase acquired when one anyon loops around another). However, the fundamental exchange phase $\theta$ has remained out of reach.

Recent theoretical work provides a roadmap to bridge this gap. By using a specialized Fabry-Perot interferometer with a quantum antidot (a small region of excluded charge), researchers have identified a way to extract this exchange phase. They do this by watching how electrical signals shift as a gate voltage is tuned.

The difficulty of isolating the exchange phase

Measuring the exchange phase $\theta$ is difficult due to mathematical ambiguity and geometric interference. Existing experiments typically observe the braiding phase ($2\theta$). This only reveals the phase modulo $\pi$. Consequently, the true exchange phase $\theta$ remains undetermined. Previous proposals often relied on fitting complex data to theoretical models. In those models, $\theta$ is entangled with non-universal parameters. This makes it hard to isolate the pure statistical signal.

Any attempt to measure these phases must contend with geometric contributions. As an anyon moves, it picks up a Berry phase (a phase resulting from the changing geometry of the path). This can mimic or mask the statistical phase. Earlier semiclassical models struggled to decouple these two effects. A new theory was needed to account for how anyons move through a device. It also had to manage real-world realities like level broadening (the spreading of energy states) and non-equilibrium occupation (the particle count when the system is driven by voltage).

Decoding statistics through antidot resonances

The authors model a Fabry-Perot interferometer with a quantum antidot embedded in one arm .

The setup relies on the interplay between two tunneling processes:

1. Direct Tunneling: When the antidot level is empty, an anyon tunnels directly from the upper edge to the lower edge.
2. Co-tunneling: When the level is occupied, an anyon tunnels out of the dot to the lower edge. Simultaneously, a new one tunnels in from the upper edge.

The core insight is that these two processes differ by the exchange phase $\theta$. To handle anyonic complexity, the authors use a perturbative Keldysh treatment (a math framework for systems driven out of equilibrium) and bosonization (describing particle interactions as collective waves).

They incorporate "Klein factors," which are mathematical operators that preserve anyonic statistics during tunneling. Their work shows that the transmission phase does not climb steadily like it does for ordinary electrons. Instead, for anyons, the phase undergoes a non-monotonic evolution. It rises, dips, and settles as the gate voltage $V_g$ tunes the antidot through resonance .

Non-monotonic signatures and phase plateaus

The paper's primary finding is that the exchange phase $\theta = \pi\nu$ is encoded in the difference between transmission-phase "plateaus." Unlike electrons, which show a steady $0 \to \pi$ phase evolution, anyons display a distinct signature.

The authors report that the transmission phase $\delta$ evolves non-monotonically near the lead chemical potentials, $V_g = \pm\Delta V/2$ . This happens due to three factors: resonant tunneling enabled by level broadening, rapid changes in antidot occupation, and the anyonic density-of-states anomaly. The antidot occupation behaves unexpectedly. Rather than following a simple Fermi distribution, the non-equilibrium coupling causes the occupation $n$ to rise above $1/2$ at one potential and dip below $1/2$ at the other .

By measuring the phase at stable plateaus away from these resonance points, the authors show that the difference between these plateaus yields the bare exchange phase. This provides a direct metric. It remains robust against temperature fluctuations and minor changes in device geometry.

Constraints on experimental implementation

There are significant practical constraints for researchers. First, the method is sensitive to how the antidot's charge is handled. The statistical phase is only observable if the antidot charge is "screened" by an external gate. For example, a graphite back gate can be used. If the charge is screened by the quantum Hall edges instead, the electrostatic phase will cancel the statistical phase. This would make the effect invisible.

Second, the utility depends on which resonance is targeted. Tuning through a single resonance allows for the extraction of $\theta$. However, tuning between two consecutive levels is unsuitable. As shown in, the phase evolution between levels is dominated by "dynamical phases." These depend heavily on the specific distances $a$ and $b$ within the device.

This makes it impossible to isolate the universal statistical signal. Finally, the precision of the lineshape near the lead chemical potentials is limited. If the coupling is too strong, the results become qualitative rather than quantitative.

A verdict on anyonic measurement

Is this a viable path to measuring the exchange phase? The answer is yes, provided experimentalists follow strict electrostatic rules. The theory offers a concrete prediction: look for the non-monotonic "overshoot" in the phase. Then, measure the difference between the resulting plateaus.

This non-monotonic signature provides a clear target for future experiments. However, success hinges on implementing precise gate-screening. This prevents the cancellation of the very phase one seeks to measure. If done correctly, this approach turns the exchange phase into a measurable spectroscopic reality.

Figures from the paper