🔬 mesoscale physics

Entropy of Non-Abelian Anyons from Slow Quasiparticle Dynamics in Quantum Hall Interferometers

This paper proposes a scheme to extract the characteristic entropy of non-Abelian anyons in fractional quantum Hall states by inferring the charge of an antidot embedded in an interferometer through time-dependent interference phase switching, thereby overcoming the limitations of conventional equilibrium charge measurements.

Original authors: Eran Sela, Mitali Banerjee

Published 2026-07-03

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CC BY 4.0

Original authors: Eran Sela, Mitali Banerjee

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 tiny, invisible world where particles don't just bounce off each other like billiard balls. Instead, they have a secret "identity" that changes depending on how they move around one another. These are called Non-Abelian Anyons. Think of them not as simple marbles, but as magical keys. If you swap two normal keys, nothing happens. But if you swap these magical keys, the universe remembers the swap, and the keys themselves change their internal state.

This paper is about a new way to "weigh" these magical keys to prove they exist and measure their unique properties.

The Problem: The Invisible Weight

Scientists know these keys exist in a special state of matter called the Quantum Hall effect (think of it as a super-flat, super-cold electronic highway). These keys carry a hidden "weight" called entropy. In simple terms, this entropy is a measure of how many different secret identities a single key can have.

For a normal electron, this weight is zero. But for these magical keys, the weight is a specific, non-zero number (mathematically, it's related to the square root of 2).

The problem is that trying to measure this weight directly is incredibly hard. It's like trying to weigh a single grain of sand using a scale that is designed for elephants. The usual tools (charge detectors) are too clumsy; they disturb the sand so much that the measurement fails.

The Solution: The "Traffic Light" Detector

The authors propose a clever new trick. Instead of trying to weigh the grain of sand directly, they watch how it affects the traffic around it.

The Trap (The Antidot): Imagine a small, isolated island (an "antidot") in the middle of the electronic highway. This island can catch one of these magical keys.
The Interference Loop: Around this island, electrons are flowing in a loop, creating an interference pattern. Think of this like ripples in a pond. When the ripples meet, they create a pattern of light and dark bands.
The Switch: When a magical key hops onto the island, it doesn't just sit there; it changes the "phase" of the ripples. It's like a single pebble changing the timing of the waves, causing the light and dark bands to suddenly flip or "switch."

How They Measure the "Weight"

Here is the magic part of their proposal:

The Slow Dance: The key doesn't hop on and off instantly. It takes a long time to tunnel (jump) onto the island. This creates a "telegraph signal"—a slow, rhythmic switching between two states (key present, key absent), like a lighthouse blinking.
The Temperature Test: The scientists propose changing the temperature slightly.
- If the key were a normal particle, the switching speed would change in a predictable way.
- Because the key is a Non-Abelian Anyon, it has that extra "hidden weight" (entropy). This weight makes the key slightly more likely to be on the island at certain temperatures.
The Result: By watching how the "blinking" pattern shifts as they warm up the system, they can calculate the hidden weight. It's like deducing the weight of a person by watching how fast they jump on a trampoline when the air gets warmer.

Why This Matters

The paper claims that by using this "traffic light" method (watching the interference pattern switch), they can finally measure the specific entropy of a single magical key.

For the famous $\nu = 5/2$ state: They predict they will see a specific signature (an entropy of $\frac{1}{2} k_B \log 2$ ) that proves the keys are indeed the magical, non-Abelian kind.
For the "boring" states: If they try this on a state with only normal keys, the signal will be different (no extra weight).

The Bottom Line

This paper doesn't build a computer or a new device yet. Instead, it provides a theoretical blueprint for a very specific experiment. It says: "If you build this interferometer, watch the blinking pattern, and change the temperature, you will be able to see the invisible 'soul' (entropy) of these magical particles for the first time." It turns a difficult direct measurement into a clever observation of how the particle affects its surroundings.

Technical Summary: Entropy of Non-Abelian Anyons from Slow Quasiparticle Dynamics in Quantum Hall Interferometers

Problem Statement
Non-Abelian anyons, predicted to exist in fractional quantum Hall (FQH) states such as the $\nu = 5/2$ Moore-Read state, carry a characteristic topological entropy $\Delta S = k_B \log d$ , where $d$ is the quantum dimension of the anyon. For the $\nu = 5/2$ state, $d = \sqrt{2}$ , yielding an entropy of $\frac{1}{2}k_B \log 2$ . While theoretical proposals, such as charging spectroscopy by Ben-Shach et al., suggest that this entropy can be extracted via Maxwell relations from charge measurements in quantum dots or antidots, experimental realization has been hindered by technical limitations. Specifically, conventional electrostatic charge detectors require weak coupling to the antidot to preserve fractional charge definitions, leading to tunneling times that exceed device stability. Furthermore, in a global system, adding a quasiparticle to an antidot creates a compensating excitation elsewhere, potentially obscuring the single-anyon entropy contribution.

Methodology
The authors propose a two-part theoretical framework to overcome these challenges:

Thermodynamic Protocol (Idealized): The paper first establishes a theoretical protocol based on the Maxwell relation. By modeling an antidot embedded in an FQH liquid with a charging energy Hamiltonian $H_c = E_c(N - n_g)^2$ , the authors analyze the thermal average charge $\langle N \rangle_{T, n_g}$ as a function of gate voltage $n_g$ and temperature $T$ . They demonstrate that the shift in the charge stability diagram (the "charging curve") with temperature is directly proportional to the entropy difference between charge states. Specifically, the shift $\Delta E$ relates to the degeneracy ratio $g(N+e^*/e)/g(N)$ via $\Delta E = -k_B \Delta T \log [g(N+e^*/e)/g(N)]$ .
Conformal Field Theory (CFT) Analysis: To isolate the entropy of a single anyon from the compensating excitation in the bulk/edge, the authors employ CFT. They model the system as a "long edge" (the interferometer loop) and a "short edge" (the antidot). They argue that in an intermediate temperature regime where $2\pi/L_{edge} \ll k_B T \ll 2\pi/L_{antidot}$ , the long edge acts as a thermal reservoir with many accessible states, while the antidot is restricted to its ground state due to large level spacing. Under these conditions, the total entropy change upon adding a quasiparticle to the antidot is shown to be exactly $\Delta S = k_B \log d$ , effectively isolating the non-Abelian contribution.
Experimental Scheme (Interferometric Detection): Recognizing the difficulty of direct charge sensing, the authors propose utilizing the Aharonov-Bohm interference phase in a Mach-Zehnder type interferometer containing a gated antidot. Recent experiments have observed slow, telegraphic switching in interference phases due to quasiparticle tunneling. The authors propose that this time-dependent phase switching serves as a non-local charge detector. By monitoring the dwell times of the interference phase (corresponding to the quasiparticle being inside or outside the antidot) at various temperatures, one can reconstruct the charge curve $\langle N \rangle(n_g, T)$ without a conventional charge sensor.

Key Results

Entropy Extraction: The authors derive that for the $\nu = 5/2$ state, the charge curve exhibits an even-odd effect. When the antidot contains an unpaired non-Abelian anyon (charge $e/4$ ), the system possesses an extra entropy of $\frac{1}{2}k_B \log 2$ compared to the paired state.
Distinct Signatures: The calculated entropy curve shows peaks of $k_B \log(1+\sqrt{2})$ at charge degeneracy points, separating plateaus of $\frac{1}{2}k_B \log 2$ . In contrast, for Abelian states like $\nu = 1/3$ (where $d=1$ ), the entropy curve shows only standard $k_B \log 2$ peaks associated with charge degeneracy, with no fractional topological entropy.
Feasibility of Non-Local Sensing: The paper demonstrates that the telegraphic noise observed in interferometer currents can be quantitatively analyzed to extract the relative occupation probabilities of the antidot. By varying the temperature and measuring the shift in the resulting charge curve, the topological entropy can be extracted via the Maxwell relation.

Significance and Claims
The paper claims to provide a viable theoretical pathway to experimentally measure the $O(1)$ topological entropy of a single non-Abelian anyon, a long-standing challenge in condensed matter physics.

Theoretical Advance: It establishes that charge measurements, when interpreted through the lens of Maxwell relations and CFT, can isolate the entropy of a single anyon despite the presence of compensating excitations, provided the antidot level spacing is sufficiently large.
Experimental Perspective: The work proposes a concrete experimental method using existing interferometric setups. By leveraging the recently observed slow dynamics of interference phases, the authors suggest that the need for sensitive, local charge detectors (which are difficult to implement in FQH systems) can be bypassed.
Scope: The authors modestly frame this as a proposal for "equilibrium thermodynamics of non-Abelian anyons." They do not claim to have performed the experiment but rather provide the theoretical justification and the specific protocol (interference-based charge sensing combined with temperature variation) required to achieve the measurement. The work positions entropy as a promising spectroscopy method for distinguishing Abelian from non-Abelian topological orders.