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arxiv: 2605.00669 · v1 · submitted 2026-05-01 · ❄️ cond-mat.mes-hall · cond-mat.str-el

Experimental Evidence of Fractional Entropy in Critical Kondo Systems

C. Piquard , A. Veillon , Y. Sato , F. Zanichelli , A. Aassime , A. Cavanna , U. Gennser , A. K. Mitchell
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A. Anthore F. Pierre
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Pith reviewed 2026-05-09 18:49 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.str-el
keywords fractional entropyKondo effectnon-Abelian anyonsMajorana zero modesFibonacci anyonsquantum critical pointsthermodynamic measurements
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The pith

Experiments measure fractional entropy in Kondo critical points matching non-Abelian anyon predictions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes experimental evidence for fractional entropy associated with single anyons in two- and three-channel Kondo systems. A micrometre-scale metallic island is coupled to two or three leads and tuned to quantum-critical states arising from frustrated interactions. Entropy is obtained from island charge data via a Maxwell relation, producing values that match the quantum dimensions of Majorana zero modes and Fibonacci anyons. This thermodynamic signature implies non-Abelian character directly through the relation between entropy and quantum dimension.

Core claim

In metal-semiconductor quantum circuits realizing two-channel and three-channel Kondo critical points, the low-temperature entropy associated with the emergent anyons takes the fractional values Delta S = kB ln(sqrt(2)) and Delta S = kB ln((1 + sqrt(5))/2), respectively, consistent with theoretical predictions for a Majorana zero mode and a Fibonacci anyon.

What carries the argument

The Maxwell relation that extracts entropy from the temperature dependence of the measured island charge at the tuned Kondo critical points.

If this is right

  • Thermodynamic measurements via charge provide a route to characterize non-Abelian anyons independent of transport signatures.
  • The scaling dimensions extracted align with those required for protected non-local encoding in topological quantum computing.
  • Critical states in multi-channel Kondo systems offer a controllable platform for studying anyonic thermodynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Charge-based entropy extraction could be applied to other candidate anyonic platforms to test consistency across different realizations.
  • The method may reduce reliance on difficult transport measurements that have so far hindered unambiguous anyon detection.
  • If the fractional entropy persists under additional gating or lead configurations, it would strengthen the case for using these circuits in anyon-based information processing.

Load-bearing premise

The entropy change extracted from charge measurements is assumed to be dominated by the anyonic contribution at the critical points, with negligible interference from background or lead effects.

What would settle it

Observation of entropy values that are integer multiples of kB ln(2) or that deviate substantially from the predicted fractional amounts in the same two- and three-lead setups.

Figures

Figures reproduced from arXiv: 2605.00669 by A. Aassime, A. Anthore, A. Cavanna, A. K. Mitchell, A. Veillon, C. Piquard, F. Pierre, F. Zanichelli, U. Gennser, Y. Sato.

Figure 1
Figure 1. Figure 1 view at source ↗
Figure 2
Figure 2. Figure 2: Conductance benchmarks and Kondo critical￾ity. (a) Conductance vs plunger gate voltage at different temper￾atures for a 2CK circuit tuned either in the regime of weak coupling (τ ≃ 0.25, blue symbols) or strong coupling (τ ≃ 0.993, green sym￾bols). Lines represents analytical predictions in the corresponding limits. (b,c) Universal Kondo conductance scaling at δVpl = 0 vs T/TK in two-channel (b) and three-… view at source ↗
Figure 3
Figure 3. Figure 3: Entropy measurement. Determination of the en￾tropy for a device tuned to 2CK by measurement of the charge N along sweeps in Vpl at different temperatures, for either weak (a,c,e) or strong (b,d,f) coupling τ. Symbols with different shades correspond to different temperatures. Predictions (Methods) are displayed as continuous black lines. (a,b) Charge measurements vs gate voltage (∆E = 2ECδVpl/∆). (c,d) Fin… view at source ↗
Figure 4
Figure 4. Figure 4: Entropy renormalization flow ∆S vs T/TK for 2CK (a) and 3CK (b). Universal NRG predictions for Simp(0) at the critical point are shown as solid lines. Colored symbols represent experimental data points where we expect the measured ∆S to closely approximate the Kondo impurity entropy Simp(0). These data are selected using the small conductance criterion Gi(∆E = EC)/Gi(0) < 0.075 which indicates a near-froze… view at source ↗
Figure 5
Figure 5. Figure 5: Crossover flow from quantum criticality at 2CK (a,c) and 3CK (b,d). a,b Conductance measurements at T = 9.3 mK are plotted as symbols versus δVpl/∆ (left panel) or versus Tco/T (right panel) for different τ, chosen such that T/TK < 0.02 to ensure well-developed criticality at charge degeneracy. The pre￾factor λco in Eq. (4) is adjusted for each τ to match universal crossover predictions for the conductance… view at source ↗
read the original abstract

Unconventional quantum states defying the ubiquitous Fermi-liquid paradigm can emerge in the presence of strong electronic correlations. Among these, non-Abelian anyons - such as Majorana zero modes and Fibonacci anyons - are of particular interest for topological quantum computing due to their non-integer quantum dimensions d>1, which allows for protected non-local encoding and processing of quantum information. However, despite considerable efforts, the unambiguous characterisation of such anyons via transport measurements has proved challenging. Instead, here we provide experimental evidence for the low-temperature fractional entropy Delta S associated with a single anyon, which directly implies its non-Abelian character through the relation Delta S = kB ln(d). This thermodynamic signature is measured in metal-semiconductor quantum circuits engineered to realize quantum-critical states from frustrated interactions. Using a micrometre-scale metallic island coupled to two or three electronic leads, we tune the system to two-channel and three-channel Kondo critical points. By measuring the island charge and exploiting a thermodynamic Maxwell relation, we estimate the entropy associated with the anyons that emerge in these critical states. Our observations reveal fractional values, exposing non-Abelian anyons. The corresponding scaling dimensions are consistent with theoretical predictions for a Majorana zero mode Delta S = kB ln(sqrt(2)) and a Fibonacci anyon Delta S = kB ln(1 +sqrt(5))/2 for two and three channels. These findings establish entropy measurements as a powerful tool for characterizing exotic quantum states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript reports experimental measurements of fractional entropy changes Delta S in a micrometre-scale metallic island coupled to two or three electronic leads, tuned to two-channel and three-channel Kondo critical points. By measuring the island charge Q(V_g, T) and applying a thermodynamic Maxwell relation, the authors extract entropy values that match k_B ln(sqrt(2)) for Majorana zero modes (2CK) and k_B ln((1 + sqrt(5))/2) for Fibonacci anyons (3CK), claiming this provides direct evidence of non-Abelian anyons via their quantum dimensions.

Significance. If the entropy extraction reliably isolates the anyonic contribution, this would be a significant result as it supplies a thermodynamic signature complementary to transport measurements for characterizing non-Abelian statistics. The approach of using a Maxwell relation on charge data to access fractional entropy is a strength, offering a potentially falsifiable test of theoretical predictions for the scaling dimensions without requiring direct braiding operations.

major comments (2)
  1. [Entropy extraction and background discussion (near the Maxwell-relation application and Fig. 3 or equivalent)] The central attribution of the observed fractional Delta S to anyons requires that lead density-of-states, charging-energy fluctuations, and non-critical Kondo channels contribute negligibly (<< 0.2 k_B) at the critical points. The manuscript states this assumption but provides no quantitative bound, subtraction protocol, or temperature-dependence analysis demonstrating suppression of these terms relative to the reported values ~0.35 k_B and ~0.48 k_B.
  2. [Methods and results on thermodynamic relation] The Maxwell relation converts the measured charge response into total system entropy; without an explicit error budget or control measurements (e.g., off-critical detuning or lead-only devices) showing that background terms do not shift integer/zero entropy into the observed fractions, the isolation of the anyonic Delta S = k_B ln(d) remains unverified and load-bearing for the non-Abelian claim.
minor comments (3)
  1. [Methods] Clarify the precise integral form of the Maxwell relation employed (e.g., whether Delta S = integral (partial Q / partial T)_V dV_g with explicit limits) and any assumptions about equilibrium or integration cutoffs.
  2. [Figures presenting Q(V_g, T) data] Add error bars, raw data traces, and fitting details to the charge-vs-gate-voltage plots at multiple temperatures to allow assessment of the precision of the extracted fractional values.
  3. [Results summary] Include a short table comparing the measured Delta S values with the exact theoretical predictions, including uncertainties, to strengthen the consistency claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the entropy extraction and its attribution to anyonic contributions. We agree that explicit quantitative bounds and controls are needed to robustly isolate the fractional entropy and will incorporate these in the revision.

read point-by-point responses

  1. Referee: The central attribution of the observed fractional Delta S to anyons requires that lead density-of-states, charging-energy fluctuations, and non-critical Kondo channels contribute negligibly (<< 0.2 k_B) at the critical points. The manuscript states this assumption but provides no quantitative bound, subtraction protocol, or temperature-dependence analysis demonstrating suppression of these terms relative to the reported values ~0.35 k_B and ~0.48 k_B.

    Authors: We agree that the manuscript would benefit from explicit quantitative bounds on background contributions. While the original text relies on the dominance of the critical Kondo physics at the tuned points, we will add in revision a dedicated error-budget section. This will include temperature-scaling analysis, off-critical detuning data, and estimates showing that lead DOS, charging fluctuations, and non-critical channels contribute <0.1 k_B, well below the reported fractional values. revision: yes

  2. Referee: The Maxwell relation converts the measured charge response into total system entropy; without an explicit error budget or control measurements (e.g., off-critical detuning or lead-only devices) showing that background terms do not shift integer/zero entropy into the observed fractions, the isolation of the anyonic Delta S = k_B ln(d) remains unverified and load-bearing for the non-Abelian claim.

    Authors: We concur that an explicit error budget and control measurements are required to verify isolation of the anyonic term. In the revised manuscript we will include additional data from off-critical detuning and lead-only reference devices, together with a full propagation of uncertainties through the Maxwell relation, demonstrating that background shifts cannot produce the observed fractional entropies. revision: yes

Circularity Check

0 steps flagged

No significant circularity; entropy extraction is independent of anyonic interpretation

full rationale

The derivation proceeds by measuring island charge Q(Vg,T) as a function of gate voltage and temperature, then applying the Maxwell relation (∂S/∂Vg)T = −(∂Q/∂T)Vg to obtain the entropy change ΔS at the two- and three-channel Kondo critical points. This thermodynamic step is a direct consequence of the measured charge data and does not presuppose the value of ΔS or the anyonic character. The resulting fractional ΔS values are subsequently compared to independent theoretical predictions (kB ln√2 for Majorana, kB ln((1+√5)/2) for Fibonacci) rather than being fitted or defined to equal those numbers. No self-definitional equations, fitted-input predictions, load-bearing self-citations, or ansatz smuggling appear in the chain; the central claim rests on experimental data plus external theory comparison and is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on experimental charge measurements interpreted through a standard thermodynamic Maxwell relation and compared to existing anyon entropy formulas; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Entropy can be obtained from charge via a thermodynamic Maxwell relation at the Kondo critical points.
    Invoked to convert measured island charge into entropy without additional fitting.

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Reference graph

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