Switching noise as a probe of statistics in the fractional quantum Hall effect

Ady Stern; Eytan Grosfeld; Steven H. Simon

REVIEW

Switching noise as a probe of statistics in the fractional quantum Hall effect

Eytan Grosfeld, Steven H. Simon, Ady Stern

Mesoscale and Nanoscale Physics · Submitted 27 Feb 2006 · 10 references

Pith's verdict: A clever noise-based proposal for FQH statistics with a real even-odd effect at ν=5/2, but the odd-n0 prediction rests on an unstated fast-collapse assumption and a probability assignment that looks reversed.

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arXiv IDcond-mat/0602634 (version 1)

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CitationAdy Stern, Eytan Grosfeld, Steven H. Simon. "Switching noise as a probe of statistics in the fractional quantum Hall effect." arXiv:cond-mat/0602634 (2006). https://pith.science/paper/OQIJI3C2

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Editor's note

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Colleague — this is a genuine and clever proposal: use the telegraph noise from thermally fluctuating quasiparticle number in a two-point-contact FQH interferometer to read out statistics. The abelian case is straightforward Markov-chain noise, and the extension to ν=5/2 with the even-odd effect is clearly new. The paper does well to distinguish this switching noise from shot noise and 1/f noise, and the idea that the noise spectrum carries statistical information is solid.

The machinery is standard, and for ν=1/q and even n0 at 5/2 the derivation checks out. The odd-n0 case, however, has two issues. First, the probability assignment in the text appears reversed: when n0 is odd, the no-interference conductance G0 should come with probability P_out, not P_in, and the two interfering values with P_in/2 each. As written, Eq. (9) doesn't follow from the stated assignments. That's a fixable typo, but it needs correcting before publication.

Second, the non-abelian signature rests on the assumption that a conductance measurement collapses the island into an eigenstate of Γ, and that this collapse is fast compared to quasiparticle hops. The paper imports this from [4,9] without deriving it or bounding the relevant rates. If the measurement only partially dephases the r degree of freedom, the classical three-state Markov chain is not justified and Eq. (10) (S3=0) could acquire corrections. The stress-test note is on target: the proposal needs either a derivation of the projective regime from the measurement setup or an explicit statement that the predicted even-odd noise requires that regime. This is a load-bearing risk, not a cosmetic omission.

There's also a minor point: the large-fluctuation case is sketched quickly, and the claim that S3 vanishes there would benefit from a few lines of justification.

Overall, the central idea is sound, the abelian analysis is solid, and the even-odd effect for 5/2 is a valuable new prediction worth testing. The odd-n0 section needs revision, and the measurement-collapse assumption needs to be stated as a condition rather than silently assumed. I'd send it to a serious referee; with those fixes it could be a well-cited paper. For me, I'd want to see the revised version before relying on the 5/2 odd-n0 noise formula.

Referee report

Major objections (2)

[Section 'We now analyze the way this parameter varies...' and the review of the ν=5/2 state] The odd-n0 prediction, including the three-state Markov chain in Fig. 2b and Eqs. (9)-(10), rests on the assertion that 'the measurement of the conductance collapses the system into an eigenstate of Γ [9].' The manuscript does not derive this projective limit from the two-point-contact measurement dynamics, nor does it state the required separation of time scales between the measurement-induced collapse rate and the quasiparticle hopping rate α. If the conductance measurement only partially dephases the r degree of freedom, or if collapse is slower than α, the system may retain coherence between Γ eigenstates, and the classical Markov reduction to three states is not justified. In that case Eq. (10) could acquire nonzero corrections and the parity-dependent denominator in Eq. (9) would not follow. Because this is the central non-Abelian signature, the paper should either derive the projective regime from the measurement setup (tunneling amplitudes, bias voltage, and noise bandwidth) or explicitly state that the prediction applies only in that regime and give a bound on the collapse rate.
[Eq. (9) and the surrounding text (three-state Markov chain for n0 odd)] There is an inconsistency in the probability assignments. The text says: 'one value, with a probability Pin, is G0. The other two values, with equal probabilities Pout/2, are G0[1 ± β cos(φ)].' For n0 odd, the quasiparticle inside the island makes nis even, producing the split conductance values, so the split states should each have probability Pin/2 and the G0 state should have probability Pout. The printed sentence has Pin and Pout reversed. In addition, Eq. (9) as written appears to lack a factor of (1−Pin) in the numerator. A direct calculation from the three-state Markov chain with rates α_in and α_out and detailed balance gives S2^b(ω) = 2α Pin(1−Pin) δG^2 / ((1−Pin)^2 α^2 + ω^2) when α is defined as α_in + α_out and α_out = α(1−Pin). Please reconcile Eq. (9) with the stated rates and probabilities, and correct the probability assignment in the text.

Minor objections (3)

[Section 'We start with the ν=1/q state'] The phrase 'hopping rates αP in and αP out' is unclear; this appears to be a typesetting artifact for α_in and α_out. Please use clear subscripts throughout.
[Reference [9]] The citation for the measurement-collapse statement, Overbosch and Bais, may not be the most direct reference for the projective measurement of Γ in this interferometric context; consider citing a quantum-measurement or FQHE interferometry reference that explicitly discusses the Zeno limit.
[Concluding discussion of 1/f noise] The claim that tuning to ν=2 makes the statistical noise disappear while other noise sources remain could be strengthened by a short explanation of why the switching mechanism is absent at integer filling in this geometry.

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The paper

The paper claims

A two-point-contact Hall bar can reveal quasiparticle statistics through current noise.

The reading

The paper proposes that in a fractional quantum Hall bar with two point contacts, thermal hopping of quasiparticles into and out of the island between the contacts makes the backscattered conductance switch in time, and the resulting telegraph current noise carries information about quasiparticle statistics. For abelian $\nu=1/q$ states, a single hopping quasiparticle gives a Lorentzian second cumulant and a nonvanishing third cumulant, both set by the conductance difference between the two states. For the non-abelian $\nu=5/2$ state, the noise acquires a parity effect: when the average island occupancy $n_0$ is even the conductance switches between two values, but when $n_0$ is odd it switches among three values and the three-time correlation vanishes. The authors argue this even-odd distinction is a direct consequence of the non-abelian degeneracy, and they show how the statistical noise can be separated from shot noise and $1/f$ noise.

What carries the argument

The argument runs on a Markov chain over island occupancy states. Its transition-rate matrix $M$ determines the conditional probabilities through matrix exponentials $e^{Mt}$, from which the second and third cumulants of the backscattered current are written as path sums over conductance values. The statistics enter through the operator $\Gamma=\prod_{i=1}^{n_{\mathrm{is}}}(\gamma_{\mathrm{edge}}\gamma_i)$, built from the Majorana operators of the island quasiparticles and the back-reflected quasiparticle; its two eigenvalues give the two possible interference phases $r$, and the commutator $[\Gamma,\gamma_{n_0+1}\gamma_{\mathrm{edge}}\Gamma]$ decides whether a hop preserves or randomizes $r$. The combination of the Markov-chain cumulant formulas and this commutator is what turns a generic telegraph noise calculation into a statistics measurement.

Extended reading notes

Core claim

For a fixed configuration of localized quasiparticles, the interference conductance of the $\nu=5/2$ state is $G_0$ for odd island occupancy and $G_0[1+\beta\cos(\varphi_0+\pi n_{\mathrm{is}}/4+r\pi)]$ for even occupancy, where $r$ is a binary phase that is set by a conductance measurement and stays fixed while the occupancy does not change. The paper's central discovery is that when thermal fluctuations make a single quasiparticle hop in and out, the fate of $r$ depends on the parity of the starting occupancy $n_0$. If $n_0$ is even, the back-reflection operators $\gamma_{n_0+1}\gamma_{\mathrm{edge}}\Gamma$ commute with $\Gamma=\prod_{i=1}^{n_{\mathrm{is}}}(\gamma_{\mathrm{edge}}\gamma_i)$, so $r$ survives and the conductance performs a two-state Markov switch described by the same formulas as the abelian case. If $n_0$ is odd, the commutator does not vanish, $r$ is randomized, and the conductance switches among three values, making the three-time correlator vanish. This parity-dependent noise is the predicted experimental signature of non-abelian statistics.

Load-bearing premise

The even-odd noise signature rests on the assumption that a conductance measurement collapses the island's ground state into an eigenstate of the interference operator $\Gamma$, and that the commutator with the quasiparticle-hop operator decides whether the interference phase stays fixed; if actual back-reflection does not collapse the state in that way, the predicted parity effect fails.

Editorial extensions

If this is right

For $\nu=1/q$, a single hopping quasiparticle yields a Lorentzian second cumulant with width set by the hopping rate and a nonzero third cumulant whenever the occupation probability is not balanced; the statistical noise dominates shot noise when the visibility is near unity and the hopping rate is below the quasiparticle traversal rate.
For $\nu=5/2$ with even $n_0$, the conductance switches between two levels and the cumulants are the abelian formulas (7)-(8) with $\delta G=G_0\beta\cos\varphi$.
For $\nu=5/2$ with odd $n_0$, the conductance switches among three levels, the second cumulant is Eq. (9), and the three-time correlation vanishes.
When fluctuations of $n_{\mathrm{is}}$ are large, the abelian $\nu=1/q$ spectrum has its peak shifted by $\alpha[1-\cos(2\pi/q)]$, while the $\nu=5/2$ spectrum has half the amplitude of the two-level case and a vanishing third cumulant.
Because the statistical noise scales with the square of the average backscattered current and needs both point contacts, it can be separated from shot noise, and tuning to an integer filling factor removes it, isolating it from generic $1/f$ noise.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension the authors do not spell out is to run the same Markov-chain calculation for other proposed non-abelian states such as $\nu=12/5$; the structure of the corresponding $\Gamma$ operator would predict whether an even-odd noise signature survives there.
If the collapse assumption is correct, the even-odd noise asymmetry offers a practical diagnostic for non-abelian statistics without performing a full braiding protocol: compare the two- and three-time correlators at tuned parities of the island occupancy.
A side-gate sweep that changes $n_0$ across the parity boundary should show the predicted switch in the three-level structure; mapping that boundary would test the commutator mechanism directly.

Original abstract

We propose an experiment to probe the unconventional quantum statistics of quasi-particles in fractional quantum Hall states by measurement of current noise. The geometry we consider is that of a Hall bar where two quantum point contacts introduce two interfering amplitudes for back-scattering. Thermal fluctuations of the number of quasi-particles enclosed between the two point contacts introduce current noise, which reflects the statistics of the quasi-particles. We analyze abelian $\nu=1/q$ states and the non-abelian $\nu=5/2$ state.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,564 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.06 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We propose an experiment to probe the unconventional quantum statistics of quasi-particles in fractional quantum Hall states by measurement of current noise. The geometry we consider is that of a Hall bar where two quantum point contacts introduce two interfering amplitudes for back-scattering. Thermal fluctuations of the number of quasi-particles enclosed between the two point contacts introduce current noise, which reflects the statistics of the quasi-particles. We analyze abelian $\nu=1/q$ states and the non-abelian $\nu=5/2$ state.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We propose an experiment to probe the unconventional quantum statistics of quasi-particles in fractional quantum Hall states by measurement of current noise.	contribution 0.96	0.34	0.21	asserted 0.65
s1	The geometry we consider is that of a Hall bar where two quantum point contacts introduce two interfering amplitudes for back-scattering.	method 0.98	0.28	0.03	asserted 0.99
s2	Thermal fluctuations of the number of quasi-particles enclosed between the two point contacts introduce current noise, which reflects the statistics of the quasi-particles.	method 0.90	0.24	0.27	asserted 1.00
s3	We analyze abelian $\nu=1/q$ states and the non-abelian $\nu=5/2$ state.	contribution 0.91	0.79	0.08	asserted 0.86

What kind of paper the classifier reports this is: method or tool 0.63 · evidence from argument 0.54.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
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R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: Experimental setup shared by recently proposed in

**Figure 2.** FIG. 2: The state space of the Markov chains considered in

Figures reproduced from arXiv:cond-mat/0602634 by the authors.

References

Works cited (10)

[9]

B.J. Overbosch, F.A. Bais, Phys. Rev. A 64, 062107 (2001)

View on Pith 2001
[1]

G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991)

View on Pith 1991
[2]

C. de C. Chamon, D. E. Freed, S. A. Kivelson, et al., Phys. Rev. B 55, 2331-2343 (1997)

View on Pith 1997
[3]

S. Das Sarma, M. Freedman, and C. Nayak, Phys. Rev. Lett. 94, 166802 (2005)

View on Pith 2005
[4]

A. Stern and B. Halperin, cond-mat/0508447

View on Pith Pith review arXiv
[5]

P. Bonderson, A. Kitaev, and K. Shtengel, cond-mat/0508616

View on Pith Pith review arXiv
[6]

P. Bonderson, K. Shtengel, J.K. Slingerland, cond-mat/0601242

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[7]

S.B. Chung, M.Stone, cond-mat/0601594

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[8]

N. Read and D. Green, Phys. Rev. B 61, 10267 (2000)

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[10]

Y. Imry, Introduction to Mesoscopic Physics, 2nd ed., Oxford, New York, 2002

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [9]

B.J. Overbosch, F.A. Bais, Phys. Rev. A 64, 062107 (2001)

View on Pith 2001

[2] [1]

G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991)

View on Pith 1991

[3] [2]

C. de C. Chamon, D. E. Freed, S. A. Kivelson, et al., Phys. Rev. B 55, 2331-2343 (1997)

View on Pith 1997

[4] [3]

S. Das Sarma, M. Freedman, and C. Nayak, Phys. Rev. Lett. 94, 166802 (2005)

View on Pith 2005

[5] [4]

A. Stern and B. Halperin, cond-mat/0508447

View on Pith Pith review arXiv

[6] [5]

P. Bonderson, A. Kitaev, and K. Shtengel, cond-mat/0508616

View on Pith Pith review arXiv

[7] [6]

P. Bonderson, K. Shtengel, J.K. Slingerland, cond-mat/0601242

View on Pith Pith review arXiv

[8] [7]

S.B. Chung, M.Stone, cond-mat/0601594

View on Pith Pith review arXiv

[9] [8]

N. Read and D. Green, Phys. Rev. B 61, 10267 (2000)

View on Pith 2000

[10] [10]

Y. Imry, Introduction to Mesoscopic Physics, 2nd ed., Oxford, New York, 2002

View on Pith 2002