A bosonic matrix product state description of Read-Rezayi states and its application to quasi-hole spins

Alexander Fagerlund; Eddy Ardonne

REVIEW 4 major objections 5 minor 1 cited by

A bosonic matrix product state description of Read-Rezayi states and its application to quasi-hole spins

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For the k=3 Read-Rezayi state, quasi-hole exchange statistics follow from local density profiles alone, without braiding.

desk verdict A genuinely new bosonic MPS construction for k=3 Read-Rezayi that yields a plausible but not airtight demonstration of vanishing Berry phase; the numerics need error bars and a less hand-picked averaging scheme before the central claim can be taken as quantitative. read the letter →

arxiv 2412.14889 v3 pith:MDHYQJ6N submitted 2024-12-19 cond-mat.str-el cond-mat.mes-hallcond-mat.quant-gas

Alexander Fagerlund , Eddy Ardonne This is my paper · ORCID

classification cond-mat.str-elcond-mat.mes-hallcond-mat.quant-gas PACS 73.43.-f

keywords Read-Rezayistatesmatrixproductquasi-holespinsspin-statisticsrelationZ3parafermionsfractionalquantumHalleffectentanglementspectrumBerryphase

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to show that the exchange statistics of quasi-holes in the $k=3$ Read-Rezayi fractional quantum Hall state can be determined from purely local information: the density profile around a single quasi-hole. It develops a purely bosonic matrix product state description of the state, using three free chiral bosons, and computes the density profiles of six quasi-hole types ($\sigma_1$, $\sigma_2$, $\psi_1$, $\psi_2$, $\epsilon$, and the Laughlin hole). From these profiles it extracts each quasi-hole's local spin $J$, and through a spin-statistics relation converts the spins into exchange statistics parameters. The numerically obtained spins converge to the values predicted under the assumption of a vanishing Berry phase, corroborating earlier explicit braiding calculations and supporting the conclusion that holonomy equals monodromy for these quasi-holes. A reader should care because this makes braiding statistics accessible from local density measurements rather than from multi-anyon interference experiments.

What carries the argument

The machinery is a purely bosonic matrix product state built from three free chiral bosons $\varphi,\chi_1,\chi_2$, with distinct vertex operators $V_a,V_b,V_c$ for electrons in the three Read-Rezayi clusters and $H_a,H_b,H_c$ for the basic quasi-holes. A spread-out background charge and an imaginary-time evolution factor produce an exponential suppression of large auxiliary quantum numbers, giving a controlled truncation $P_{\max}$ of the bond spaces. The argument is carried by the spin-statistics identity $\kappa^c_{ab}=J_a+J_b-J^c_{ab} \pmod{1}$, where $J=\int (r^2/2\ell_B^2-1)(\rho_{\mathrm{qh}}-\rho_0)\,d^2r$ is the local quasi-hole spin; the left side is computed independently from the operator product expansion of the minimal $\mathbb{Z}_3$ parafermion fields. Matching the two sides for the six quasi-hole types is what establishes the vanishing Berry phase.

What would settle it

Braiding two $\sigma_1$ quasi-holes around each other using the same bosonic MPS wave functions and accumulating the Berry phase would settle the claim: if the directly obtained phase differs from the monodromy exponent predicted by the $\mathbb{Z}_3$ description, the spin-statistics-derived parameters would not reflect the true exchange statistics. A less expensive check is to push the plateau-averaged spin computation to larger circumferences and cutoffs for the $\psi_2$ hole, whose predicted spin is exactly zero, and test whether the numerical value remains pinned to zero rather than drifting with $P_{\max}$.

Watch

Extended reading notes

Core claim

The central claim is that for the $k=3$ Read-Rezayi state, the exchange statistics of all six types of quasi-holes can be read off from a local quantity, the quasi-hole spin, computed directly from density profiles. The paper constructs a matrix product state for the Read-Rezayi wave function using only three free chiral boson fields, inserts a single quasi-hole of each type at the center of a finite cylinder, and evaluates the density profile $\rho_{\mathrm{qh}}(\tau,x)$ for each case. The local spin is obtained from the integral $J=\int (r^2/2\ell_B^2-1)(\rho_{\mathrm{qh}}-\rho_0)\,d^2r$, with $\rho_0$ the background density, and no assumption is made about the Berry phase. Applying the spin-statistics relation $\kappa^c_{ab}=J_a+J_b-J^c_{ab} \pmod{1}$ gives braiding parameters that match the monodromy exponents of the minimal $\mathbb{Z}_3$ parafermion description; for the fermionic $M=1$ state the six spins converge to $1/5$, $2/5$, $-1/5$, $3/5$, $1/5$, and $0$. The paper takes this agreement as evidence that the Berry phase vanishes, so the exchange statistics is fully contained in the monodromy of the wave functions.

Load-bearing premise

The load-bearing premise is that the spin-statistics relation of ref. [30] applies to non-Abelian Read-Rezayi quasi-holes and that the spin $J$ computed from finite-cylinder density profiles is exactly the topological spin entering that relation, so if either part fails, the inferred braiding parameters and the vanishing-Berry-phase conclusion do not follow.

Editorial extensions

If this is right

Exchange statistics of Read-Rezayi quasi-holes can be obtained from a single quasi-hole's density profile, with no explicit braiding simulation.
The vanishing Berry phase for the $k=3$ Read-Rezayi state is corroborated by local spin data alone, strengthening the monodromy-only description of these anyons.
The purely bosonic MPS formulation reproduces the $\mathbb{Z}_3$ parafermion content of the state, as verified by entanglement-spectrum state counting.
The same local-spin route could be applied to other multi-boson trial states where explicit braiding is computationally prohibitive.

Reading between the lines

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

If the spin-statistics route is robust, Berry-phase conclusions for more complicated parafermionic states, including $k>3$ Read-Rezayi states, could be drawn from single-quasi-hole density measurements, bypassing expensive braiding simulations.
The convergence of plateau-averaged spins with the MPS cutoff could serve as a practical truncation diagnostic, complementing entanglement-spectrum counting as a check that a simulation has captured the topological sector.
The clear separation of predicted spins across sectors (for example $2/5$ for $\sigma_2$ versus $-1/5$ for $\psi_1$) suggests that local density measurements around a pinned quasi-hole could distinguish topological sectors without interferometric braiding.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper delivers what it says on the tin: a purely bosonic MPS formulation for the k=3 Read-Rezayi state, with explicit matrix elements for electrons and quasi-holes, and a direct computation of quasi-hole spins from density profiles. The three-boson construction is a real methodological step up from the existing Z3 parafermion MPS work, and the derivation in sections VII and Appendix A is careful and largely self-contained. The charge computations converge nicely to the CFT values, which gives you some confidence the implementation is right, and the entanglement spectrum check showing the Z3 counting is a sensible validation. I believe the central claim—that the exchange statistics can be read off from local spins via a spin-statistics relation, so the Berry phase vanishes—is very plausible and consistent with what explicit braiding calculations found. The method is also genuinely lighter: no multi-anyon braiding needed, which matters for potential k>4 extensions.

The soft spots are where the reader and the stress-test note point, but I would calibrate them a bit differently. The worry that the spin-statistics relation of ref. [30] is unvalidated for parafermionic anyons is a legitimate caveat, but it is not a circularity: the spins in section XI are computed straight from the MPS density, with no fitting to the predicted values. If the relation were badly wrong, the comparison would not have worked as well as it does. The more concrete problem is that the numerical evidence is presented without error bars, and the plateau averaging intervals in section XI are chosen after inspecting the curves. The convergence plots do show values moving toward the predictions, but the patterns are erratic (the authors themselves note this), and for the (ψ2, 4/5) quasi-hole the predicted spin is exactly zero, which weakens the test. Also, a Moore-Read sanity check would have strengthened the case that the density integral J really is the topological spin. Finally, no code or data is released, which makes it hard to re-examine the plateau choice or to quantify the uncertainty.

Who gets value from this? People working on MPS representations of non-Abelian FQH states, and anyone interested in extracting topological data from local densities. It is a solid paper that deserves a serious referee. My recommendation is to send it to review, but with the request that the authors provide uncertainty estimates, a more principled plateau selection (or at least a sensitivity analysis), and ideally a benchmark on a state with known spin values.

Referee Report

4 major / 5 minor

Summary. The paper develops a purely bosonic matrix product state (MPS) formulation of the k=3 Read-Rezayi quantum Hall state, using three free chiral boson fields, and applies it to compute density profiles, charges, local spins, and entanglement spectra for six types of quasi-holes. The charges converge to the expected CFT values, and the entanglement spectra reproduce the expected Z3 parafermion state counting. The central physical claim is that the quasi-hole spins computed directly from the MPS density profiles agree with the values predicted by the spin-statistics relation (SSR) of Ref. [30] under the assumption of vanishing Berry phase, and that this agreement corroborates the statement that holonomy equals monodromy for these anyons, i.e., that the Berry phase vanishes. The paper argues that exchange statistics can therefore be read off from local density information alone.

Significance. If the central claim is correct, the paper provides a significant methodological advance: it offers a free-boson MPS framework for the full Read-Rezayi series and a route to exchange statistics from local quasi-hole data, avoiding explicit multi-anyon braiding. The derivation of the MPS matrix elements, the time-evolution factor U, and the entanglement-spectrum verification are careful and valuable contributions. The charge results and the Z3 CFT state counting are concrete, reproducible checks that lend credibility to the numerical setup. The significance is tempered, however, by the fact that the main physical conclusion rests on the external SSR and on an identification of the finite-size density integral J with the topological spin, neither of which is independently tested in this work; the reported numerical convergence also lacks error bars and an extrapolation protocol.

major comments (4)

[Section XI, Fig. 5] The claimed convergence of the plateau-averaged spins to the predicted values is not quantitative. The averaging intervals rmax ∈ [30,40], [26,36], [26,30] for L = 20, 22, 24 are selected after inspecting the curves, no error bars or standard deviations are quoted, and no extrapolation in Pmax or L is attempted. Some curves are visibly non-monotonic (for example the σ1, L=20 panel moves away from the prediction between Pmax=8 and Pmax=9, as the text itself notes). To support the statement that the results 'converge' to the predicted values, the authors should provide a systematic plateau-selection criterion, error estimates, and a finite-Pmax extrapolation or at least a convincing scaling argument.
[Section X, Eqs. (68)-(69)] The central inference chain depends on two identifications that are not independently validated in the manuscript: (i) the spin-statistics relation κ^c_ab = J_a + J_b - J_ab mod 1 holds for non-Abelian k=3 Read-Rezayi anyons, and (ii) the quantity J defined in Eq. (69) and evaluated numerically on a finite cylinder equals the topological spin appearing in that relation. The charge convergence shown in Fig. 3 tests only the zeroth moment of the density deviation and does not test the r^2-weighted moment entering J. A direct way to remedy this would be to apply the same MPS-density spin extraction to a state whose Berry phase is known independently, such as the Laughlin state or the Moore-Read state, and to demonstrate that the method reproduces the known spin values; without such a check, the agreement reported for the six RR quasi-holes could be coincidental.
[Sections II and XI, Eqs. (6)-(11) and (73)] For the ψ1, ψ2, and ε quasi-holes, the single-quasi-hole wave functions are constructed by sending other quasi-holes to the edge of the cylinder, and the density profiles are computed from these finite-cylinder states. The edge contamination from those auxiliary quasi-holes affects the density deviation ρ_qh - ρ_0 and hence the integral in Eq. (73), but its magnitude is not quantified. Since the spin predictions for these sectors are exactly the cases where the deviations from the prediction in Fig. 5 are largest (for some L), the authors should estimate the systematic error due to the edge-construction procedure, for example by varying the number of orbitals between the bulk quasi-hole and the edge and checking the stability of J.
[Section XIII, Fig. 8] The choice to use the one-dimensional line integral Eq. (73) rather than the two-dimensional integral Eq. (74) for the spin is justified by the fact that the line integral gives values closer to the expected ones. This post-hoc selection risks confirmation bias, because the same data are used both to select the integration method and to test the prediction. The authors should provide an a priori criterion for choosing the integration method, or show that the line-integral and area-integral results converge to the same value in the limit of large L where the self-interference is negligible.

minor comments (5)

[Section VII, Eq. (52)] The origin of the charge-dependent signs in the quasi-hole matrix elements is explained only verbally; a short derivation or a reference to the counting of same-type electrons in the auxiliary state would improve reproducibility.
[Section XI, Fig. 4] For the ψ2 quasi-hole, the predicted spin is zero and the dotted line coincides with the rmax axis, which makes the comparison harder to read; a slight offset or an inset would help.
[Section IX A] The text says the quasi-hole 'would touch itself' for L ≲ 20, but the plotted profiles are for L=20; a brief explanation of how this touching affects the density shown in Figs. 1-2 would clarify the finite-size effects.
[Appendix A, Eq. (A4)] The statement 'Here, we have used that q0 = 3(3M+2), q1 = 12, q2 = 4 for minor simplifications' could be expanded by one line, since q2 = 4 follows from the rewriting in Eq. (24) and is not an independent convention.
[Throughout] A few typographical and formatting issues remain (e.g., 'n ¨ıvely' in Section VII and inconsistent use of 'in charges' in Section XII); these do not affect the results.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: quasi-hole spins are computed from MPS density integrals independently of the CFT spin predictions, and the external spin-statistics relation cited is parameter-free and does not contain the target Berry-phase claim.

full rationale

The derivation chain is not circular. The predicted spins in Table IV are obtained by combining OPE monodromy exponents from the minimal Z3 CFT with the spin-statistics relation (68) from Ref. [30], under an explicit assumption of zero Berry phase ('The crucial assumption in doing so is that all the statistics is contained in the monodromy'). The MPS computation of the spins uses eq. (73), a direct r^2-weighted integral of the density profiles obtained from the free-boson MPS; no parameter is fitted to Table IV, and the Berry-phase assumption is not used in constructing or evaluating the density. The shared input (quasi-hole charges and vertex-operator content) does not make the r^2 moment of the density equal to the monodromy exponent by construction: the two quantities are related only through the external SSR. Although Ref. [30] shares an author (Ardonne), it is a separately published, parameter-free general relation for FQH systems and does not itself assert the target result (vanishing Berry phase for k=3 Read-Rezayi); per the independence rule it is real external support rather than a self-citation loop. The main caveats—that the plateau intervals in Sec. XI are chosen after inspecting the curves and that no error bars or extrapolation in Pmax and L are given—are correctness/robustness concerns, not circularity. Section X itself flags the 'highly nontrivial assumption' that the Berry phase vanishes, so the conditional nature of the inference is stated rather than hidden.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central computation relies on standard free-boson CFT identities, the known symmetrized form of the Read-Rezayi wave function, a fusion-rule-based identification of quasi-hole operators, and the external spin-statistics theorem. No new physical entities are introduced; the three boson fields are a computational representation of the known state. Numerical hyperparameters Pmax, L, and plateau intervals affect the extracted spin values but are not fitted to the predicted spins.

free parameters (3)

MPS cutoff Pmax = 7 to 12, with maximum 12 used for main results
Truncation of the auxiliary Hilbert space. The spin results are shown to converge as Pmax increases, but Pmax is a numerical convergence parameter, not fitted to the target spins.
Cylinder circumference L = 20, 22, 24 in units of lB
Finite-size choice. The paper studies L in this range because larger L slows MPS convergence and smaller L causes quasi-hole self-interference around the cylinder.
Plateau averaging intervals for spins = [30,40] for L=20, [26,36] for L=22, [26,30] for L=24
Chosen by inspecting where J(rmax) has a plateau for all quasi-hole types. This post hoc choice affects the reported spin values and is not derived from a first-principles criterion.

assumptions (5)

standard math Free-boson CFT correlation identity, eq. (15), and independent charge neutrality for each boson field.
Used throughout sections III and IV to represent Jastrow factors and vertex-operator products as expectation values.
domain assumption The k=3 Read-Rezayi wave function is the symmetrized product form of eq. (4), following the construction of ref. [20].
This form is the basis for assigning three electron types and for constructing the free-boson electron operators in section IV.
domain assumption The identification in table I between Z3 parafermion quasi-hole sectors and products of the bosonic operators Ha, Hb, Hc is correct.
This map fixes which quasi-hole operators are inserted in the MPS for each of the six sectors. It is motivated by fusion rules and wave-function zeros in section V.
domain assumption The spin-statistics relation of ref. [30], eq. (68), applies to non-Abelian Read-Rezayi quasi-holes.
The paper uses this relation to convert the measured spins into braiding parameters. If the relation does not apply in this setting, the conclusion about the Berry phase does not follow.
domain assumption The spin predictions in table IV are derived under the nontrivial assumption that the Berry phase vanishes.
The paper explicitly labels this assumption in section X. The measured spins do not use this assumption, so it functions as a hypothesis being tested rather than an input to the numerical spin values.

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Pith. "Pith review of A bosonic matrix product state description of Read-Rezayi states and its application to quasi-hole spins." pith.science (2026). https://pith.science/paper/MDHYQJ6N

@misc{pith2026241214889,
  author       = {Pith},
  title        = {Pith review of: A bosonic matrix product state description of Read-Rezayi states and its application to quasi-hole spins},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MDHYQJ6N}},
  note         = {Machine review of arXiv:2412.14889}
}

abstract

We study the $k=3$ Read-Rezayi quantum Hall state by means of a purely bosonic matrix product state formulation, which is described in detail. We calculate the density profiles in the presence of bulk quasi-holes of six different types: one for each $\mathbb{Z}_3$ parafermion sector. From the density profiles, we calculate the (local) spins of these quasi-holes. By employing a spin-statistics relation, we obtain the exchange statistics parameters. Our results, which are entirely based on local properties of the quasi-holes, corroborate previous results obtained by explicitly braiding quasi-holes, showing that the exchange statistics can be read off from the monodromy properties of the wave functions, i.e., that the associated Berry phase vanishes. We also discuss the entanglement spectrum, to show that our bosonic matrix product state formulation correctly captures the $\mathbb{Z}_3$ parafermionic structure of the $k=3$ Read-Rezayi states.

Figures

Figures reproduced from arXiv: 2412.14889 by the authors.

**Figure 2.** FIG. 2. Scaled density profiles at the quasi-hole center [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** FIG. 3. Charges [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 5.** Figure 5: fig. 5 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 4.** Figure 4: FIG. 4. Spins [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

**Figure 5.** Figure 5: FIG. 5. Plateau-averaged values for the spins of various different quasi-hole types, against three different cylinder circumferences [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: FIG. 6. The low-lying part of the entanglement spectrum for [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 8.** Figure 8: FIG. 8. Comparison between the line integral method eq. (64) and the double integral method eq. (65) for the charge, and [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Spin fractionalization at the edge of quantum Hall fluids induced by bulk quasiparticles
cond-mat.str-el 2024-12 conditional novelty 6.0 of 10

A straight-edge spin observable is defined for quantum Hall fluids and shown to fractionalize with bulk quasiparticles through a reference-dependent density dipole moment.

Reference graph

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