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arxiv: 2512.04649 · v2 · submitted 2025-12-04 · 🪐 quant-ph · cond-mat.str-el· hep-th

Recognition: no theorem link

Probing chiral topological states with permutation defects

Yarden Sheffer , Ruihua Fan , Ady Stern , Erez Berg , Shinsei Ryu
Authors on Pith no claims yet

Pith reviewed 2026-05-17 01:57 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-elhep-th
keywords permutation defectschiral topological phasesentanglement measureschiral central chargebulk wavefunctionbulk-edge correspondenceconformal field theory
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The pith

Permutation defects between wavefunction replicas extract the chiral central charge directly from the bulk.

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

The paper constructs multipartite entanglement measures by applying different permutations to replicas of the ground-state wavefunction in neighboring spatial regions. These create permutation defects whose contributions reveal chirality that is otherwise hidden in the bulk. A field-theoretic calculation shows the measures equal the partition function of a chiral conformal field theory on high-genus surfaces. The construction follows from the requirement that any regularization of the defects must introduce gapless boundary modes. The resulting quantities give the chiral central charge and Hall conductance with only a finite number of replicas, opening them to Monte Carlo sampling and noisy intermediate-scale quantum hardware. Checks on free-fermion and interacting models match the expected values.

Core claim

Applying different permutations to replicas of the ground state in adjacent spatial regions creates permutation defects whose entanglement measures equal the chiral CFT partition function on high-genus Riemann surfaces. This correctly captures the chiral anomaly missed by standard topological field theory prescriptions.

What carries the argument

permutation defects at boundaries between regions with different replica permutations, which introduce gapless boundary modes whose contribution is the chiral CFT partition function

If this is right

  • The chiral central charge becomes extractable from a finite number of wavefunction replicas.
  • The Hall conductance follows from the same finite-replica construction.
  • The measures apply equally to free-fermion and strongly interacting chiral states.
  • Monte Carlo techniques can now compute these chiral quantities.
  • Noisy intermediate-scale quantum devices can implement the extraction.

Where Pith is reading between the lines

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

  • The replica-permutation construction may generalize to probe other bulk anomalies or higher-dimensional topological invariants.
  • Time-dependent or Floquet versions of the same defects could diagnose dynamical chiral properties.
  • The approach offers an entanglement-based diagnostic that could be implemented on quantum simulators without requiring full tomography.

Load-bearing premise

Any regularization at the permutation defects introduces gapless boundary modes whose contribution yields the chiral CFT partition function on high-genus surfaces.

What would settle it

A numerical evaluation of the proposed entanglement measure on the ground state of a chiral p+ip superconductor that yields a value different from the known chiral central charge of 1/2.

Figures

Figures reproduced from arXiv: 2512.04649 by Ady Stern, Erez Berg, Ruihua Fan, Shinsei Ryu, Yarden Sheffer.

Figure 1
Figure 1. Figure 1: FIG. 1. The geometry considered in this work. Three permu [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Permutation defect (black) and the enclosing [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Calculation of the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: ). Consequently, characterizing Σv amounts to constructing an analytic ‘uniformization” map f : C¯ → C¯, which maps the R-sheet cover onto the base sphere, and maps punctures on Σv to those on Yv at z = 0, 1, ∞. The structure of f near the punctures is constrained by the permutation. Specifically, near a pre-image z0 of one of the three reference points, we have f(z) ≈ f(z0) + (z − z0) n, the ramification … view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Illustration of the seams used to calculate [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The covering surfaces Σ [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The covering surfaces Σ [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The covering surface use to calculate [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. (a) Analytical predictions on the angles of [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: shows the phases of Jn, Φr across the transition from the toric code (TC) to the Ising phase of the model. We find that these phases remain zero inside the toric code phase and jump to the predicted values across the transition, in good agreement with the analytical results. B. Chern Insulator We verify our result (14) for the Hall conductance using a U(1) symmetric state, the Chern insulator, which has a… view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. (a) The phase of [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. (a) The relative error [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. The neighborhood of the vertex [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. (a) The POPs Σ [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. The vertex surfaces Σ [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. The geomtry used for the free-fermion numerical [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. The geometry used for the calculation of the LeSME [PITH_FULL_IMAGE:figures/full_fig_p025_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Lifting of paths surrounding the defects [PITH_FULL_IMAGE:figures/full_fig_p027_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. The cover of the Riemann sphere obtained by gluing [PITH_FULL_IMAGE:figures/full_fig_p027_20.png] view at source ↗
read the original abstract

The hallmark of two-dimensional chiral topological phases is the existence of anomalous gapless modes at the spatial boundary. Yet, the manifestation of this edge anomaly within the bulk ground-state wavefunction itself remains only partially understood. In this work, we introduce a family of multipartite entanglement measures that probe chirality directly from the bulk wavefunction. Our construction involves applying different permutations between replicas of the ground state wavefunction in neighboring spatial regions, creating "permutation defects" at the boundaries between these regions. We provide general arguments for the robustness of these measures and develop a field-theoretical framework to compute them systematically. While the standard topological field theory prescription misses the chiral contribution, our method correctly identifies it as the chiral conformal field theory partition function on high-genus Riemann surfaces. This feature is a consequence of the bulk-edge correspondence, which dictates that any regularization of the theory at the permutation defects must introduce gapless boundary modes. We numerically verify our results with both free-fermion and strongly-interacting chiral topological states and find excellent agreement. Our results enable the extraction of the chiral central charge and the Hall conductance using a finite number of wavefunction replicas, making these quantities accessible to Monte-Carlo numerical techniques and noisy intermediate-scale quantum devices.

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

1 major / 1 minor

Summary. The paper introduces multipartite entanglement measures constructed by applying permutations between replicas of the ground-state wavefunction in neighboring spatial regions, thereby creating permutation defects. It develops general arguments and a field-theoretic framework to show that these measures capture the chiral CFT partition function on high-genus Riemann surfaces (via bulk-edge correspondence forcing gapless modes at the defects), even though standard TFT misses the chiral contribution. Numerical checks on free-fermion and strongly interacting chiral topological states are reported to agree with the predictions, enabling extraction of the chiral central charge and Hall conductance from a finite number of replicas.

Significance. If the central identification holds, the work supplies a concrete route to extract chiral invariants directly from bulk wavefunctions, making them accessible to Monte Carlo sampling and NISQ devices. The numerical agreement across both free-fermion and interacting models is a positive feature, and the permutation-defect construction offers a new probe of chirality that standard entanglement measures do not isolate.

major comments (1)
  1. [Field-theoretical framework] The central claim that the permutation-defect measure equals the chiral CFT partition function on high-genus surfaces rests on the assertion that any regularization at the defects introduces gapless boundary modes whose contribution is precisely chiral. The construction applies permutations between replicas in neighboring bulk spatial regions rather than at a physical edge; an explicit derivation showing how this regularization enforces the chiral (as opposed to full) CFT contribution, without extra terms or additional tuning, is required to substantiate the identification.
minor comments (1)
  1. [Numerical results] The abstract and numerical section state 'excellent agreement' but do not report quantitative error bars, system sizes, or the precise lattice regularizations used for the interacting case; adding these details would strengthen the numerical support.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its significance. We address the single major comment below and have revised the manuscript to strengthen the presentation of the field-theoretic framework.

read point-by-point responses

  1. Referee: The central claim that the permutation-defect measure equals the chiral CFT partition function on high-genus surfaces rests on the assertion that any regularization at the defects introduces gapless boundary modes whose contribution is precisely chiral. The construction applies permutations between replicas in neighboring bulk spatial regions rather than at a physical edge; an explicit derivation showing how this regularization enforces the chiral (as opposed to full) CFT contribution, without extra terms or additional tuning, is required to substantiate the identification.

    Authors: We agree that a more explicit derivation of the regularization procedure would improve clarity. In the field-theoretic treatment (Section III), the permutation defects are introduced as interfaces where the replica wavefunctions are glued according to a permutation operator. Because the underlying bulk theory is chiral, the bulk-edge correspondence requires that any regularization of these interfaces (implemented via a short-distance cutoff or infinitesimal gap) must host gapless chiral modes whose anomaly cancels that of the bulk. The permutation gluing projects the boundary theory onto the chiral sector; anti-chiral modes are absent from the start and do not appear. We have added a new subsection (III.C) and Appendix B that explicitly compute the regularized partition function on the high-genus surface obtained by the defect network, confirming that it reduces to the chiral CFT partition function with no additional non-chiral terms or parameter tuning. The numerical results already reported remain unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external bulk-edge principle

full rationale

The paper defines permutation defects via replica permutations in the bulk wavefunction and invokes the standard bulk-edge correspondence (an established external principle in topological phases) to argue that regularization at defects introduces gapless modes yielding the chiral CFT partition function on high-genus surfaces. This is not a self-definitional reduction, fitted input renamed as prediction, or load-bearing self-citation chain; the abstract explicitly attributes the chiral identification to bulk-edge correspondence rather than deriving it from the paper's own equations or prior self-work. Numerical verification on free-fermion and interacting states provides independent external benchmarks. No quoted step reduces the central claim to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the bulk-edge correspondence as the mechanism that supplies gapless modes at defects and on the field-theoretic identification of the resulting quantity with the chiral CFT partition function.

axioms (1)
  • domain assumption Bulk-edge correspondence requires that regularization at permutation defects introduces gapless boundary modes.
    Invoked to explain why the measures capture the chiral contribution missed by standard TFT.

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Forward citations

Cited by 1 Pith paper

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Reference graph

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