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REVIEW 4 major objections 5 minor 2 cited by

Majorana edge modes in isolated wires

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper establishes that in number-conserving, interacting Richardson-Gaudin-Kitaev chains, Majorana zero modes leave sharp fingerprints in the single-particle correlation matrix, not only in mean-field spectra.

desk verdict A useful correlation-based diagnostic and solid DMRG evidence for edge modes in the open RGK chain, but the Majorana identification itself rests on an unvalidated ansatz; worth refereeing. read the letter →

arxiv 2509.00158 v2 pith:G32MFBFZ submitted 2025-08-29 cond-mat.str-el cond-mat.supr-con

Jaden Thomas-Markarian , Kartiek Agarwal , Ivar Martin This is my paper · ORCID
classification cond-mat.str-elcond-mat.supr-con
keywords Majoranazeromodesparticle-numberconservationtopologicalsuperconductivitycorrelationmatrixlong-rangeinteractionsRichardson-Gaudin-Kitaevchaindensity-matrixrenormalizationgroupedge
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

Most theoretical evidence for Majorana zero modes comes from BCS mean-field Hamiltonians that do not conserve particle number, but real wires are finite and isolated, where number conservation matters. This paper asks whether Majorana edge modes survive in a fully interacting, number-conserving model and how one would see them. In the Richardson-Gaudin-Kitaev chain with long-range interactions, the authors find three concordant signatures in open chains: the odd-even parity gap vanishes as 1/L, the single-particle correlation matrix develops large parity-dependent inter-edge entries, and the difference of odd- and even-parity correlation matrices has eigenvalues ±1/2 with edge-localized eigenvectors. These are assembled into a correlation-based method that reconstructs the Majorana operators from ground-state correlations alone, and the resulting MZM parity operator is nearly quantized. The method also applies to short-range interacting models, opening a route to diagnosing MZMs in realistic finite systems.

What carries the argument

The central object is the odd-even correlation difference matrix ΔM = M_o - M_e, where M_e is the ground-state single-particle density matrix in the even sector and M_o is the symmetrized average of neighboring odd sectors. In the topological phase ΔM is low-rank with eigenvalues ≈ ±1/2; the sum and difference of the corresponding eigenvectors give the amplitudes α ± β that define left and right Majorana wavefunctions. The argument is carried by the particle-hole symmetric ansatz of Eq. (5), which asserts that adding or removing a fermion across the parity switch is an equal-weight superposition of creating a particle and creating a Cooper pair, and by the requirement that nonlocal inter-edg

What would settle it

Evaluate the two normalization conditions in Eqs. (6)-(7) with high accuracy at L = 64, 96, 128: if the first quantity drifts away from 2 rather than toward it while the gap still decays, the ansatz connecting ΔM to Majorana wavefunctions is wrong. A simpler check: measure ⟨N|iγ_Rγ_L|N⟩ for L > 48; the claimed ≈ 0.5 × (-1)^N quantization should become sharper, not degrade.

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Extended reading notes

Core claim

The paper claims that Majorana zero modes are a real property of fully interacting Hamiltonians that conserve particle number, not a mean-field artifact. Using the integrable Richardson-Gaudin-Kitaev chain with long-range interactions, the authors identify the topological phase in an open chain by three concordant signatures: the symmetrized odd-even ground-state energy splitting |ΔE| falls off as 1/L; the single-particle correlation matrix develops large, parity-dependent inter-edge entries; and the difference ΔM between odd- and even-parity correlation matrices has two eigenvalues ≈ ±1/2 whose eigenvectors combine into left- and right-edge wavefunctions. To interpret these, they conjecture

Load-bearing premise

The identification of the ΔM eigenmodes as Majorana wavefunctions rests on the conjecture (Eq. 5) that the odd- and even-parity ground states are connected exactly by an equal-weight particle-hole superposition with edge-localized vectors, a relation that is checked numerically but not proven.

Editorial extensions

If this is right

  • In finite, isolated number-conserving wires, Majorana zero modes can be identified from correlation functions rather than from broken-symmetry mean-field Hamiltonians.
  • The odd-even parity gap in the topological phase decays as 1/L, with the MZM contribution isolated by higher discrete derivatives of the ground-state energy.
  • The same correlation-difference method works for short-range interacting models, where topological superconductivity is fragile because of the absence of a bulk excitation gap.
  • Because the Majorana operators contain Cooper-pair creation, their action changes density uniformly by 1/L, so odd and even parity ground states become locally indistinguishable in the thermodynamic limit.
  • The correlation matrix is measurable in cold-atom simulators through quenches, so these signatures are experimentally accessible.

Reading between the lines

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

  • The ΔM eigenvector construction may serve as a model-independent detector of edge zero modes in any finite many-body system, not just integrable chains, since it uses only ground-state correlation matrices.
  • If the no-teleportation equality between normal and anomalous inter-edge correlators is general, it gives a criterion for topological ground-state degeneracy that could be checked without computing energy gaps.
  • For short-range-interacting wires, the 1/L scaling tied to long-range interactions implies MZM signatures may be buried under charging energy at accessible sizes; higher-order discrete derivatives of E0(N) should expose the same parity oscillations there.
  • The operator d† and its Majorana combinations suggest a concrete route to constructing parity-changing operators for braiding in number-conserving simulators, though the delocalized hole piece would require corrections to naive edge-only protocols.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The manuscript studies the Richardson-Gaudin-Kitaev chain with open boundary conditions, a number-conserving, long-range interacting model, using DMRG. The authors report three signatures of Majorana zero-modes: (i) the symmetrized odd-even ground-state energy gap decays as ~1/L in the topological phase (Fig. 1); (ii) the single-particle correlation matrix develops large, parity-dependent inter-edge correlations (Fig. 2), with the odd-even difference matrix ΔM showing two eigenvalues ≈±1/2 whose eigenvectors are edge-localized (Fig. 3); and (iii) a ‘Majorana parity’ operator ⟨N|iγ_Rγ_L|N⟩ is found to be ≈0.5×(−1)^N. To interpret these observations, the authors conjecture a particle-hole symmetric ansatz connecting even- and odd-parity ground states (Eq. 5), from which they derive normalization conditions (Eqs. 6–7) and construct an effective fermionic mode d† (Eq. 8) and Majorana operators Γ_L,Γ_R. The Supplemental Material provides additional tests, including projected-BCS analysis, anomalous correlators, and an estimate of the MZM overlap energy via fourth differences of E_0(N).

Significance. If the central ansatz were rigorously validated, the paper would introduce a practical correlation-based diagnostic for Majorana zero-modes in fully interacting, number-conserving wires—an advance over mean-field approaches and relevant to cold-atom emulators where the correlation matrix is measurable. The direct numerical signatures—gap decay, inter-edge correlations, and spectral structure—are compelling and clearly presented. The paper also honestly discusses limitations (e.g., the anomalously half-integer parity) and provides a detailed supplement. However, the Majorana identification rests on a conjectural ansatz whose numerical validation is incomplete and partly circular, and the effective mode d is never shown to obey canonical fermion algebra. These are load-bearing gaps: without them the results establish exotic nonlocal edge correlations but not the existence of Majorana modes. The work is therefore significant but not yet conclusive.

major comments (4)
  1. [Particle-hole symmetric ansatz, Eq. (5)-(7); SM.E] The ansatz in Eq. (5) is the central theoretical assumption, but it is not derived from the Hamiltonian or from an independent wavefunction construction. The vectors α and β are extracted from the eigenmodes of ΔM, the same data used to validate the ansatz via Eqs. (6)-(7). This circularity is not fatal by itself, but the numerical check in SM.E gives 2.38 instead of 2 (−19% error) and −0.03 instead of 0, with deviations attributed to finite size, particle-number difference, and DMRG truncation without any scaling analysis. The authors should show systematic convergence of these quantities with L and with bond dimension, and, ideally, demonstrate that Eq. (5) reproduces the actual ground states (e.g., by comparing overlaps of the constructed states with the exact DMRG states). Without this, the identification of the ΔM eigenmodes as Majorana wavefunctions is not established.
  2. [MZM operators, Eq. (8) and following] The operator d† in Eq. (8) contains the nonlocal Cooper-pair creation operator C†, and the paper never verifies that d obeys canonical fermionic algebra: {d,d†}=1 and d²=0 on the relevant low-energy subspace. The relations |N+1⟩_o = d†|N⟩_e and |N⟩_e = d|N+1⟩_o are asserted from Eq. (5), but because Eq. (5) itself is unvalidated, this is insufficient. The authors should directly compute the anticommutator and the operator norm of d on the ground states, or at least show that d†d and dd† have the expected expectation values. Similarly, the reported parity ⟨N|iγ_Rγ_L|N⟩ ≈ 0.5×(−1)^N is not the ±1 expected for a Majorana parity operator; the missing half is attributed to an uncomputed anomalous contribution. This missing contribution must be calculated or the quantization claim should be revised.
  3. [No-teleportation argument, around Eqs. (6)-(7)] The argument that nonvanishing inter-edge correlators would imply instantaneous teleportation, and therefore that ΔM and ΔP must have equal amplitudes (M̃=P̃), is heuristic. It is not derived from a rigorous information-theoretic or locality principle, and the equality is not directly measured. The numerical support given in SM.E is the 2.38/−0.03 check, which is too imprecise to confirm the equality. The authors should provide a more quantitative test: e.g., directly compare the long-distance behaviors of ΔM_{1,L} and ΔP_{1,L} and show they converge to the same value as L grows. Without this, the derivation of M̃=1/2, which is central to linking the correlation-matrix eigenvalues to Majorana modes, remains a conjecture.
  4. [Definition of ΔM and ΔP, main text after Fig. 3] In the topological phase the authors define M_o = (M_7+M_9)/2, whereas in the trivial phase M_o = M_9. This asymmetric choice is justified only by producing a low-rank ΔM; however, it could bias the extracted eigenvalues and eigenvectors. Since the ±1/2 eigenvalues and the edge-localized modes are the key objects used to infer Majorana wavefunctions, the authors should test how the spectrum of ΔM depends on alternative prescriptions (e.g., using M_o = (M_{N−1}+M_{N+1})/2 in both phases, or other symmetrizations) and show that the large eigenvalues and their eigenvectors are robust. Otherwise the ‘two large eigenvalues’ could be an artifact of the averaging choice.
minor comments (5)
  1. [Eq. (5), second line] There is a typo: the second superposition uses α_j c_i instead of α_j c_j. Please correct.
  2. [Abstract and main text] The abstract speaks of ‘off-diagonal two-point correlation functions’ while the main results use the single-particle density matrix ⟨c†_i c_j⟩. Please clarify the terminology to avoid confusion with four-fermion correlators.
  3. [SM.E] The numerical values 2.38 and −0.03 are quoted without specifying the DMRG bond dimension or convergence criteria. Please include these details so the deviations can be assessed.
  4. [MZM parity discussion] Calling ⟨N|iγ_Rγ_L|N⟩ ≈ 0.5×(−1)^N ‘nearly quantized’ is misleading, since 0.5×(−1)^N is not a ±1 eigenvalue. Please rephrase, e.g., ‘half-integer valued’ or ‘consistent with a missing anomalous contribution.’
  5. [Eq. (8)] The Cooper-pair creation operator C† is defined in the text, but it would help to state explicitly that it acts in the full Fock space and to specify its commutation relation with c_j (or note that it is nonlocal).

Circularity Check

3 steps flagged · score 6.0 of 10

Majorana identification in open chain is built from the ΔM eigenvectors: Eq. (6) is satisfied by choosing ΔP, and the γ operators are constructed from the fitted α,β; direct gap/correlation signatures remain independent, but the Majorana interpretation is partially circular.

  1. fitted input called prediction [Particle-hole symmetric ansatz, Eqs. (5)-(7); SM.E]
    "Given the observed low-rank of ∆ M and the eigenvalues ±1/2, it is natural to consider the ansatz ∆ M jk = ( βjβk − αjαk) ˜M with ˜M ≈ 1/2. Eq. (6) is then satisfied by choosing ∆P jk = ( βjαk − βkαj) ˜P , with ˜P = 1 /2."

    The vectors α and β are the combinations of the two eigenvectors of the numerically observed ΔM (Fig. 3d). The statement ΔM = (ββ − αα)/2 is just the spectral decomposition of that rank-2 matrix, not a testable prediction. Eq. (6) is then made to hold by choosing ΔP, and the later 'numerical verification' in SM.E (2.38 vs 2) checks a residual of this construction, not an independent consequence of the Hamiltonian. The ansatz is therefore validated with the same data used to define its parameters.

  2. self definitional [MZM operators, Eq. (8) and following; Fig. 3(d)]
    "Due to the bimodal support of d and the special relationship between the vectors α and β, the operators Γ† L = d† + dC † = 1/√2 P j(αj − βj)(c† j + cj C † ) and Γ † R = i(d† − dC † ) = i/√2 P j(αj + βj)(c† j − cj C † ) appear to be localized to a single edge each. Thus, the combinations αj ± βj can be thought of as MZM wavefunctions (shown as green and red lines in Fig. 3d)."

    Γ_L/R and γ_L/R are defined using the same α,β taken from the ΔM eigenvectors, so their edge localization is put in by hand rather than derived. The subsequent 'nearly quantized' parity ⟨N|iγ_Rγ_L|N⟩ ≈ 0.5(−1)^N is an output of this fitted ansatz, not an independent observable, and the paper concedes the value is only half-integer because 'it misses the anomalous contribution of the same magnitude.' Thus the central identification of these modes as Majorana modes is the ansatz restated in operator form, not a consequence of the Hamiltonian.

1 more flagged steps
  1. self citation load bearing [Particle-hole symmetric ansatz, text preceding Eq. (5)]
    "In correspondence with [20], we anticipate αj = αL+1−j and βj = −βL+1−j, which are significant only near the edges."

    Reference [20] is Sajith, Agarwal, and Martin, i.e. the same group as the present paper. This citation supplies the symmetric/antisymmetric ansatz for α and β that is then used to interpret αj±βj as left/right MZM wavefunctions. Since the ansatz is the load-bearing step that turns the correlation-matrix eigenvectors into Majorana modes, and it is justified here by a self-citation rather than by an independent derivation or external theorem, the identification rests on a self-citation chain.

full rationale

The paper is not wholly circular: the 1/L decay of |ΔE| and the parity-dependent inter-edge correlations in M are direct numerical observations, and the number-projected BCS ansatz provides an independent consistency check for the trivial/closed-chain cases. However, the central claim that the open topological chain hosts Majorana edge modes is built on the particle-hole ansatz Eq. (5). The vectors α,β are extracted from the ΔM eigenmodes, the normalization condition Eq. (6) is forced by choosing ΔP, and the MZM operators d, Γ_L, Γ_R, γ_L, γ_R are then constructed from those same fitted vectors. Consequently the reported edge-localized 'Majorana wavefunctions' and the approximate parity quantization are consequences of the construction, not independent predictions. The ansatz's symmetry is also imported from the authors' own prior work [20]. These elements make the Majorana identification partially circular; the direct spectroscopic and correlational signatures would remain even if Eq. (5) were wrong, but they would only establish nonlocal edge correlations, not Majorana modes.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The paper's interpretive framework rests on the prior phase diagram of [1], assumed DMRG convergence, a projected-BCS mean-field picture, and an ad hoc ansatz (Eq. 5). The free parameters are standard Hamiltonian couplings (t1=1, t2=0, g=1,3) and the empirically adopted ΔM eigenvalue (~1/2), which is partially fit to the data.

free parameters (3)
  • g = 1 (topological), 3 (trivial)
    Interaction strength chosen from the phase diagram of [1] to represent topological and trivial phases; not fitted to the target claims.
  • t1, t2 = 1, 0
    Hopping parameters set to the nearest-neighbor-only limit; fixed by model choice.
  • M~ (edge correlation amplitude) = ≈ 1/2
    Adopted from the numerically observed eigenvalues of ΔM; used as input to the ansatz (Eq. 6), so this parameter is partially fit to the data.
assumptions (5)
  • domain assumption RGK model integrability and phase classification (topological vs trivial) from Ortiz et al. [1]
    Used to label g=1 as topological and g=3 as trivial; the paper does not re-derive the phase diagram.
  • domain assumption DMRG converges to the exact ground state within small truncation error
    No bond dimension or truncation error is reported; deviations from expected identities (SM.E) are attributed to DMRG truncation.
  • domain assumption Number-projected BCS ansatz for correlation matrices (SM C)
    The interpretation of the doubly degenerate spectra and the 'pair unbinding' picture relies on this mean-field-like wavefunction.
  • ad hoc to paper Ground-state relation ansatz Eq. (5) with edge-localized α, β
    Conjectured in the main text; no derivation, only approximate numerical checks.
  • ad hoc to paper No-teleportation principle linking normal and anomalous inter-edge correlators
    Used to justify M~=P~=1/2; a physical heuristic, not a mathematical theorem.
invented entities (1)
  • Non-local fermionic mode operator d and edge Majorana operators Γ_L, Γ_R, γ_L, γ_R
    purpose: To represent the parity-switching quasiparticle and the Majorana zero modes in the number-conserving many-body ground states.
    Constructed from the eigenvectors of ΔM; internal numerical checks (quantization of ⟨iγ_Rγ_L⟩ at ~95%/99.9% accuracy) provide evidence, but no external experimental handle is proposed.

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Cite this review

Pith. "Pith review of Majorana edge modes in isolated wires." pith.science (2026). https://pith.science/paper/G32MFBFZ

@misc{pith2026250900158,
  author       = {Pith},
  title        = {Pith review of: Majorana edge modes in isolated wires},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G32MFBFZ}},
  note         = {Machine review of arXiv:2509.00158}
}
read the original abstract

Topological superconductors are believed to host exotic quasiparticle excitations known as Majorana zero-modes (MZMs), with much of the evidence based on BCS mean-field theory. The direct application of mean-field arguments is tenuous in finite, isolated systems relevant in some experiments. Here, we develop a new correlation-based method for identifying MZMs in interacting, number-conserving systems. Using DMRG, we study fermion number-conserving models with long-range interactions, which under periodic boundary conditions exhibit robust topological and non-topological superconductivity, tuned by the strength of interaction [1]. We find evidence that, on the topological side, Majorana edge modes appear in open chains, manifesting as the vanishing of the energy splitting between odd- and even-parity ground states with increasing system size. Additionally, off-diagonal two-point correlation functions show nonlocal, parity-dependent edge effects. These correlations reveal the spatial structure of Majorana modes in the many-body wavefunction. We show that the correlation diagnostic applies broadly, including to short-range interacting models, where topological superconductivity is more fragile due to the absence of a bulk excitation gap.

Figures

Figures reproduced from arXiv: 2509.00158 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Odd-even excitation gap vs system size for open [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Heat maps of the two point correlation matrix [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Spectra of correlation matrices (a) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Even- and odd-parity ground state energies as a func [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Odd-even excitation gap vs system size for periodic [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a) [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The correlation spectrum for the even and odd states [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Eigenvalue-1 and (b) eigenvalue-0 mode (eigen [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. (a) Heat map of ∆ [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]

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Forward citations

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    We discuss the nature of these modes in Sec. D. It turns out that the latter (fully occupied) mode is a direct descendant of the k = 0 mode in a closed system, while the empty mode descends from the k = π mode (see Fig. 8). Zak phase.— The span of the eigenvalues λl differs in...

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.