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REVIEW 2 major objections 4 minor 34 references

In a number-conserving gapless Luttinger liquid of spinless fermions with short-range attraction, the two-point correlator revives from edge to edge with a sign fixed by fermion-number parity, giving a sharp diagnostic of Majorana edge phys

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2026-07-10 22:03 UTC pith:667GN6CX

load-bearing objection Solid DMRG evidence for a parity-dependent edge-to-edge correlator revival in the attractive t-V Luttinger liquid, with a usable cold-atom protocol; the Majorana label is interpretive but the numerics hold. the 2 major comments →

arxiv 2607.06751 v1 pith:667GN6CX submitted 2026-07-07 cond-mat.quant-gas cond-mat.str-elquant-ph

Majorana physics in a Luttinger liquid with attractive interactions

Francesco Debortoli , Nitya Cuzzuol , Luca Barbiero , Fabian Grusdt This is my paper
classification cond-mat.quant-gas cond-mat.str-elquant-ph
keywords Majorana edge modesLuttinger liquidnumber-conserving modelsattractive interactionstwo-point correlator revivalultracold dipolar gasesbeam-splitter interferometryparticle-hole ansatz
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Majorana edge modes are usually discussed in gapped, particle-number-nonconserving models such as the Kitaev chain. This paper claims that clear signatures of the same edge physics survive in the conceptually simplest number-conserving, gapless setting: a one-dimensional chain of spinless fermions with nearest-neighbor attraction, still inside the Luttinger-liquid phase. The diagnostic is ordinary two-point correlators: when one operator sits at one edge, the correlator decays into the bulk yet revives to a finite value at the opposite edge; the sign of that revival flips with the parity of the total fermion number. The revival remains finite in the thermodynamic limit, appears at other fillings, and is still present in low-lying excited states. A simple particle-hole ansatz that relates the even- and odd-parity ground states works across a wide range of interaction strengths, interpolating between free fermions and a strongly interacting Majorana regime. Because the model is number-conserving and can be realized with dipolar atoms or molecules, the authors also sketch a beam-splitter interferometry protocol that would measure the edge-to-edge correlator directly.

Core claim

Signatures of Majorana edge physics persist in a number-conserving, gapless Luttinger liquid of spinless fermions with short-range attractive interactions. The two-point correlator evaluated on the ground state (and on low-lying excited states) decays into the bulk but revives at the opposite edge with a sign fixed by fermion-number parity; the revival is thermodynamically robust and is captured by a particle-hole ansatz that interpolates between free-fermion and strongly interacting limits.

What carries the argument

The edge-to-edge revival of the normal two-point correlator ⟨c†_1 c_j⟩ (and of ⟨c†_d c_{L+1-d}⟩), whose sign tracks fermion parity. This is interpreted, via a particle-hole ansatz for the odd-parity ground state, as the number-conserving analogue of the non-local Majorana bilinear that encodes parity in the Kitaev chain.

Load-bearing premise

That a finite, parity-dependent edge-to-edge revival of the ordinary two-point correlator is enough, by itself, to identify underlying Majorana edge modes rather than a generic correlation effect of strong attraction near the phase-separation point.

What would settle it

Measure or compute the two-point correlator ⟨c†_1 c_L⟩ (or ⟨c†_d c_{L+1-d}⟩) on large open chains of the attractive t-V model at fixed filling as V approaches -2 from above; if the edge revival vanishes in the thermodynamic limit or loses its strict parity dependence, the claimed Majorana diagnostic fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript studies the number-conserving attractive t-V chain of spinless fermions (Eq. 1) in the gapless Luttinger-liquid regime V > −2. Using DMRG, it shows that the ground-state two-point correlator ⟨c†₁ cⱼ⟩ decays into the bulk but revives to a finite value at the opposite edge, with a sign fixed by fermion-number parity. The revival is argued to be thermodynamically robust (Appendix A 1/L extrapolations), present at other fillings (Appendix B ΔCᵢⱼ maps), and preserved under low-lying density excitations (Appendix C bosonization). A particle-hole ansatz (Eq. 5) built from the rank-2 difference matrix ΔC interpolates continuously between edge-localized (strong-V) and free-fermion (weak-V) regimes. An ultracold-molecule/atom horseshoe geometry plus beam-splitter interferometry is proposed to measure the edge-to-edge correlator.

Significance. If the numerical diagnostics hold, the work supplies a concrete, experimentally accessible signature of Majorana-like edge physics inside a short-range, number-conserving, gapless Luttinger liquid—outside the conventional gapped, mean-field Kitaev setting. The combination of finite-size scaling, parity dependence, filling dependence, excited-state robustness, and a simple interpolating ansatz is a solid package. The proposed cold-atom protocol (horseshoe lattice + beam-splitter occupancy) is realistic and falsifiable, and the observation that bulk correlations can be suppressed by tuning near V_c = −2 is a practical experimental advantage. These elements make the paper a useful contribution to the ongoing search for number-conserving Majorana physics.

major comments (2)
  1. The interpretive step that equates the parity-dependent revival of ⟨c†₁ c_L⟩ with underlying Majorana edge modes (via self-adjoint operators γ_L, γ_R satisfying P(N) = −ic ⟨N|γ_L γ_R|N⟩) is imported from Ref. [13] and is not re-derived for the short-range gapless model. While the paper carefully phrases its claim as “signatures,” a short explicit construction or numerical check of the edge operators for Eq. (1) would strengthen the central diagnostic claim and remove residual ambiguity with generic strong-attraction effects near phase separation.
  2. Appendix A reports linear 1/L fits that yield nonzero intercepts (q ≈ 0.020 for V = −1.9, q ≈ 0.013 for V = −1.5). The manuscript should state the range of L used, the goodness-of-fit, and whether higher-order corrections (e.g., 1/L² or oscillatory terms) remain consistent with a finite thermodynamic limit; a single linear fit on a limited window is load-bearing for the “robust in the thermodynamic limit” claim.
minor comments (4)
  1. Fig. 2a panels for weaker V still show residual bulk oscillations; a quantitative definition of “revival magnitude” (e.g., max |⟨c†₁ cⱼ⟩| for j > L/2 minus bulk envelope) would make the V-dependence clearer.
  2. Appendix C assumes continuum bosonization with periodic boundaries; a brief remark that open-boundary edge modes are not spoiled by the long-wavelength modulation would help non-specialist readers.
  3. The experimental section mentions that dipolar 1/r³ interactions yield the same phenomenology; a single supplemental figure or sentence quantifying the revival for a truncated 1/r³ potential would make this claim more concrete.
  4. Notation: the constant c appearing after Eq. (3) is never specified; a short definition or reference would avoid confusion.

Circularity Check

0 steps flagged

No significant circularity: edge-to-edge revival is a direct DMRG observable of the t-V model; Majorana interpretation is imported from independent prior work, not forced by self-definition or fit.

full rationale

The central diagnostic—the parity-dependent revival of ⟨c†_{1} cⱼ⟩—is obtained by exact diagonalization/DMRG of the number-conserving Hamiltonian (1) and is not constructed from any fitted parameter or self-referential definition. Thermodynamic robustness is an ordinary linear extrapolation of finite-size data (App. A); persistence in excited states follows from a standard bosonization modulation that multiplies the ground-state correlator by a long-wavelength factor (App. C, Eq. 21). The particle-hole ansatz (5) and the operators γ_L, γ_R that link revival sign to parity are taken from Ref. [13] (different author set) and are verified a posteriori by high numerical overlaps; they are not derived circularly inside the present manuscript. No uniqueness theorem, self-citation chain, or renaming of a known result carries the load of the claim. The interpretive step that equates the revival with “Majorana edge physics” is therefore an external mapping, not an internal circularity. Score 1 reflects only the ordinary reliance on a prior construction for language, which does not reduce any equation of the paper to its own input.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 1 invented entities

The central numerical claim (revival of ⟨c†_1 c_j⟩) rests on standard many-body assumptions and DMRG practice; the Majorana interpretation additionally imports the number-conserving edge-operator construction of Ref. [13]. No new particles or forces are postulated. Free parameters are ordinary simulation choices (V, L, N), not fitted to force the revival.

free parameters (2)
  • Interaction strength V (simulation scans) = e.g. V = −1.9, −1.7, −1.5, −0.8
    Chosen by hand in the window −1.9 ≲ V ≲ −0.2 to approach the V_c = −2 transition; not fitted to data, but the visibility of the revival depends on proximity to V_c.
  • System size L and filling N = L = 100 (main figures); L = 4n+2 series for scaling
    Finite open chains (L ~ 100, half-filling and other N) used for DMRG; scaling in 1/L is used to extrapolate the revival.
axioms (5)
  • domain assumption For V > −2 the t-V model of spinless fermions is a gapless Luttinger liquid with power-law correlators controlled by K(V); at V_c = −2 it phase-separates (Giamarchi textbook).
    Used throughout to interpret bulk decay and the enhancement of the revival as K → ∞ near V_c.
  • domain assumption In number-conserving models one can construct edge-localized self-adjoint operators γ_L, γ_R such that ground-state parity satisfies P(N) = −ic ⟨N|γ_L γ_R|N⟩, and ΔC_ij has approximately rank two (Thomas-Markarian et al. [13]).
    Load-bearing for the claim that the revival sign is a Majorana signature; invoked after Eq. (3) and in the ansatz section.
  • domain assumption DMRG ground states and correlators for open chains of length L ≲ 100 are sufficiently accurate for the reported revival magnitudes and 1/L extrapolations.
    All numerical figures rest on SyTen DMRG; no bond-dimension or truncation-error data are shown.
  • ad hoc to paper Low-lying excitations of the Luttinger liquid are long-wavelength density modes a†_p |N⟩; continuum bosonization with periodic boundaries still captures the short-distance revival envelope (Appendix C).
    Simplifying assumptions stated a posteriori; used to argue the revival survives in excited states and thermal mixtures.
  • domain assumption Dipolar 1/r^3 interactions in 1D are effectively short-ranged enough that the same revival physics holds (numerically checked, not shown in detail).
    Needed for the experimental proposal with dipolar molecules/atoms.
invented entities (1)
  • Particle-hole ansatz |N+1⟩_ansatz ∝ ∑ α_j c†_j |N⟩ + ∑ β_j c_j |N+2⟩ with α,β from rank-2 ΔC no independent evidence
    purpose: Approximate the odd-parity ground state and interpolate free-fermion and Majorana regimes.
    Construction taken from [13]; here applied and validated by overlap ~0.85 across V. Not a new physical particle; an approximate state ansatz.

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Majorana zero modes are the hallmark of topological superconductivity. In one-dimensional systems, these zero modes are usually introduced in the context of gapped, mean-field models that do not conserve particle number, such as the Kitaev chain. By non-locally encoding a conventional fermion across spatially separated Majorana zero modes, these systems become inherently immune to local decoherence. In this work, we show that signatures of Majorana edge physics persist in a number-conserving, gapless Luttinger liquid of spinless fermions with short-range attractive interactions. We identify the two-point correlator as a sharp diagnostic, revealing an edge-to-edge revival whose sign depends on the fermion-number parity. This revival is robust in the thermodynamic limit, and persists in the excited states of the system and at different fillings. A simple particle-hole ansatz for the ground state of the system with an odd number of fermions captures the physics of the system for a wide range of interaction strengths, interpolating between the free-fermion limit and the strongly interacting Majorana regime. Finally, we propose a concrete protocol to realize this model with ultracold dipolar molecules or atoms in an optical lattice, and to detect the revival via beam-splitter interferometry, opening an experimental route to Majorana physics beyond the conventional gapped-superconductor paradigm.

Figures

Figures reproduced from arXiv: 2607.06751 by Fabian Grusdt, Francesco Debortoli, Luca Barbiero, Nitya Cuzzuol.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: , we display the matrix ∆Cij for four different fill￾ings N/L. Since the two ground states |N⟩ and |N + 1⟩ differ by the parity of the number of fermions, the revival has opposite signs in the two ground states (see Fig. 2b). For this reason, we expect the matrix ∆Cij to have a peak when i and j are located at the opposite edges of the chain. This peak is visible in blue color in [PITH_FULL_IMAGE:figures/… view at source ↗

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