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arxiv: 2606.10043 · v1 · pith:KVCMTUU7new · submitted 2026-06-08 · ❄️ cond-mat.supr-con

Distribution of Majorana modes in the extended-range Kitaev chain

Pedro B. Widniczck , Gerardo Mart\'inez This is my paper

Pith reviewed 2026-06-27 14:32 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords Kitaev chainMajorana modestopological phasesextended-range interactionswinding numberfermion parityedge states
0
0 comments X

The pith

Extended Kitaev chains develop one distinct topological phase for each coupled neighboring site in truncated range.

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

The paper studies the Kitaev chain when hopping and pairing terms reach beyond nearest neighbors but fall off as power laws. In the version where interactions stop after a finite number of neighbors, the system enters exactly as many separate topological phases as there are sites that interact. An analytical expression is supplied for the winding number that labels these phases and for the locations of the transitions between them. The authors also define a Majorana average position to track where the zero-energy edge modes sit and show that this position moves in step with switches in the ground-state fermion parity.

Core claim

In the truncated-range scenario, there are as many distinct topological phases as the number of coupled neighboring sites. An explicit analytical formulation is provided to evaluate the topological invariant and the phase transitions that emerge in these systems. The spatial distribution of the edge modes, quantified by the Majorana average position, correlates directly with the occupation of edge-to-edge non-local fermion states and with changes in the ground-state fermion parity.

What carries the argument

The topological winding number (preserved under algebraic decay of pairing and hopping) together with the Majorana average position that measures the spatial spread of zero-energy edge modes.

If this is right

  • The count of topological phases rises directly with the number of retained neighbors.
  • Phase boundaries can be located by solving the analytical winding-number expression without diagonalizing the full Hamiltonian.
  • The Majorana average position localizes or delocalizes exactly when the effective edge parity flips.
  • Ground-state fermion parity switches are accompanied by measurable changes in edge-mode occupation numbers.

Where Pith is reading between the lines

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

  • Varying the cutoff range offers a direct experimental knob to set the number of protected edge modes in a nanowire.
  • The parity-position correlation could be used as an alternative signature of topological order when energy spectroscopy is noisy.
  • The same counting and distribution analysis may apply to other one-dimensional models with power-law interactions once the winding number is preserved.

Load-bearing premise

The pairing and hopping terms decay algebraically with distance, and the winding number stays well-defined and unchanged when the range is extended.

What would settle it

A numerical diagonalization of a finite-range extended Kitaev chain that finds a number of topological phases different from the number of coupled neighbors would disprove the central counting claim.

Figures

Figures reproduced from arXiv: 2606.10043 by Gerardo Mart\'inez, Pedro B. Widniczck.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Evolution of the winding vector [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Schematic representation of the Kitaev chain [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Topological phase diagrams of the extended Kitaev [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Colored phase diagram showing the effective parity [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Schematic example of a finite chain with [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Colored diagram for a chain with [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. These plots show a rich structure of the Majorana average position [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The plots in (a) for [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (a) These diagrams show the dependence of (a) the IPR and (b) [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
read the original abstract

The topological properties of the Kitaev chain model with extended-range interactions are investigated, focusing on cases where the topological winding number is preserved. We assume that the pairing and hopping terms decay algebraically in space with exponents $\alpha$ and $\beta$, respectively. We show that in the truncated-range scenario, there are as many distinct topological phases as the number of coupled neighboring sites. In addition, an explicit analytical formulation is provided to evaluate the topological invariant and the phase transitions that emerge in these systems. Besides the analytical description, we introduce a new physical insight into the topological excitations of the ground-state by measuring the spatial distribution of the edge modes with the Majorana average position. Taking the next-nearest neighbor Kitaev chains as a probe, various numerical calculations of Majorana edge states were performed in finite-size clusters to determine the sensitivity of the topological zero energy modes to the parameters of interest. The occupation of edge-to-edge non-local fermion states is computed and defined as an effective parity. Such an effective parity exhibits new interesting features beyond the energetic exchange from the ground-state fermion parity switches, which are related to the distribution of the respective edge modes. Our calculations show a direct correlation between the ground-state fermion parity and the edge occupation numbers, which are translated into localization and delocalization of the Majorana average position.

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 / 2 minor

Summary. The manuscript studies the Kitaev chain with algebraically decaying hopping (exponent β) and pairing (exponent α) terms, restricting attention to parameter regimes in which the standard winding number remains the topological invariant. It claims that, in the truncated-range case, the number of distinct topological phases equals the number of coupled neighboring sites, supplies an explicit analytical expression for the invariant and the associated phase boundaries, and introduces the Majorana average position as a diagnostic of edge-mode localization. Numerical results on finite clusters for the next-nearest-neighbor truncation are used to relate ground-state fermion parity, effective edge parity, and the spatial distribution of the Majorana modes.

Significance. If the preservation of the winding number is placed on a firm footing with explicit bounds on α and β, the work would provide a systematic counting of topological phases in extended-range Kitaev models together with a new, physically motivated observable for Majorana edge-state distribution. The analytical formulation and the parity–localization correlation are potentially useful for both theory and numerics in long-range topological superconductors.

major comments (1)
  1. [Abstract] Abstract (and the central claim): the assertion that the truncated-range model supports exactly as many topological phases as the number of coupled neighbors rests on the preservation of the conventional winding number. No explicit bounds on the decay exponents α and β are supplied. For α, β ≲ 2 the long-range tails are known to alter correlation decay and can invalidate the standard bulk-boundary correspondence; without a stated range of validity the claimed phase count does not necessarily follow from the usual winding-number formula.
minor comments (2)
  1. The definition and normalization of the “Majorana average position” should be stated explicitly (including any averaging procedure over the two Majorana components) so that the numerical results can be reproduced.
  2. The manuscript should clarify whether the effective parity is computed from the full many-body ground state or from the single-particle BdG spectrum, and how finite-size effects are controlled when relating parity switches to the Majorana average position.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to strengthen the statement on the validity of the winding number. We agree that explicit bounds on α and β are required and will revise the manuscript to address this point directly.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the central claim): the assertion that the truncated-range model supports exactly as many topological phases as the number of coupled neighbors rests on the preservation of the conventional winding number. No explicit bounds on the decay exponents α and β are supplied. For α, β ≲ 2 the long-range tails are known to alter correlation decay and can invalidate the standard bulk-boundary correspondence; without a stated range of validity the claimed phase count does not necessarily follow from the usual winding-number formula.

    Authors: We agree that the central claim requires a clear statement of the regime in which the conventional winding number remains the topological invariant. The manuscript already restricts attention to cases where this invariant is preserved, but we did not supply explicit bounds. In the revised version we will (i) add a sentence to the abstract specifying the range of validity and (ii) insert a short paragraph (with citations) in the introduction that recalls the known conditions under which the bulk-boundary correspondence holds for algebraically decaying Kitaev chains (typically α, β > 2). For the truncated-range models that form the core of the work, the interactions are strictly finite-range after truncation, so the standard winding-number formula applies without qualification once the truncation is imposed. We will also clarify that the algebraic exponents α and β enter only through the choice of which neighbors are retained before truncation. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained from standard winding number

full rationale

The paper asserts that the topological winding number remains the invariant under algebraic decay of interactions (with unspecified bounds on α, β) and derives an explicit analytical form for the invariant plus phase count equal to the number of coupled neighbors in the truncated-range case. No equations, self-citations, fitted parameters renamed as predictions, or ansatzes are visible in the provided text that reduce the central claims to their own inputs by construction. The Majorana average position and effective parity are presented as additional numerical insights without load-bearing self-reference. The derivation chain is therefore independent of the patterns that would trigger circularity flags.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The model rests on the algebraic-decay assumption for the interaction terms and introduces one new diagnostic quantity.

axioms (1)
  • domain assumption Pairing and hopping terms decay algebraically in space with exponents α and β
    Explicitly stated as the setup for the extended-range model.
invented entities (1)
  • Majorana average position no independent evidence
    purpose: Measure spatial distribution of edge modes
    Introduced as new physical insight into topological excitations

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