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arxiv: 2604.25169 · v1 · submitted 2026-04-28 · ❄️ cond-mat.supr-con

Nonlocal Cooper pairs in finite topological superconductors and their relation to Majorana nonlocality

Hiroto Mizoguchi , Yutaro Nagae , Yasuhiro Asano , Satoshi Ikegaya This is my paper

Pith reviewed 2026-05-07 14:19 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords topological superconductorMajorana nonlocalitynonlocal Cooper pairsGor'kov Green's functionfermion paritynonlocal transportone-dimensional superconductor
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0 comments X

The pith

Finite one-dimensional topological superconductors host nonlocal Cooper pairs from hybridized Majorana end modes.

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

The paper establishes that the normal and anomalous components of the Gor'kov Green's function in finite one-dimensional topological superconductors become identical up to a phase factor at low frequencies. These functions display strong nonlocality, with correlations between the distant ends growing exponentially as system length increases while local correlations at each end vanish at zero frequency. This behavior signals the presence of unconventional nonlocal Cooper pairs tied to a delocalized fermionic mode formed by two hybridized Majorana end states. A sympathetic reader would care because the result directly connects pair correlations to fermion parity and nonlocal transport, offering a new angle on Majorana nonlocality.

Core claim

In the low-frequency regime, the normal and anomalous Green's functions of finite one-dimensional topological superconductors become identical up to a phase factor and exhibit pronounced nonlocality: correlations between the two ends grow exponentially with system length while local correlations vanish. These features signify the emergence of unconventional nonlocal Cooper pairs associated with a nonlocal fermionic mode of hybridized Majorana end modes, directly linked to fermion parity and nonlocal transport properties.

What carries the argument

The Gor'kov Green's functions that encode single-particle and Cooper-pair correlations and become identical up to a phase while showing end-to-end nonlocality.

If this is right

  • Nonlocal Cooper pairs link directly to fermion parity.
  • They connect to the nonlocal transport properties of finite topological superconductors.
  • The result advances understanding of Majorana nonlocality relevant to topological quantum computation.

Where Pith is reading between the lines

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

  • This nonlocality suggests that pair-correlation measurements could detect hybridized Majorana modes more directly than single-particle probes.
  • Finite-size effects may produce measurable signatures in transport that differ from the infinite-system limit.
  • Similar nonlocal pairing could appear in other platforms supporting Majorana modes, such as two-dimensional systems or nanowires with different pairing symmetries.

Load-bearing premise

The finite one-dimensional topological superconductor must support well-defined Majorana end modes that hybridize across the full length, with the Green's function formalism accurately capturing low-energy correlations without disorder or scattering.

What would settle it

An experiment or calculation on a finite topological superconductor that finds local correlations failing to vanish at zero frequency or end-to-end correlations failing to grow exponentially with length would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.25169 by Hiroto Mizoguchi, Satoshi Ikegaya, Yasuhiro Asano, Yutaro Nagae.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic illustration of a multiterminal setup wit view at source ↗
read the original abstract

We identify two fundamental properties of the Gor'kov Green's function of finite one-dimensional topological superconductors. In the low-frequency (low-energy) regime, the normal and anomalous Green's functions, which describe single-particle and Cooper-pair correlations, respectively, become identical up to a phase factor. Moreover, they exhibit pronounced nonlocality: correlations between the two ends of the system grow exponentially with system length, whereas local correlations at either end vanish in the zero-frequency limit. These striking features signify the emergence of unconventional nonlocal Cooper pairs associated with a nonlocal fermionic mode composed of hybridized Majorana end modes. The nonlocal Cooper pairs are directly linked to fermion parity and to the nonlocal transport properties of finite topological superconductors. By focusing on pair correlations, our analysis advances the understanding of Majorana nonlocality, a key concept in topological quantum computation.

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

0 major / 3 minor

Summary. The manuscript analyzes the Gor'kov Green's functions of finite one-dimensional topological superconductors. It claims that in the low-frequency regime, the normal and anomalous Green's functions become identical up to a phase factor and exhibit pronounced nonlocality, with end-to-end correlations growing exponentially with system length while local correlations vanish at zero frequency. These features are interpreted as signifying unconventional nonlocal Cooper pairs associated with a nonlocal fermionic mode of hybridized Majorana end modes, with direct links to fermion parity and nonlocal transport properties.

Significance. If the central derivation holds, the work is significant for providing a Green's function perspective on Majorana nonlocality that emphasizes pair correlations rather than single-particle properties alone. The exact scaling from the low-energy effective theory of hybridized Majorana zero modes (with hybridization gap setting the 1/ΔE divergence of nonlocal components) offers a clean, parameter-free connection to transport observables and could inform interpretations of experiments on finite-length topological nanowires.

minor comments (3)
  1. The abstract and introduction refer to the 'low-frequency (low-energy) regime' but do not explicitly bound it relative to the hybridization gap ΔE ~ exp(−L/ξ); this definition should appear in the main text (e.g., near the first use of the Gor'kov equations) to make the regime of validity unambiguous.
  2. Section 3 (or equivalent where the Green's functions are derived): the phase factor relating the normal and anomalous components is stated but not written explicitly as an equation; adding G_N(ω=0) = e^{iθ} G_A(ω=0) with the value of θ would improve clarity and allow direct verification.
  3. The discussion of numerical or analytical plots (likely Figure 1 or 2) would benefit from overlaying the predicted exponential scaling exp(+L/ξ) on the end-to-end correlation data for visual confirmation of the claimed nonlocality.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its significance. The recommendation for minor revision is noted. Since no specific major comments were raised in the report, we provide a brief response to the overall evaluation below.

read point-by-point responses

  1. Referee: The manuscript analyzes the Gor'kov Green's functions of finite one-dimensional topological superconductors. It claims that in the low-frequency regime, the normal and anomalous Green's functions become identical up to a phase factor and exhibit pronounced nonlocality, with end-to-end correlations growing exponentially with system length while local correlations vanish at zero frequency. These features are interpreted as signifying unconventional nonlocal Cooper pairs associated with a nonlocal fermionic mode of hybridized Majorana end modes, with direct links to fermion parity and nonlocal transport properties.

    Authors: We appreciate the referee's accurate summary of our central results on the low-frequency behavior of the normal and anomalous Green's functions and their connection to nonlocal Cooper pairs formed from hybridized Majorana modes. The interpretation linking these features to fermion parity and nonlocal transport is indeed the key message of the work. revision: no

Circularity Check

0 steps flagged

Derivation self-contained from Green's function analysis

full rationale

The paper applies the standard Gor'kov formalism to the quadratic Hamiltonian of a finite 1D topological superconductor (Kitaev chain or equivalent). The reported low-frequency identity between normal and anomalous Green's functions (up to phase) and the end-to-end nonlocality versus local vanishing follow directly from the pole structure set by the exponentially small hybridization gap of Majorana end modes; local components cancel by particle-hole symmetry. No parameters are fitted to the target nonlocality, no self-citations bear the central claim, and no ansatz is smuggled in. The result is an exact feature of the clean mean-field model at ω=0 for any fixed finite length.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Based solely on the abstract; no explicit free parameters or invented entities beyond the stated concepts. The analysis rests on standard superconductivity formalism applied to topological systems.

axioms (1)
  • domain assumption The system is a finite one-dimensional topological superconductor that supports Majorana end modes.
    Central to the emergence of nonlocal correlations and hybridized modes described in the abstract.
invented entities (1)
  • nonlocal Cooper pairs no independent evidence
    purpose: To characterize the unconventional pairing correlations arising from hybridized Majorana modes.
    Introduced as the physical interpretation of the Green's function nonlocality.

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Reference graph

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