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arxiv: 2510.08171 · v1 · submitted 2025-10-09 · ❄️ cond-mat.supr-con · cond-mat.mes-hall

Odd-frequency Pairing in Josephson Junctions Coupled by Magnetic Textures

Ignacio Sardinero , Jorge Cayao , Rub\'en Seoane Souto , Pablo Burset This is my paper

Pith reviewed 2026-05-18 08:35 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.mes-hall
keywords odd-frequency superconductivityMajorana bound statesJosephson junctionsmagnetic texturestopological superconductivitytriplet pairingGreen function
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The pith

Josephson junctions with magnetic textures host odd-frequency triplet pairing alongside Majorana states in the topological phase.

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

The paper examines Josephson junctions connected through magnetic textures using a tight-binding Green function approach. It finds that in the topologically trivial regime even-frequency singlet pairing dominates, but the topological phase features coexistence of Majorana bound states and robust odd-frequency equal-spin triplet pairing at the interface edges. This odd-frequency pairing shows a divergent 1/ω behavior at zero frequency when the Majorana states are decoupled, a feature tied directly to their self-conjugation property. The work positions these junctions as a controllable platform where the odd-frequency correlations can be tuned via geometry, phase difference, and magnetic profiles.

Core claim

In the topological phase the system exhibits coexistence of Majorana bound states and robust odd-frequency equal-spin triplet pairing at the interface edges, with the polarized triplets displaying a divergent 1/ω behavior when the Majorana states are decoupled, intrinsically connected to their self-conjugation property.

What carries the argument

Tight-binding Green function calculation of induced pair correlations in Josephson junctions coupled by magnetic textures.

Load-bearing premise

The tight-binding model with chosen magnetic texture profiles and superconducting pairing terms accurately captures the low-energy physics of real devices without significant effects from disorder, interface roughness, or higher-order interactions omitted from the Hamiltonian.

What would settle it

Measurement of the frequency dependence of the odd-frequency equal-spin triplet correlations showing or failing to show a 1/ω divergence exactly when the Majorana states are decoupled from each other.

Figures

Figures reproduced from arXiv: 2510.08171 by Ignacio Sardinero, Jorge Cayao, Pablo Burset, Rub\'en Seoane Souto.

Figure 1
Figure 1. Figure 1: FIG. 1. Josephson junction coupled by a magnetic texture. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Different setup configurations: (a,b) Uninterrupted [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Interface density of states and pairing. (a,b) Total [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Wide junction in the topological regime. (a) LDoS along the interface as a function of the energy. The Majorana edge [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Narrow junction in the topological regime. (a) LDoS along the interface as a function of the energy. The wave function [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Topological phase diagram. (a) Zero-energy total DoS at the interface as a function of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Phase bias effect on Majorana hybridization. (a,b,c) [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Two topological segments separated by a nonmagnetic region. (a,b) LDoS as a function of the energy for [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Phase stabilization of inner Majorana modes. (a) [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
read the original abstract

Josephson junctions coupled through magnetic textures provide a controllable platform for odd-frequency superconductivity and Majorana physics. Within a tight-binding Green function framework, induced pair correlations and spectral properties are analyzed under various magnetic and geometric conditions. When the junction is in the topologically trivial regime, even-frequency singlet pairing is dominant, whereas the topological phase is characterized by the coexistence of Majorana bound states and robust odd-frequency equal-spin triplet pairing at the interface edges. The odd-frequency polarized triplets reveal a divergent $1/\omega$ behavior when the Majorana states are decoupled, which is intrinsically connected to their self-conjugation property. The zero-frequency divergence evolves into shifted resonances and linear low-frequency behavior once hybridization occurs. A nonmagnetic interruption in the texture separates the topological superconductor into two topological segments and generates additional inner Majorana modes. When the nonmagnetic barrier is comparable to the inner Majorana states localization length, they hybridize and modify their associated odd-frequency triplet pairing, while the outer edge modes preserve their self-conjugated nature. Tuning the superconducting phase difference further controls the onset of the topological regime and the stability of localized Majorana states. The results highlight the central role of odd-frequency triplet correlations as a probe of topological superconductivity in magnetically engineered Josephson junctions.

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

2 major / 3 minor

Summary. The manuscript uses a tight-binding Green-function formalism to study Josephson junctions coupled by magnetic textures. It reports that the topologically trivial regime is dominated by even-frequency singlet pairing, while the topological regime exhibits coexistence of Majorana bound states with robust odd-frequency equal-spin triplet pairing localized at the interface edges. The polarized triplet component is shown to display a 1/ω divergence when the Majorana states are decoupled, which the authors link to the self-conjugate property of the zero modes; hybridization converts this into shifted resonances and linear low-frequency behavior. Additional results address the effect of a nonmagnetic interruption that creates inner Majorana modes and the tuning of the superconducting phase difference.

Significance. If the numerical observations hold under systematic checks, the work supplies a concrete microscopic platform in which odd-frequency triplet correlations serve as a diagnostic of topological superconductivity engineered by magnetic textures. The reported connection between the self-conjugation of decoupled Majoranas and the 1/ω singularity in the anomalous Green function is a clear, falsifiable prediction that could be tested in devices with controllable textures. The ability to separate outer and inner Majorana modes via a nonmagnetic barrier and to tune the topological regime with phase difference adds practical value for future experiments.

major comments (2)
  1. [§4] §4 (topological regime results): the statement that odd-frequency equal-spin triplet pairing is 'robust' at the interface edges is supported only by selected parameter sets; a quantitative scan over magnetic texture strength (a free parameter) and junction length is needed to establish that the 1/ω divergence survives generic perturbations rather than appearing only for finely tuned profiles.
  2. [Methods] Methods section (Green-function implementation): the frequency discretization, imaginary broadening η, and finite-size convergence criteria used to extract the 1/ω divergence and the resonance shift upon hybridization are not reported. Without these, it is impossible to judge whether the reported low-frequency behavior is numerically stable or an artifact of the cutoff.
minor comments (3)
  1. [Figures] Figure captions should explicitly state the value of the superconducting phase difference Δφ and the magnetic texture parameters used for each panel.
  2. [Methods] The decomposition of the anomalous Green function into even- and odd-frequency components is introduced without an explicit formula; adding the standard Matsubara-frequency parity definitions would aid readability.
  3. [Introduction] A brief comparison to existing literature on odd-frequency pairing in Rashba nanowires or helical magnetic chains would help situate the novelty of the magnetic-texture geometry.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for the constructive comments, which will help improve the clarity and robustness of our results. We address each major comment below and will incorporate the suggested revisions.

read point-by-point responses

  1. Referee: [§4] §4 (topological regime results): the statement that odd-frequency equal-spin triplet pairing is 'robust' at the interface edges is supported only by selected parameter sets; a quantitative scan over magnetic texture strength (a free parameter) and junction length is needed to establish that the 1/ω divergence survives generic perturbations rather than appearing only for finely tuned profiles.

    Authors: We agree that demonstrating robustness requires more than representative cases. In the revised manuscript we will add a systematic parameter scan over magnetic texture strength (exchange field amplitude J) and junction length L, confirming that the 1/ω divergence in the odd-frequency equal-spin triplet component persists throughout the topological regime whenever the Majorana modes remain decoupled. These results will be shown in an updated Figure 4 together with a brief discussion of the topological phase boundary. revision: yes

  2. Referee: [Methods] Methods section (Green-function implementation): the frequency discretization, imaginary broadening η, and finite-size convergence criteria used to extract the 1/ω divergence and the resonance shift upon hybridization are not reported. Without these, it is impossible to judge whether the reported low-frequency behavior is numerically stable or an artifact of the cutoff.

    Authors: We thank the referee for highlighting this omission. The calculations employed a uniform frequency grid of 2000 points spanning [-20Δ, 20Δ], an imaginary broadening η = 0.001Δ, and system sizes up to 300 lattice sites; convergence was verified by comparing results for N = 150, 200, and 300 sites, with the low-frequency features remaining unchanged beyond N = 200. In the revised manuscript we will insert a new paragraph in the Methods section that explicitly states these numerical parameters and the convergence tests performed. revision: yes

Circularity Check

0 steps flagged

No significant circularity; numerical outputs from microscopic model

full rationale

The paper computes anomalous Green functions and spectral properties directly from a tight-binding Hamiltonian with prescribed magnetic textures and superconducting terms. The reported coexistence of Majorana states with odd-frequency triplet pairing, the 1/ω divergence in the decoupled limit, and its evolution upon hybridization are explicit numerical results of the Green-function solution rather than fitted parameters, self-definitions, or load-bearing self-citations. The connection to self-conjugation is a standard property of zero modes invoked to interpret the computed divergence, not a circular premise that defines the output. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claims rest on a standard microscopic model whose parameters are chosen to realize topological and trivial regimes; no new particles or forces are postulated.

free parameters (2)
  • magnetic texture strength and profile
    Amplitude and spatial variation of the magnetic field are selected to drive the system across the topological transition.
  • superconducting phase difference
    Phase bias across the junction is varied to control the onset of the topological regime.
axioms (1)
  • domain assumption The low-energy physics is captured by a tight-binding Hamiltonian containing kinetic hopping, superconducting pairing, and Zeeman-like magnetic terms.
    Standard modeling assumption for mesoscopic Josephson junctions in the presence of magnetic textures.

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