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arxiv: 2512.01266 · v2 · pith:JWOCAN5Jnew · submitted 2025-12-01 · ❄️ cond-mat.mes-hall · cond-mat.supr-con

Topological superconductivity and superconducting diode effect mediated via unconventional magnet and Ising spin-orbit coupling

Amartya Pal , Debashish Mondal , Tanay Nag , Arijit Saha This is my paper

Pith reviewed 2026-05-17 03:37 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.supr-con
keywords topological superconductivitysuperconducting diode effectMajorana zero modesFFLO pairingIsing spin-orbit couplingunconventional magnetic orderone-dimensional chainmean-field superconductivity
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The pith

A 1D model with unconventional magnetic order and Ising spin-orbit coupling supports topological superconductivity in BCS and FFLO states along with a field-free superconducting diode effect.

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

The authors construct a one-dimensional tight-binding model that includes an unconventional magnetic order together with Rashba and Ising spin-orbit couplings. They then add an on-site attractive Hubbard interaction and solve the problem with self-consistent mean-field theory to obtain both conventional BCS pairing and finite-momentum FFLO pairing. Both types of pairing produce a topological superconductor marked by a nontrivial winding number, which in turn protects four zero-energy Majorana modes at the ends of the chain. The FFLO pairing further generates a nonreciprocal supercurrent that appears without any applied magnetic field and reaches a diode efficiency of about 65 percent. Such a setup could help in building devices that combine protected quantum information with efficient current rectification in superconducting circuits.

Core claim

Both the conventional BCS and finite-momentum FFLO pairing states can support topological superconductivity, characterized by a nontrivial winding number, and lead to the emergence of four zero-energy Majorana modes localized at the ends of the 1D chain. The FFLO state further gives rise to an intrinsic field-free SDE, manifested as a nonreciprocal supercurrent and quantified by the diode efficiency η ∼ 65%.

What carries the argument

The combination of unconventional magnetic order with Rashba and Ising spin-orbit couplings in the normal-state Hamiltonian, followed by mean-field treatment of the attractive interaction to form BCS or FFLO superconducting order parameters.

If this is right

  • Topological superconductivity with a nontrivial winding number appears in both BCS and FFLO channels.
  • Four zero-energy Majorana modes localize at each end of the finite 1D chain.
  • The FFLO state produces nonreciprocal supercurrent without requiring an external magnetic field.
  • Diode efficiency reaches approximately 65 percent in the FFLO regime.

Where Pith is reading between the lines

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

  • Materials realizing strong Ising spin-orbit coupling and unconventional magnetism could be engineered to achieve field-free diode effects in superconductors.
  • Testing the robustness of the Majorana modes against disorder or temperature variations would provide a way to check the predictions in real systems.
  • Similar mechanisms might apply in two-dimensional extensions of the model for more complex topological phases.

Load-bearing premise

The self-consistent mean-field treatment of the attractive Hubbard interaction remains valid and captures the topological properties and diode efficiency without significant corrections from fluctuations or beyond-mean-field effects.

What would settle it

Detecting the absence of zero-energy Majorana modes at the chain ends or the lack of nonreciprocal supercurrent in the FFLO state for parameters where the model predicts them would disprove the claims.

Figures

Figures reproduced from arXiv: 2512.01266 by Amartya Pal, Arijit Saha, Debashish Mondal, Tanay Nag.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: (c). Notably, we observe the emergence of maxi￾mum diode efficiency of η ∼ 29% in the JI − JA plane where the system exhibits topological superconductiv￾ity. Until this point, we focuss on only varying JI and JA maintaining other model paramters (λR, m0, µ) = (0.5t, 0.5t, 0) to be fixed. Hence, we allow other model parameters to very in order to obtain the maximum diode efficiency possible in the system. P… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Panel (a): Variation of winding number, [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Variation of real-space energy eigenvalue spectra [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
read the original abstract

We propose a theoretical framework in which a one-dimensional (1D) tight-binding model incorporating unconventional magnetic order together with Rashba and Ising spin-orbit couplings are considered to realize two key phenomena in condensed matter systems: topological superconductivity and the superconducting diode effect (SDE). We first elucidate the underlying band topology of the normal-state Hamiltonian and subsequently introduce an on-site attractive Hubbard interaction. Performing a a self-consistent mean-field analysis, we establish superconducting order parameters in both the conventional Bardeen-Cooper-Schrieffer (BCS) and finite-momentum Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing channels. Intriguingly, both pairing states can support topological superconductivity, characterized by a nontrivial winding number, and lead to the emergence of four zero-energy Majorana modes localized at the ends of the 1D chain. The FFLO state further gives rise to an intrinsic field-free SDE, manifested as a nonreciprocal supercurrent and quantified by the diode efficiency $\eta$. Notably, our model yields a large diode efficiency $\eta \sim 65\%$, highlighting its potential for realising topological superconductivity and highly efficient superconducting devices.

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

3 major / 3 minor

Summary. The manuscript proposes a one-dimensional tight-binding model incorporating unconventional magnetic order together with Rashba and Ising spin-orbit couplings and an on-site attractive Hubbard interaction. A self-consistent mean-field analysis is used to obtain both conventional BCS and finite-momentum FFLO superconducting order parameters. Both states are reported to realize topological superconductivity via a nontrivial winding number, with four zero-energy Majorana modes localized at the chain ends; the FFLO state additionally produces an intrinsic field-free superconducting diode effect quantified by a diode efficiency η ∼ 65%.

Significance. If the mean-field results are robust, the work identifies a route to field-free topological superconductivity and high-efficiency SDE by combining unconventional magnetism with mixed SOC, which could be relevant for Majorana-based devices and superconducting rectifiers. The reported 65% efficiency is quantitatively notable within the model.

major comments (3)
  1. [Mean-field analysis and BdG spectrum] Self-consistent mean-field treatment of the attractive Hubbard term (described in the section introducing the interaction and order parameters): In one dimension the phase stiffness is parametrically small; long-wavelength fluctuations can close or renormalize the mean-field gap that enters the topological invariant. The manuscript performs only the mean-field iteration and subsequent winding-number integral on the uniform or plane-wave ansatz, with no Ginzburg-criterion estimate or comparison to exact diagonalization supplied. This directly affects the claimed nontrivial winding number and the count of four protected Majorana modes.
  2. [Topological superconductivity section] Winding-number calculation and zero-mode localization (in the topological superconductivity results): The winding number is evaluated on the mean-field Bogoliubov-de Gennes Hamiltonian; because the gap itself is an uncontrolled mean-field quantity in 1D, the topological classification and the protection of the end modes are not guaranteed once fluctuations are restored.
  3. [Superconducting diode effect] Diode-efficiency extraction (in the SDE discussion): The value η ∼ 65% is obtained from the mean-field current-phase relation of the FFLO state. Any fluctuation-induced renormalization of the gap or supercurrent would alter the nonreciprocity, so the quoted efficiency inherits the same uncontrolled approximation.
minor comments (3)
  1. [Abstract] Abstract contains a repeated article: 'Performing a a self-consistent mean-field analysis'.
  2. [Model Hamiltonian] The definition and microscopic origin of the 'unconventional magnetic order' term should be stated explicitly with its coupling constants and lattice structure, as these enter the normal-state band topology.
  3. [Numerical results] Parameter values (hopping t, SOC strengths, Hubbard U, magnetic order amplitude) used for the numerical results and the 65% efficiency should be tabulated or listed to permit reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments. The concerns regarding the mean-field approximation in one dimension are valid and we have revised the manuscript to include a more detailed discussion of these limitations while maintaining the core findings as exploratory results within the mean-field framework.

read point-by-point responses

  1. Referee: [Mean-field analysis and BdG spectrum] Self-consistent mean-field treatment of the attractive Hubbard term (described in the section introducing the interaction and order parameters): In one dimension the phase stiffness is parametrically small; long-wavelength fluctuations can close or renormalize the mean-field gap that enters the topological invariant. The manuscript performs only the mean-field iteration and subsequent winding-number integral on the uniform or plane-wave ansatz, with no Ginzburg-criterion estimate or comparison to exact diagonalization supplied. This directly affects the claimed nontrivial winding number and the count of four protected Majorana modes.

    Authors: We agree that one-dimensional systems are particularly susceptible to fluctuations that can destroy or renormalize long-range superconducting order, as per the Mermin-Wagner theorem and related considerations. Our work employs a self-consistent mean-field approach, which is commonly used in the literature to identify candidate phases for topological superconductivity and related effects. In the revised manuscript, we have added a dedicated subsection discussing the limitations of the mean-field approximation in 1D, including a qualitative estimate of the Ginzburg parameter using the model parameters (e.g., interaction strength and bandwidth). We emphasize that the results should be viewed as indicative of possible behaviors in quasi-1D realizations where interchain couplings can stabilize the order. A full fluctuation analysis or exact diagonalization for finite chains is computationally intensive for the parameter space explored and is left for future investigations. revision: partial

  2. Referee: [Topological superconductivity section] Winding-number calculation and zero-mode localization (in the topological superconductivity results): The winding number is evaluated on the mean-field Bogoliubov-de Gennes Hamiltonian; because the gap itself is an uncontrolled mean-field quantity in 1D, the topological classification and the protection of the end modes are not guaranteed once fluctuations are restored.

    Authors: The winding number and the associated Majorana modes are computed within the mean-field BdG framework, providing a topological classification in that approximation. We concur that restoring fluctuations could alter the gap and thus the invariant. To address this, we have updated the topological superconductivity section to explicitly state that the nontrivial winding number and the four Majorana modes are protected within the mean-field theory. We also discuss how the unconventional magnetism and mixed SOC may help in stabilizing the phase against some fluctuations. This clarification ensures readers understand the context of our claims. revision: yes

  3. Referee: [Superconducting diode effect] Diode-efficiency extraction (in the SDE discussion): The value η ∼ 65% is obtained from the mean-field current-phase relation of the FFLO state. Any fluctuation-induced renormalization of the gap or supercurrent would alter the nonreciprocity, so the quoted efficiency inherits the same uncontrolled approximation.

    Authors: The diode efficiency η ∼ 65% is indeed derived from the mean-field calculation of the current-phase relation in the FFLO state. Fluctuations could renormalize this value. In the revised version, we have added a note in the SDE section clarifying that the reported efficiency is a mean-field result and may be subject to renormalization in a more complete treatment. We highlight that even with possible reduction, the combination of unconventional magnetism and SOC offers a promising route to field-free SDE, and the qualitative nonreciprocity is robust within the model. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper assembles a standard 1D tight-binding Hamiltonian from hopping, Rashba+Ising SOC, and an unconventional magnetic term, then adds an on-site attractive Hubbard interaction. Self-consistent mean-field decoupling yields BCS and FFLO order parameters; the winding number is computed directly from the resulting BdG spectrum, and diode efficiency η is extracted from the supercurrent-phase relation. None of these outputs are defined in terms of themselves or fitted to reproduce the claimed topology or nonreciprocity. No load-bearing self-citations, uniqueness theorems, or ansatzes smuggled from prior work appear in the provided text. The chain is a conventional model-to-computation pipeline and remains independent of its target results.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on a mean-field decoupling of the Hubbard term and on the assumed form of the unconventional magnetic order; no new particles or forces are introduced.

free parameters (1)
  • Hubbard interaction strength U
    The attractive on-site interaction is introduced and its value is chosen or self-consistently solved to stabilize the reported superconducting states.
axioms (1)
  • domain assumption Mean-field approximation suffices to determine the superconducting order parameters and their topological character.
    Invoked when the on-site attractive Hubbard interaction is treated self-consistently.

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Reference graph

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