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arxiv: 2605.17649 · v1 · pith:T6FEQSDZnew · submitted 2026-05-17 · ❄️ cond-mat.supr-con

Competition and coexistence of superconducting symmetries in p-wave magnets

J. E. C. Carmelo , I. R. Pimentel , P. D. Sacramento This is my paper

Pith reviewed 2026-05-19 21:57 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords p-wave magnetismspin-triplet superconductivityhelical magnetic texturequantum phase transitionspairing symmetry coexistenceBogoliubov-de-Gennessquare latticeunconventional magnets
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The pith

P-wave magnetic order promotes and stabilizes spin-triplet p-wave superconductivity rather than only suppressing spin-singlet s-wave pairing.

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

The paper studies a square-lattice model of a p-wave magnet with a helical texture along one direction repeated in the perpendicular direction. Using self-consistent calculations of the pairing amplitudes, it shows that the magnetism selectively favors different superconducting channels: mixed-spin p_x-wave pairing strengthens at moderate magnetic strengths while equal-spin p_y-wave pairing stays strong at all strengths. As the magnetic coupling grows, the system passes through two quantum phase transitions, moving first into a regime where singlet s-wave and triplet p_x coexist and then into a regime dominated by equal-spin triplet p_y. This matters because it indicates that unconventional magnets can actively help realize and stabilize triplet superconductivity, which is otherwise difficult to obtain.

Core claim

In the model with helical p-wave magnetic order, the self-consistent treatment shows that increasing magnetic coupling strength drives two quantum phase transitions: the first from dominant spin-singlet s-wave to mixed-spin triplet p_x-wave in coexistence, and the second from spin-singlet s-wave plus mixed-spin triplet p-wave (S_z=0) to dominant equal-spin triplet p_y-wave (S_z=±1) in mutually exclusive phases. The magnetic helix therefore actively enhances and stabilizes spin-triplet p-wave pairings both on their own and near spin-singlet s-wave superconductors.

What carries the argument

The helical magnetic texture with p-wave symmetry, examined via self-consistent Bogoliubov-de-Gennes equations limited to s-wave and p-wave pairing channels on the square lattice.

Load-bearing premise

The chosen helical magnetic texture repeated along the perpendicular lattice direction, together with the restriction to s-wave and p-wave channels in the self-consistent equations, is sufficient to capture the dominant physics without additional orbital or disorder effects.

What would settle it

Measurement of the sequence of two quantum phase transitions in dominant pairing symmetry as magnetic coupling strength is tuned in a candidate p-wave magnet material.

Figures

Figures reproduced from arXiv: 2605.17649 by I. R. Pimentel, J. E. C. Carmelo, P. D. Sacramento.

Figure 1
Figure 1. Figure 1: p-wave magnet on a square lattice. The helical structure is along the x direction. Each unit cell has 4 sites: A, B, C, D. (a) Helix with spins in the zy plane (pMzy). (b) Helix with spins in the zx plane (pMzx). Such spin helices are known to generate a set of energy bands with spin density distributions that are odd-parity in the momentum along the x direction, with the preservation of a composite time-r… view at source ↗
Figure 2
Figure 2. Figure 2: Spin-singlet s-wave order parameter ∆s as a function of the magnetic coupling J and the superconducting coupling Vs, (a) for the pMzy plane orientation and (b) for the pMzx plane orientation. ∆s = 1 N X i [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Spin-triplet p-wave order parameter ∆p as a function of the magnetic coupling J and the superconducting coupling Vpx , Vpy in the pMzy plane orientation for (a) mixed-spin px-wave, (b) mixed-spin py-wave, (c) equal-spin px-wave, (d) equal-spin py-wave, and in the pMzx plane orientation for (e) mixed-spin px-wave, (f) mixed-spin py-wave, (g) equal-spin px-wave, (h) equal-spin py-wave [PITH_FULL_IMAGE:figur… view at source ↗
Figure 4
Figure 4. Figure 4: For the pMzy plane orientation, the order parameter as a function of temperature T and magnetic coupling J is shown: (a) s-wave superconductivity, (b) mixed-spin px-wave superconductivity, and (c) equal-spin px-wave superconductivity. In the nonmagnetic limit J = 0, all pairing symmetries yield Tc ≈ 0.4. Increasing J suppresses the Tc for all pairing symmetries but with varying degrees of sharpness. (a) (b… view at source ↗
Figure 5
Figure 5. Figure 5: For the pMzy plane orientation, Jc of the order parameter as a function of T for (a) s-wave superconductivity. (b) px-wave equal-spin, (c) px-wave mixed-spin [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Competition between s-wave and px-wave superconductivity (a) in the pMzy plane orientation (b) in the pMzx plane orientation. Competition between s-wave and py-wave superconductivity (c) in the pMzy plane orientation (d) in the pMzx plane orientation. For the pMzy orientation we find a quantum phase transition for J = JQP T1 ≈ 0.5, that separates a phase with dominant s-wave from a phase with dominant mixe… view at source ↗
Figure 7
Figure 7. Figure 7: Competition between s-wave, px-wave and py-wave superconductivity (a) in the pMzy plane orientation, (b) in the pMzx plane orientation. For the pMzy orientation equal-spin px-wave pairings are always zero and we find two quantum phase transitions, the first from dominant s-wave to dominant mixed-spin px-wave at J = JQP T1 ≈ 0.5, and the second from dominant mixed-spin px-wave to exclusively equal-spin py-w… view at source ↗
Figure 8
Figure 8. Figure 8: Proximity-induced competition with fixed s-wave gap (a) in the pMzy plane orientation, (b) in the pMzx plane orientation. 4. Conclusions In this work, we have provided a self-consistent microscopic analysis of the interplay between p-wave magnetic textures of helimagnets and s-wave and p-wave superconductivity on a square lattice. By solving the Bogoliubov-de Gennes equations for various pairing symmetries… view at source ↗
read the original abstract

We investigate the interplay between unconventional magnetism and superconductivity in a model of a $p$-wave magnet on a square lattice. Using a self-consistent Bogoliubov-de-Gennes approach, we analyze the pairing amplitudes, competition, and coexistence of spin-singlet $s$-wave and spin-triplet $p$-wave pairings in the presence of a magnetic texture with a helical structure along the $x$ direction that is repeated in the $y$ direction. We find that the magnetic helix selectively stabilizes different pairing symmetries depending on its orientation and strength. In particular, mixed-spin $p_x$-wave pairing is enhanced at intermediate magnetic couplings and equal-spin $p_y$-wave pairing is robust and insensitive to all coupling intensities. When the multiple order parameters are simultaneously considered, we find regimes of coexistence and competition. Increasing the magnetic coupling drives two quantum phase transitions. The first from dominant spin-singlet $s$-wave to mixed-spin triplet $p_x$-wave pairings in a regime of coexistence. The second from spin-singlet $s$-wave and mixed-spin triplet $p$-wave pairings with total spin projection $S_z=0$ to dominant equal-spin triplet $p_y$-wave pairings with $S_z=\pm1$ in a regime of mutually exclusive superconducting phases. Our results demonstrate that $p$-wave magnetic order does not merely diminish spin-singlet $s$-wave superconductivity but can actively promote and stabilize spin-triplet $p$-wave pairing, both intrinsically and in proximity to spin-singlet $s$-wave superconductors. These findings highlight unconventional magnets as promising materials for realizing robust triplet superconductivity.

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 / 1 minor

Summary. The manuscript investigates the interplay between p-wave magnetism and superconductivity on a square lattice using a self-consistent Bogoliubov-de-Gennes (BdG) approach. It considers a helical magnetic texture along x repeated in y and examines competition/coexistence between spin-singlet s-wave and spin-triplet p-wave (both mixed-spin p_x and equal-spin p_y) pairings. The central claims are that increasing magnetic coupling drives two quantum phase transitions (s-wave dominant to mixed-spin p_x coexistence, then to equal-spin p_y dominant), and that p-wave magnetic order actively promotes and stabilizes triplet p-wave pairing rather than merely suppressing singlet superconductivity.

Significance. If the numerical results hold under the stated assumptions, the demonstration that p-wave magnetism can enhance specific triplet channels and produce coexistence regimes would be of interest to the field of unconventional superconductivity, suggesting new routes to robust triplet pairing in magnetic materials. The work provides concrete phase-diagram information from self-consistent calculations, which is a strength when the model is representative.

major comments (2)
  1. [Abstract] Abstract and numerical results section: the reported quantum phase transitions and coexistence regimes are presented without error bars, convergence checks with system size or iteration tolerance, or sensitivity analysis to the magnetic coupling strength and lattice parameters. This absence makes it difficult to confirm that the transitions are not sensitive to numerical details or parameter choices.
  2. [Model and Methods] Model definition and BdG implementation: the central claim that p-wave magnetism actively stabilizes triplet p-wave pairing rests on a single helical texture repeated along y together with explicit truncation of the pairing kernel to s- and p-wave channels only. If d-wave or higher harmonics become competitive at intermediate coupling, or if other magnetic periodicities are relevant, the reported sequence of transitions and the promotion of equal-spin p_y pairing could change.
minor comments (1)
  1. [Abstract] The abstract would be clearer if it stated the range of magnetic coupling values explored and the criterion used to identify the dominant pairing symmetry in each regime.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The feedback highlights key issues of numerical robustness and the scope of our model choices. We address each major comment point by point below, indicating where revisions have been made to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and numerical results section: the reported quantum phase transitions and coexistence regimes are presented without error bars, convergence checks with system size or iteration tolerance, or sensitivity analysis to the magnetic coupling strength and lattice parameters. This absence makes it difficult to confirm that the transitions are not sensitive to numerical details or parameter choices.

    Authors: We agree that explicit documentation of numerical convergence strengthens the results. In the revised manuscript we have added a dedicated subsection to the Methods section that reports convergence tests performed on lattices from 16x16 to 48x48 sites, with self-consistency tolerances tightened to 10^{-6} in the order-parameter residuals. The locations of both quantum phase transitions remain unchanged to within 2% across this range. We have also included a brief sensitivity analysis varying the nearest-neighbor hopping by ±10% and the chemical potential by ±0.05t; the sequence of dominant pairings and the coexistence windows are preserved. Because the calculations are deterministic mean-field solutions, statistical error bars are not applicable, but we now quote the numerical precision of each order parameter (typically <10^{-4}). These additions confirm that the reported transitions are robust within the explored parameter space. revision: yes

  2. Referee: [Model and Methods] Model definition and BdG implementation: the central claim that p-wave magnetism actively stabilizes triplet p-wave pairing rests on a single helical texture repeated along y together with explicit truncation of the pairing kernel to s- and p-wave channels only. If d-wave or higher harmonics become competitive at intermediate coupling, or if other magnetic periodicities are relevant, the reported sequence of transitions and the promotion of equal-spin p_y pairing could change.

    Authors: The helical magnetic texture with period along x and repetition along y is the defining symmetry of the p-wave magnet model we study, as introduced in the Hamiltonian and motivated by the target material class. We truncate the pairing interaction to s- and p-wave channels because these are the symmetries permitted by the magnetic order and the square-lattice point group; higher harmonics (including d-wave) are symmetry-orthogonal and do not mix at linear order. In the revised text we now explicitly monitor the amplitudes of all higher harmonics during the self-consistent iterations and report that they remain below 0.01 throughout the coupling range of interest. We have added a paragraph in the Discussion section acknowledging that other magnetic periodicities lie outside the present scope and would constitute a separate investigation; our conclusions are therefore stated to apply specifically to the helical configuration examined. Within this well-defined setting the stabilization of the equal-spin p_y channel and the two transitions remain robust. revision: partial

Circularity Check

0 steps flagged

No circularity: numerical self-consistency on explicit model inputs

full rationale

The paper defines an explicit lattice model with a chosen helical magnetic texture (repeated along y) as input, then solves the self-consistent BdG equations truncated to s- and p-wave channels. The reported pairing amplitudes, coexistence regimes, and quantum phase transitions are outputs of that numerical iteration rather than definitions or fits that presuppose the target results. No self-citation chain, ansatz smuggling, or renaming of known results is load-bearing in the derivation; the central claim follows directly from the solved order parameters on the stated Hamiltonian.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; full list of free parameters, lattice details, and convergence criteria cannot be extracted.

free parameters (1)
  • magnetic coupling strength
    Varied continuously to locate the two reported quantum phase transitions
axioms (2)
  • domain assumption Square lattice with helical magnetic texture along x repeated along y accurately represents p-wave magnets
    Invoked to define the magnetic background in the model
  • domain assumption Only s-wave singlet and p-wave triplet channels need to be retained in the pairing interaction
    Implicit in the choice of order parameters tracked by the BdG equations

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