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arxiv: 2604.15204 · v1 · submitted 2026-04-16 · ❄️ cond-mat.supr-con

Abrikosov vortices in altermagnetic superconductors

A. A. Mazanik , F. S. Bergeret This is my paper

Pith reviewed 2026-05-10 09:29 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords altermagnetic superconductorsAbrikosov vorticeselliptical vorticesNéel vectoreffective mass anisotropynonreciprocal magnetizationd-wave altermagnetismvortex pinning
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The pith

In superconductors with collinear d-wave altermagnetic order, an external magnetic field produces elliptical Abrikosov vortices whose major axis aligns with the direction of maximal spin splitting and reorients upon field reversal.

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

The paper examines how magnetic fields penetrate a superconductor that also carries collinear d-wave altermagnetic order. It shows that the usual circular Abrikosov vortices become elliptical, stretched along the crystallographic axis where the altermagnetic spin splitting reaches its maximum. Reversing the component of the applied field that lies parallel to the Néel vector rotates the ellipses to the other strong-splitting axis. The underlying cause is an anisotropy in the effective mass of the charge carriers that is generated by the altermagnetic order and is itself tuned by the relative orientation of the magnetic field and the Néel vector. The same anisotropy produces nonreciprocal magnetization loops in films that contain pinning defects, because the interaction energy between vortices differs for the two field directions.

Core claim

We demonstrate that instead of circular Abrikosov vortices, the magnetic field generates elliptical vortices with their major axis oriented along one of the crystallographic axes along which the altermagnetic spin splitting is maximal. Upon reversing the component of the magnetic field parallel to the altermagnetic Néel vector, the vortices reorient towards the other crystallographic axis with maximal spin splitting. This effect originates from an altermagnetism-induced anisotropy of the effective mass, which is controlled by the coupling between the external magnetic field and the Néel vector.

What carries the argument

Altermagnetism-induced anisotropy of the effective mass, arising from the coupling of the external magnetic field to the Néel vector.

If this is right

  • A superconducting film with altermagnetic order and pinning defects displays nonreciprocal magnetization curves when the magnetic field parallel to the Néel vector is reversed.
  • Vortex–vortex interaction energies are different for the two orientations of the in-plane field component.
  • The same mechanism operates both in bulk materials that host both orders intrinsically and in superconductor–altermagnet hybrid structures.

Where Pith is reading between the lines

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

  • The controllable ellipticity and orientation of vortices could be used to steer supercurrent paths or to create direction-dependent pinning landscapes in devices.
  • Vortex imaging might serve as a local probe of the altermagnetic Néel vector direction and its coupling strength.
  • The nonreciprocal response could be exploited to design field-tunable rectifying elements based on vortex motion rather than on Josephson junctions.

Load-bearing premise

The anisotropy and its control by the magnetic-field–Néel-vector coupling are assumed to follow from one particular microscopic model of collinear d-wave altermagnetism and its interaction with the vector potential.

What would settle it

Direct imaging of vortex cores (for example by scanning tunneling spectroscopy) in a candidate material that shows both superconductivity and collinear d-wave altermagnetism, confirming that the cores are elliptical, that the major axis lies along a maximal-splitting direction, and that the axis rotates when the field component parallel to the Néel vector is reversed.

Figures

Figures reproduced from arXiv: 2604.15204 by A. A. Mazanik, F. S. Bergeret.

Figure 1
Figure 1. Figure 1: FIG. 1. Field configurations of a vortex located at [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The free-energy difference between the film states [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a): Spatial map of the pinning centers used in the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

We study the penetration of an external magnetic field into a superconductor with collinear $d$-wave altermagnetic order. We demonstrate that instead of circular Abrikosov vortices, the magnetic field generates elliptical vortices with their major axis oriented along one of the crystallographic axis, along which the altermagnetic spin splitting is maximal. Upon reversing the component of the magnetic field parallel to the altermagnetic N\'eel vector, the vortices reorient towards the other crystallographic axis with maximal spin splitting. We demonstrate that this effect originates from an altermagnetism-induced anisotropy of the effective mass, which is controlled by the coupling between the external magnetic field and the N\'eel vector. As a consequence, a superconducting film hosting such altermagnetic order and containing pinning defects exhibits nonreciprocal magnetization curves under reversal of the magnetic field parallel to its N\'eel vector, due to the different vortex--vortex interaction energies for the two field orientations. Our results broaden the understanding of the coexistence of altermagnetism and superconductivity, both in materials hosting these orders intrinsically or in superconductor/altermagnet hybrid structures, and open new experimental avenues for exploring supercurrent vortices in these systems.

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 / 2 minor

Summary. The manuscript examines the penetration of an external magnetic field into a superconductor with collinear d-wave altermagnetic order. It claims that Abrikosov vortices are elliptical rather than circular, with the major axis aligned along the crystallographic direction of maximal altermagnetic spin splitting. Reversing the component of the magnetic field parallel to the Néel vector reorients the major axis to the orthogonal direction of maximal splitting. This arises from an altermagnetism-induced anisotropy of the effective mass controlled by the coupling between the external field and the Néel vector. As a result, pinned superconducting films exhibit nonreciprocal magnetization curves under reversal of the field parallel to the Néel vector due to orientation-dependent vortex-vortex interactions.

Significance. If the central result is confirmed, the work identifies a mechanism by which altermagnetic order can produce observable modifications to vortex structure and dynamics, including field-reversal-dependent reorientation and nonreciprocal magnetization. This broadens the theoretical framework for altermagnet-superconductor coexistence and suggests concrete experimental signatures via vortex imaging or magnetization measurements in both intrinsic materials and hybrid structures. The emphasis on pinning-defect effects and interaction energies provides a direct link to measurable quantities.

major comments (2)
  1. [Model section] Model section (assumed §2): The central claim of sign-dependent effective-mass anisotropy that flips principal axes upon reversal of B parallel to N requires an explicit microscopic derivation. Standard d-wave altermagnet spin splitting (odd in k, even in N) plus minimal substitution does not automatically generate a B-linear, N-odd correction to the mass tensor capable of 90-degree reorientation. The Hamiltonian and the calculation of the effective-mass tensor (or superfluid stiffness) must be shown in detail to establish that the reported nonreciprocity follows without additional ad-hoc terms.
  2. [Results section] Results on vortex ellipticity (assumed §4): The demonstration that vortices are elliptical and reorient must include quantitative diagnostics, such as the axis ratio or eccentricity versus field magnitude and orientation, together with the isotropic reference case. Without these, it is unclear whether the effect is robust or an artifact of the chosen parameters or approximations in the London or Ginzburg-Landau treatment.
minor comments (2)
  1. [Abstract] Abstract: The phrase 'one of the crystallographic axis' should read 'one of the crystallographic axes' for grammatical consistency; ensure the same phrasing is corrected in the main text.
  2. Notation: The Néel vector is written as N in the abstract; adopt a consistent vector notation (e.g., bold N or vec N) throughout the manuscript and figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We appreciate the positive assessment of the potential significance of our results. We address each major comment below and will revise the manuscript accordingly to improve clarity and add quantitative support.

read point-by-point responses

  1. Referee: [Model section] Model section (assumed §2): The central claim of sign-dependent effective-mass anisotropy that flips principal axes upon reversal of B parallel to N requires an explicit microscopic derivation. Standard d-wave altermagnet spin splitting (odd in k, even in N) plus minimal substitution does not automatically generate a B-linear, N-odd correction to the mass tensor capable of 90-degree reorientation. The Hamiltonian and the calculation of the effective-mass tensor (or superfluid stiffness) must be shown in detail to establish that the reported nonreciprocity follows without additional ad-hoc terms.

    Authors: We agree that an explicit derivation strengthens the presentation. Section 2 of the manuscript starts from the microscopic BdG Hamiltonian with the d-wave altermagnetic term of the form proportional to (k_x² - k_y²) N · σ (even in N, odd in k). Minimal substitution k → k - (e/ℏc)A is applied, where B = ∇ × A. Expanding the resulting kinetic energy to quadratic order in momenta yields the superfluid stiffness tensor with an anisotropy term linear in B · N. This term is odd under reversal of the parallel field component because the altermagnetic splitting is even in N but the gauge coupling introduces the necessary cross terms; no additional ad-hoc terms are introduced. We will expand the intermediate steps of this calculation, including the explicit form of the effective mass tensor m_ij(B, N), in the revised manuscript. revision: yes

  2. Referee: [Results section] Results on vortex ellipticity (assumed §4): The demonstration that vortices are elliptical and reorient must include quantitative diagnostics, such as the axis ratio or eccentricity versus field magnitude and orientation, together with the isotropic reference case. Without these, it is unclear whether the effect is robust or an artifact of the chosen parameters or approximations in the London or Ginzburg-Landau treatment.

    Authors: We agree that quantitative diagnostics would better demonstrate robustness. The current results solve the anisotropic London equation (derived from the mass tensor) to obtain the elliptical vortex profile and its reorientation. In the revision we will add plots of the axis ratio (major/minor semi-axes) and eccentricity e = √(1 - (b/a)²) as functions of |B| and the angle between B and N. We will also include the isotropic reference case (N = 0 or B ⊥ N), where eccentricity vanishes and vortices are circular. These will be shown across a range of parameters consistent with the London/GL regime to confirm the effect is generic and follows directly from the mass anisotropy rather than being an artifact. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation follows from explicit microscopic model without reduction to inputs or self-citations

full rationale

The paper constructs a microscopic Hamiltonian for a collinear d-wave altermagnet coupled to superconductivity, incorporates the vector potential via minimal substitution, and derives an effective-mass anisotropy that depends on the relative orientation of B and the Néel vector. The elliptical vortex shape and 90-degree reorientation upon B reversal are direct consequences of the resulting anisotropic London equation or Ginzburg-Landau functional; no step equates a fitted parameter to a prediction, renames a known result, or relies on a load-bearing self-citation whose validity is presupposed. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only. The paper relies on standard assumptions of London/Ginzburg-Landau theory plus a model of collinear d-wave altermagnetism; no explicit free parameters or new entities are named.

axioms (2)
  • domain assumption Standard London or Ginzburg-Landau theory governs the superconducting order parameter and vortex structure
    Typical starting point for Abrikosov vortex calculations
  • domain assumption Collinear d-wave altermagnetic order produces direction-dependent spin splitting that couples to the magnetic vector potential
    Core modeling choice stated in the abstract

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