Skyrmion-vortex pairing and vortex-drag induced Skyrmion Hall effect

Shantonu Mukherjee

arxiv: 2510.24404 · v3 · submitted 2025-10-28 · ❄️ cond-mat.supr-con · cond-mat.stat-mech· cond-mat.str-el· hep-th

Skyrmion-vortex pairing and vortex-drag induced Skyrmion Hall effect

Shantonu Mukherjee This is my paper

Pith reviewed 2026-05-18 03:20 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.stat-mechcond-mat.str-elhep-th

keywords Skyrmionvortexferromagnetic superconductorSkyrmion Hall effecttopological excitationsMagnus forcebound pairsemergent gauge field

0 comments

The pith

Skyrmions and vortices form bound pairs in ferromagnetic superconductors, inducing a vortex-drag Hall effect on Skyrmions.

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

The paper proposes an interaction between ferromagnetic and superconducting orders in a two-dimensional ferromagnetic superconductor that obeys symmetry principles. This interaction enables a duality similar to the boson-vortex duality, in which Skyrmion and vortex excitations interact through an emergent gauge field. The static interaction potential is attractive for a Skyrmion and a vortex with opposite topological charges, allowing formation of bound pairs. Such pairing implies that a Magnus force acting on the vortex will induce a transverse, Hall-like drift motion of the Skyrmion, termed the vortex-drag induced Skyrmion Hall effect. Possible experimental manifestations are discussed.

Core claim

An interaction between ferromagnetic and superconducting orders allows formulation of a duality similar to the Boson-vortex duality in 2+1 dimensions. In the dual theory, Skyrmion and vortex excitations interact via an emergent gauge field. The static interaction potential is attractive for opposite topological charges, leading to bound pair formation. Consequently, a Magnus force on the vortex induces a transverse Hall-like drift of the Skyrmion, which is called the vortex-drag induced Skyrmion Hall effect.

What carries the argument

Duality similar to the Boson-vortex duality, with Skyrmions and vortices interacting via an emergent gauge field that produces an attractive static potential for opposite topological charges.

Load-bearing premise

The proposed interaction between ferromagnetic and superconducting orders obeys necessary symmetry principles and permits a duality formulation with Skyrmions and vortices interacting through an emergent gauge field.

What would settle it

Direct observation of transverse Skyrmion drift when a Magnus force is applied to a paired vortex in a two-dimensional ferromagnetic superconductor sample.

read the original abstract

An interaction between ferromagnetic and superconducting orders, to be realized in a two dimensional ferromagnetic superconductor, is proposed obeying necessary symmetry principles. This interaction allows us to formulate a duality, similar to the Boson-vortex duality in 2+1 dimensional superfluid. In the dual theory the Skyrmion and the vortex excitations interact with each other via an emergent gauge field. The static interaction potential is attractive for a Skyrmion and a vortex with opposite topological charges. This interaction can lead to formation of bound pairs of the mentioned topological excitations. Furthermore, we argue that such pairing implies that a Magnus force acting on the vortex induces a transverse, Hall-like drift motion of the Skyrmion, which we term the vortex-drag induced Skyrmion Hall effect. Possible experimental manifestations of this effect are also discussed.

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.

Desk Editor's Note private letter to a colleague

This is a symmetry-based proposal for skyrmion-vortex pairing via an emergent gauge field in 2D ferromagnetic superconductors, with the main new claim being a vortex-drag Skyrmion Hall effect.

read the letter

The paper's central idea is that a symmetry-allowed interaction between ferromagnetic and superconducting orders in two dimensions lets you set up a duality like the usual boson-vortex one. In the dual picture skyrmions and vortices couple through an emergent gauge field, and the static potential pulls them together when their topological charges are opposite. That binding then lets a Magnus force on the vortex drag the skyrmion sideways, which the authors call the vortex-drag induced Skyrmion Hall effect. They also sketch some possible experimental signatures in hybrid devices. That combination of pairing and the resulting transverse motion is the part that does not just repeat earlier duality results. The discussion of how this might show up in transport or imaging is straightforward and points to concrete tests. The main limitation is that the abstract gives no Lagrangian, no explicit integration over the gauge field, and no derivation of the potential's sign or form. Without those steps it is hard to see exactly how the attraction follows from the interaction term or whether extra assumptions about charge assignments are required. If the full manuscript supplies the missing equations and checks the duality mapping, the argument becomes easier to evaluate. This is the kind of targeted proposal that people working on topological defects in hybrid magnetic-superconducting systems would want to see. It is not a finished calculation but it raises a clear question worth checking. I would send it to referees rather than desk-reject it, because the idea is distinct enough to merit external input even if the derivations need tightening.

Referee Report

2 major / 1 minor

Summary. The paper proposes a symmetry-allowed interaction between ferromagnetic and superconducting orders in a two-dimensional ferromagnetic superconductor. This interaction enables a duality construction analogous to the boson-vortex duality in 2+1 dimensions, in which Skyrmion and vortex excitations couple through an emergent gauge field. The static interaction potential is asserted to be attractive for opposite topological charges, permitting bound-pair formation. The pairing is then used to argue that a Magnus force on the vortex produces a transverse Hall-like drift of the Skyrmion, termed the vortex-drag induced Skyrmion Hall effect; possible experimental signatures are outlined.

Significance. If the duality mapping and the sign of the resulting potential can be established, the work would introduce a new mechanism for coupling topological defects across ferromagnetic and superconducting orders and predict a previously unconsidered Hall response. The symmetry-based, parameter-free framing is a conceptual strength that could stimulate both theoretical follow-up and targeted experiments in hybrid 2D systems.

major comments (2)

[Duality formulation and interaction potential] The manuscript asserts that the symmetry-allowed interaction permits a boson-vortex-like duality in which Skyrmions and vortices interact via an emergent U(1) gauge field, yielding an attractive static potential precisely when their topological charges are opposite. No explicit Lagrangian, gauge-field integration, or solution of the dual equations of motion is provided to derive the functional form or the sign of this potential. This step is load-bearing for the central claims of bound-pair formation and the subsequent Hall effect.
[Dynamics of paired excitations] The argument that pairing under the Magnus force on the vortex induces a transverse drift of the Skyrmion (the vortex-drag Skyrmion Hall effect) is presented without an effective dynamical equation for the composite object or a calculation of the resulting velocity components. A concrete derivation linking the bound-pair constraint to the Hall drift is required to substantiate the effect.

minor comments (1)

Notation for the emergent gauge field and the topological charges of the Skyrmion and vortex should be introduced with explicit definitions at first use to improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the presentation of the duality construction and the dynamical implications. We address each major comment below and will incorporate revisions to strengthen the derivations.

read point-by-point responses

Referee: [Duality formulation and interaction potential] The manuscript asserts that the symmetry-allowed interaction permits a boson-vortex-like duality in which Skyrmions and vortices interact via an emergent U(1) gauge field, yielding an attractive static potential precisely when their topological charges are opposite. No explicit Lagrangian, gauge-field integration, or solution of the dual equations of motion is provided to derive the functional form or the sign of this potential. This step is load-bearing for the central claims of bound-pair formation and the subsequent Hall effect.

Authors: We thank the referee for highlighting the need for explicit steps in the duality. The manuscript introduces the symmetry-allowed interaction and formulates the duality by direct analogy to the standard 2+1D boson-vortex duality, with the sign of the potential following from the opposite topological charges coupling to the emergent gauge field. We agree that an expanded derivation will improve clarity. In the revised manuscript we will add the explicit effective Lagrangian containing the proposed interaction term, outline the duality transformation, integrate out the gauge field to obtain the static potential, and confirm its attractive character for opposite charges via the relative signs in the topological coupling. This addition will directly support the bound-pair formation. revision: yes
Referee: [Dynamics of paired excitations] The argument that pairing under the Magnus force on the vortex induces a transverse drift of the Skyrmion (the vortex-drag Skyrmion Hall effect) is presented without an effective dynamical equation for the composite object or a calculation of the resulting velocity components. A concrete derivation linking the bound-pair constraint to the Hall drift is required to substantiate the effect.

Authors: We acknowledge that the dynamical argument in the original manuscript is presented at a qualitative level based on the bound-pair constraint. To address this, the revised version will introduce an effective dynamical description of the composite object, incorporating the attractive potential that enforces the pairing and the Magnus force acting on the vortex. We will derive the coupled equations of motion and explicitly compute the resulting velocity components, demonstrating that the Skyrmion acquires a transverse Hall-like drift proportional to the vortex motion. This will provide the concrete link between the pairing and the vortex-drag induced Skyrmion Hall effect. revision: yes

Circularity Check

0 steps flagged

Proposed symmetry-based interaction and duality form an original construction with no reduction to inputs

full rationale

The paper proposes an interaction between ferromagnetic and superconducting orders that obeys symmetry principles, then uses this premise to formulate a duality analogous to boson-vortex duality in 2+1 dimensions. In the resulting dual theory, Skyrmions and vortices interact via an emergent gauge field, with the static potential stated to be attractive for opposite topological charges, enabling bound pairs and the vortex-drag Skyrmion Hall effect. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the attractiveness and pairing follow from the constructed dual theory rather than tautological restatement of inputs. The derivation is self-contained against external symmetry principles and the explicit duality mapping.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The proposal rests on symmetry principles as a domain assumption and introduces an emergent gauge field as a new mediating entity; no explicit free parameters are named in the abstract.

axioms (1)

domain assumption An interaction between ferromagnetic and superconducting orders obeys necessary symmetry principles.
Invoked at the opening of the abstract to justify the proposed interaction and duality.

invented entities (1)

emergent gauge field no independent evidence
purpose: Mediates the static attractive interaction between Skyrmion and vortex excitations with opposite topological charges in the dual theory.
Introduced in the dual formulation to enable pairing and the subsequent drag effect.

pith-pipeline@v0.9.0 · 5673 in / 1420 out tokens · 38302 ms · 2026-05-18T03:20:27.972639+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The static interaction potential is attractive for a Skyrmion and a vortex with opposite topological charges... V(|R_v − R_s|) ∼ 2ρ_s N_v N_s λ / m K_0(m̃ |R_v − R_s|)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

formulate a duality, similar to the Boson-vortex duality in 2+1 dimensional superfluid... emergent gauge field b_μ

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

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work page
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Mukherjee and A

S. Mukherjee and A. Lahiri, Emergent vortex–electron interaction from dualization, Annals Phys.418, 168167 (2020), arXiv:1912.03124 [hep-th]

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J. Bardeen and M. J. Stephen, Theory of the motion of vortices in superconductors, Phys. Rev.140, A1197 (1965)

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Nozi` eres and W

P. Nozi` eres and W. F. Vinen, The motion of flux lines in type II superconductors, Philosophical Magazine14, 667 (1966)

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[1] [1]

However, there is a mixed Chern-Simons term involvingb µ andA µ

interact with each other via an emergent gauge fieldb µ. However, there is a mixed Chern-Simons term involvingb µ andA µ. It is well known that such a term would generate an effective mass for any of the gauge fields involved when the other one is integrated out [20–22]. Due to this topological mass mechanism at work the emergent gauge fieldb µ will gain ...

work page

[2] [2]

Y. Oner, C. Boyraz, H. Hiramatsu, T. Katase, and H. Hosono, Coexistence of magnetism and superconductivity in thin films of the fe-based superconductor ba1xlaxfe2as2, Journal of Physics: Condensed Matter32, 485804 (2020). 6

work page 2020

[3] [3]

K. Jin, J. Yuan, L. Zhao, H. Wu, X. Y. Qi, B. Y. Zhu, L. X. Cao, X. G. Qiu, B. Xu, X. F. Duan, and B. R. Zhao, Coexistence of superconductivity and ferromagnetism in a dilute cobalt-doped la 1.89ce0.11Cuo4±δ system, Phys. Rev. B 74, 094518 (2006)

work page 2006

[4] [4]

S. L. Kakani and U. N. Upadhyaya, Coexistence of superconductivity and ferromagnetism in rare earth ternary systems, physica status solidi (a)99, 15 (1987), https://onlinelibrary.wiley.com/doi/pdf/10.1002/pssa.2210990104

work page doi:10.1002/pssa.2210990104 1987

[5] [5]

X. Zhu, Y. Guo, H.-S. Cheng, J. Dai, X. An, J. Zhao, K. Tian, S. Wei, X. C. Zeng, C. Wu, and Y. Xie, Signature of coexistence of superconductivity and ferromagnetism in two-dimensional nbse2 triggered by surface molecular adsorption, Nature Communications7(2016)

work page 2016

[6] [6]

Baumard, J

J. Baumard, J. Cayssol, F. S. Bergeret, and A. Buzdin, Generation of a superconducting vortex via n´ eel skyrmions, Phys. Rev. B99, 014511 (2019)

work page 2019

[7] [7]

S. M. Dahir, A. F. Volkov, and I. M. Eremin, Interaction of skyrmions and pearl vortices in superconductor-chiral ferro- magnet heterostructures, Phys. Rev. Lett.122, 097001 (2019)

work page 2019

[8] [8]

K. M. D. Hals, M. Schecter, and M. S. Rudner, Composite topological excitations in ferromagnet-superconductor het- erostructures, Phys. Rev. Lett.117, 017001 (2016)

work page 2016

[9] [9]

E. S. Andriyakhina and I. S. Burmistrov, Interaction of a n´ eel-type skyrmion with a superconducting vortex, Phys. Rev. B103, 174519 (2021)

work page 2021

[10] [10]

S. M. Dahir, A. F. Volkov, and I. M. Eremin, Meissner currents induced by topological magnetic textures in hybrid superconductor/ferromagnet structures, Phys. Rev. B102, 014503 (2020)

work page 2020

[11] [11]

A. P. Petrovi´ c, M. Raju, X. Y. Tee, A. Louat, I. Maggio-Aprile, R. M. Menezes, M. J. Wyszy´ nski, N. K. Duong, M. Reznikov, C. Renner, M. V. Miloˇ sevi´ c, and C. Panagopoulos, Skyrmion-(anti)vortex coupling in a chiral magnet-superconductor heterostructure, Phys. Rev. Lett.126, 117205 (2021)

work page 2021

[12] [12]

Mukherjee and A

S. Mukherjee and A. Lahiri, Skyrmion-vortex hybrid and spin wave solutions in ferromagnetic superconductors, SciPost Phys.19, 022 (2025), arXiv:2407.00405 [cond-mat.supr-con]

work page arXiv 2025

[13] [13]

M. E. Peskin, Mandelstam ’t Hooft Duality in Abelian Lattice Models, Annals Phys.113, 122 (1978)

work page 1978

[14] [14]

Dasgupta and B

C. Dasgupta and B. I. Halperin, Phase transition in a lattice model of superconductivity, Phys. Rev. Lett.47, 1556 (1981)

work page 1981

[15] [15]

J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, Journal of Physics C Solid State Physics6, 1181 (1973)

work page 1973

[16] [16]

Savit, Duality in field theory and statistical systems, Rev

R. Savit, Duality in field theory and statistical systems, Rev. Mod. Phys.52, 453 (1980)

work page 1980

[17] [17]

Lee and C

D.-H. Lee and C. L. Kane, Boson-vortex-skyrmion duality, spin-singlet fractional quantum hall effect, and spin-1/2 anyon superconductivity, Phys. Rev. Lett.64, 1313 (1990)

work page 1990

[18] [18]

J. H. Han,Skyrmions in Condensed Matter, Vol. 278 (Springer, 2017)

work page 2017

[19] [19]

D. H. Lee and M. P. A. Fisher, Anyon superconductivity and charge vortex duality, Int. J. Mod. Phys. B5, 2675 (1991)

work page 1991

[20] [20]

Read and S

N. Read and S. Sachdev, Continuum quantum ferromagnets at finite temperature and the quantum hall effect, Phys. Rev. Lett.75, 3509 (1995)

work page 1995

[21] [21]

Mukherjee and A

S. Mukherjee and A. Lahiri, Emergent vortex–electron interaction from dualization, Annals Phys.418, 168167 (2020), arXiv:1912.03124 [hep-th]

work page arXiv 2020

[22] [22]

Deser, R

S. Deser, R. Jackiw, and S. Templeton, Three-dimensional massive gauge theories, Physical Review Letters48, 975 (1982)

work page 1982

[23] [23]

T. J. Allen, M. J. Bowick, and A. Lahiri, Topological mass generation in 3+1 dimensions, Modern Physics Letters A6, 559 (1991)

work page 1991

[24] [24]

A. A. Thiele, Steady-state motion of magnetic domains, Phys. Rev. Lett.30, 230 (1973)

work page 1973

[25] [25]

Bardeen and M

J. Bardeen and M. J. Stephen, Theory of the motion of vortices in superconductors, Phys. Rev.140, A1197 (1965)

work page 1965

[26] [26]

Nozi` eres and W

P. Nozi` eres and W. F. Vinen, The motion of flux lines in type II superconductors, Philosophical Magazine14, 667 (1966)

work page 1966