Pairing-induced Momentum-space Magnetism and Its Implication In Optical Anomalous Hall Effect In Chiral Superconductors

Bin Geng; Qian Niu; Yang Gao

arxiv: 2506.18005 · v1 · submitted 2025-06-22 · ❄️ cond-mat.supr-con · cond-mat.mtrl-sci

Pairing-induced Momentum-space Magnetism and Its Implication In Optical Anomalous Hall Effect In Chiral Superconductors

Bin Geng , Yang Gao , Qian Niu This is my paper

Pith reviewed 2026-05-19 08:28 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.mtrl-sci

keywords chiral superconductorsoptical anomalous Hall effectmomentum space magnetismpairing potentialsspin-orbit couplingnon-unitary pairingunitary pairing

0 comments

The pith

Effective momentum-space magnetism from pairing enables optical anomalous Hall effect in single-orbital chiral superconductors.

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

The paper shows that in generic single-orbital spinful models of chiral superconductors, pairing can induce effective magnetism in momentum space that produces optical anomalous Hall conductivity. It generalizes Onsager's relation to find the necessary conditions and employs a down-folding method to distinguish two mechanisms. Non-unitary pairing gives rise to magnetism from the Cooper pair's angular momentum. Unitary pairing produces magnetism through normal-state spin-orbit coupling, an effect previously overlooked. This reveals the important role of spin degrees of freedom and suggests the effect can occur without multi-orbital complexity.

Core claim

We generalize the Onsager's relation to obtain the necessary conditions for the optical anomalous Hall effect in a generic single-orbital and spinful Hamiltonian. Using the down-folding method, we identify two types of effective momentum-space magnetism responsible for the optical anomalous Hall conductivity from non-unitary and unitary pairing potentials respectively. The former is due to the angular momentum of Cooper pair, while the latter requires the participation of the spin-orbit coupling in the normal state and has been largely overlooked previously. Using concrete examples, we show that the unitary pairing can lead to both ferromagnetism and complicated antiferromagnetic spin textu

What carries the argument

Down-folding method to identify effective momentum-space magnetism from pairing potentials in single-orbital spinful Hamiltonians.

If this is right

Non-unitary pairing induces momentum-space magnetism due to the angular momentum carried by Cooper pairs.
Unitary pairing, assisted by normal-state spin-orbit coupling, generates ferromagnetic or antiferromagnetic spin textures in momentum space.
These textures produce an in-plane optical anomalous Hall effect with the effective magnetism aligned parallel to the Hall deflection plane.
The spin degree of freedom plays an essential role in enabling the optical anomalous Hall effect in chiral superconductors.

Where Pith is reading between the lines

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

This suggests that magneto-optical Kerr signals could be used to probe pairing symmetry in simpler band structures with spin-orbit coupling.
Previous explanations of optical AHE in chiral superconductors may need to include this unitary pairing mechanism alongside multi-orbital contributions.
Testable predictions include specific in-plane Hall responses in materials like certain heavy-fermion or transition-metal superconductors with known pairing types.

Load-bearing premise

The down-folding method faithfully extracts the effective momentum-space magnetism from the pairing without requiring multi-orbital degrees of freedom.

What would settle it

An experimental observation of the absence of in-plane optical anomalous Hall effect in a chiral superconductor confirmed to have unitary pairing, spin-orbit coupling, and single-orbital character would falsify the mechanism.

Figures

Figures reproduced from arXiv: 2506.18005 by Bin Geng, Qian Niu, Yang Gao.

**Figure 2.** Figure 2: FIG. 2. Momentum-space antiferromagnetism. (a) The ori [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

read the original abstract

The intrinsic mechanisms of the magneto-optical Kerr signal in chiral superconductors often involve multi-orbital degree of freedom. Here by considering a generic single-orbital and spinful Hamiltonian, we generalize the Onsager's relation to obtain the necessary conditions for the optical anomalous Hall effect. Using the down-folding method, we identify two types of effective momentum-space magnetism responsible for the optical anomalous Hall conductivity from non-unitary and unitary pairing potentials respectively. The former is due to the angular momentum of Cooper pair, while the latter requires the participation of the spin-orbit coupling in the normal state and has been largely overlooked previously. Using concrete examples, we show that the unitary pairing can lead to both ferromagnetism and complicated antiferromagnetic spin texture in the momentum space, resulting in an in-plane optical anomalous Hall effect with the magnetism parallel to the Hall-deflection plane. Our work reveals the essential role of spin degree of freedom in the optical anomalous Hall effect.

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

The paper flags an overlooked unitary-pairing route to k-space magnetism via normal-state SOC that can produce in-plane optical AHE in single-orbital chiral superconductors.

read the letter

The central new point is that unitary pairing plus normal-state spin-orbit coupling can generate effective momentum-space magnetism and an in-plane optical anomalous Hall effect even in a single-orbital spinful model. Earlier work apparently missed this because it leaned on multi-orbital setups or non-unitary pairing. The authors generalize Onsager reciprocity to get the necessary conditions, then use down-folding to separate the two magnetism channels: one tied to Cooper-pair angular momentum and the other to SOC in the unitary case. Concrete examples then show both simple ferromagnetic and more intricate antiferromagnetic textures in k-space, with the magnetism aligned to produce the in-plane Hall response. That framing is useful and gives a clearer picture of how spin degrees of freedom enter the optical signal. The examples are the strongest part; they make the abstract claim tangible without obvious parameter fitting. The down-folding step is where things get softer. The stress-test concern about interband virtual processes is reasonable on its face, since optical conductivity draws on current-operator matrix elements that can involve high-energy states. If the projection does not preserve the correct renormalization or mixing with remote bands, the extracted magnetism could under- or over-estimate the Hall conductivity at finite frequency. The paper asserts the procedure is faithful, but without a direct side-by-side check against the full Kubo formula in the examples, it is difficult to judge how large the error actually is. Minor gaps in that validation do not sink the work, but they do limit how far one can push the quantitative claims right now. This is aimed at people working on topological superconductivity and magneto-optics in chiral materials. Readers who care about single-orbital models or SOC routes will get the most out of it. The paper is coherent enough and the claim specific enough that it deserves a serious referee, even if revisions on the optical-response checks are likely. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper claims that generalizing Onsager reciprocity to a generic single-orbital spinful Hamiltonian yields necessary conditions for the optical anomalous Hall effect (AHE). A down-folding procedure then extracts two distinct forms of effective momentum-space magnetism: one induced by non-unitary pairing and traceable to Cooper-pair angular momentum, and a second induced by unitary pairing that requires normal-state spin-orbit coupling and has been overlooked. Concrete examples are presented in which unitary pairing produces both ferromagnetic and antiferromagnetic spin textures in k-space, each generating an in-plane optical AHE with the effective magnetism lying parallel to the Hall-deflection plane. The work emphasizes the essential role of spin degrees of freedom in single-orbital models.

Significance. If the central derivation and down-folding approximation hold, the result is significant because it supplies a mechanism for optical AHE that does not rely on multi-orbital degrees of freedom and identifies a previously neglected unitary-pairing channel mediated by normal-state SOC. The explicit mapping from pairing symmetry to k-space magnetism offers a falsifiable framework that could be tested in candidate chiral superconductors and may guide future magneto-optical experiments.

major comments (2)

[§3] §3 (down-folding procedure) and the subsequent Kubo-formula implementation: the projection onto the low-energy subspace must be shown to preserve the correct renormalization of the current operator and the pairing-induced virtual mixing with remote bands. The optical Hall conductivity is obtained from interband matrix elements; if these are truncated by the down-folding, the extracted effective magnetism may not reproduce the full response even in a single-orbital BdG model.
[Concrete examples] Concrete-examples section (presumably §4 or §5): direct numerical comparison between the optical Hall conductivity computed from the full Hamiltonian and from the down-folded effective model is required across the relevant frequency window. Without this benchmark, it remains unclear whether projection errors are negligible for the claimed in-plane AHE signals.

minor comments (2)

[Abstract] The abstract states that 'concrete examples exist' but does not name the model Hamiltonians or parameter regimes; a single sentence identifying them would improve readability.
[Notation] Notation for the effective momentum-space magnetization (e.g., m(k) versus M_eff) should be defined once and used consistently in all figures and equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the presentation of our down-folding procedure and its validation. We address each major comment below and will incorporate the requested clarifications and benchmarks into a revised manuscript.

read point-by-point responses

Referee: [§3] §3 (down-folding procedure) and the subsequent Kubo-formula implementation: the projection onto the low-energy subspace must be shown to preserve the correct renormalization of the current operator and the pairing-induced virtual mixing with remote bands. The optical Hall conductivity is obtained from interband matrix elements; if these are truncated by the down-folding, the extracted effective magnetism may not reproduce the full response even in a single-orbital BdG model.

Authors: We agree that an explicit demonstration is necessary to confirm that the projected current operator retains the renormalization and virtual-mixing contributions relevant to the optical Hall response. In the revised manuscript we will add a dedicated subsection deriving the effective current operator under the down-folding projection, showing that the interband matrix elements responsible for the anomalous Hall conductivity are preserved to leading order in the pairing amplitude. We will also state the validity conditions under which truncation errors remain negligible for the frequency window of interest. revision: yes
Referee: [Concrete examples] Concrete-examples section (presumably §4 or §5): direct numerical comparison between the optical Hall conductivity computed from the full Hamiltonian and from the down-folded effective model is required across the relevant frequency window. Without this benchmark, it remains unclear whether projection errors are negligible for the claimed in-plane AHE signals.

Authors: We acknowledge that a direct numerical benchmark is the most convincing way to quantify projection accuracy. In the revised version we will include plots comparing the real and imaginary parts of the optical Hall conductivity obtained from the full single-orbital BdG Hamiltonian versus the down-folded effective model for both the ferromagnetic and antiferromagnetic examples, over the frequency range where the in-plane AHE is predicted. These comparisons will be presented alongside the existing analytic results to demonstrate that the effective-magnetism description reproduces the full response within the stated approximations. revision: yes

Circularity Check

0 steps flagged

Derivation from generic Hamiltonian via Onsager generalization and down-folding is self-contained

full rationale

The paper begins with a generic single-orbital spinful Hamiltonian, generalizes Onsager reciprocity to derive necessary conditions for optical anomalous Hall effect, and then applies the down-folding procedure to extract effective momentum-space magnetism induced by non-unitary and unitary pairing. No step reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation; the central claims rest on explicit model construction and standard projection techniques rather than renaming or smuggling prior results. The derivation chain is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are listed. The model implicitly assumes a generic single-orbital spinful Hamiltonian and the validity of the down-folding approximation.

axioms (1)

domain assumption Onsager's relation can be generalized to obtain necessary conditions for optical anomalous Hall effect in this Hamiltonian
Invoked to derive conditions for the effect.

pith-pipeline@v0.9.0 · 5703 in / 1335 out tokens · 51100 ms · 2026-05-19T08:28:18.299685+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.

Using the down-folding method, we identify two types of effective momentum-space magnetism responsible for the optical anomalous Hall conductivity from non-unitary and unitary pairing potentials respectively. ... mUN(k) = iδ0[E + A0(k)][d(k) × d*(k)] ... mU(k) = iδ0A(k) × [d0(k)d*(k) − d*0(k)d(k)]
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

We first employ the time reversal operation T ... σHxy(ω; Δ̂(k)) = −σHxy(ω; Δ̂*(k))

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

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A. Y. Kitaev, Phys. Usp. 44, 131 (2001)

work page 2001

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D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001)

work page 2001

[3] [3]

R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010)

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Asahi and N

D. Asahi and N. Nagaosa, Phys. Rev. B 86, 100504 (2012)

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Nagaosa, J

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