Spin-orbit-enabled Fermi-surface splitting in noncollinear antiferromagnetic SmBi

Aifeng Wang; Dong-Hui Xu; Honghao Wan; Huixia Fu; Jingyu Li; Long Zhang; Ming Cheng; Mingquan He; Xiaoyuan Zhou; Xuan Luo

arxiv: 2606.22049 · v1 · pith:G7TW6AC7new · submitted 2026-06-20 · ❄️ cond-mat.mtrl-sci · cond-mat.str-el

Spin-orbit-enabled Fermi-surface splitting in noncollinear antiferromagnetic SmBi

Long Zhang , Ming Cheng , Jingyu Li , Honghao Wan , Xiaoyuan Zhou , Mingquan He , Aifeng Wang , Yuping Sun

show 5 more authors

Dong-Hui Xu Huixia Fu Youguo Shi Xuan Luo Yisheng Chai

This is my paper

Pith reviewed 2026-06-26 11:52 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cond-mat.str-el

keywords spin-orbit couplingnoncollinear antiferromagnetFermi surface splittingSmBiquantum oscillationsmagnetic symmetryantiferromagnetic transitions

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The pith

Noncollinear antiferromagnetic order in SmBi splits its Fermi surfaces only when combined with spin-orbit coupling.

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

The paper shows that spin-split Fermi surfaces in compensated antiferromagnets need not arise from nonrelativistic magnetism alone. In SmBi, noncollinear order breaks global parity-time symmetry yet leaves residual spin-space symmetries that keep bands twofold degenerate without spin-orbit coupling. Only when SOC is added do these protections disappear, locking spin to the lattice and producing measurable Fermi-surface changes. Quantum oscillations detect new branches below the first antiferromagnetic transition and further reconstruction below the second, while the isostructural compound SmSb shows no such evolution. This establishes a cooperative relativistic route to spin splitting that extends unconventional magnetism to systems lacking nonrelativistic degeneracy lifting.

Core claim

For the candidate noncollinear orders of SmBi, breaking global parity-time symmetry is insufficient in the nonrelativistic limit because residual spin-space symmetries protect twofold band degeneracy; conversely, SOC alone cannot lift the degeneracy of the centrosymmetric paramagnetic phase. Only the coexistence of noncollinear order and SOC locks spin to the lattice and removes the residual protection. This is observed through new oscillation branches that emerge below TN and undergo further reconstruction below T*, detected via ultrahigh-sensitivity ac magnetostriction, with supporting magnetic-symmetry analysis and first-principles calculations.

What carries the argument

Magnetic symmetry analysis of candidate noncollinear antiferromagnetic structures, which identifies residual spin-space symmetries that persist after parity-time breaking and are removed only by adding spin-orbit coupling.

If this is right

New quantum-oscillation branches appear below the antiferromagnetic transition TN in SmBi.
Further Fermi-surface reconstruction occurs below the second transition T*.
No comparable Fermi-surface changes occur in isostructural SmSb across its magnetic transitions.
Spin splitting requires the joint presence of noncollinear order and SOC rather than either ingredient alone.

Where Pith is reading between the lines

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

The same cooperative mechanism may operate in other rare-earth pnictides whose magnetic structures leave analogous residual symmetries in the nonrelativistic limit.
Materials discovery could target compounds with known noncollinear order to engineer spin-split surfaces without net magnetization.
Neutron diffraction that confirms or rules out the candidate structures would directly test the symmetry analysis.

Load-bearing premise

The assumed candidate noncollinear magnetic structures for SmBi are correct, and the residual spin-space symmetries identified in the nonrelativistic limit are the only protections that remain after breaking global parity-time symmetry.

What would settle it

First-principles calculations including SOC for the proposed noncollinear structures of SmBi show no lifting of twofold band degeneracy, or quantum-oscillation measurements detect no new branches or reconstruction across the magnetic transitions.

read the original abstract

Spin-split electronic structures in compensated antiferromagnets are commonly sought in the nonrelativistic limit, where magnetic order lifts spin degeneracy without spin-orbit coupling (SOC). Whether SOC can instead be the indispensable symmetry-breaking ingredient remains largely unexplored. Here we combine quantum oscillations detected by ultrahigh-sensitivity ac magnetostriction, magnetic-symmetry analysis and first-principles calculations to resolve the bulk Fermi-surface evolution of SmBi across two successive antiferromagnetic (AFM) transitions. New oscillation branches emerge below TN and undergo a further reconstruction below T*, whereas isostructural SmSb shows no comparable change. For the candidate noncollinear orders of SmBi, breaking global parity-time symmetry is insufficient in the nonrelativistic limit because residual spin-space symmetries protect twofold band degeneracy; conversely, SOC alone cannot lift the degeneracy of the centrosymmetric paramagnetic (PM) phase. Only the coexistence of noncollinear order and SOC locks spin to the lattice and removes the residual protection. SmBi therefore realizes a cooperative, relativistic route to spin-split Fermi surfaces, broadening unconventional magnetism beyond systems whose splitting is already present in the nonrelativistic limit.

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

SmBi shows Fermi-surface reconstruction at its AFM transitions that the authors tie to a SOC-dependent splitting enabled only by noncollinear order, but the argument rests on unconfirmed candidate structures.

read the letter

The main thing to know is that SmBi develops new quantum-oscillation branches below TN and T* that SmSb lacks, and the authors link this to spin splitting that needs both the noncollinear AFM order and SOC because residual spin-space symmetries otherwise protect degeneracy.

They combine ultrahigh-sensitivity magnetostriction data, symmetry analysis on candidate structures, and DFT to make the case. The direct comparison to the isostructural SmSb is useful for showing the effect is tied to the specific magnetism in SmBi rather than generic AFM order.

The symmetry point that global PT breaking is not enough in the nonrelativistic limit is laid out plainly and matches the candidate orders they consider. That framing is the clearest part of the work.

The soft spot is exactly what the stress test flags: the structures are labeled candidates, and the oscillation data confirm reconstruction without independently fixing propagation vectors or moment directions. If a different noncollinear PT-breaking order keeps extra symmetries, the cooperative SOC requirement does not follow. Without the full fitting details or calculation parameters it is also hard to judge how robust the DFT matches are.

This is for groups working on spin-split bands in compensated antiferromagnets. The experimental observation plus the symmetry argument is enough to send it to referees, though the structure assumption will need checking.

Referee Report

1 major / 2 minor

Summary. The manuscript reports quantum-oscillation measurements via ac magnetostriction on SmBi that reveal new Fermi-surface branches below TN and further reconstruction below T*, absent in isostructural SmSb. Magnetic-symmetry analysis and DFT calculations on candidate noncollinear antiferromagnetic structures are used to argue that global PT breaking is insufficient in the nonrelativistic limit because residual spin-space symmetries protect twofold degeneracy; only the joint presence of noncollinear order and SOC lifts this protection, realizing a cooperative relativistic route to spin-split Fermi surfaces.

Significance. If substantiated, the work identifies a distinct mechanism for spin splitting in compensated antiferromagnets that requires both magnetic order and SOC, thereby extending the landscape of unconventional magnetism beyond purely nonrelativistic cases. The experimental detection of Fermi-surface reconstruction supplies an external benchmark against which the symmetry and DFT interpretations can be tested.

major comments (1)

[Symmetry analysis and candidate structures] The central claim that residual spin-space symmetries protect degeneracy for the candidate noncollinear orders in the nonrelativistic limit (and that SOC is required to remove it) is load-bearing. The quantum-oscillation data establish Fermi-surface reconstruction but do not independently constrain the magnetic propagation vectors or moment directions; the structures remain labeled “candidate” in the abstract. If an alternative PT-breaking noncollinear structure preserves additional spin-space symmetries even without SOC, the cooperative mechanism is not required. A dedicated discussion or additional experimental constraint on the realized magnetic structure is needed.

minor comments (2)

[Experimental methods] Clarify in the methods or supplementary information the precise fitting procedures and background subtraction used for the ac-magnetostriction oscillation data, including how frequencies were extracted and assigned to Fermi-surface sheets.
[Discussion] The comparison with SmSb is useful; a brief statement on whether the same candidate-structure analysis was performed for SmSb (or why it was not) would aid the reader.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review. The major comment raises an important point about the load-bearing role of the symmetry analysis and the need for clarity on the candidate magnetic structures. We address this below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Symmetry analysis and candidate structures] The central claim that residual spin-space symmetries protect degeneracy for the candidate noncollinear orders in the nonrelativistic limit (and that SOC is required to remove it) is load-bearing. The quantum-oscillation data establish Fermi-surface reconstruction but do not independently constrain the magnetic propagation vectors or moment directions; the structures remain labeled “candidate” in the abstract. If an alternative PT-breaking noncollinear structure preserves additional spin-space symmetries even without SOC, the cooperative mechanism is not required. A dedicated discussion or additional experimental constraint on the realized magnetic structure is needed.

Authors: We agree that the quantum-oscillation data establish Fermi-surface reconstruction but do not independently constrain the magnetic propagation vectors or moment directions; this is why the structures are labeled “candidate” in the abstract and throughout the text. The candidates we analyze are the noncollinear antiferromagnetic orders consistent with the two observed transitions (TN and T*) in SmBi and with the rock-salt lattice symmetry. Using magnetic space-group and spin-space-group analysis, we show that each of these candidates breaks global PT symmetry yet retains residual spin-space symmetries (e.g., combined spin rotations and lattice translations) that enforce twofold degeneracy in the nonrelativistic limit. SOC is required to break these residual symmetries. While an exhaustive enumeration of every conceivable PT-breaking noncollinear structure is impractical, no alternative structure that would preserve additional spin-space symmetries without SOC has been reported for this material class or is compatible with the experimental observations (including the absence of analogous reconstruction in isostructural SmSb). To make this reasoning fully transparent, we will add a dedicated subsection that (i) lists the candidate structures, (ii) tabulates their residual spin-space symmetries in the nonrelativistic limit, and (iii) demonstrates why the cooperative SOC + noncollinear order mechanism is required. This addresses the request for a dedicated discussion. revision: yes

Circularity Check

0 steps flagged

No significant circularity; symmetry analysis and DFT interpret external quantum-oscillation data

full rationale

The paper's derivation chain consists of (i) experimental detection of new oscillation branches below TN and T* via ac magnetostriction, (ii) standard magnetic-symmetry analysis applied to explicitly labeled 'candidate' noncollinear structures, and (iii) first-principles DFT to interpret the observed Fermi-surface reconstruction. The key statement that 'only the coexistence of noncollinear order and SOC locks spin to the lattice' follows deductively from group-theoretic enumeration of residual spin-space symmetries in the nonrelativistic limit versus the SOC-inclusive case; it does not reduce to a fitted parameter, a self-citation chain, or an ansatz smuggled from prior work. Quantum-oscillation frequencies supply an independent external benchmark that is not reproduced by construction from the symmetry assumptions. No load-bearing step equates its output to its input by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.1-grok · 5774 in / 1084 out tokens · 23233 ms · 2026-06-26T11:52:54.549633+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

31 extracted references · 1 linked inside Pith

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[26]

Wang, J.-F. et al. Quantum oscillations in the magnetic Weyl semimetal NdAlSi arising from strong Weyl fermion-4f electron exchange interaction. Phys. Rev. B 108, 024423 (2023)

2023
[27]

Sakhya, A. P. et al. Behavior of gapped and ungapped Dirac cones in the antiferromagnetic topological metal SmBi. Phys. Rev. B 106, 085132 (2022)

2022
[28]

Kushnirenko, Y. et al. Rare-earth monopnictides: Family of antiferromagnets hosting magnetic Fermi arcs. Phys. Rev. B 106, 115112 (2022)

2022
[29]

Yuan, M. et al. Collective resonance of superconducting/normal domain walls in the intermediate state of type-I superconductor. Preprint at https://arxiv.org/abs/2604.17333 (2026)

Pith/arXiv arXiv 2026
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Song, Z. et al. Constructions and applications of irreducible representations of spin-space groups. Phys. Rev. B 111, 134407 (2025)

2025
[31]

benchmark

Huan, S. et al. Magnetotransport evidence for the nontrivial topological states in the fully spin-polarized Kondo semimetal CeBi. J. Alloys Compd. 875, 159993 (2021). Methods Crystal growth Single crystals of SmBi were grown by a self-flux method. Ac magnetostriction measurements Ac magnetostriction-coefficient measurements were performed using a composit...

2021

[1] [1]

Baltz, V. et al. Antiferromagnetic spintronics. Rev. Mod. Phys. 90, 015005 (2018)

2018

[2] [2]

& Jungwirth, T

Šmejkal, L., Sinova, J. & Jungwirth, T. Beyond conventional ferromagnetism and antiferromagnetism: a phase with nonrelativistic spin and crystal rotation symmetry. Phys. Rev. X 12, 031042 (2022)

2022

[3] [3]

& Jungwirth, T

Šmejkal, L., Sinova, J. & Jungwirth, T. Emerging research landscape of altermagnetism. Phys. Rev. X 12, 040501 (2022)

2022

[4] [4]

Krempaský, J. et al. Altermagnetic lifting of Kramers spin degeneracy. Nature 626, 517– 522 (2024)

2024

[5] [5]

Zhu, Y.-P. et al. Observation of plaid-like spin splitting in a noncoplanar antiferromagnet. Nature 626, 523–528 (2024)

2024

[6] [6]

Jiang, B. et al. A metallic room-temperature d-wave altermagnet. Nat. Phys. 21, 754–759 (2025)

2025

[7] [7]

Yamada, R. et al. A metallic p-wave magnet with commensurate spin helix. Nature 646, 837–842 (2025)

2025

[8] [8]

Liu, Y. et al. Symmetry classification of magnetic orders using oriented spin space groups. Nature 652, 869–873 (2026)

2026

[9] [9]

Wu, Z. et al. Fermi surface of RuO₂ measured by quantum oscillations. Phys. Rev. X 15, 031044 (2025)

2025

[10] [10]

Long, M. et al. 3D bulk-resolved g-wave magnetic order parameter symmetry in the metallic altermagnet CrSb. Preprint at https://arxiv.org/abs/2601.14526 (2026). 10

arXiv 2026

[11] [11]

Terashima, T. et al. Altermagnetic spin-split Fermi surfaces in CrSb revealed by quantum oscillation measurements. Preprint at https://arxiv.org/abs/2601.19105 (2026)

arXiv 2026

[12] [12]

Naduvile Thadathil, S. et al. Fermi-surface studies of altermagnetic CrSb from Shubnikov– de Haas oscillations. Preprint at https://arxiv.org/abs/2602.24033 (2026)

arXiv 2026

[13] [13]

Duan, X. et al. Tunable electronic structure and topological properties of LnPn (Ln=Ce, Pr, Sm, Gd, Yb; Pn=Sb, Bi). Commun. Phys. 1, 71 (2018)

2018

[14] [14]

Wu, Z. et al. Probing the origin of extreme magnetoresistance in Pr/Sm mono- antimonides/bismuthides. Phys. Rev. B 99, 035158 (2019)

2019

[15] [15]

Wu, F. et al. Anomalous quantum oscillations and evidence for a non-trivial Berry phase in SmSb. npj Quantum Mater. 4, 20 (2019)

2019

[16] [16]

Zhang, W. et al. Tunable non-Lifshitz–Kosevich temperature dependence of Shubnikov– de Haas oscillation amplitudes in SmSb. npj Quantum Mater. 8, 55 (2023)

2023

[17] [17]

P., Paulose, P

Sakhya, A. P., Paulose, P. L., Thamizhavel, A. & Maiti, K. Evidence of nontrivial Berry phase and Kondo physics in SmBi. Phys. Rev. Mater. 5, 054201 (2021)

2021

[18] [18]

Zhang, Y. et al. Observation of enhanced ferromagnetic spin-spin correlations at a triple point in quasi-two-dimensional magnets. Phys. Rev. B 107, 134417 (2023)

2023

[19] [19]

Mi, X. et al. Precisely tracking critical spin fluctuations using the ac magnetostriction coefficient: a case study on the Kitaev spin liquid candidate Na₃Co₂SbO₆. Phys. Rev. B 111, 014417 (2025)

2025

[20] [20]

Zhang, L. et al. Magnetic phase diagram of Cr₂Te₃ revisited by ac magnetostrictive coefficient. Appl. Phys. Lett. 127, 012413 (2025)

2025

[21] [21]

Zhang, L. et al. Comprehensive investigation of quantum oscillations in semimetal using an ac composite magnetoelectric technique with ultrahigh sensitivity. npj Quantum Mater. 9, 11 (2024)

2024

[22] [22]

Lu, Y. et al. Fermi surface structure revealed by quantum oscillations in half-Heusler semimetal LuAuSn. Rare Met. 45, e70109 (2026)

2026

[23] [23]

& Nagamiya, T

Yamamoto, Y. & Nagamiya, T. Spin arrangements in magnetic compounds of the rocksalt crystal structure. J. Phys. Soc. Jpn. 32, 1248 (1972)

1972

[24] [24]

& Rüegg, K

Hulliger, F., Natterer, B. & Rüegg, K. Low-temperature properties of Samarium monopnictides. Z. Phys. B 32, 37 (1978). 11

1978

[25] [25]

Kouvel, J. S. & Kasper, J. S. Long-range antiferromagnetism in disordered Fe-Ni-Mn alloys. J. Phys. Chem. Solids 24, 529 (1963)

1963

[26] [26]

Wang, J.-F. et al. Quantum oscillations in the magnetic Weyl semimetal NdAlSi arising from strong Weyl fermion-4f electron exchange interaction. Phys. Rev. B 108, 024423 (2023)

2023

[27] [27]

Sakhya, A. P. et al. Behavior of gapped and ungapped Dirac cones in the antiferromagnetic topological metal SmBi. Phys. Rev. B 106, 085132 (2022)

2022

[28] [28]

Kushnirenko, Y. et al. Rare-earth monopnictides: Family of antiferromagnets hosting magnetic Fermi arcs. Phys. Rev. B 106, 115112 (2022)

2022

[29] [29]

Yuan, M. et al. Collective resonance of superconducting/normal domain walls in the intermediate state of type-I superconductor. Preprint at https://arxiv.org/abs/2604.17333 (2026)

Pith/arXiv arXiv 2026

[30] [30]

Song, Z. et al. Constructions and applications of irreducible representations of spin-space groups. Phys. Rev. B 111, 134407 (2025)

2025

[31] [31]

benchmark

Huan, S. et al. Magnetotransport evidence for the nontrivial topological states in the fully spin-polarized Kondo semimetal CeBi. J. Alloys Compd. 875, 159993 (2021). Methods Crystal growth Single crystals of SmBi were grown by a self-flux method. Ac magnetostriction measurements Ac magnetostriction-coefficient measurements were performed using a composit...

2021