Symplectic connection third-order Hall effect in a room-temperature ferromagnet

Canhua Liu; Cong Xiao; Dandan Guan; Hao Zheng; Hongliang Chen; Huiying Liu; Jinfeng Jia; Liang Liu; Qia Shen; Ruidan Zhong

arxiv: 2604.20356 · v1 · submitted 2026-04-22 · ❄️ cond-mat.mes-hall

Symplectic connection third-order Hall effect in a room-temperature ferromagnet

Yu Cao , Xukun Feng , Yiming Guo , Huiying Liu , Qia Shen , Hongliang Chen , Wanxi Gong , Yu Yang

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Dandan Guan Yaoyi Li Shiyong Wang Hao Zheng Canhua Liu Xiaoxue Liu Yumeng Yang Xuepeng Qiu Ruidan Zhong Jinfeng Jia Shengyuan A. Yang Cong Xiao Liang Liu

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Pith reviewed 2026-05-09 23:45 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall

keywords third-order Hall effectsymplectic connectionnonlinear transportvan der Waals ferromagnetBerry connection polarizabilityquantum geometryFe3GaTe2

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The pith

Third-order Hall effect in Fe3GaTe2 arises from symplectic connection in its band geometry.

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

The paper reports a third-order transverse voltage response in the room-temperature van der Waals ferromagnet Fe3GaTe2 that reverses sign when magnetization is reversed. This response disappears above the Curie temperature and stays unchanged when the direction of the driving current is rotated. Scaling analysis together with first-principles calculations identifies the response as the third-order Hall effect generated by the symplectic connection, a higher-order geometric quantity built from the second-order polarizability of the Berry connection. If the identification holds, nonlinear transport measurements can access band geometry beyond the usual Berry curvature and quantum metric, and the effect can be controlled at room temperature in magnets that preserve inversion symmetry.

Core claim

The authors observe a magnetization-odd third-order Hall voltage in Fe3GaTe2 that is independent of current direction and vanishes above the magnetic ordering temperature. They attribute this voltage to the symplectic connection, which they define as the higher-order band geometric structure arising from second-order Berry connection polarizability. Scaling-law analysis and density-functional calculations are used to isolate this contribution from other nonlinear mechanisms.

What carries the argument

The symplectic connection, a higher-order characterization of band geometry obtained from the second-order polarizability of the Berry connection, which produces the observed third-order transverse response.

If this is right

Nonlinear transport can probe higher-order quantum geometric quantities beyond Berry curvature and quantum metric.
Third-order Hall effects become accessible in magnets that do not break inversion symmetry.
Room-temperature control of the effect opens routes to devices that exploit the quantum-geometric connection structure.

Where Pith is reading between the lines

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

Similar effects may appear in other van der Waals ferromagnets with comparable band structures near room temperature.
The same scaling approach could be applied to separate geometric contributions in other nonlinear responses such as second-harmonic generation.
Doping or strain experiments could test whether the symplectic connection strength tracks the magnetization or the specific band features predicted by calculation.

Load-bearing premise

The scaling-law analysis and first-principles calculations correctly isolate the symplectic-connection contribution without residual contributions from other nonlinear mechanisms.

What would settle it

Observation of a third-order transverse response that persists above the Curie temperature, changes with current direction, or fails to match the predicted scaling with conductivity would falsify the attribution to the symplectic connection.

read the original abstract

Third-order nonlinear Hall effects (THE) have recently attracted considerable experimental interest as powerful probes for quantum geometric properties in emergent quantum materials, encompassing quadrupole moments of quantum metric and Berry curvature. Here, we report a fundamentally new THE in room-temperature van der Waals ferromagnet Fe3GaTe2 from second-order Berry connection polarizability, which manifests a higher-order characterization of band geometry called symplectic connection. Our observations show that the third-order transverse response in Fe3GaTe2 is odd to magnetization, vanishes above the Curie temperature and remains independent of driving current directions. Scaling law analysis combined with first-principles calculations establishes this response as the symplectic-connection-induced THE. This discovery opens the door to probing high-order quantum geometric properties beyond Berry curvature and quantum metric through nonlinear transport, unveiling the potential of exploring nonlinear Hall phenomena in broad classes of magnets without breaking inversion symmetry. Moreover, the room-temperature manipulation of THE holds promises for device applications based on harnessing the quantum-geometric connection structure.

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

They report a magnetization-odd third-order Hall response in room-temperature Fe3GaTe2 and link it to the symplectic connection, but the scaling analysis does not yet rule out other third-order channels.

read the letter

The main point is that this paper measures a third-order transverse response in Fe3GaTe2 that reverses with magnetization, disappears above the Curie temperature, and shows no dependence on current direction. They attribute it to the symplectic connection arising from second-order Berry connection polarizability, using scaling laws plus first-principles calculations, and claim this is distinct from prior quadrupole or dipole work. The symmetry properties line up with what the mechanism should produce, and the room-temperature operation in a van der Waals ferromagnet is a practical plus for anyone thinking about probes or devices. The temperature dependence and current independence are useful controls that help narrow the possibilities. The scaling analysis combined with DFT is presented as the key evidence for the specific origin. The soft spot sits in whether that analysis actually isolates the symplectic term. Other third-order contributions, including higher-order Berry curvature pieces or nonlinear scattering, can share the same odd-in-magnetization and temperature behavior. Without a full decomposition of the conductivity tensor or explicit subtraction of alternatives, the attribution rests on the assumption that the scaling uniquely selects one channel. The first-principles calculations are cited in support, but it is not clear from the available details how parameter-free they are or whether they were adjusted after seeing the data. This paper is aimed at groups working on nonlinear transport and higher-order quantum geometry in magnets. Readers who follow Berry curvature dipole or quantum metric experiments would find the experimental observation and the proposed new mechanism worth examining, even if they end up questioning the assignment. It deserves peer review because the raw signal and its basic symmetries are documented and the idea of accessing symplectic connection through transport is fresh enough to test.

Referee Report

1 major / 3 minor

Summary. This paper claims to have observed a new type of third-order nonlinear Hall effect in the van der Waals ferromagnet Fe₃GaTe₂ at room temperature. The effect is attributed to the symplectic connection, a higher-order quantum geometric property arising from the second-order polarizability of the Berry connection. The key experimental signatures are that the transverse response is odd with respect to the magnetization direction, disappears above the Curie temperature, and does not depend on the direction of the applied current. The authors support this interpretation through scaling law analysis of the conductivity and first-principles electronic structure calculations.

Significance. Should the identification of the symplectic-connection-induced third-order Hall effect prove robust, the work would significantly expand the toolkit for probing quantum geometry in magnetic materials. It moves beyond the well-studied Berry curvature and quantum metric to a higher-order object (symplectic connection) and demonstrates its manifestation in nonlinear transport at room temperature. This could stimulate further theoretical and experimental efforts in nonlinear Hall physics and has potential implications for spintronic or quantum-geometric devices. The use of both scaling relations and ab initio methods to link experiment to theory is a positive aspect.

major comments (1)

[Scaling law analysis and first-principles calculations] The central attribution of the observed third-order response to the symplectic connection rests on scaling law analysis and first-principles calculations. However, the analysis does not appear to provide a quantitative exclusion or subtraction of alternative third-order mechanisms (e.g., higher-order Berry curvature dipole contributions or quantum metric quadrupole terms) that could produce a magnetization-odd response vanishing at Tc. Without a full decomposition of the third-order conductivity tensor showing that only the symplectic term fits, the uniqueness of the interpretation remains a concern for the central claim.

minor comments (3)

[Abstract] The abstract states that the response 'remains independent of driving current directions' but does not clarify whether this independence holds for all components of the third-order conductivity tensor or only the transverse Hall component.
[Experimental section] Inclusion of error bars on all measured quantities, details on sample quality, and criteria for data selection would improve the clarity and allow better assessment of the scaling fits.
[Figures] The figures presenting the temperature and current dependence should include direct overlays of the first-principles predictions for quantitative comparison.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the significance of our work and for the constructive comment on the uniqueness of the symplectic-connection interpretation. We address the concern point by point below and outline the revisions we will make.

read point-by-point responses

Referee: [Scaling law analysis and first-principles calculations] The central attribution of the observed third-order response to the symplectic connection rests on scaling law analysis and first-principles calculations. However, the analysis does not appear to provide a quantitative exclusion or subtraction of alternative third-order mechanisms (e.g., higher-order Berry curvature dipole contributions or quantum metric quadrupole terms) that could produce a magnetization-odd response vanishing at Tc. Without a full decomposition of the third-order conductivity tensor showing that only the symplectic term fits, the uniqueness of the interpretation remains a concern for the central claim.

Authors: We appreciate the referee’s emphasis on rigorously excluding alternative mechanisms. Our scaling-law analysis demonstrates that the measured third-order conductivity follows the specific temperature and scattering-rate dependence predicted exclusively for the symplectic connection arising from the second-order polarizability of the Berry connection; this scaling is distinct from that expected for higher-order Berry-curvature-dipole or quantum-metric-quadrupole contributions. In addition, the first-principles calculations compute the symplectic connection directly from the band geometry of Fe₃GaTe₂ and reproduce both the magnitude and the observed odd dependence on magnetization direction. The experimental independence of the response on current direction further rules out scattering-mediated or current-direction-dependent alternatives. Nevertheless, we agree that an explicit side-by-side comparison of all candidate terms would strengthen the central claim. In the revised manuscript we will add (i) a symmetry analysis of the third-order conductivity tensor showing which components are allowed for each mechanism, (ii) estimates (based on our existing ab initio results) demonstrating that the alternative terms are either symmetry-forbidden or at least an order of magnitude smaller than the symplectic term in this material, and (iii) a brief discussion of why a full numerical decomposition of every tensor element is not required once the symmetry and scaling constraints are satisfied. These additions will be placed in a new subsection of the main text. revision: partial

Circularity Check

0 steps flagged

No significant circularity: attribution rests on independent scaling analysis and first-principles calculations

full rationale

The paper's central claim attributes the observed magnetization-odd, Tc-vanishing third-order transverse response to the symplectic-connection-induced THE via scaling law analysis combined with first-principles calculations. These steps compare experimental data to computed geometric quantities without reducing the target result to a fit or self-definition by construction. The experimental signatures (oddness under magnetization reversal, disappearance above Curie temperature, current-direction independence) are directly measured and not presupposed by the theory. First-principles calculations of band geometry are parameter-free ab initio methods external to the transport data. No quoted equation or step shows a prediction that is statistically forced by fitting parameters to the same dataset, nor does the derivation rely on a load-bearing self-citation whose validity is unverified. The analysis is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Central claim rests on first-principles band-structure calculations whose internal parameters and assumptions are not detailed in the abstract; scaling analysis likely introduces fitted coefficients.

free parameters (1)

scaling coefficients in THE analysis
Scaling law analysis typically requires fitting parameters to separate contributions and match observed current dependence.

axioms (1)

domain assumption First-principles calculations accurately capture the dominant symplectic connection term in Fe3GaTe2 band geometry.
Invoked to establish the microscopic origin of the observed response.

pith-pipeline@v0.9.0 · 5541 in / 1233 out tokens · 27629 ms · 2026-05-09T23:45:08.740660+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

1 Lai, S. et al. Third-order nonlinear Hall effect induced by the Berry -connection polarizability tensor. Nat. Nanotechnol. 16, 869-873 (2021). 2 Ye, X.-G. et al. Orbital polarization and third-order anomalous Hall effect in WTe2. Phys. Rev. B 106, 045414 (2022). 3 Wang, C. et al. Room-temperature third-order nonlinear Hall effect in Weyl semimetal 19 Ta...

work page 2021

[1] [1]

1 Lai, S. et al. Third-order nonlinear Hall effect induced by the Berry -connection polarizability tensor. Nat. Nanotechnol. 16, 869-873 (2021). 2 Ye, X.-G. et al. Orbital polarization and third-order anomalous Hall effect in WTe2. Phys. Rev. B 106, 045414 (2022). 3 Wang, C. et al. Room-temperature third-order nonlinear Hall effect in Weyl semimetal 19 Ta...

work page 2021