2026年3月1日日曜日

Suspicious Topological Hall Signals

Introduction

   There is a phenomenon known as the topological Hall effect (THE). When a spin texture becomes noncoplanar and acquires a finite scalar spin chirality, conduction electrons experience an emergent (fictitious) magnetic field, which can generate an additional Hall response beyond the ordinary and anomalous contributions. In several materials, experimental observations are consistent with this picture.

   Motivated by such examples, many studies have measured Hall responses in various magnetic materials and attempted to extract a THE component to claim that it is a "clear" indication of unconventional, complicated, and more often topologically nontrivial magnetic structures (, and they are happy because they can publish papers). At the same time, a growing number of works have raised cautions. It would be highly plausible that many reported “THE” signals can be explained without hypothesizing noncoplanar spin configurations. When such possibilities are not carefully examined, the field becomes a kind paradise of “inverse Occam’s razors”. In this informal layperson's blog, I try to summarize practical pitfalls and provide a list of materials where interpretation deserves re-examination.

Topological Hall ansatz

   Let me start with a summary of common (, but not all) origins of “fake” topological Hall signals. This can be better understood by looking at the widely used empirical anomalous Hall effect ansatz for the Hall resistivity \(\rho_{H}\) (, well I won't go into the details of the notorious misconception about \(\rho_{xy}\) vs. \(\rho_{yx}\) as I expect the readers of the blog are smart enough to understand the difference between them):

\(\rho_{H}(H_{\text{ext}})=\rho_{H}^O+\rho_{H}^A+\rho_{H}^T=R_0B+R_sM_z+\rho_{H}^T\),

where \(H_{\text{ext}}\) is the external magnetic field, \(R_0\) is the \(B\)-linear (, not \(H_{\text{ext}}\)-linear) ordinary Hall coefficient, \(B\) is the magnetic flux density in the sample, \(R_s\) is an \(M\)-linear anomalous Hall coefficient, \(M_z\) is the out-of-plane magnetization component, and \(\rho_{H}^T\) is the residual component not captured by the first two terms. In literature, \(\rho_{H}^T\) is sometimes referred to as the topological Hall contribution, a nonlinear Hall component, or an unconventional Hall effect, depending on the context. 

1. Demagnetization effect

   In general, the relevant internal field differs from the applied field due to demagnetization. For an ellipsoidal approximation, 

\(H_{\text{int}}=H_{\text{ext}}-N_{\text{D}}M\),

and the flux density is

\(B=\mu_0(H_{\text{int}}+M)=\mu_0H_{\text{ext}}-N_{\text{D}}M+M\),

where \(N_{\text{D}}\) is the demagnetization factor, which can be approximately estimated from the dimensions of the sample when the sample shape is simple. If \(M\) and \(\rho_{yx}\) are compared using different samples of different shapes, \(B\) can be different at the same \(H_{\text{ext}}\), producing systematic discrepancies that may appear in \(\rho_{H}^T\) as an "extra" Hall signal.

   Practically, this discrepancy can be suppressed by using the same sample for magnetization and transport measurements. We note that the magnetization and transport measurements are two independent experiments and it's quite hard to avoid systematic discrepancies in measurement conditions such as field misalignment, temperature differences, and sample degradation. We may not be able to expect that the simulation of ordinary and anomalous Hall contributions by the first two terms in the above ansatz perfectly matches with the measured \(\rho_{H}\) when \(\rho_H^T\) is absent. It is insane to magnify the subtracted data \(\rho_{H}-\rho_{H}^O-\rho_{H}^A\), claiming the unconventional, nontrivial, and/or topological mechanism hiding behind it. No, it is highly likely an artifact.

2. \(R_0\) is not a constant

   Assuming \(R_0\) as a constant is often unjustified. Field-dependent ordinary Hall effect with curvature can arise from, e.g., multicarrer effect, Fermi-surface anisotropy, mobility and/or carrier density changes (across metamagnetism), magnetic breakdown, Landau quantization, polaron formation, or metal-insulator crossovers.

3. Choice of scaling ansatz of \(R_s\)

   Practically, \(R_s\) is often expected to be modelled by a simple function of longitudinal resistivity (\(\rho\)), i.e., a scaling ansatz. For example, under an intrinsic AHE-dominated case, one may write

\(R_s=\rho_{xx}^2S_H\),

with a constant \(S_{H}\) (, well this is also an assumption, and sometimes not reliable, see #4). This is based on the assumption that the systems is isotropic with respect to the \(xy\) plane. In the Hall channel \(\rho_{xz}\) in a uniaxial system, for example, one should use

\(R_s=\rho_{xx}\rho_{zz}S_H\),

that requires independent measurements of \(\rho_{xx}\) and \(\rho_{zz}\) and involves the demagnetization correction for samples in different shape.

   When multiple mechanisms including intrinsic and extrinsic skew/side-jump contributions coexist, a single scaling ansatz for \(R_s\) is not valid and analysis becomes ambiguous by the increasing number of fitting parameters.

4. \(R_s\) is a function of magnetic structure

   It is also a too simplified assumption that \(\rho_{H}^A\) is a function of \(\rho_{xx}\) and only scales with the \(z\) component of \(\vec{M}\). AHE highly depends on the electronic structure and magnetic structures. It is expected that different magnetic structures (or different spin cant angles) would provide different \(R_s\). When the system has antiferromagnetic, noncollinear, or modulated orders, the magnetic structural change across the magnetization process potentially modifies the electronic structure, and thus \(R_s\) varies as a function of magnetic field, which cannot be resolved only by considering the scaling in terms of longitudinal \(\rho\).

Pitfalls of "reverse engineering" of noncoplanar spin texture by \(\rho_{H}^T\)

   Claiming the discovery of the noncoplanar spin texture solely from the "observation" of the THE often ended with misassignment of the underlying spin structure in the corresponding field region. A number of materials eventually were shown not to host a noncoplanar spin configuration such as MnSi thin film (doi.org/10.1103/PhysRevLett.112.059701), MnGe thin film (doi.org/10.1103/PhysRevB.96.220414), and Eu(Al,Ga)4 (doi.org/10.1103/PhysRevB.103.L020405, doi.org/10.1103/PhysRevB.111.165136).

Cant-field hump

Most trivial artifact due to misconception of the relationship between magnetism and anomalous Hall effect as described in B. Rai et al., PRB 112, 035110 (2025).

  • Fe3GeTe2: Angular dependence of the topological Hall effect in the uniaxial van der Waals ferromagnet Fe3⁢GeTe2, Y. You, G. Xu et al., doi.org/10.1103/PhysRevB.100.134441 (2019).

    Magnetoresistance effect

    Magnetoresistance is too large to apply THE ansatz with a field-independent constant \(R_0\). See e.g., Blog2026#4, Blog2025#102

    • EuAs: Above-ordering-temperature large anomalous Hall effect in a triangular-lattice magnetic semiconductor, doi.org/10.1126/sciadv.abl5381 (2021), hump type

    Over-subtraction

    The background curve is not reliable. A tiny residual signal after over-subtraction does not carry meaningful physical information.

    • YMn6Sn6: Competing magnetic phases and fluctuation-driven scalar spin chirality in the kagome metal YMn6Sn6, doi.org/10.1126/sciadv.abe2680 (2020), mathematically unvalidated procedure for verifying the presence of a net scalar spin chirality (particularly, Eq. (S29-31) are mathematically incorrect). 
    • YMn6Sn6: Field-induced topological Hall effect and double-fan spin structure with a 𝑐-axis component in the metallic kagome antiferromagnetic compound Y⁢Mn6⁢Sn6, doi.org/10.1103/PhysRevB.103.014416, same with the above, verification of the scalar spin chirality was ambiguous.
    • GdRhPb, doi.org/10.1103/dx66-ywt9, large MR -60%, intrinsic AHE is not correctly subtracted. \(\rho_{xx}\rho_{zz}S_HM\) should be used instead of \(\rho_{xx}^2S_HM\)


    Unknown

    It's too early to decide. Simple explanations (Occam's razor, as proposed above) are not enough to discard the THE hypothesis, and careful magnetic structure analysis is still needed.

    • Fe3Ge: Giant anomalous Hall effect in the kagome nodal surface semimetal Fe3⁢Ge, S.-X. Li, Z.-A. Xu et al., doi.org/10.1103/f9rs-dswt (2025).

    • Fe3Ge: Large Anomalous and Topological Hall Effect and Nernst Effect in a Dirac Kagome Magnet Fe3Ge, doi.org/10.1002/adfm.202511059 (2025), demagnetization effect was excluded by measuring transport and magnetization with the same sample, however, topological spin configuration is unlikely since residual \(\rho_{yx}\) remains above magnetization saturation.

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      Suspicious Topological Hall Signals