Topological Hall Signal in Magnetism

• Topological Hall signal is an electrical transport phenomenon arising from Berry phases acquired by electrons traversing noncoplanar spin textures.
• It is distinguished from ordinary and anomalous Hall effects by isolating contributions from emergent magnetic fields in systems like skyrmion-hosting magnets.
• Applications include probing emergent electrodynamics in correlated systems and advancing topological spintronic device development.

The topological Hall signal is an emergent electrical transport phenomenon observed in a broad array of magnetic systems possessing noncoplanar spin textures. In materials systems such as skyrmion-hosting magnets, topologically nontrivial chiral spin orders, or even fluctuating spiral states, conduction electrons traversing the magnetic landscape accumulate real-space Berry phases. This geometric phase imparts to the electrons a fictitious Lorentz force analogous to a genuine magnetic field, generating a measurable transverse (Hall) voltage termed the topological Hall effect (THE). The topological Hall signal is rigorously separated from the ordinary Hall effect (OHE) and the anomalous Hall effect (AHE), with the former arising from the applied magnetic field and the latter from the net magnetization. Beyond serving as an indirect electronic signature of skyrmionic and related textures, the topological Hall signal is a sensitive probe of emergent electrodynamics in correlated electron systems, and its reliable identification is essential for the development and implementation of topological spintronic applications.

1. Theoretical Foundations: Real-Space Berry Curvature and Scalar Spin Chirality

The topological Hall effect originates from the coupling of conduction electrons to noncoplanar spin textures, through a real-space Berry curvature—a geometric property of the local spin arrangement. For a triad of spins $S_i, S_j, S_k$ subtending a finite solid angle $\Omega_{ijk}$, one associates a scalar spin chirality

$$\chi_{ijk} = \mathbf{S}_i \cdot (\mathbf{S}_j \times \mathbf{S}_k)\,,$$

such that an electron hopping around this triangle acquires a Berry phase $\Phi_{ijk} = \Omega_{ijk}/2$. In lattice or continuum form, the emergent magnetic field is expressed as

$$\mathbf{B}_\text{eff}(\mathbf{r}) = \frac{\hbar}{2e} \epsilon_{\alpha\beta\gamma} \mathbf{n} \cdot (\partial_\alpha \mathbf{n} \times \partial_\beta \mathbf{n}) \hat{\gamma}\,,$$

where $\mathbf{n}(\mathbf{r})$ is the unit vector field of magnetization. In two-dimensional skyrmion lattices this field is quantized in units of the flux quantum $\Phi_0 = h/e$ per integer winding number. For noncoplanar spin configurations, including conical spirals and chiral magnetic bubbles, a nonzero $\langle \chi_{ijk} \rangle$ produces a finite emergent field and hence a topological Hall signal.

Importantly, this signal is independent of—or additive to—the ordinary and anomalous Hall components, leading to the general decomposition

$$\rho_{xy}(H) = R_0 B + R_s M + \rho_{xy}^\text{T}\,,$$

where $R_0$ is the ordinary Hall coefficient, $R_s$ the anomalous Hall coefficient, $B$ the external field, $M$ the magnetization, and $\rho_{xy}^\text{T}$ the topological Hall contribution (Schlitz et al., 2018, Porter et al., 2013, Meng et al., 2017).

2. Measurement Protocols and Signal Decomposition

Establishing the presence of a topological Hall signal requires careful transport measurement and data analysis to disentangle $\rho_{xy}^\text{T}$ from the background Hall effects. The protocol typically involves:

• Measuring the transverse resistivity $\rho_{xy}(H)$ and longitudinal $M(H)$ over wide field and temperature windows, often simultaneously.
• Antisymmetrizing Hall data to remove misalignment artifacts.
• Fitting the linear (OHE) and monotonic (AHE) parts at high field (well above magnetic saturation), such that $\rho_{xy}(H \gg H_c)$ is dominated by $R_0 B + R_s M$.
• Subtracting this background to extract $\rho_{xy}^\text{T}(H)$ as the residual, commonly observed as a “hump” (positive or negative) localized in a field window associated with noncoplanar spin structures (Li et al., 2021, Meng et al., 2017, He et al., 2020, Baral et al., 18 Feb 2025).

Advanced protocols include angular sweeps (rotating field with respect to current), temperature-dependent maps, and, where available, simultaneous thermoelectric measurements to extract Nernst analogs of the Hall signal (Schlitz et al., 2018). For microstructured or nanopatterned samples, “all-electrical” approaches normalize the Hall and Nernst signals to eliminate the need for independent magnetometry, enhancing the reliability of the topological identification.

Table: Decomposition of Transverse Resistivity

Contribution	Physical Origin	Typical Extraction Method
$\rho_{xy}^{\mathrm{OHE}}$	Lorentz force from external $B$	High-field linear fit
$\rho_{xy}^{\mathrm{AHE}}$	Magnetization, Berry curvature in $k$-space	Correlation with $M(H)$ or square-loop fit
$\rho_{xy}^{\mathrm{T}}$	Real-space Berry curvature (texture)	Residual after OHE+AHE subtraction

3. Material Systems and Microscopic Mechanisms

Skyrmion Crystals and Chiral Magnets

In chiral B20 helimagnets (e.g., FeGe, MnSi, Fe$_{1-x}$Co$_x$Si), Dzyaloshinskii–Moriya interactions stabilize skyrmion lattices over a finite field/temperature window. The topological Hall signal tracks the skyrmion density, scaling as

$$\rho_{xy}^\text{T} = P R_0 B_\text{eff} = P R_0 \Phi_0 n_{sk}\,,$$

where $P$ is the conduction-electron spin polarization, $n_{sk}$ the areal skyrmion density (Porter et al., 2013, Gupta et al., 7 Nov 2025). Direct confirmation is provided by combining Hall measurements with magnetic imaging (Lorentz-TEM, SANS), enabling unambiguous correlation between transport anomalies and underlying topological textures.

Thin-film disorder, epitaxial strain, and compositional tuning all serve as powerful parameters for controlling skyrmion size, density, and thus the magnitude and robustness of $\rho_{xy}^\text{T}$ for technologically relevant devices (Gupta et al., 7 Nov 2025, Porter et al., 2013).

Noncollinear Lattices, Spiral Magnets, and Fluctuation-Driven Effects

Topological Hall signals are also present in systems lacking long-range, static skyrmion order:

• Noncoplanar Spin Spirals: In Fe$_5$Sn$_3$, a conical spin-spiral state generated by competing exchange, DMI, and anisotropy gives rise to a “planar” PTHE. The emergent scalar chirality is maximized when the applied field is nearly parallel to the current, yielding a distinct angular signature and monotonic temperature enhancement owing to increased noncoplanarity at elevated $T$ (Li et al., 2021).
• Fluctuation-Induced Scalar Chirality: For single-$\mathbf{k}$ spiral states such as in Fe$_3$Ga$_4$ or ErMn$_6$Sn$_6$, dynamical breaking of time-reversal symmetry by thermal magnon populations creates a finite $\langle \chi_{ijk}\rangle$ even without multi-$\mathbf{k}$ order. The resulting topological Hall signal shows a strong, often linear $T$-dependence and a phase-space “dome” in $H$–$T$ diagrams (Fruhling et al., 30 Jan 2024, Baral et al., 18 Feb 2025).

Interface and Proximity Phenomena: Insulators and Heterostructures

Electrical detection of topological spin structures in magnetic insulators has been realized via two main mechanisms:

• Spin-Hall Topological Hall Effect (SH-THE): In Pt/ferrimagnetic insulator (TmIG) heterostructures, a spin current from Pt is torqued by the itinerant-skyrmion flux in TmIG, generating a transverse voltage in Pt—the SH-THE. The effect is distinguished by its independence from magnetization-induced proximity AHE in Pt, evident in control studies (Ahmed et al., 2019).
• Interfacial Topological Hall Effect (ITHE): Here, proximity-induced magnetization and chirality at the interface between a heavy metal (Pt) and a noncoplanar magnetic insulator (h-LuFeO$_3$) imprint a persistent topological Hall signal on Pt. Notably, the ITHE is robust over a wide field range and features a Hall-conductivity/magnetization ratio exceeding 2 V$^{-1}$, distinctly larger than conventional AHE (Li et al., 16 Sep 2025).

4. Signal Characteristics: Angular, Temperature, and Gate Dependencies

The topological Hall signal exhibits characteristic dependencies revealing its underlying microscopic origin:

• Angular Dependence: PTHE in Fe$_5$Sn$_3$ peaks near field–current alignment, following $\sin \theta$ behavior below saturation, with maxima at the hard axis where scalar chirality is large (Li et al., 2021). In gate-tunable or field-tilt experiments, the amplitude and even sign of $\rho_{xy}^{\mathrm{T}}$ change predictably with magnetic geometry and control parameters (Wong et al., 2022, Cai et al., 10 Mar 2025).
• Temperature Dependence: The signal can be highly sensitive to $T$, with peak amplitude increasing with $T$ in systems where thermal fluctuations enhance noncoplanarity (Li et al., 2021, Baral et al., 18 Feb 2025), or displaying a narrow $T$-window tracking the topologically ordered phase in clean skyrmion crystals (Gupta et al., 7 Nov 2025).
• Gate and Chemical Control: In proximitized or heterostructured systems (e.g., TI/FI, graphene–CGT-graphene), electrical gating modulates both carrier type/sign and spin polarization, thus enabling electrically sign-reversible THE and direct control over skyrmion density and emergent field (Wong et al., 2022, Cai et al., 10 Mar 2025).

5. Artifacts, Diagnostics, and the Role of Material Inhomogeneity

A central challenge in the field is distinguishing genuine topological Hall signals from artifacts arising from two-component anomalous Hall effects or domain wall scattering. Several rigorous criteria have emerged:

• Minor-Loop and History-Dependence Tests: Genuine THE is strictly reproducible in minor and major loops, whereas two-AHE superpositions display partial loops that break outside the main loop or present history dependence (Gerber, 2018, Tai et al., 2021).
• Temperature and Gate Dependence: True THE sits inside the AHE transition and is insensitive to smooth sign reversals with $T$ or $V_g$, while artifact humps/dips track changes in overall AHE loop sign or merge/disappear across phase transitions (Tai et al., 2021, Sabri et al., 11 Nov 2024).
• Microscopic Imaging and Correlation: Co-registration of $\rho_{xy}^{\mathrm{T}}$ with direct images of real-space spin textures via Lorentz TEM, MFM, or MOKE microscopy remains essential for unambiguous assignment (Gerber, 2018, Tai et al., 2021).

Domain-wall resistance in multidomain ferromagnets, or compositional inhomogeneity yielding spatially distinct AHE coefficients, can fully reproduce THE-like signatures in resistor-network models—all without any real-space Berry phase (Sabri et al., 11 Nov 2024, Wu et al., 2018, Gerber, 2018). Thus, comprehensive experimental protocols and multipronged diagnostic analysis are required to conclusively establish a topological Hall origin.

6. Applications and Outlook in Spintronics

The topological Hall effect serves both as a probe of emergent electrodynamics and as a functional element for spintronic technologies:

• Skyrmion-Based Memory and Logic: The ability to electrically detect, manipulate, and control skyrmion populations via $\rho_{xy}^{\mathrm{T}}$ underpins proposals for racetrack memories, logic devices, and neuromorphic applications (Gupta et al., 7 Nov 2025, Meng et al., 2017).
• Insulating Systems: SH-THE and ITHE mechanisms expand the technological landscape to ferrimagnetic and antiferromagnetic insulators, offering reduced losses and enhanced speed/efficiency for device integration (Ahmed et al., 2019, Li et al., 16 Sep 2025).
• Dynamic and Fluctuation-Driven Phenomena: Harnessing fluctuation-induced THE opens avenues for magnonic devices, energy harvesting, and topologically driven electronic response at and above room temperature (Baral et al., 18 Feb 2025, Fruhling et al., 30 Jan 2024).

Device implementation requires precise engineering of material parameters (DMI, anisotropy, carrier concentration, disorder) to optimize the magnitude, robustness, and controllability of the topological Hall signal, as inferred from systematic experimental investigation and sophisticated theoretical modeling across a wide variety of materials platforms.

7. Summary Table: Key Topological Hall Signal Characteristics in Notable Systems

System/Phenomenon	Key Origin	Typical Max $\rho_{xy}^{\text{T}}$	Distinctive Features
B20 Chiral Magnet (FeGe, Fe$_{0.7}$Co$_{0.3}$Si)	Skyrmion lattice	80–820 n$\Omega$·cm (Porter et al., 2013, Gupta et al., 7 Nov 2025)	Finite $T$/$H$ window, sensitive to disorder
Fe$_5$Sn$_3$	Conical spiral-PTHE	T-dependent peak (Li et al., 2021)	Maximal near $H\parallel I$; monotonic $T$ dependence
Mn$_{1.8}$PtSn, MnBi	Noncoplanar texture/skyrmion bubbles	8 n$\Omega$·m, 0.02 $\mu\Omega$·cm (Schlitz et al., 2018, He et al., 2020)	All-electrical diagnostic possible; size/thickness crossover
Pt/TmIG, Pt/h-LuFeO$_3$	SH-THE/ITHE, MPE-induced chirality	SH-THE: 3.9 n$\Omega$·cm (Ahmed et al., 2019); ITHE: $0.5\%~\rho_{xx}$ (Li et al., 16 Sep 2025)	Interfacial nature, insulating substrate, robust over wide field
Fluctuation-driven (Fe$_3$Ga$_4$, ErMn$_6$Sn$_6$)	Chiral magnon imbalance	0.03–0.04 $\mu\Omega$·cm (Baral et al., 18 Feb 2025, Fruhling et al., 30 Jan 2024)	Linear $T$ dependence, no static skyrmions needed

In conclusion, the topological Hall signal is a highly sensitive, yet nontrivial, signature of real-space magnetic topology. Accurate measurement, decomposition, and interpretation—grounded in direct structural and magnetic characterization—are essential for progress in topological spintronics, materials design, and the broader understanding of Berry-phase physics in correlated electron systems.