## Introduction

Magnetic skyrmions are topologically charged nanoscale spin textures that form due to competition between spin-rotating and spin-aligning magnetic interactions. In thin-film heterostructures, these magnetic interactions can be finely tuned via the multilayer geometry and composition, rendering skyrmion-hosting films ideally suited for spintronic applications1,2,3,4,5. One promising route towards functionalizing skyrmions in metallic systems is to utilize their intrinsic magnetoelectric coupling, which is manifested by a topological Hall effect (THE)6,7,8. Charge carriers moving through a skyrmion spin texture experience an emergent magnetic field (Beff) associated with the spin winding of a skyrmion. The transverse deflection of charge carriers interacting with Beff results in the THE7. Provided that each skyrmion remains stationary with respect to incident charge carriers, an array of skyrmions will exhibit a topological Hall resistivity ρTHE, given by

$${\rho }^{\mathrm{THE}}=P\cdot {R}_{0}^{\prime}\cdot ({n}_{\mathrm{sk}}\cdot {{{\Phi }}}_{0}),$$
(1)

where P is the spin polarization of the charge carriers, $${R}_{0}^{\prime}$$ is the Hall coefficient representing the effective charge density contributing to the THE (usually taken as the classical Hall coefficient R0), Beff ≡ nsk Φ0 is the emergent field associated with a given skyrmion density nsk, and Φ0 = h/e is the magnetic flux quantum, with h Planck’s constant and −e the electron charge. This phenomenon is distinct from the classical and anomalous Hall effects, which are proportional to the applied magnetic field H and magnetization M(H), respectively7. Experimentally, ρTHE(H) can be identified as the residual between the total measured Hall resistivity ρyx(H) and a fit to the classical R0(H) and anomalous RSM(H) Hall resistivities, namely $${\rho }_{yx}^{\mathrm{fit}}(H)={R}_{0}H+{R}_{{\rm{S}}}M(H)$$9 (where R0 and RS are the classical and anomalous Hall coefficients, respectively). The anomalous Hall resistivity ρAHE ≡ RSM(H) can be described by a superposition of terms with linear and quadratic dependences on the longitudinal resistivity ρxx, corresponding to the skew scattering and side jump terms, respectively8. However, the variation of ρxx(H) in these multilayers is extremely small (<0.01% for fields up to magnetic saturation at HS and <0.17% for fields up to ±5 T5,9). Consequently, the treatment of ρAHE as $$[a\cdot {\rho }_{xx}(H)+b\cdot {\rho }_{xx}^{2}(H)]M(H)$$ or simply RSM(H) has negligible influence on the estimated ρTHE9. A detailed scaling analysis of ρAHE with ρxx as a function of temperature T (5–300 K) and H for these multilayers can be found in refs. 5,9, together with a discussion of the validity and reproducibility of the estimated ρTHE.

Using Eq. (1) and our experimentally-determined ρTHE, one can hence estimate nsk from an electrical transport measurement as:

$${n}_{\mathrm{sk}}({\mathrm{THE}})=| {\rho }^{\mathrm{THE}}|\,\div\,|(P\cdot {R}_{0}\cdot {{{\Phi }}}_{0})| .$$
(2)

nsk may also be measured directly using real-space imaging techniques such as magnetic force microscopy (MFM)9,10,11, magnetic transmission X-ray microscopy4,5, or Lorentz transmission electron microscopy12. Comparing these transport and imaging approaches can yield evidence for adiabatic transport if nsk(THE) ≈ nsk(MFM)6,7,8, non-adiabaticity if nsk(THE) < nsk(MFM)13,14, or alternatively reveal enhanced transverse scattering mechanisms if nsk(THE) > nsk(MFM)13,15,16,17.

In bulk non-centrosymmetric materials which exhibit stable skyrmion lattices, the values of nsk estimated from transport and imaging are in good agreement6,7,8. However, this is not the case for thin-film multilayers with an interfacial Dzyaloshinskii–Moriya interaction (DMI), which can be tuned to exhibit isolated or dense skyrmion configurations. Large, conflicting values for ρTHE have been reported in magnetic thin films, corresponding to nsk(THE) orders of magnitude larger than nsk(MFM)5,10,11,12,18,19.

Determining the mechanism leading to this extraordinary disagreement is crucial for understanding the electrical response of chiral spin textures and their detection in devices. Recently, a valuable clue has been provided by predictions15,16,17 and observations20,21 of spin fluctuation-induced effects on charge transport in non-coplanar magnets. It is hence plausible that quantum or thermal fluctuations may influence the Hall response of materials with a chiral instability.

In this work, using temperature-dependent Hall transport and MFM, we track the evolution of the THE across the transition between isolated skyrmions and a disordered skyrmion lattice. We find that isolated skyrmion configurations produce a larger THE than dense arrays of skyrmions, with an enhancement of up to three orders of magnitude at the transition. Our data reveal a power-law behavior in nsk(THE)/nsk(MFM), which we interpret in terms of chiral spin fluctuations. Following universal scaling laws, we extract the critical exponents governing this phase transition22,23,24.

## Results

Figure 1 shows measurements of ρTHE(H) at 300 K for a set of Ir/Fe(x)/Co(y)/Pt multilayers. MFM images acquired at H maximizing ρTHE(H) display spin configurations ranging from isolated skyrmions (Fig. 1a–c) to dense, disordered skyrmion lattices (Fig. 1d–g) (see Section 1 of the Supplementary Information for the evolution of magnetic textures). This evolution in skyrmion configuration is driven by the T-dependent stability parameter $$\kappa \equiv \,\pi D/4\sqrt{A{K}_{\mathrm{eff}}}$$, which describes the competition between the three key magnetic interactions: exchange coupling (A), magnetic anisotropy (Keff), and DMI (D). Our multilayer Ir/Fe(x)/Co(y)/Pt stacks allow us to systematically tune κ (and hence nsk) via the Fe/Co layer thickness ratio: x/y < 0.5 yields κ < 1, whereas κ ≥ 1 for x/y ≥ 0.55. Varying T provides an additional handle to tune κ: for a given Fe/Co ratio, κ increases with T due to T-dependent magnetic interactions (see Section 2 of the Supplementary Information). At H = 0, these chiral magnetic films exhibit a labyrinthine stripe domain phase. Under finite H, this transforms into a metastable skyrmion phase if κ < 1 (Fig. 1a–c), or a disordered skyrmion lattice phase if κ ≥ 1 (Fig. 1d–f). For κ ≥ 1, the skyrmion lattice dissolves into isolated skyrmions before a uniformly polarized ferromagnetic (FM) phase develops for H > HS. The isolated skyrmion phase emerges between a lattice and a FM phase; its appearance can be regulated by κ, T, H, or a combination of these parameters (see Section 1 of the Supplementary Information). Indeed, in chiral magnetic films, the transformation of a polarized FM phase into an array of isolated skyrmions and ultimately a skyrmion lattice occurs via a nucleation-type second-order phase transition22,23,24.

We find that isolated skyrmion configurations (Fig. 1a–c) consistently generate a larger ρTHE(H), despite their small nsk(MFM). Contrary to expectations from Eq. (1), dense skyrmion arrays (Fig. 1d–f) with larger nsk(MFM) typically display a smaller ρTHE(H). The generic large ρTHE(H) for isolated skyrmions can be further confirmed using the evolution of a dense skyrmion array with increasing H, as shown in Fig. 1d–f. Here, ρTHE(H) rises as the skyrmion lattice dissolves into isolated skyrmions before reaching a FM phase with ρTHE(H) ≈ 0. To further probe this discrepancy between transport and imaging experiments, we evaluate Δρ/M ≡ nsk(THE)/nsk(MFM)9,11,12,18,19,25, which quantifies the effective topological charge contributing to the measured THE, then compare this quantity for different skyrmion configurations imaged by MFM. As nsk(MFM) increases, Δρ/M is systematically reduced (Fig. 1g). Hence for a skyrmion lattice, we have nsk(THE) ≈ nsk(MFM) and for isolated skyrmions nsk(THE) > nsk(MFM). The strong influence of the skyrmion configuration on ρTHE underlines the crucial role of κ in determining the magnitude of ρTHE(H). We therefore examine the transition between isolated and dense arrays of skyrmions, and the influence of κ on the measured THE.

Figure 2a depicts the T − κ space and hence the wide range of spin textures, which we can experimentally access by varying T (5–300 K), the Fe/Co thickness ratio, and the number of repeats of [Fe/Co] in a multilayer stack. Using our MFM images, we identify the transition (blue shaded region in Fig. 2a) between isolated skyrmions and a dense, disordered skyrmion lattice5 and track its dependence on T and κ (see Section 1 of the Supplementary Information for the evolution of magnetic textures across T and κ). We explore the impact of this transition on the THE by correlating the evolution of ρTHE across the T − κ phase space. Figure 2b displays ρTHE(T, κ) curves for individual multilayers studied as a function of T, while Fig. 2c shows ρTHE(κ) curves at a fixed T with varying Fe/Co. The location of the maximum in ρTHE (Fig. 2b, c) tracks the phase boundary between isolated skyrmions and a disordered skyrmion lattice tuned by T, κ. This confirms that the degree of THE enhancement is closely linked to proximity to the phase boundary.

Here we note that the sign of THE remains unchanged across the T − κ phase diagram. However, R0 changes from positive to negative with T due to multiband transport9. This crossover consistently follows the local maximum in ρTHE and the transition from isolated skyrmions to the lattice phase (for details see Sects. 46 of the Supplementary Information). Such correlation between the local enhancement of ρTHE, skyrmion configuration, and sign reversal of R0 suggests that these factors are influenced by systematic changes in the occupancy of the electronic bands while varying T and the Fe/Co composition. To understand the mechanism responsible for the R0 sign reversal, we employed a tight-binding model together with ab-initio calculations of the electronic band structure of our multilayer stack. When electron-like spin-up states and hole-like spin-down states are both present near the Fermi energy, a T induced change in the individual carrier densities can lead to a sign change of R0 due to compensation of the normal Hall signal from carriers with opposite charge. However, due to their opposing spin alignment, the interaction of these carriers with Beff from the skyrmions gives rise to a THE which maintains the same sign from T = 5–300 K. A complete quantitative analysis of the band structure is beyond the scope of this work. However, these experimental observations provide valuable insight into the links between thermodynamic stability and charge transport in chiral magnetic textures.

We now return to the enhancement of the THE. Within the critical region surrounding a second-order phase transition, fluctuations of the incipient order parameter (η) dominate the material response. Hence, chiral spin fluctuations at all length scales may develop at the skyrmion lattice-phase boundary7. The spin chirality contribution to the THE originates from a non-zero scalar triple product [Si (Sj × Sk)], where Si, Sj, and Sk form a cluster consisting of three fluctuating spins. Such clusters can generate Hall signals via an unconventional skew-scattering mechanism, which can greatly exceed the contribution from a stable skyrmion configuration15,16,17,26. The size of these clusters can be as small as several atomic spacings, which is much smaller than the skyrmion radius determined by the macroscopic magnetic interaction parameters in our films. Upon approaching the phase boundary, such fluctuation-induced clusters are expected to proliferate throughout the material, hence generating considerable topological charge15,16,17. The correlation of these fluctuating spin clusters is associated with the formation/destruction of long-range order, i.e., a stable skyrmion lattice configuration. Within this scenario, the spin correlation length of the skyrmion lattice (which for an infinite thermodynamic system would diverge at the transition) is distinct from the short length-scale spin fluctuations responsible for a non-zero scalar spin chirality. A detailed microscopic understanding of this fluctuating spin chirality in relation to the phase transitions in chiral magnetic systems remains to be established theoretically26. As we approach the transition, the influence of such fluctuations should be revealed by power-law behavior in material properties, including the spin susceptibility and hence THE20. The maximum observable THE magnitude is expected to saturate at a value corresponding to the maximal chiral spin cluster density (imposed by the atomic spacing). Consequently, the power-law enhancement is truncated close to the transition and there is no THE singularity.

Our experimental observations reveal that the phase transition between isolated skyrmions and a disordered skyrmion lattice is sensitive to κ, T, and H. In the following, we consider an effective temperature $${T}^{\prime}$$, which accounts for the role of varying T and κ in controlling this phase transition. Figure 2d illustrates how the critical temperature Tc and stability κc defining the transition vary with κ and T, respectively, in accordance with experiments (Fig. 2a). The phase boundary between isolated skyrmions and a skyrmion lattice may be considered a smooth function Tc(κ)22,23,24. In the vicinity of a phase transition (κ = κc ~ 1, within the transition region, Fig. 2a) the boundary can be described as Tcκ ≈ constant and hence,

$$\frac{{\mathrm{d}}\left({T}_{\mathrm{c}}\cdot \kappa \right)}{{\mathrm{d}}\kappa }=\frac{{\mathrm{d}}\left({\mathrm{constant}}\right)}{{\mathrm{d}}\kappa }\Rightarrow \kappa \frac{{\mathrm{d}}{T}_{\mathrm{c}}}{{\mathrm{d}}\kappa }+{T}_{\mathrm{c}}=0\Rightarrow \frac{{\mathrm{d}}{T}_{\mathrm{c}}}{{\mathrm{d}}\kappa }\approx -{T}_{\mathrm{c}},$$

which in turn suggests an effective temperature $$T^{\prime} =T\cdot \kappa$$ with a critical value $$T^{\prime} ={T}_{\mathrm{c}}\cdot {\kappa }_{\mathrm{c}}$$ defining the transition. One may also consider the phase boundary in terms of varying H, however, the interdependence of κ and H, and its variation with T remain unclear. We therefore adopt the simpler picture presented in Fig. 2d, which allows a straightforward interpretation of our experimental observations in Figs. 1 and 2a–c.

Figure 3a shows a power-law behavior in Δρ/M surrounding a critical value of $${T}_{\mathrm{c}}^{\prime}\approx 110\pm 15$$ K. This is consistent with the presence of a second-order phase transition driven by T and κ. According to Landau theory, the amplitude of η grows as a power-law on the low-symmetry side of any second-order transition (which in our case corresponds to $${T}^{\prime} \, > \, {T}_{\mathrm{c}}^{\prime}$$):

$$\eta \propto | T^{\prime} -{T}_{\mathrm{c}}^{\prime}{| }^{\beta },$$

whereas fluctuations of η on both sides of the transition scale as:

$$\langle {({{\Delta }}\eta )}^{2}\rangle \propto \frac{1}{| T^{\prime} -{T}_{\mathrm{c}}^{\prime}{| }^{\gamma }}.$$

Extracting the critical exponents β, γ from our data requires the identification of η for the skyrmion lattice configuration. In the ordered phase of a chiral magnet, a helical wavevector that describes the spin rotation period emerges, capturing the symmetry broken by η at the transition. Discrete Fourier transforms of a skyrmion lattice yield a strong peak in the structure factor at k = l−1, where $$l\propto 1/\sqrt{{n}_{\mathrm{sk}}({\mathrm{MFM}})}$$ is the skyrmion lattice parameter and k rises continuously from zero with the emergence of a lattice. We, therefore, expect η to scale with $$\sqrt{{n}_{{\mathrm{sk}}}({\mathrm{MFM}})}$$, allowing analysis of the transition using the conventional scaling approach

$${n}_{\mathrm{sk}}({\mathrm{MFM}})\sim | T^{\prime} -{T}_{{\mathrm{c}}}^{\prime}{| }^{2\beta },\qquad T^{\prime} \, > \, {T}_{{\mathrm{c}}}^{\prime}.$$
(3)

Fluctuations of η in the vicinity of the phase transition create a topological charge and hence contribute an effective skyrmion density nfl ≈ 〈(Δη)2〉 (nflnsk(MFM)) to the total measured nsk(THE). Consequently,

$${{{\Delta }}}_{\rho /{\mathrm{M}}}=\frac{{n}_{{\mathrm{sk}}}({\mathrm{THE}})}{{n}_{{\mathrm{sk}}}({\mathrm{MFM}})}=\frac{{n}_{{\mathrm{sk}}}({\mathrm{MFM}})+{n}_{{\mathrm{fl}}}}{{n}_{{\mathrm{sk}}}({\mathrm{MFM}})}\approx \frac{{n}_{{\mathrm{fl}}}}{{n}_{{\mathrm{sk}}}({\mathrm{MFM}})}.$$

Therefore, we may model the power-law rise in Δρ/M as follows:

$${{{\Delta }}}_{\rho /{\mathrm{M}}} \sim \frac{1}{| T^{\prime} -{T}_{{\mathrm{c}}}^{\prime}{| }^{\gamma }},\qquad T^{\prime} \, < \, {T}_{{\mathrm{c}}}^{\prime}$$
(4)
$${{{\Delta }}}_{\rho /{\mathrm{M}}} \sim \frac{1}{| T^{\prime} -{T}_{{\mathrm{c}}}^{\prime}{| }^{2\beta +\gamma }},\qquad T^{\prime} \, > \, {T}_{{\mathrm{c}}}^{\prime}$$
(5)

where the extra 2β in the exponent above $${T}_{{\mathrm{c}}}^{\prime}$$ originates from the appearance of a stable skyrmion lattice as indicated in Eq. (3).

Fits to Eqs. (3) and (4) to extract γ and β are shown as red lines in Fig. 3a (for $${T}^{\prime} \, < \, {T}_{{\mathrm{c}}}^{\prime}$$) and 3b, respectively, where we use two independent data sets to obtain exponents γ ≈ 1.88 ± 0.43, and β ≈ 0.31 ± 0.05. These values suggest a three-dimensional Heisenberg spin system in which the exponents are modified by competing spin interactions in a quasi-two-dimensional environment. This is in agreement with earlier studies of thin-film magnets27, which consistently reveal an increased γ and a reduced β with respect to the three-dimensional Heisenberg values 1.39 and 0.36, respectively28. To independently crosscheck our estimate of the exponents γ and β, we also fit the discrepancy in the skyrmion lattice regime using Eq. (5) (Fig. 3a for $${T}^{\prime} \,> \, {T}_{c}^{\prime}$$). We obtain the combination (γ + 2β) = 2.61 ± 0.44, which is consistent with the γ and β extracted individually from our fluctuation analysis, adding credence to the validity of these critical exponents.

As discussed above, the finite size of the short length-scale fluctuating spin clusters imposes an upper limit on the magnitude of nfl, which is expected to saturate as the system approaches $${T}_{{\mathrm{c}}}^{\prime}$$. Experimentally, we can only modulate $${T}^{\prime}$$ in discrete steps, so we cannot tune $${T}^{\prime}$$ continuously through $${T}_{{\mathrm{c}}}^{\prime}$$ to reveal the expected saturation in nfl and hence THE. However, our experimental Δρ/M clearly demonstrates the anticipated power-law scaling on either side of $${T}_{{\mathrm{c}}}^{\prime}$$, systematically varying from 1–103. Using Eq. (1) and the values of R0 ≈ 0.5–16 nΩ cm/T measured in our multilayer films, we estimate that Δρ/M should saturate at a value 104–105 as $$| {T}^{\prime}-{T}_{{\mathrm{c}}}^{\prime}| \to 0$$. From the power-law trends shown in Fig. 3a, we deduce that this saturation is only visible for $$| {T}^{\prime}-{T}_{{\mathrm{c}}}^{\prime}| \, < \, 20$$ K, which lies beyond the minimum $$| {T}^{\prime}-{T}_{{\mathrm{c}}}^{\prime}| \approx 40\pm 15$$ K accessed during our experiments. It may therefore be possible to engineer a further THE increase of at least another order of magnitude, by tuning skyrmion-hosting multilayers more closely towards their isolated skyrmion/disordered lattice transition. Finally, we note that the presence of a finite population of skyrmions for $${T}^{\prime} \, < \,{T}_{{\mathrm{c}}}^{\prime}$$ is consistent with the nucleation-type transition by which the skyrmion lattice is established22,23,24. Some of these isolated skyrmions may also be stabilized by local variations in the magnetic interactions due to disorder in our films.

## Discussion

Our results are summarized in Fig. 3c, which highlights the transition between isolated skyrmions and a disordered skyrmion lattice, determined by real-space imaging of the spin textures as well as magnetotransport. The critical parameter governing the transition Tc(κ) is identified by three separate methods: the maximum in ρTHE, MFM imaging, and the fluctuation-induced rise in Δρ/M at $${T}_{\mathrm{c}}^{\prime}=110\pm 15$$ K. All three data sets display considerable overlap, indicating the active role of critical spin fluctuations in determining the magnitude of THE in chiral magnetic films. The ensuing discrepancy of up to three orders of magnitude between nsk(THE) and nsk(MFM) in the vicinity of the phase transition may account for the widely-varying magnitudes of ρTHE values previously reported in technologically-relevant chiral magnetic films9,10,11,12,18,25,29. Our material platform allows the THE to be tuned from nsk(THE) ≈ nsk(MFM), which is typical for skyrmion crystals of B20 compounds6,7,8, to nsk(THE) > nsk(MFM) in dilute skyrmion configurations characteristic of interfacial systems9,10,11,12,18,25,29. This acute sensitivity of ρTHE to the magnetic skyrmion configuration indicates the crucial role of chiral spin fluctuations and opens a promising avenue towards controllable topological spintronics.

## Methods

### Film deposition

Thin-film multilayers consisting of Ta(30)/Pt(100)/[Ir(10)/Fe(x)/Co(y)/Pt(10)]N/Pt(20) (numbers in the parentheses are layer thickness in Å and N refers to the number of times the Ir/Fe/Co/Pt stack is repeated, x, y are varied between 0 and 6 Å and 5–10 Å, respectively) were deposited on thermally oxidized Si wafers by dc magnetron sputtering at room temperature, using a Chiron ultra-high vacuum multi-source system. The optimal growth rates for individual layers are Ta: 0.55 Å/s, Pt: 0.47 Å/s, Ir: 0.12 Å/s, Fe: 0.13 Å/s, Co: 0.2 Å/s. The base vacuum of the sputter chamber is 1 × 10−8 Torr and an argon gas pressure of 1.5 × 10−3 Torr is maintained during sputtering.

### Electrical transport

The magnetotransport measurements were performed using a custom-built variable-temperature insert (VTI) housed in a high-field magnet, complemented by a Quantum Design Physical Property Measurement System (PPMS). Current densities as low as 104 A/m2 at 33 Hz were used to avoid current-driven perturbation of spin textures. Detailed analysis for the extraction of the topological Hall resistivity (ρTHE(H)) can be found in our earlier works5,9. The non-zero value of ρTHE at H > HS serves as a conservative estimate of the error bar in the extracted ρTHE(H), which is ≤2 nΩ cm. This includes the systematic errors involved in data analysis.

### Magnetization

Magnetization measurements were performed in the range T = 5–300 K and a magnetic field of ±4 T, using superconducting quantum interference device (SQUID) magnetometry, in a Quantum Design Magnetic Property Measurement System (MPMS) (see Section 2 of the Supplementary Information). M(H) loops were recorded with the applied field in-plane (hard axis) and out of the film plane (easy axis). The saturation magnetization MS and the difference in saturation fields HS along the easy and hard axes were used to estimate the effective magnetic anisotropy Keff5,9.

### Magnetic interactions

MS and Keff were determined from SQUID magnetometry measurements. The detailed methods for estimatingthe exchange stiffness A and DMI were reported in our previous work5. In this work, we use the scaling laws $$\frac{A(T)}{A(T=5\,{\mathrm{K}})}={\left[\frac{{M}_{\mathrm{S}}(T)}{{M}_{\mathrm{S}}(T = 5\,{\mathrm{K}})}\right]}^{1.5}$$ and $$\frac{D(T)}{D(T=5\,{\mathrm{K}})}={\left[\frac{{M}_{\mathrm{S}}(T)}{{M}_{\mathrm{S}}(T = 5\,{\mathrm{K}})}\right]}^{1.5}$$ involving the T-dependent saturation magnetization MS(T) to estimate A(T), D(T), and the resulting κ(T). Details can be found in Section 2 of the Supplementary Information.

### Magnetic force microscopy (MFM)

Room temperature MFM experiments were carried out using a Veeco Dimension 3100 Scanning Probe Microscope. The MFM tips used (Nanosensors SSS-MFMR) were ≈30 nm in diameter, with low coercivity (≈12 mT) and ultralow magnetic moment (≈80 emu/cc). Samples were initially saturated in the out of plane (OP) direction using fields up to H = −0.5 T. The measurements were performed in OP fields starting from H = 0 and incrementally approaching H = +HS, with a typical tip height of 20 nm. The field evolution of the spin textures is presented in Section 1 of the Supplementary Information. Low T (5–200 K) MFM imaging is carried out using a cryogenic frequency-modulated MFM system9. We used two commercial probes by Team Nanotec, model ML3 (35–40 nm Co alloy coating), with f0 ≈ 75 kHz and k0 ≈ 1 N/m. The sample was first stabilized at a given temperature and then magnetized in the OP direction by applying H > HS. After saturation, MFM images were acquired at various field values as H was swept from −HS to +HS. The details of the MFM image analysis are reported in our earlier reports5,9,30.

### Calibration of applied magnetic field

Extensive field calibrations were performed using a reference Hall sensor and a reference Palladium sample to minimize the field offsets between different experimental setups. Sensor details and the calibration data together with analysis for the extraction of ρTHE(H) are reported in our earlier works5,9. An additional field-calibration crosscheck is performed for our MFM images by estimating the magnetization from the image itself. Here the magnetization of the image is estimated as the ratio of the effective area of the imaged surface polarized along the applied field to the total area of the image. The magnetization of the MFM image is then located on M(H) data recorded by the SQUID magnetometer.