## Introduction

Scalar spin chirality, $${{{{{{\bf{S}}}}}}}_{i}\cdot ({{{{{{\bf{S}}}}}}}_{j}\times {{{{{{\bf{S}}}}}}}_{k})$$, is a quantity that corresponds to a solid angle subtended by three spins Si, Sj, and Sk on neighboring triangular sites1,2,3. Electrons hopping through non-coplanar spin textures with finite scalar spin chirality acquire Berry phase equivalent to a fictitious magnetic field, leading to topological Hall effect (THE)4,5 that is distinct from other Hall effects. Among various non-coplanar spin textures, a swirling spin texture called the magnetic skyrmion4,5,6 is particularly interesting because it can not only behave like an individual particle-like object but also can form an ordered skyrmion crystal (SkX) under the delicate competition of magnetic interactions and external perturbations. Understanding the stability and controllability of SkX phase is currently at the frontier of condensed matter research.

The established approach for the formation of SkX phase is to use noncentrosymmetric bulk crystals4,5 in which finite contributions of spin-orbit coupling (SOC) induce the Dzyaloshinskii-Moriya interaction (DMI)7,8, expressed as $${H}_{{{{{{\rm{DMI}}}}}}}=-{{{{{\bf{D}}}}}}\cdot ({{{{{{\bf{S}}}}}}}_{i}\times {{{{{{\bf{S}}}}}}}_{j})$$ with D being the DMI vector. This interaction favors a relative twist between otherwise parallel or anti-parallel spins, giving rise to complex magnetic states such as canted ferromagnetism7,8, non-coplanar and helical/spiral spin states4,5. In thin-film heterostructures, the role of SOC in bulk crystals can be activated by fabricating an asymmetric interface with individual magnetic and SOC layers, which is known as the interfacial DMI9,10. In various interface systems including metals11,12,13,14,15,16, perovskite-type oxides17,18, and topological insulators18,19, the formation of isolated skyrmions and the detection of THE have been reported. However, the two-dimensional SkX has rarely been achieved by the interfacial DMI mechanism20. Given the crucial role of crystal symmetry in the bulk approaches, a choice of specific lattice systems that enable the tuning of complex magnetic interactions is essential.

With this perspective, we focused on kagome-lattice magnets with a triangular-based lattice, which is suitable for inducing non-coplanar magnetic states required for the SkX phase. We selected a ferromagnetic intermetallic compound Fe3Sn with the D019 structure. Bulk Fe3Sn is ferromagnetic below Curie temperature Tc of 743 K with in-plane magnetic anisotropy in the Fe3Sn kagome plane (ab plane)21. Very recently, it has been reported that Fe3Sn can be epitaxially grown as a thin film on Pt(111) (ref. 22). The combination of Fe3Sn and Pt offers the following advantages for spin chirality engineering via the interfacial DMI: (i) Fe3Sn consists of ideal two-dimensional kagome planes (Fig. 1a), (ii) the epitaxial interface across Pt with strong SOC effectively activates the interfacial DMI, (iii) large magnetization derived from Fe enables the characterization of magnetic properties in the ultrathin films.

Here, we report a comprehensive study that strongly suggests the formation of a two-dimensional SkX phase in Fe3Sn/Pt epitaxial bilayers (Fig. 1a). By tuning the magnetic anisotropy and the interfacial DMI via the reduction of the Fe3Sn thickness, t, we attempt to achieve the non-coplanar spin state with finite scalar spin chirality (Fig. 1b, c). The evolution of magnetic anisotropy and the occurrence of THE revealed by the systematic t-dependent measurements are consistent with the formation of SkX in a theoretical model based on a ferromagnetic kagome lattice.

## Results and discussion

### Fabrication of kagome-lattice Fe3Sn/Pt bilayers

We fabricated t-nm-thick Fe3Sn/10-nm-thick Pt bilayers on Al2O3(0001) substrates by radio-frequency magnetron sputtering (see Methods). Figure 1d displays a cross-sectional transmission electron microcopy (TEM) image of a bilayer sample with t = 8 nm. The selected-area electron diffraction, shown in Fig. 1e, together with the macroscopic X-ray diffraction (Supplementary Fig. 1), identifies the crystallization of the high-temperature Fe3Sn phase (JCPDS PDF 01-074-5857) with an epitaxial orientation relationship of Fe3Sn$$[01\bar{1}0]$$(0001)/Pt$$[11\bar{2}]$$(111)/Al2O3$$[11\bar{2}0](0001)$$ (see Supplementary Fig. 2 for the analysis of fast Fourier transformation diffraction patterns). The well-crystallized and homogeneous growth of Fe3Sn was also observed for t = 1.6 nm (Supplementary Fig. 3). Because it is difficult to apply the same analysis to much thinner films (t < 1 nm) shown below, we hereafter adopt the nominal t value based on the sputtering rate assuming uniform film deposition. As shown in Fig. 1f, M versus magnetic field μ0H curves (where μ0 is vacuum permeability and H is magnetic field strength) measured for t = 0.80 nm (~2 unit cells, the bulk c-axis length = 0.436 nm) in an out-of-plane H show hard-axis behavior without remanent M (see Supplementary Fig. 4 for the data of other t values). The saturation magnetization Msat of 1.2 × 106 J m−3 T−1 agrees with the bulk value reported in the literature21,23. The t dependence of Msat at temperature T = 400 K (Fig. 1g) shows that the Msat decreases below t = 0.80 nm. However, the bulk-like Msat value at low T (see Supplementary Fig. 5 for the T dependence of Msat) persists down to t = 0.32 nm, indicating that Fe spins order ferromagnetically even in these ultrathin samples. The Msat significantly decreases at t = 0.24 nm. The clear qualitative change in the magnetic behavior occurs between t = 0.32 nm and 0.24 nm. Combined with the TEM results (Fig. 1d and Supplementary Fig. 3), this implies that the nominal t value corresponds to the actual film thickness even in these ultrathin films. The estimated Tc is significantly higher than 400 K for t ≥ 0.48 nm and becomes comparable to 400 K for 0.32 nm ≤ t ≤ 0.40 nm. A more detailed analysis of the T dependence of magnetization across the ferromagnetic transition will clarify the accurate TC and possible role of disorder on the magnetic behavior24.

### Thickness-dependent magnetic anisotropy in Fe3Sn/Pt bilayers

By comparing M versus μ0H curves in out-of-plane and in-plane configurations, we detected the expected variation of magnetic anisotropy, as displayed in Fig. 2a–c. For t = 0.64 nm (Fig. 2a), the M saturates easily under the application of in-plane H (blue curve), showing the in-plane magnetic anisotropy consistent with the bulk behavior21. In the sample with t = 0.48 nm (Fig. 2b), however, the M saturates at almost comparable μ0H values in the two measurement configurations. The even thinner condition of t = 0.24 nm appears to favor perpendicular magnetic anisotropy (Fig. 2c). From the $$M/{M}_{{{{{{\rm{sat}}}}}}}$$ versus μ0H curves, we calculated the magnetic anisotropy energy as the effective anisotropy field μ0HA,eff in the unit of T (μ0HA,eff > 0 for perpendicular magnetic anisotropy and < 0 for in-plane magnetic anisotropy). As plotted in Fig. 2d, μ0HA,eff gradually varies from negative to positive with decreasing t, changing its sign at approximately 0.5 nm (~1 unit cell). This result captures that the inherent in-plane magnetic anisotropy turns to perpendicular magnetic anisotropy at a crossover thickness of tMA ~0.5 nm. To gain insight into spin textures in these ultrathin Fe3Sn bilayer samples, we performed a numerical simulation for a single kagome plane that took into account the ferromagnetic Heisenberg interaction J, magnetic anisotropy A (in-plane and perpendicular magnetic anisotropy for A < 0 and > 0, respectively), and DMI (see Methods). As schematically shown in Fig. 2e, this model considers spin interactions between the neighboring spins on the kagome plane in an out-of-plane magnetic field. Bulk and interfacial DMIs contribute to the out-of-plane and in-plane components of D (Dǁ and D), respectively, which modifies the local spin interactions to produce a finite spin solid angle. Figure 2f displays the total spin solid angle per the unit cell of Fe3Sn, $$|{\varOmega }_{{{{{{\rm{tot}}}}}}}|$$, as a function of A and D (J = Dǁ = 1 and out-of-plane magnetic field BH = 0.05, where g is the g-factor and μB is the Bohr magnetron). In the upper-left region of large negative A and small D, the spins basically lie in the kagome plane with slightly tilting to the out-of-plane H direction (see Supplementary Fig. 6 for the simulated magnetic moments). This corresponds to a ferromagnetically ordered phase with in-plane magnetic anisotropy in the kagome plane as in the bulk21. Increasing A to the positive side stabilizes a different ordered phase where spins fully point to the out-of-plane H direction. Although these ordered phases do not generate $$|{\varOmega }_{{{{{{\rm{tot}}}}}}}|$$, we found that a moderate contribution of D stabilizes a two-dimensional SkX phase with a finite $$|{\varOmega }_{{{{{{\rm{tot}}}}}}}|$$ (inset, also see Supplementary Fig. 7). By a further increase of D, the spiral phase without a finite $$|{\varOmega }_{{{{{{\rm{tot}}}}}}}|$$ is induced. This magnetic phase diagram highlights the critical importance of the interface-driven modification of DMI for the stabilization of SkX phase in the ferromagnetic kagome lattice. In fact, the clear switching of magnetic anisotropy shown in Fig. 2d corroborates that our t-controlled Fe3Sn/Pt bilayers are an excellent arena for experimentally verifying scalar spin chirality via the THE analysis.

### Verification of interfacial DMI

We characterized three sample structures schematically shown in Fig. 3a–c: 0.80-nm-thick Fe3Sn/Pt (thick-bilayer, t > tMA), 0.40-nm-thick Fe3Sn/Pt (thin-bilayer, t < tMA) and Pt/0.40-nm-thick Fe3Sn/Pt (thin-trilayer, t < tMA; see Supplementary Fig. 8 for the M data). In the thick-bilayer and thin-bilayer structures, the interfacial DMI is activated by the bottom Pt layer, whereas in the thin-trilayer structure, the effective interfacial DMI would be canceled by opposite contributions from the top Pt/Fe3Sn and bottom Fe3Sn/Pt interfaces. Although there is no direct correspondence between the actual samples and simulation conditions, the thick-bilayer, thin-bilayer, and thin-trilayer structures can be compared with the simulation conditions of negative A and small D, small positive A and large D, and small positive A and small D, respectively. Note here that a large part of conduction in these heterostructures is governed by the highly conducting Pt layer (See Supplementary Figs. 9 and 10 for the t-dependence of sheet resistance and the magnetoresistance data, respectively). Figure 3d shows Hall resistance Ryx versus μ0H curves of the thick-bilayer structure at T = 400 K. In addition to the linear ordinary Hall effect of Pt (Supplementary Fig. 11k), a nonlinear response in the Ryx is clearly discernable. A fit using an empirical relation15,16,17,18,19, $${R}_{yx}={R}_{0}{{\hbox{'}}}{\mu }_{0}H+{R}_{{{{{{\rm{A}}}}}}}{{\hbox{'}}}M+{R}_{yx}^{{{{{{\rm{T}}}}}}}$$, where $${R}_{0}^{{\prime} }$$ and $${R}_{{{\rm{A}}}}^{{\prime} }$$ are the ordinary and anomalous Hall coefficients in the unit of Ω T−1, and $${R}_{{{{{{\rm{A}}}}}}}^{{\prime} }M\,(={R}_{yx}^{{{{{{\rm{A}}}}}}})$$ and $${R}_{yx}^{{{{{{\rm{T}}}}}}}$$ are the anomalous and topological Hall resistances, reveals the dominant contribution of $${R}_{yx}^{{{{{{\rm{A}}}}}}}$$ to the Ryx (Fig. 3e inset), which is attributed to the anomalous Hall effect (AHE) of Fe3Sn. The negligibly small residual $${R}_{yx}^{{{{{{\rm{T}}}}}}}$$ (Fig. 3e) indicates the absence of THE in the thick-bilayer structure. This is consistent with that the bulk-like in-plane magnetic anisotropy favors the coplanar spin state without scalar spin chirality (Fig. 1b). Contrastingly, in the thin-bilayer structure, the extracted $${R}_{yx}^{{{{{{\rm{T}}}}}}}$$ (Fig. 3g) at T = 400 K ~Tc (Supplementary Fig. 5) overwhelms both ordinary and anomalous Hall resistances (Fig. 3f and the inset of Fig. 3g), indicating the presence of finite scalar spin chirality that contributes to THE. The addition of top Pt layer in the thin-trilayer structure, as intended, completely diminishes the THE (Fig. 3h, i). On the other hand, the comparable AHE for the thin-trilayer and thin-bilayer structures at T = 300 K (the insets of Fig. 3f, h; also see Supplementary Fig. 11 for the data at various T) indicates their macroscopically similar ferromagnetic states. In recent studies on SrRuO3 ultrathin films and SrRuO3-based perovskite multilayers and superlattices24,25,26,27,28, the impact of inhomogeneity on the occurrence of THE-like Ryx anomalies has been argued. When AHE changes its sign depending on thickness as in the present system (Supplementary Fig. 11), local thickness fluctuation could give rise to hump-like Ryx behavior via the superposition of AHE components with different signs. To understand the origin of the observed $${R}_{yx}^{{{{{{\rm{T}}}}}}}$$, we evaluated asymmetric Ta/Fe3Sn/Pt and W/Fe3Sn/Pt trilayer structures (Supplementary Fig. 12 and Supplementary Note 1). The results were consistently explained by considering different magnitudes of the interfacial DMI contributions from the top and bottom interfaces. In conjunction with the insignificant thickness fluctuation suggested by the TEM and magnetization measurements, this strongly supports that the interfacial DMI plays a more decisive role in the occurrence of $${R}_{yx}^{{{{{{\rm{T}}}}}}}$$ than inhomogeneity. Furthermore, the superposition of Ryx versus μ0H curves for different t values at T = 400 K (Supplementary Fig. 11) cannot reproduce the sharp Ryx peaks detected in the thin-bilayer structure (Fig. 4f). These observations are fully consistent with the generation of finite scalar spin chirality owing to the local modification of spin interactions by the interfacial DMI.

### Analysis of THE and possible formation of SkX phase

Having verified the interfacial DMI origin, we demonstrate the t-controlled variation of scalar spin chirality via the detection of THE. As shown in Fig. 4a–c, a slight reduction of t effectively lowers the T range where THE appears (see Supplementary Fig. 13 for the corresponding $${R}_{yx}^{{{{{{\rm{T}}}}}}}\,$$data). Upon comparing the Ryx data including those of other t values (Supplementary Fig. 11), we noticed the sign reversal of Ryx at approximately T = 370 K for t = 0.40 nm (Fig. 4b) and at T = 340 K for t = 0.32 nm (Fig. 4c). Figure 4d shows a contour plot of $$\varDelta {R}_{yx}(\pm 3\,{{{{{\rm{T}}}}}})={R}_{yx}(+3\,{{{{{\rm{T}}}}}})-{R}_{yx}(-3\,{{{{{\rm{T}}}}}})$$ on tT plane; its sign reversal occurs in close proximity to the THE region (surrounded by dashed lines). According to recent band structure calculation29, Fe3Sn is classified as a magnetic Weyl semimetal with Weyl nodes30 near the Fermi level. The T-induced shift of the Fermi level and the resulting change in the intrinsic AHE contribution may play a role in the T-induced sign reversal of AHE.

More importantly, as shown in Fig. 4e, the magnitude of $${R}_{yx}^{{{{{{\rm{T}}}}}}}$$ increases with increasing T (up to our measurement limit of 400 K), concomitantly with the increase of the peak magnetic field at which Ryx shows local maxima/minima due to THE, μ0Hpeak (black circles). Using the standard linear-response theory for a Kondo-lattice model, we performed the analysis of finite-temperature Hall conductivity $$|{\sigma }_{{{{{{\rm{H}}}}}}}|$$ of the intrinsic Berry phase mechanism (Fig. 4f, also see the $$|{\varOmega }_{{{{{{\rm{tot}}}}}}}|$$ result for Supplementary Fig. 14). Overall, the simulation results reproduce the observed experimental trend, indicating the contribution of thermal fluctuation18,31 to the intrinsic (topological) Hall effect. These excellent agreements between the experiments and simulation suggest the formation of SkX phase in the THE region. To support this, we attempted to extract real-space features from the THE data using the relation5,16,17: $${\rho }_{yx}^{{{{{{\rm{T}}}}}}}=P{R}_{0}{n}_{{{{{{\rm{sk}}}}}}}{\phi }_{0}$$, where $${\rho }_{yx}^{{{{{{\rm{T}}}}}}}$$ is the topological Hall resistivity, P is the spin polarization, R0 is the ordinary Hall coefficient, nsk is the skyrmion density, and $${\phi }_{0}$$ is one magnetic flux quantum (= h/e with h being the Planck constant and e being the elementary charge). The $${n}_{{{{{{\rm{sk}}}}}}}^{-0.5}$$ corresponds to the average separation of skyrmions. Assuming a parallel circuit consisting of two conducting layers, we calculated the resistivity and Hall resistivity (Supplementary Fig. 15) of the Fe3Sn layer from the 0.40-nm-thick bilayer data and reference Pt monolayer data at T = 350 K, yielding $${\rho }_{yx}^{{{{{{\rm{T}}}}}}}$$ = 37.4 nΩ cm and R0 = 8.48 × 10−5 cm3 C−1. These values give $${n}_{{{{{{\rm{sk}}}}}}}^{-0.5}$$ = 9.68 nm and 30.6 nm for P = 0.1 and 1, respectively, which are reasonable as compared with the size of skyrmions reported for other bilayer systems (Supplementary Table 1). We therefore think that densely arranged skyrmions like SkX exist in the Fe3Sn/Pt bilayer samples. Direct observation of the spin textures using microscopy will be an interesting future study.

## Conclusion

The epitaxial interface of kagome-lattice ferromagnet Fe3Sn and Pt enables the rational control of magnetic and electrical properties based on the interfacial DMI. Considering the rich variety of kagome-lattice magnets such as Fe3Sn2, FeSn, and Co3Sn2S2 that have been discovered from the aspect of topological physics, the development of heterointerfaces and superlattices is worthy of investigation. These fascinating features of kagome-lattice magnets will offer tremendous opportunities for exploring new functionalities of SkX-based phenomena.

## Methods

### Thin-film growth

The films were fabricated on Al2O3(0001) substrates by radio-frequency magnetron sputtering at an Ar gas pressure of 0.5 Pa. The Pt, Fe3Sn, and SiOx layers were in situ deposited at 600, 400, and 100 °C with Pt, Fe-Sn [ref. 32], and SiOx targets, respectively. The 2-nm-thick Pt top layer of the thin-trilayer structure (Fig. 3c) was deposited at 100 °C before the SiOx capping. The crystal structure of the films was characterized by TEM and X-ray diffraction using Cu Kα radiation. The t values were calculated based on the sputtering rate that was calibrated with the cross-sectional TEM image shown in Fig. 1d.

### Measurements

The magnetization was measured with a superconducting quantum interference device magnetometer (MPMS3, Quantum Design) upon decreasing μ0H from 7 T to −7 T and increasing μ0H from −7 T to 7 T. By subtracting a diamagnetic contribution from Al2O3 substrate, which was estimated from the high-field data at μ0H = 4–7 T, the M was calculated. By anti-symmetrizing the decreasing-field and increasing-field M data, the two anti-symmetrized M curves shown in Figs. 1f and 2a–c and Supplementary Figs. 4 and 8 were obtained. The electrical properties were measured with a physical property measurement system (PPMS, Quantum Design). The films were patterned into a Hall-bar shape by mechanical scratch, and electrical contacts were made with indium solder. For the analysis of THE, the decreasing-field and increasing-field Ryx data were anti-symmetrized against μ0H to eliminate spurious contributions arising from thermoelectric effect and misalignment of the Hall voltage probes.

### Simulation

We considered a classical Heisenberg model on the kagome lattice to understand the magnetism of Fe3Sn thin films. The Hamiltonian reads

$$H=\mathop{\sum}\limits_{\langle i,j\rangle}[-J{{{{{{\bf{S}}}}}}}_{i}\cdot {{{{{{\bf{S}}}}}}}_{j}-{{{{{{\bf{D}}}}}}}_{ij}\cdot {{{{{{\bf{S}}}}}}}_{i}\times {{{{{{\bf{S}}}}}}}_{j}]+\mathop{\sum}\limits_{i}[-A{({S}_{i}^{z})}^{2}-g{\mu }_{{{{{{\rm{B}}}}}}}H{S}_{i}^{z}],$$
(1)

where $${{{{{{\bf{S}}}}}}}_{i}$$ represents a classical spin with fixed length $$|{{{{{{\bf{S}}}}}}}_{i}|=1$$ on ith site. The first sum of $$\langle i,j\rangle$$ runs over all the nearest neighbor sites; J and $${{{{{{\bf{D}}}}}}}_{ij}$$ represent the ferromagnetic Heisenberg coupling and the DMI, respectively. The second sum represents the single ion anisotropy A and the Zeeman coupling to the external magnetic field H perpendicular to the kagome plane. We set the direction of $${{{{{{\bf{D}}}}}}}_{ij}$$ as shown in Fig. 2e from the symmetry point of view8: The out-of-plane component Dǁ is inherent to the inversion symmetry breaking on the bond centers on the kagome lattice (bulk DMI), while in-plane component D arises from the breaking of mirror symmetry due to the attached Pt layer (interfacial DMI). Note that the latter satisfies C3 and C6 rotational symmetries around the center of the triangular and hexagonal plaquettes of the kagome lattice, respectively. We adopted $$J=\,{D}_{||}$$; similar conditions have been used in previous studies33,34. The magnetic field H was set to be weak so that the magnetization is not forced to be parallel to the applied H direction. In the finite-temperature analysis shown in Fig. 4f, we set $${D}_{\perp }=0.25\, < \, J={D}_{||}$$. Because D is a contribution that is effective only at the interface, this assumption is qualitatively valid. Positive and negative D gave the identical simulation results. As for the single-ion anisotropy A, we set $$A=0.1$$ by considering the saturation field ~0.5 T and TC ≥ 400 K (i.e., A << J) in the experimental magnetization data.

To obtain the ground-state spin configuration of (1), we used a combined method of simulated annealing and local optimization. The annealing was performed from the temperature 1 to 0.001 with 100 steps in the logarithmic scale. In each step, we spent 1000 Monte Carlo (MC) sweeps. After the annealing, we optimized the spin directions one by one so as to minimize the local energy with fixed surrounding spins. We repeated 20000 sweeps of this optimization process. For analyzing the finite-temperature properties, we performed MC simulations with 100000 MC sweeps after 100000 thermalization at each temperature. In all the calculations, we considered $$N=3{L}^{2}$$ spins with $$L=48$$ under the periodic boundary condition.

To detect the non-coplanar spin structure in the SkX, we computed the solid angle defined as

$$\begin{array}{c}{\varOmega }_{{{{{{\rm{tot}}}}}}}=\frac{1}{{L}^{2}}\left[\mathop{\sum}\limits_{(i,j,k)}{\varOmega }_{ijk}+\mathop{\sum}\limits_{(i,j,k,l,m,n)}({\varOmega }_{ijk}+{\varOmega }_{ikl}+{\varOmega }_{ilm}+{\varOmega }_{imn})\right]\end{array}$$
(2)

with the solid angle35

$${\varOmega }_{ijk}=2{\mathrm arctan}\left(\frac{{{{{{{\bf{S}}}}}}}_{i}\cdot {{{{{{\bf{S}}}}}}}_{j}\times {{{{{{\bf{S}}}}}}}_{k}}{1+{{{{{{\bf{S}}}}}}}_{i}\cdot {{{{{{\bf{S}}}}}}}_{j}+{{{{{{\bf{S}}}}}}}_{j}\cdot {{{{{{\bf{S}}}}}}}_{k}+{{{{{{\bf{S}}}}}}}_{k}\cdot {{{{{{\bf{S}}}}}}}_{i}}\right),$$
(3)

which is defined in the range of $$(-2{{{{{\rm{\pi }}}}}},2{{{{{\rm{\pi }}}}}})$$. The sums of $$(i,j,k)$$ and $$(i,j,k,l,m,n)$$ run over all the triangular and hexagonal plaquettes of the kagome lattice, respectively, in which i, j, … were assigned in the counterclockwise order when viewed from the z direction. We also computed the magnetizations (Supplementary Fig. 6):

$${m}_{z}=\frac{1}{N}{\mathop {\sum} \limits_{i}^{\,\!}}\langle{S}_{i}^{z}\rangle,\,{m}_{xy}=\frac{1}{N}\sqrt{\left\langle{\left({\mathop {\sum}\limits _{i}^{\,\!}}{S}_{i}^{x}\right)}^{2}\right\rangle+\left\langle{\left({\mathop {\sum} \limits_{i}^{\,\!}}{S}_{i}^{y}\right)}^{2}\right\rangle}.$$
(4)

We calculated the Hall conductivity σH for the spin configurations obtained by the MC simulations. For this purpose, we introduced the Kondo lattice model;

$$\begin{array}{c}{ {\mathcal H} }_{{{{{{\rm{KLM}}}}}}}=-{t}_{{{{{{\rm{K}}}}}}}\mathop{\sum}\limits_{\langle i,j\rangle,s}[{c}_{is}^{{{\dagger}} }{c}_{js}+{{{{{\rm{h}}}}}}{{{{{\rm{.c}}}}}}.]-{J}_{{{\rm{H}}}}\mathop{\sum}\limits_{i,s,{s}^{\prime}}({c}_{is}^{{{\dagger}} }{{{{{{\boldsymbol{\sigma }}}}}}}_{s{s}^{\prime}}{c}_{i{s}^{\prime}})\cdot {{{{{{\bf{S}}}}}}}_{i}\end{array}$$
(5)

with the given spin configurations Si, and computed σH by using the standard Kubo formula. The first term represents the kinetic energy of itinerant electrons with the nearest neighbor hopping $${t}_{{{{{{\rm{K}}}}}}}$$, and the second term represents the Hund’s coupling $${J}_{{{{{{\rm{H}}}}}}}$$ between itinerant electron spins defined by the Pauli matrices σ and localized spins Si. In the calculations, we took $${J}_{{{{{{\rm{H}}}}}}}=4{t}_{{{{{{\rm{K}}}}}}}$$ and fixed the electron density at $${n}_{{{{{{\rm{el}}}}}}}=\mathop{\sum}\limits_{i,s}\langle{c}_{is}^{{{\dagger}} }{c}_{is}\rangle/N=0.1$$. This low $${n}_{{{{{{\rm{el}}}}}}}$$ is regarded as a dilute limit36,37, which is appropriate to capture the generic feature of the single kagome plane. For simplicity, we set the temperature of the itinerant electrons sufficiently low as $${T}_{{{{{{\rm{KLM}}}}}}}={t}_{{{{{{\rm{K}}}}}}}/40$$ to focus on the fluctuations of spins.