de Haas-van Alphen effect of correlated Dirac states in kagome metal Fe3Sn2

Primarily considered a medium of geometric frustration, there has been a growing recognition of the kagome network as a harbor of lattice-borne topological electronic phases. In this study we report the observation of magnetoquantum de Haas-van Alphen oscillations of the ferromagnetic kagome lattice metal Fe3Sn2. We observe a pair of quasi-two-dimensional Fermi surfaces arising from bulk massive Dirac states and show that these band areas and effective masses are systematically modulated by the rotation of the ferromagnetic moment. Combined with measurements of Berry curvature induced Hall conductivity, our observations suggest that the ferromagnetic Dirac fermions in Fe3Sn2 are subject to intrinsic spin-orbit coupling in the d electron sector which is likely of Kane-Mele type. Our results provide insights for spintronic manipulation of magnetic topological electronic states and pathways to realizing further highly correlated topological materials from the lattice perspective.


Supplementary Note 1 | Torque profile in broad angular and temperature range
In Supplementary Fig.1(a) we show the field profile of magnetic torque ( ) up to applied fields of # = 65 T at the base temperature (T = 0.45 K ~ 0.6 K) for ' = -85 o to 85 o for sample A ( ' = 0 corresponds to H parallel to the normal of the kagome plane). de Haas-van Alphen (dHvA) oscillations are seen to onset above approximately 20 T. At low angles (≤ 35°) a distinct peak appears before a gradual decay while at high angles (≥ 40°) a broad shoulder covers the whole field range, as exemplified by 15 o and 60 o data shown in main text Fig.1(c). This behavior is also observed in low field capacitive torque experiments performed in a superconducting magnet (see main text Fig. 4).
With increasing H, we identify distinct regimes for the observed overall torque response of Fe3Sn2. At low field (< 1 T) the magnetization evolves sharply, consistent with motion of soft ferromagnetic domains to be polarized along the field direction and ∼ 0 ; as described in the main text the moment is aligned to better than 0.12 1 /f.u. by 2 T. This is followed by an approximately flat torque response up to 10 T, corresponding to further alignment of the moments along H to within 0.01 1 /f.u. Above this field scale we see the onset of dHvA oscillations and  the appearance of a torque of opposite sign to that at low field (this appears as the small deviation from H -1 for 8 ( ) in main text Fig. 4(b)). Plotting the torque response versus H 2 makes it clear that this high field deviation corresponds to a linear susceptibility. This is naturally associated with the electronic susceptibility that itself is the source of the dHvA oscillations. Here for this contribution to torque ' we define an effective volume susceptibility 566 as the coefficient in linear in ( # ) 0 where | ' | = 566 ( # ) 0 . Here ' = × # = # ( => ? − ? => ) = # 0 ( => => ? − ? ? => ) = ( # ) 0 ( => sin cos − ? cos sin ) = ( # ) 0 ( => − ? ) × sin2 /2 , which gives 566 = ( => − ? )sin2 /2 . This angular dependence is seen in Supplementary Fig. 1(d) and the resulting Δ = => − ? ∼ 4 × 10 KL is on the same order of Pauli paramagnetism/Landau diamagnetism expected for a free electron system with comparable carrier density with Fe3Sn2 [1].
We estimate the anisotropy energy ' using the angular dependent torque response at 10 T (Supplementary Fig. 1(c)) assuming the planar type anisotropy to the first order = ' sin 0 . ' = −0.3 × 10 S J/m 3 for sample A which may include a considerable sample-dependent shape anisotropy in addition to the easy-plane type ferromagnetism at low temperatures.
In Supplementary Fig. 2(a) we show the low field torque response at an extended T range measured in a superconducting magnet. No hysteresis is observed. The sign change of the torque with T is a further indication of the bulk spin-reorientation [2,3]. At high T, the torque profiles close to the easy c axis (Supplementary Fig. 2(b)) resembles that observed in uniaxial antiferromagnetic systems with H close to the Ising axis where the peak location concurs with the spin-flop field [4]. This suggests a potentially complex evolution of magnetism at low fields particularly at elevated T. An interesting direction for further study is to investigate the energetics of the massive Dirac and neighboring bands and their evolution in this process, as this may be a key to understanding the low energy scale for these changes.

Supplementary Note 2 | Angular dependence of the dHvA oscillations
In Supplementary Fig. 3  close to the kagome plane, the FFT spectra is composed of two peaks (a weak third frequency is also resolved at some angles) that are maximized at the in-plane direction, and we assign these to ' and 0 .

Supplementary Note 3 | Comparison of torque response in different in-plane directions
In sample A we have measured the torque response with rotation of H from c towards the two inequivalent in-plane directions ( ' and 0 rotations as defined in the main text Fig. 1(b)). The results are shown in Supplementary Figs. 5(a) and (b), respectively. The behaviors are similar in both the overall torque and dHvA response (for the frequency behaviors see Fig. 2

Supplementary Note 4 | Anisotropic magnetoresistance in Fe3Sn2
We show the anisotropic magnetoresistance (AMR) in Where A1 describes the two-fold anisotropy defined by the current and A2 describes the six-fold anisotropy related to the in-plane ferromagnetic order. Typical fitting curves to Supplementary Eqn. 1 can be seen in Supplementary Fig. 6(b) and the temperature dependence of the fitting parameters are shown in Supplementary Fig. 6(c) where the two-fold symmetry defined by the current direction is dominating over a wide temperature range and comparable with the six-fold symmetry at the lowest temperature.

Supplementary Note 5 | Temperature/Field dependences of dHvA oscillations
In Supplementary Fig. 7 we show the Lifshitz-Kosevich (LK) fitting for the normalized FFT peak intensities at various angles (see main text). The blue circles correspond to ' oscillations and red circles to 0 oscillations. Supplementary Fig. 8 shows the field dependence of the oscillation amplitude from which we may extract the Dingle temperature TD. For sample B (Supplementary Fig. 8(a)) we find `=  (4.6 ± 0.8) K for sample C, and (4.9 ± 0.6) K for D with a fit shown in Supplementary Fig. 8(b) where oscillations are seen to onset at approximately 10 T.

Supplementary Note 6 | Comparison with the Yamaji model
An alternative scenario to describe the faster than 1/cos evolution of the observed frequencies is a quasi-2D Fermi surface with kz-warping. We construct here for comparison a simple Yamaji model with a sinusoidal warping for in-plane massive Dirac dispersions [6]. The  is the Kronecker delta. Due to the lattice symmetry, t11 = t22 = t and t12 = t21 = t′. We introduce an additional parameter γ to characterize the imperfection of kagome lattice within each plane (γ = 1 corresponds to the perfect kagome lattice) [8]. In the second sum in Supplementary Eqn. 2, λ is the strength of spin-orbital coupling, Eij is the local electric field in the hopping path along Rij direction. Bi is the local magnetic field, which we employ to simulate the effect of ferromagnetic order along different directions. We use and to represent the direction of the electric field: is the polar angle defined away from z axis (as shown in Supplementary Fig. 10(a)) and is the azimuth angle in the plane defined away from each in-plane nearest neighbor bonding Rij. We note that on the bilayer kagome lattice both in and out-of-plane electric fields are symmetrically allowed.
The set of bands shown in Supplementary Fig. 10 are calculated using the following parameters: = OE = 1, = 1.01 while the , , values are noted in each panel. Without spinorbit coupling nor magnetic order the system shows a massless Dirac crossing at K ( Supplementary   Fig. 10(b)), which is protected by the crystallographic symmetries [8]. With the introduction of spin-orbit coupling a gap opens at this Dirac point ( Supplementary Fig. 10(c)). When the Fermi level is placed within the gap this results in a time-reversal symmetric topological insulating phase.
Further introducing the ferromagnetic order gives rise to a moment-orientation-dependent gap ( Supplementary Figs. 10(d,e)) where the gap is opened exclusively when the moment is out-ofplane. This reflects the selection rule ⋅ • wz × wz ‚ dedicated by the microscopic form of spinorbit coupling [9]. We note that the present model only takes into account the lattice and the conclusions regarding band topology hold in a general sense. Further consideration of the orbital