Anomalous electrons in a metallic kagome ferromagnet

Ordinary metals contain electron liquids within well-defined ‘Fermi’ surfaces at which the electrons behave as if they were non-interacting. In the absence of transitions to entirely new phases such as insulators or superconductors, interactions between electrons induce scattering that is quadratic in the deviation of the binding energy from the Fermi level. A long-standing puzzle is that certain materials do not fit this ‘Fermi liquid’ description. A common feature is strong interactions between electrons relative to their kinetic energies. One route to this regime is special lattices to reduce the electron kinetic energies. Twisted bilayer graphene1–4 is an example, and trihexagonal tiling lattices (triangular ‘kagome’), with all corner sites removed on a 2 × 2 superlattice, can also host narrow electron bands5 for which interaction effects would be enhanced. Here we describe spectroscopy revealing non-Fermi-liquid behaviour for the ferromagnetic kagome metal Fe3Sn2 (ref. 6). We discover three C3-symmetric electron pockets at the Brillouin zone centre, two of which are expected from density functional theory. The third and most sharply defined band emerges at low temperatures and binding energies by means of fractionalization of one of the other two, most likely on the account of enhanced electron–electron interactions owing to a flat band predicted to lie just above the Fermi level. Our discovery opens the topic of how such many-body physics involving flat bands7,8 could differ depending on whether they arise from lattice geometry or from strongly localized atomic orbitals9,10.

noteworthy is the prediction from 2011 based on a tight binding model that a kagomé material which is both ferromagnetic and metallic could display interesting collective phenomena such as the fractional quantum Hall effect in the absence of an external magnetic field 3 .
Rhombohedral (space group R-3m, SG-166) Fe3Sn2 is precisely such a substance 17 .Its building blocks are bilayers of distorted, three-fold (not six-fold) symmetric kagomé layers of Fe atoms alternating with single layers of stanene, as shown in Fig. 1a.Of long-standing interest have been the ferromagnetic properties, including a high Curie temperature of 640 K 1 and a spin reorientation transition near 120 K, which was recently confirmed to be of first order 5,6 .The latter follows from a crossing of different free energies associated with subtly different band structures and resulting Fermi surfaces for the different moment directions 4,6 (see figure SI8 in supplementary information (SI)).The electronic band structure itself has been directly investigated by photoemission 4,[18][19][20] , scanning tunneling microscopy 21 , and bulk transport measurements 2,5,[22][23][24] .The simplest tight-binding calculations (single orbital per site of kagomé lattice) for idealized two-dimensional slabs reveal flat bands and Dirac points, and there are claims of their observation in angle-resolved photoemission (ARPES), bulk Hall effect, and de Haas-van Alphen measurements 7,18 .In contrast, density functional theory (DFT) for the bulk material yields neither obvious flat bands nor Dirac nodes (see figure SI10) but does reveal Weyl points near the Fermi surface 4 .Nonetheless, it accounts for the seeming appearance of Dirac nodes in ARPES in terms of superposed bulk and surface bands 4,20 .The ARPES data collected so far do not display sharply resolved Weyl nodes.However, there is another feature that unambiguously distinguishes the tight-binding calculations from DFT predictions, namely, only DFT predicts electron pockets surrounding the zone centers (see figure 1c and figure SI7 -SI9 4 ).Furthermore, as shown in figure 1c, the pockets are highly sensitive to electron correlation effects as parametrized by the effective Coulomb interaction U in DFT + U calculations, and so could be exploited for measurement of U. We have therefore set out to search for and characterize these pockets experimentally.
Here, we report the first detailed experimental study of the distinct electron pockets, present in lower resolution and more surface-sensitive measurements at the Brillouin zone (BZ) center of Fe3Sn2 4 , by employing state-of-the-art micro-focused laser ARPES (µ-ARPES) to overcome the presence of crystallographic twinning.Our results are consistent with the same density functional calculations which suggest that the Dirac-like cones reported in conventional synchrotron-based ARPES 4,18 derive from a coincidence between minima and maxima of surface and bulk bands 4,20 , respectively.At the lowest temperature (6 K), there are very sharp quasiparticles which define small Fermi surfaces around the Brillouin zone centre, and their inverse lifetimes show the rapid linear increase with binding energy characteristic of strongly interacting fermions.On warming, the most sharply resolved band crossing the Fermi surface effectively dissolves, possibly due to hybridization and subsequent localization involving a flat band placed by DFT just above the chemical potential.

Motivation for spatially resolved ARPES experiments
ARPES is the natural technique for determining the electronic band structure and has been applied on numerous occasions to Fe3Sn2 4,[18][19][20] .However, there are four potential difficulties in assessing ARPES data, beyond the usual challenge of distinguishing surface from bulk contributions.A first challenge relates to the large number of atoms per unit cell (18 Fe and 12   Sn), leading to closely spaced bands which, due to magnetic order, will also be spin-split.A second difficulty is the possibility of crystal twinning originating during the crystal growth and which corresponds to rotation by  =  × 60°,  = ±(1,3,5) (figure 1a).A third problem is the possibility of different surface terminations, which would yield different ARPES results.A fourth aspect is the presence of ferromagnetism where, due to spin-orbit coupling, different magnetic domains are associated with different electronic structures 4 .
The first problem can only be remedied by ultra-high-resolution spectroscopy, while the remaining three are all linked to different types of multidomain structures over which ordinary ARPES measurements perform averages.For instance, in the case of crystal twinning, the complication is caused by the "breathing" kagomé motif of Fe3Sn2 consisting of two different corner-triangle sizes denoted by triangles with different colors in figure 1a, resulting in a threefold rotation symmetry, rather than six-fold, in the absence of magnetic order 4 .The twinning will be clearly observed when the isoenergy surfaces are away from the high symmetry   = (0,   ) planes as shown by the results of the DFT calculations in figure 1b ( 1c) as it can reproduce the electron pockets found by ARPES near the Γ point, which will be discussed in detail later.However, the ARPES results reported thus far are predominantly six-fold symmetric 4,[18][19][20] , which can only be attained at exactly the high symmetry   = (0,   ) planes for a single domain phase.Tanaka et al. 20 interpreted their ARPES results as being three-fold symmetric for   ≠ (0,   ), but the data quality is not sufficient to reach that conclusion unambiguously.
Here, we address the question of discriminating domains with different crystallographic twins, surface termination, and magnetic orientation, by using three forms of photoemission microscopy to characterize Fe3Sn2, namely, spatially-resolved X-ray photoelectron spectroscopy (XPS), X-ray absorption spectroscopy (XAS), and X-ray magnetic circular dichroism (XMCD), which are used to distinguish physical domains related to surface termination and magnetism, respectively, and show that in fact they are uncorrelated.
Separately, by using -focused laser ARPES, we probe at very high energy and momentum resolution the band structure close to the Brillouin zone center Γ.In addition to discovering electron pockets, we also observe directly that there are different electronic domains in Fe3Sn2 rotated around the c-axis by  =  × 60°,  = ± (1,3,5).By varying the temperature and by comparing the evolving magnetic domain pattern, we conclude that the observed differences in band structure are solely due to twinning and not due to different magnetic states.Direct comparison of the laser  -ARPES data with synchrotron ARPES data that averages over much larger areas (presented in the SI) shows that the data from synchrotron ARPES corresponds to the average over crystallographic twins and therefore cannot be interpreted without help from either calculations or a spatially resolved technique like  -ARPES.

Spatially-resolved XMCD, XAS, and XPS
We first consider the magnetic domain configuration and different surface terminations, which typical ARPES averages over, by using x-ray photoemission electron microscopy (XPEEM) to obtain spatially resolved XAS and XPS maps of the sample.In our case, the XAS is monitored using photoelectrons with kinetic energy below ~ 1 -2 eV (low energy secondary electron, 'bulk' sensitive to a depth of a few nm from the surface) while the XPS photoelectrons have a kinetic energy of 96 eV (surface sensitive); the sample was cleaved in vacuum and investigated at 80 K, the base temperature of the cryostat.Figure 2a shows the XMCD map, which is obtained from the differences of XAS collected for left circular (CL) and right circular (CR) polarized photons at the Fe L3-edge, while figure 2b shows the map of the photoelectron yield summed over both polarizations (CL+CR).Figure 2a displays the magnetic domains of the system while figure b confirms that the iron in the bulk of the sample is homogeneous.
Moving to the topic of surface termination, figure 2c shows the XPS map for the same region as in figure 2a and b, obtained for the Sn 3d levels and indicating variations in the Sn intensity.Similar results are observed for a different cleaved surface, shown in figure SI2. Figure 2d displays the intensity histogram from the red square in figure 2c: it is bimodal (in this case well characterized as the sum of two Gaussians), which indicates that there are two possible Sn populations for the surfaces formed upon cleaving Fe3Sn2.The ratio between the mean intensities for the two Gaussians is  1 / 2 = 1.8 ± 0.6, where  1 is the brighter intensity and  2 is the darker intensity mean value (follow the derivation in SI).The quantization of the Sn intensity suggested by the bimodal histogram is also apparent from the line scan, also shown in figure 2d, revealing a sharp step in intensity.These phenomena follow because the material consists of (kagomé) Fe bilayers alternating with stanene monolayers.In the first two top-most layers one expects either one or two Fe layers (no stanene layer).Thus, the clear contrast in Sn distribution (figure 2c) arises from stanene-termination for the brighter areas and kagométermination for the smaller lower intensity islands.Furthermore, given the much bigger area associated with the larger Sn photoemission intensity, we conclude that the majority of the sample is stanene-terminated, agreeing with the suggestion made in reference 4 .Comparison of figures 2c and 2a also demonstrates that the magnetic domain pattern is uncorrelated with the surface termination.More discussions about the XPEEM and its interpretation can be found in the SI.
The XMCD map reveals a magnetic domain pattern with a hierarchy of characteristic features ranging from 50 to < 1 µm, indicating the need for  -ARPES to avoid averaging of momentum and energy-resolved states associated with different magnetization directions at 80 K.At the same time, the different surface terminations as established by XPS are characterized by length scales in the same range, implying that -ARPES could also detect differences in surface states should they matter for the photon energy chosen.

Laser 𝜇-ARPES
We performed laser -ARPES (detailed setup described in SI) to probe the electronic structure at high energy and spatial resolution to determine inhomogeneities in the electronic structure linked to magnetic, chemical, and crystallographic domains.The 6.01 eV photon energy limits the Brillouin zone probed to near the  point, which nonetheless is of great interest because it enables detailed imaging of the electron pocket seen previously at low resolution with a large beam 4 , and predicted by DFT 4,19,20 but not by other band theories 18 .Figure 3a shows our experimental data at 6 K, which reveal this electron pocket with unprecedented clarity for two laser beam positions on the sample.
The Fermi surfaces are located on three rings in (  ,   ), with the inner ring clearly three-fold modulated, and the second and third less obviously so, but nonetheless with their most intense feature 60 degrees out-of-phase with the inner ring.The data presented in figure 3a are superpositions of data from both linear laser polarizations.Comparison of the data in figure 3a(i) and (ii) immediately indicates a relative rotation of 180°, which is equivalent to 60° for this rhombohedral material.We thus come to the important finding that the electronic structure of Fe3Sn2 at 6 K shows two distinct yet equivalent areas (rotated from each other).The positions of these rings are located at the center of the BZ as shown in figure 3b(i).The dispersion curves in figure 3a(iii) and (iv) show the intercepts of three bands with the three Fermi surfaces, and confirms the same 180° rotation in (  ,   ) which relates to their respective Fermi surfaces measured in figure 3a(i) and (ii).Two of the bands cross the inner and outermost Fermi surfaces and are labeled as  and  in figure 3a(iii) and (iv), respectively, and merge to form a parabola centered at Γ ̅ with 0.1 eV minimum binding energy.There is also a very sharp feature between  and  labelled , which abruptly loses intensity through  around   = 0.025 eV, as can be seen from the energy-momentum cut image (figure 3a(iii) and (iv)).We also notice a very weak broad band lying at higher binding energy labeled as , which has asymmetric intensity similar to that of the  and  bands, indicating that it is three-fold symmetric as well.
The two distinct areas rotated by 180° with respect to each other, i.e., twinned domains, can be mapped in real-space by tracing the intensity differences between the left versus right side of the  band in the ARPES cut [figure 3a(iii) and (iv), where data for LH and LV polarizations are summed].Figure 3c is the resulting domain image, where for  band intensity larger on the left side, we label the sample position in red, otherwise we color it green.In the gradation from green to red, white indicates no intensity difference.In the upper panel of figure 3c, we see that there is a relatively abrupt transition from the "red" to "green" regions, indicating that the twinned domain is rotated discontinuously.A rough measurement of the domain wall angle gives a value of 60°, in agreement with the twin rotation for the crystal structure.
The bulk electronic properties of Fe3Sn2 undergo pronounced evolution on cooling below 80 K, where the spin reorientation to the low temperature phase is complete 5 .Particularly dramatic are the modulation of the carrier density depending on the magnetization direction below 80 K 24 and the temperature dependence of the anisotropic magnetoresistance and planar Hall effect 23 , which features a threefold (not sixfold) antisymmetric term as a function of in-plane field direction.To determine whether any of the ARPES features which we see at low temperature could be related to these transport phenomena, we have collected  -ARPES data at 80 K, which is at the boundary of these characteristic temperatures but still in the regime where the magnetism is dominated by domains with moments along the kagomé planes 6 , although with smaller domains compared to 6 K.The lower panel of Figure 3c shows the intensity asymmetry image for 80 K, which is not appreciably different from the 6 K image above it.This is not surprising since based on magnetometry 6 , magnetoresistance 5 , and magnetic domain imaging 6 , the magnetization is in the kagome plane for temperatures below 80 K.The major effect of warming in this case is a broadening of the spectra, and in particular the loss of the very sharp  bands as seen in figure 4a, which we discuss in greater detail below.In the SI, we show that the earlier synchrotron ARPES data are consistent with an average over the twin domain data obtained from laser µ-ARPES.We attribute the three-fold pattern itself to originate from probing off high-symmetry   plane, yet still close to   = 0 plane (see SI for further explanation), confirming the bulk nature of the spectral data; high symmetry probing at   = 0 plane or   =  plane will generate a six-fold pattern regardless of the "breathing" pattern of the kagomé.

Comparison to DFT calculations
We now take advantage of the high real and reciprocal space resolution of our single domain ARPES to investigate the effect of correlations on the electronic band structure of Fe3Sn2.The first question concerns the effective on-site Coulomb potential U, which should be included in the DFT + U band structure calculations.Electron pockets centered on Γ and close to the Fermi level exist for a large range of U values (U = 0.0 -3.0 eV) (see figure SI7).However, the distance Δ to the Fermi energy from the bottom of the bands for the pocket varies strongly with U, as do the Fermi vectors   , which can be seen in figure 1c  , where Δk is the half width of the momentum distribution curve (MDC) peaks (at 6 K and at the Fermi level).
The area of the electron pockets is calculated by tracing the visible peaks of the corresponding band at the Fermi level and assuming a circular extrapolation to make closed contours.More details on the EDC of the bands from   to band minimum can be found in figure SI5.
For the  band, λ =155 ± 5 Å at 6 K, a remarkably long mean free path for a topological (bulk) metal and comparable to that for the much simpler surface Dirac quasiparticles for the n-type Bi2Te3 topological insulator (~190 Å) 27 ; for , λ is reduced by a factor of two to ∼ 85 Å but is still long by the same standards.These values are consistent with the residual ( = 2 K) resistivity   (0) = 4.35 Ω. and a carrier density of 1.8 × 10 28  −3 from transport data 5,24 for samples similar to ours, grown by the same method by the same group.In particular, applying the Drude formula  0 =  2   for conductivity, we obtain a scattering time ~4.5 × 10 −14 s by assuming that the carriers have an effective mass  =   .For an averaged Fermi velocity of ~1.2 eV-Å,  converts to a mean free path  =    ∼ 82 Å, which is of the order of the ARPES results for the  and  bands.
It is also interesting to ask whether there is bulk evidence for small Fermi surfaces similar to those which we have discovered.The answer is yes: the calculated de Haas van Alphen (dHvA) frequency 254 T of the  band, whose Fermi surface is centered on Γ , is in reasonable agreement with that measured for reference 7 (~200 T) and attributed to the "Dirac ring" emanating from the putative Dirac point at the K ̅ point and 0.1 eV below the Fermi surface.
Given that DFT accounts for electron pockets centered on Γ while explaining the Dirac-like features in terms of an overlap of surface and bulk bands 20  depends strongly on the band chosen, with resulting values from 0.35 to 0.63 30 , and for the ferropnictides  varies throughout the range between 0 and 1.7 31,32 .
Eq.( 1) implies that a temperature increase from   = 6 to 70 K, as in figure 4a should cause, for small  and The conclusion is that / scaling is roughly obeyed in the sense that marginal quasiparticles for both the  and the  bands are distinguishable or not near the Fermi surface to an extent consistent with Eq(1).However, the value for '/ is unusually large, and the energymomentum diagram is very peculiar in that the much broader  band seems much more associated with the minimum near Γ than the short, 20 meV stretch of  band visible near the Fermi surface, which simply looks like a sharp copy of the  band displaced upwards by the same 20 meV towards   .These observations, not easily accounted for by DFT, together suggest a many-body origin for the  band.A possible scenario is that there is strong hybridization between flat and  band states near the bare Fermi surface which results in interaction-induced (Mott-like) localization that reduces the free carrier density;   for the electron pockets seen here would then be correspondingly reduced by   , and the resulting () band will be raised towards   by Δ = ℏ    .There will of course still be broad spectral weight from the unrenormalized () bands at the original   , and for binding energies in excess of Δ , the localized electrons rejoin the Fermi sea and the spectral weight will reside primarily near the  band as might be calculated using DFT.Warming would similarly act to delocalize the carriers, and thus cause the loss of the  "band".

Conclusions
We This means that we are able to measure scattering lengths as well as Fermi surface areas.The former is in semiquantitative agreement with the mean free path obtainable from residual resistance data, while the Fermi surface areas are near de Haas van Alphen results previously interpreted to derive from the Dirac cones not found by DFT.
At low temperature, once the quasiparticle Green's functions are examined away from the regime where their widths are dominated by elastic scattering, it appears that their widths in momentum space scale with their momenta, implying marginal rather than ordinary Fermi liquid behaviour, and associated strong correlation physics.However, the abrupt disappearance of the sharpest band at the very modest binding energy of 20 meV, and quasiparticle broadening greatly in excess of    on warming to 70 K indicate that the kagome ferromagnet Fe3Sn2 is hosting more than "conventional" strong correlations.The flat band seen in DFT just above the Fermi energy is a distinguishing feature: virtual and thermal occupancy of this band could greatly enhance electron-electron scattering that we see directly in our data, and also account for formation of the peculiarly sharp and displaced  band which appears below 70 K together with numerous anomalous magnetotransport properties.We close by noting that there is an enormous literature dealing with the Anderson and Kondo lattices where dispersive bands coexist with bands which are flat on account of small underlying atomic orbitals 33 .Our  band results, which we can only explain at a qualitative level, suggest further theoretical and experimental studies to examine interaction effects where dispersive bands cross bands which are flat for reasons of lattice geometry, as exhibited for example by kagome systems, rather than small orbital extent 34 .This shows that the magnetic domain has no correlation to the surface termination.Panel d.
shows that the intensity difference found in panel c. is step-like with intensity ratio of 1.8 ± 0.6.Fe3Sn2 crystalizes in a rhombohedral structure with space group 3 ̅ , with crystal axis ,  = 5.34 Å,  = 19.80Å, and  = 120°.Fe3Sn2 single crystals were grown by a vapor transport method.Stoichiometric iron powder (Alfa Aesar, ＞99.9%) and tin powder (Alfa Aesar, ＞ 99.9%) were placed into an evacuated quartz tube.Then the tube was annealed at 800℃ for 7 days before quenching in icy water.The crystal structure of the polycrystalline Fe3Sn2 precursor is confirmed by the x-ray diffraction (XRD) data shown in Fig. SI1a.The obtained polycrystalline Fe3Sn2 precursor was thoroughly ground and sealed with I2 (~ 4 mg/cm 3 ) in a quartz tube 1 cm in diameter and 16 cm in length.Fe3Sn2 single crystals were obtained under a temperature gradient of 650 ℃ (source) to 720 ℃ (sink) for two weeks.
The cleaved surface for laser ARPES is confirmed to be (001), the ab plane, from low energy electron diffraction (LEED) measurements on a freshly cleaved surface, as shown in Fig. SI1b.The typical cleaved surface is displayed in figure SI1.c indicating a flat surface suitable for ARPES measurements.

B. Magnetic domain and surface termination probed by X-ray photoemission electron microscopy (XPEEM)
The surface termination of the Fe3Sn2 samples is probed by mapping the Sn 3d3/2 and Fe 2p3/2 local photoemitted electron intensities using the spectroscopic capabilities of the SPELEEM III instrument (Elmitec GmbH) at the Surface/Interface Microscopy (SIM) beamline of the Swiss Light Source.The photon excitation energy was varied such that the kinetic energy of the detected electrons was fixed at 96.8 eV in order to achieve a high surface sensitivity (~2 Å according to the universal curve 1 ).The photoemission peaks are determined by recording XPEEM image sequences with x rays, linearly polarized in the plane of the sample, with varying photon energy.The samples are cleaved in situ prior to the XPEEM measurements at 80 K, base temperature of the cryostat.A typical result is shown in Fig. SI2a, displaying a clear cleaved surface and an area with glue residue and a crack.The XPS intensity distributions for the Sn 3d3/2 and Fe 2p3/2 peaks are shown in Fig. SI2b, c, respectively.Electrons emitted from the Sn 3d3/2 orbitals have two distinct intensities on the spatially resolved map in Fig. SI2b, revealing a large region of higher intensity together with smaller islands of lower intensity.The iron emission map shown in Fig. SI2c is more homogeneous and has a much lower signal to noise ratio, expected from the lower non-resonant x-ray absorption of Sn, Z = 50, when compared to Fe, Z = 26, which prevents the observation of the smaller islands as in panel (b).
The XPS spectra displayed in panel (c) are obtained by integrating over an area larger than that for the Sn XPS peak shown in panel (b).We assign the bright area in Fig. SI2b as stanineterminated and the darker area as kagomé-terminated.However, we cannot determine the exact kagomé termination, i.e., whether it is one or two Fe layers (Fig. SI2d.i or ii) within the intensity resolution of our measurements.The cleaved surface discussed here is different from the photoemission experiment presented in figure 2 of the main manuscript.A comparison between these two sets of data shows that the area attributed to kagomé-terminated can vary in size and is not limited to only small areas as shown in Fig. 2b.
The magnetic domain configuration was determined using the x-ray magnetic circular dichroic (XMCD) effect.XPEEM images were recorded by exciting the sample with circularly polarized x-rays with the photon energy tuned to the Fe L3 edge and measuring the intensity of the low energy secondary electrons (with a probing depth of about 3-5 nm).Magnetic contrast maps were finally obtained by pixelwise division of two XPEEM images recorded with opposite x-ray helicities.

Intensity ratio of XPS signal
The PEEM micrographs showing different terminations in Fe3Sn2 can be numerically analysed as following: 1. We assume that the photon penetration depth is much longer than the escape depth of the most energetic photo-electron, i.e., an electron at the Fermi level.Thus, we ignore the exponential decay component of the photon intensity.
2. Assumption 1 leads to an expression of photo-electron intensity from a given atom called , on layer called , with distance  from the surface as where  is the escape depth of the electron,   is the proportionality constant that contains the photon cross section of atom  at layer of type , and   is the atom  density at layer of type .
3. Generalizing (2), we can express the total intensity from atoms a from all layers of type b as a summation where we assume the proportionality constant   to be the same for all similar layers at different depths.
With these three points we can start to write down the expression of XPS intensity from both Sn and Fe in total.First, we notice that there are three possible terminations and call them double kagome termination (  ), single kagome termination (  ), and stanene termination (  ) as shown in figure SI2d.

Double kagome termination (𝑺 𝒌𝒌 )case
The total Sn and Fe intensity from this termination can be expressed as Comparing with the experimental result of  ℎ /  = 1.84 ± 0.57 from figure 2 in the main text, while we cannot tell if the darker region is coming from the single or the bi-layer kagome, we can safely infer that the bright regions have stanene termination.

C. Micro-focused Laser Angle Resolved Photoemission Spectroscopy (µ-ARPES)
For the µ-ARPES measurements, we use a 6.01 eV 4 th -harmonic generation continuous laser from LEOS as photon source, a custom-built micro-focusing lens to focus the beam spot diameter to ~3 m, and an MB-Scientific analyzer equipped with a deflection angle mode to map the dispersion relation while keeping the area of interest intact (i.e., not changing due to sample rotation).Typical energy and angular resolution were 3 meV / 0.2°.The pressure during the measurement is kept at < 10 −10 mbar.The sample is mounted on a conventional 6-axes ARPES manipulator described in Ref. 2 and sample position is scanned with an xyz stage of 100 nm resolution and < 1 μm bidirectional reproducibility.A more detailed explanation can be found in Ref. 3 .During the measurement, the temperature is first lowered to 6 K, where the sample is cleaved in situ, and the sample drift at subsequent higher temperatures is tracked by using the edges of the sample as reference.The samples are pre-aligned to the high symmetry cut with low energy electron diffraction (LEED) after cleaving.where   * is the effective mass of the electron,   = ℎ −  − |  | is the kinetic energy of the electron, ℎ is the photon energy, |  | is the binding energy of the electron,  is the work function of the detector,   is the inner potential of the material, and  is the analyzer slit angle (more details can be found in these references 4,5 ).The laser photon energy used, 6.01 eV, corresponds to a perpendicular momentum close to the center of the Brillouin zone   ≈  × 2  ,  6.01 ≈ 5.93 ≈ 6, as shown in Fig. SI3, whose inner potential used is taken from Ref. 6 .The synchrotron photon energy used is 48 eV, which also lies roughly in the same position at the center of the Brillouin zone,  48 ≈ 11.98 ≈ 12. From Fig. SI3 we can see that those two photon energies differ by two Brillouin zones.
We can also estimate the uncertainty in   determination (  ) from the escape depth of the electron.In this case, we can focus on the electron close to the Fermi energy (  ≈ 1.64 eV), which has an escape depth around  ≈ 60 Å according to the universal curve These values give us an estimation of   ranging   ≈ (0.02 − 0.08) Γ agreeing (by the order) with the estimation from the universal curve.

Synchrotron vs Laser ARPES
Twinned domains exist in Fe3Sn2 and will both contribute to the ARPES data if the beam spot is bigger than the size of the domains.For example, we show in Fig. SI4a an example of ARPES data measured with 48 eV photon energy, temperature 17 K, and a spot size of ~35  collected at the SIS beamline, Swiss Light Source, equipped with Scienta R4000 hemispherical analyser.The overall feature shows a 6-fold pattern with no indication of a 3fold pattern.A closer look around the center also shows a rather blurred circular shape, which can be attributed to a combination of all twinned domains that it may cover.This central feature can in fact be reconstructed from the laser ARPES results by combining data from both twinned domains and different polarizations (LH+LV) to give us the picture shown in Fig. SI4b.Indeed, we reproduce the synchrotron ARPES data but with higher momentum resolution.

Fitting momentum distribution curve (MDC)
The momentum distribution curves (MDC) are fitted to obtain the half peak width (Δk) which can be related to the averaged scattering length of the quasiparticle as λ = 1 Δk .We obtain this peak by fitting the MDC peak with a Voigt lineshape, i.e., a Lorentzian convolved with a Gaussian, where the Gaussian simulates the instrument response.For momentum scans (MDC), the latter combines angular and energy resolutions to yield a full-width-at-half-maximum (FWHM) δk ≈ √(   )

D. Density Functional Theory Calculations
Electronic structure calculations are performed within density functional theory (DFT) using the projector augmented wave method (PAW) 7 as implemented in the Viena ab initio simulation package (VASP) 8,9 .The exchange-correlation potential is described using the Perdew-Burke-Ernzerhof (PBE) 10 functional within the generalized-gradient approximation (GGA).Plane waves are used as a basis set with a kinetic energy cutoff of 450 eV.To sample the Brillouin zone, we employ a 24 × 24 × 24 Monkhorst-Pack grid.The spin-orbit coupling (SOC) is included in our calculations as described in Ref. 11 .We take into account the p and d semi-core states for the Fe and Sn PAW datasets used, respectively, and the simplified on-site Hubbard U correction 12 is added for the Fe d orbitals.We determined a Hubbard U value of 1.3 eV to reproduce the ARPES band structure around the Γ point at the Fermi level.Band structures calculated using several different U values with magnetic moments pointing in the kagomé plane are shown in Fig. SI7 (notation and labeling of Brillouin zone points are shown in Fig. SI6).In Fig. SI8, the band structure evolution as a function of magnetic moment direction ( ∥ ,  = 70°, and  ∥ ) is also plotted with selected U values displayed (0, 0.5 and 1.3 eV).Fig. SI9 simulates the band at ARPES energy window by plotting bands in the range of   = 0 − 0.05 ΓZ.

"Dirac points" in DFT calculations at various U values
We attempted to reproduce the "Dirac points" predicted in the tight-binding model by changing the U value in the DFT calculations as shown in Fig. SI10.Our results show that the "Dirac points" are not reproduced for U values between 0 and 1.3 eV.One may see that the closest resemblance of the band dispersion to the linear crossing is achieved at a U value of 0.5 eV or above, where the gap is narrow but never fully closes.

References for the SI
and SI7.The best description of the  and  bands, taking into account both the band bottom and   , occurs for U = 1.3 eV, differing from the values U = 0.5 eV4 or U > 2 eV25 selected for previous DFT calculations.We display the DFT cut along M ̅ Γ ̅ M′ ̅̅̅ , for   = 0 − 0.05 ,  = 1.3 eV in figure3b(ii).The cut is dominated by parabolic bands coexisting with a relatively flat band located around 15 meV above   , and supports the assertion that  and  originate from bulk states smeared due to   broadening26 , and are asymmetric (3-fold instead of 6-fold) due to slight off   = 0 probing.We have labelled the features in the cut accordingly: the   broadening yields two visible branches cutting the Fermi level along the M ̅ Γ ̅ trajectory, while there is a broader, merged like dispersion along Γ ̅ M′ ̅̅̅ .The Fermi velocities do not precisely match those obtained by ARPES but can be tuned via small adjustments to the chemical potential on account of the apparent hybridization with the flat band very close to the Fermi level.We note that irrespective of the choice of U significant differences remain between ARPES and DFT.None of the existing DFT calculations reproduces the very small gap between the bottom of the  and  bands and the top of ; additionally, the sharp  band found by ARPES near the Fermi level is not seen in calculations.Sharp but marginal quasiparticles at low temperaturesKey parameters of quasiparticles which include their Fermi surface areas, mass, and scattering rates, are calculated from our experimental data.We display the representative dispersion cuts for the individual linear photon polarizations in figure4a(i) and (iii) for horizontal (LH) polarization at 6 and 70 K, respectively, and in figure4a(ii) and (iv) for vertical (LV) polarization at 6 and 70 K, respectively.LH displays mostly the  band, indicating it is an even orbital with respect to the incident plane of the experiment, while the , , and  bands are odd parity orbitals.In TableIwe summarize for the three pockets the effective mass at the Fermi level calculated as  * = ℏ  (  ) −1 , where the Fermi velocity is   = of the electron pocket converted into the de Haas-van Alphen (dHvA) frequency as () = ℏ 2 × , and the scattering lengths of the quasiparticle defined as λ = 1 Δk

1 𝜋,𝑏= 1 .
, a re-interpretation of the observed dHvA frequency is needed.Moving below   , the  band is significantly broadened and effectively disappears for binding energies in excess of approximately 20 meV.Below this energy, the half-width at half-maxima Δ() of the peaks in the MDC increases linearly with |() −   |, with slope   = 0.361 ± 0.062 , as shown in figure 4. For the much steeper  band, there is a similar finding: Δ() rises linearly with |() −   | .The fitted slope   = 0.353 ± 0.071 is indistinguishable from both that for the  band, and is also not far from   = for which the relative quasiparticle wavelength 2/|() −   | is precisely twice the quasiparticle scattering length 1/Δ().Because of the linearity of the quasiparticle dispersions (upper frame of Fig. 4b) over the same momentum ranges, the low temperature (6 K) results are all consistent with a marginal Fermi liquid hypothesis 28,29 , for which the imaginary part of the self-energy is Im(Σ) = Δ = max(|  |,   T) = max(ℏ  |() −   | , ′  T) , which translates to Δ() = max(|() −   | , ′  T/ℏ  ) (1), where  and ′ are coupling constants of order unity, where the initial theoretical Ansatz was  ′ Experiments have yielded a variety of values for  in strongly correlated systems.For example, for the cuprate superconductor Bi2Sr2CaCu2O8+δ (BSCCO),  BSCCO = 0.75 29 , while for BaCr2As2,  BaCr 2 As 2

1 ,
Δ() to increase by  ′   ( −   )/ℏ  = 1.3 × 10 −3 Å −1 and 2.7 × 10 −3 Å −1 for the  and  bands respectively.This is much more modest than what we actually observe for Fe3Sn2, where at   , Δ for  grows by (8 ± 1) × 10 −3 Å −1 , which is '/ = 6 times larger than the theoretically expected ratio of unity and 2.4 times bigger than '/ observed for BSCCO31,32 , while corresponding to a very large scattering rate, Δ = ℏΔ *   ≈ 20 meV ≈ 3.3   T. At the same time, any distinction between the  and  bands, which near the Fermi level are simply displaced by 20 meV with respect to each other, vanishes, which is consistent with the Δ measured for the  band at 70 K, and also the fact that at 6 K, the  band becomes invisible for binding energies above a similar energy.
have discovered electron pockets labeled as , , and  centered at the  point in Fe3Sn2 by employing state-of-the-art -focused laser ARPES to overcome the averaging effects of crystallographic twinning in Fe3Sn2 crystals.These electron pockets are not found in the simple tight-binding models proposed for Fe3Sn2, which also posit doubled Dirac cones near the K point, while they can be reproduced by DFT calculations, where no Dirac cones are predicted (see SI for detailed discussion).The  and  electron pockets are very sensitive to the Coulomb interaction parameter U in DFT+U calculations, and are well reproduced by  = 1.3 eV.The same calculations also yield a flat band about 20 meV above the Fermi level for the pockets.Our combination of high-resolution spectroscopy with single domain selection provides an unprecedented opportunity to examine the quasiparticles in a strongly correlated kagome metal.

Figure 1 .
Figure 1.a. Crystal structure of Fe3Sn2 displaying a stanene layer sandwiched between kagomé bilayers.  denotes the angle between the magnetic moment M ⃗⃗⃗ and the c-axis.Twin domains may occur when the system is rotated around the c-axis.b.Calculated Fermi surface at   = 0.05/ for U =1.3 eV showing a three-fold pattern (left) and its twinned image (right) when the magnetic moment is pointing off the c-axis (  = 90°) and along the c-axis (  = 0°).c. Results of DFT calculations for   = 0 with representative  = 0, 0.5, and 1.3 eV values showing the presence of an electron pocket (denoted by asterisks), which is investigated by μ-ARPES (marked by the area within the dashed-line).

Figure 2 .
Figure 2. a. XPEEM magnetic contrast (CR-CL intensity) map taken at the Fe L3 peak, whose signal is 'bulk'-sensitive, showing a complex magnetic domain configuration in an area where b. panel (CR+CL intensity) shows a homogeneous contrast, indicating a homogeneous phase of the sample.c.XPEEM XPS map of the sample at the Sn 3d3/2 peak, whose signal is surface sensitive, showing a domain structure distinct from the magnetic domain shown in panel a., which is attributed to different cleaving terminations, i.e., two different surface terminations.

Figure 3 .
Figure 3. a. Laser ARPES data for Sn-terminated Fe3Sn2.(i)-(ii) Fermi surfaces showing a three-fold pattern with an electron pocket at the Γ ̅ point and three petals, which are rotated relative to each other distinguishing the crystal twin domains.(iii)-(iv) Energy versus momentum cut of one of the petals (dashed line at (i) and (ii), M ̅ Γ ̅ M′ ̅̅̅ direction) showing the electron pocket band labeled , a band labeled , which abruptly disappears through , a petal labeled  sharing a bottom with the  band, and lastly a broad and weak band referred to as .b (i) Illustration of the reciprocal space probed with laser ARPES, which spans a small portion of the total BZ.b (ii) DFT calculation along M ̅ Γ ̅ M′ ̅̅̅ direction for   = 0 − 0.05  as illustration of   broadening and asymmetrization, agreeing tentatively on the formation of  and  bands.c.Twin domains showing a relatively unchanged shape at 6 K as compared to 80 K.The magnetisation points along the ab-plane at both temperatures but the magnetic domain size is known to be different at these two temperatures, with large domains at temperatures below 64 K and with relatively smaller domains at 80 K 6 .

Figure 4 .
Figure 4. (a) M ̅ Γ ̅ M′ ̅̅̅ cut from LH polarization displays mostly band labeled  at 6 K (i) and 70 K (iii), with MDC at the Fermi energy and the fitting result.M ̅ Γ ̅ M′ ̅̅̅ cut from LV polarization reveals mostly the band labeled  and  at 6 K (ii) and 70 K (iv), with MDC at the Fermi energy and the fitting result.(b) (i) Plot of the fitted peak position from MDC for  (blue 6 K and red 70 K) and  (green 6 K) with resulting linear fit super-imposed.(ii) Plot of peak width Δ vs  −   showing a same-gradient linear behaviour close to the Fermi level for  (blue 6 K and red 70 K) and  (green 6 K) indicating a marginal Fermi liquid.Upon heating to 70 K we can see that the line is shifted as described by Eq. (1) in text.

Fig
Fig. SI1 a. Powder diffraction pattern of polycrystalline Fe3Sn2 precursor, showing a good agreement with the previously reported 3 ̅  structure, with crystal axes ,  = 5.34 Å,  = 19.80Å, and  = 120°.b.Low energy electron diffraction of the cleaved surface of an Fe3Sn2 flake showing the correct symmetry pattern exposing (001) surface or ab-plane in conventional lattice.c.Typical cleaved surface picture of Fe3Sn2 showing a relatively flat surface for ARPES measurement.

Fig. SI2
Fig. SI2 a. Optical image of a cleaved surface of Fe3Sn2 single crystal with highlighted area where the spatially resolved XPS was taken showing the area with crack and silver glue residue.b.Real space XPS intensity distribution of the Sn photoemission line with the typical XPS spectrum shown below it.c.Real space XPS intensity distribution of the Fe 2p3/2 photoemission line with the typical XPS spectrum shown below it.d.The three possible surface terminations.

Fig. SI3
Fig. SI3 Perpendicular momentum (  ) position for 6.01 eV laser energy and 48 eV synchrotron photon energy, showing that the ARPES data were collected on planes with   ≈  × 2  .

Fig. SI5 (
Fig. SI5 (a) and (b) Energy distribution curve (EDC) of  (a) and & (b) at 6 K and 70 K.(c) Fitting of quasiparticle peak right at the EF (fitting the energy distribution curve or EDC) by convolving Lorentzian peak and gaussian peak (detector response).

Fig. SI6
Fig. SI6 Brillouin zone of Fe3Sn2 and the naming convention for this work.

Fig. SI7
Fig. SI7 Band structure plots of Fe3Sn2 for  ∥  and various U values in the direction ΓK, displaying the shift of bands, where the electron pocket close to the Fermi level is formed by different bands.

Fig. SI8
Fig. SI8 Band structure plots of Fe3Sn2 for selected U values (0, 0.5, 1.3 eV) at various moment directions ( ∥ ,  = 70°, and  ∥ ) in the direction ΓM, displaying the shift of bands and band splitting as the moment points more towards the c-axis.

Fig. SI10
Fig. SI10 Band structure calculations of Fe3Sn2 performed with different U values fail to reproduce the "Dirac points" at the K point.

Table I
Summary of the calculated quantities (at 6 K): Fermi surface area, dHvA frequency, Fermi velocity, effective mass, and scattering length, calculated at the Fermi surface of the corresponding , , and  bands.
where  Sn, is the proportionality constant for Sn atoms in the stanene layer,  Sn, is the Sn density in the stanene layers,  Sn, ,  Fe, are the proportionality constants for Sn, Fe atom in the kagome layer, respectively,  Sn, ,  Fe, are the Sn, Fe densities at the kagome layer, respectively, ℎ  is the distance of Fe layers between two adjacent kagome layers (the kagome bilayer), ℎ  is the distance of Fe layers in the kagome layer to the nearest stanene layer,   is the distance of Sn atom layers between two adjacent kagome layers (the kagome bilayer),   is the distance of Sn atom layers from a kagome layer to the nearest stanene layer.From the expression above, we can conclude that  ,  <  ,  <  ,  based on the following relation With these assumptions, we have Sn intensity relation for different terminations as  ,  <  ,  <  ,  From the intensity relation of both Sn and Fe, we can conclude that they are inversely related to each other.We investigate the Sn intensity ratio between terminations to correlate with the experimental data we obtained from XPS.In this model, we insert the numerical value   ≈ 2 Å and   ≈ 2.5 Å.The summary of the intensity (proportional to  Sn,k  Sn,k ) is given in the table below for two different  values.
The perpendicular momentum   of the electrons measured by ARPES can be obtained from the expression ≈ 0.05 ΓZ, as drawn in main figure3(b)(ii).We can estimate   from the experimental line widths using the following relation: DFTIn our data, FWHM EDC,experiment from each band at the Fermi level is (see SI5 (a),(b)) ) shows the sharp quasiparticle peak of  at the Fermi level, fitted with a Voigt function to represent the detector response.We obtain the resolution corrected Lorentzian FWHM of 2.5 meV.