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Electrons on a kagome lattice constitute a pre-eminently suited scenario for exotic quantum phenomena at all coupling scales: within the Mott limit, it is the established paradigmatic setting for spin liquids and other aspects of frustrated magnets1. For symmetric metallic states or itinerant magnets, the diversity of dispersing kagome signatures such as Dirac cones, flat bands and van Hove singularities enable a plethora of correlated electron phenomena from topological band formation and symmetry breaking to be unlocked2,3,4,5,6,7,8,9,10,11,12,13,14.

The family of XV6Sn6 kagome materials, where X is a rare-earth element, belongs to a new series (hereafter dubbed the ‘166’ family) that has been predicted to host electronic states with non-trivial topology. In particular, not only are the surface states that appear at natural cleavage planes of the crystals theoretically conceived to have a non-trivial origin15,16, but also the correlated flat band naturally arising from the kagome geometry17 is characterized by a non-zero \({{\mathbb{Z}}}_{2}\) Kane–Mele invariant for the action of spin–orbit coupling (SOC)18,19,20,21,22,23,24. If the onsite energy of such a separated flat band could be controlled, one could trigger new topological phases with potential applications in spintronics and non-volatile electronics25,26,27,28,29. Therefore, uncovering the non-trivial topological character of such a flat band would be a true milestone in the field of condensed matter physics.

The direct experimental observation of non-trivial topological properties in 166 kagome metals remains an open challenge. Although transport is unable to probe correlated flat band states below the Fermi level EF or isolate topological surface states10,11,30,31,32,33,34,35,36, in angle-resolved photoemission spectroscopy (ARPES) there are tantalizing hints that these states exist. Crucially, measurements of the spin degree of freedom are missing; when ARPES has been able to detect the surface states manifold and the flat bands in XV6Sn6 systems15,16,37, the lack of measurements for the spin degree of freedom and the action on it of time-reversal symmetry hinder the conclusive proof of their topological nature.

Here we provide the spectroscopic evidence of the non-trivial topology in the 166’ kagome family. We use spin-ARPES and density functional theory (DFT) calculations to determine the electronic structure of these systems resolved in energy, momentum and spin. We not only find a net spin polarization of the surface states in the prototypical compound TbV6Sn6, but we ultimately demonstrate the non-trivial topology of the gap between the dispersive Dirac band and the nearly flat band arising from the kagome geometry. Such a gap is a common feature of all kagome lattices with non-zero SOC, and its nature is believed to be topological, yet its demonstration has been elusive until now10,11,35,38. Notably, we detect a finite spin Berry curvature in kagome metals, and by systematically studying the whole series of (Tb,Ho,Sc)V6Sn6 compounds, we also demonstrate its resilience against the onset of a charge ordered phase, which is a distinctive feature of many recently discovered kagome metals9,39,40. As well as unveiling the interplay between many-body electronic states and topology in this class of materials, our work provides experimental measurements of a spin Berry curvature in real quantum systems. Indeed, the detection of energy- and momentum-resolved finite Berry curvature signals was hitherto limited to cold atom experiments, which unveiled the deep relationship between topology and flatness in optical lattices41. Furthermore, in that context, flat bands serve as a platform for emergent correlated phases and their simplicity can advance the understanding of the physics that occurs in argon ice, Landau levels and twisted Van der Waals bilayers42. Here we extend this context to real solid-state systems.

TbV6Sn6 (Fig. 1a) is a kagome system belonging to the 166 family of rare-earth kagome metals, along with GdV6Sn6 and HoV6Sn6 (refs. 5,37,43,44). It exhibits a uniaxial ferromagnetic transition at 4.1 K with a substantial anisotropy in the magnetic susceptibility, suggesting a ferromagnetic alignment of Tb3+ 4f moments perpendicular to the V kagome layers44. The DFT bulk electronic band structure is shown in Fig. 1b. It is characterized by prominent features hinting at a non-trivial topology. Dirac-like dispersions appear at the K points of the Brillouin zone (BZ) and contribute to the metallic character of the material. In addition, two flat bands are visible below and above the chemical potential, as highlighted by the yellow colour proportional to the band n- and momentum k-resolved density of states ρnk ≈ 1/vnk, where vnk represents the electronic velocity. Both the Dirac cones and the flat band around −1 eV are spectroscopically detectable by ARPES owing to their occupied character. As shown in Fig. 1c, which shows enlargements of the red rectangles (1) and (2) of Fig. 1b, SOC opens gaps at the Dirac cones. The SOC permits direct gaps between bands throughout the BZ, which in turn allows the topological invariant \({{\mathbb{Z}}}_{2}\) for the occupied bands to be defined using parity products at time-reversal invariant momenta45. As for GdV6Sn6 (refs. 15,16), we find \({{\mathbb{Z}}}_{2}=1\) for the bands around the Fermi level. We also highlight that, in the absence of SOC, the Dirac cones carry a finite Chern number \({{{\mathcal{C}}}}\), and are the source and sink of finite Berry curvature Ω(k), resulting in topologically protected arcs on the surface of this class of materials, as Fig. 1g shows.

Fig. 1: Crystal structure, bulk electronic properties and surface terminations of TbV6Sn6 kagome metal.
figure 1

a, Crystal structure of TbV6Sn6 showing top and side views of the unit cell. b, Bulk electronic structure along the Γ–K–M direction in the presence of SOC. The electronic states are coloured by the band and momentum-resolved density of states, with yellow highlighting a large contribution. c, Enlargement of the red boxes (1) and (2) in b. Red and blue bands refer to calculations with and without SOC, respectively. d, Sn 4d core level spectroscopy for the kagome-terminated (term.) (green) and Sn-terminated (red) surface (surf.) of TbV6Sn6. e,f, ARPES Fermi surfaces for the kagome (e) and Sn (f) termination of TbV6Sn6, respectively. g, Spectral function of the (001) surface Green’s function for the Sn termination in the absence of SOC. Boxes (1) and (2) refer to those in b.

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On cleavage, two terminations are possible, namely a Sn-terminated and a mixed V/Sn-terminated surface plane, with the latter being characterized by V atoms arranged into a kagome pattern. To determine the type of termination, we acquired the Sn core levels alongside the ARPES, finding good agreement with previous works on the sister compound GdV6Sn6 (ref. 37). Specifically, as shown in Fig. 1d, the Sn termination exhibits the presence of two extra peaks at lower binding energies in the Sn 4d core levels, compatible with the corresponding Sn-derived surface components. The kagome termination, on the other hand, shows a substantially different line shape, featuring an asymmetric profile and a shift of about 0.43 eV towards higher binding energy values. These differences are attributed to the different local atomic environment present at the two surfaces37. The measured Fermi surfaces (Fig. 1e,f) for both terminations agree well with the calculated ones (Supplementary Fig. 2) and the typical kagome motif of corner-sharing triangles is also recognizable in reciprocal space.

The surface states at the kagome termination have minimal separation from the bulk continuum15,16. This makes their spin-resolved measurements challenging, since the bulk contribution is intense and makes the spin-polarized signal too weak to be observed (see also Supplementary Fig. 3 for additional spectra). However, the Sn-terminated surface features well-separated surface states, which allow the spin-ARPES experiments to be performed more easily, offering the perfect playground to investigate the topological properties of this system. We will focus on this termination when discussing the topological character of the surface states of TbV6Sn6. In contrast, the correlated flat band topology is not termination-dependent. Such a band, in fact, is a feature inherent to the bulk kagome geometry, making this study of broad interest for kagome lattices in general and offering a comparative parallel case study for other systems. In addition, to determine the topology of the gap between the Dirac state and the flat band, the experimental measurement of the spin Berry curvature is essential, because all the electronic states are spin degenerate, and thus inaccessible by standard spin-ARPES.

The high-resolution electronic structure of the Sn termination is shown in Fig. 2. By using both linear vertical and horizontal light polarization (Es and Ep, respectively; see also the experimental setting in Fig. 2a), we detect a plethora of interesting electron features, as shown in Fig. 2b,c; multiple Dirac dispersions, van Hove singularities and the correlated flat band, which are a hallmark of kagome materials, are well identified. In particular, the van Hove singularities are located slightly above the Fermi energy and at about −0.4 eV (labelled VH1 and VH2 in Fig. 2f) and are visible in both theory and experiment. In addition, the Dirac cones form at the K point at binding energies of −1.5 eV (D1) and −0.3 eV (D2) (Fig. 2b,f). The latter set of bands evolves symmetrically across the BZ with a quadratic minimum (Fig. 2b) at the zone centre. According to a basic first-nearest-neighbour tight-binding calculation (see also Fig. 4a), such a quadratic minimum is pinned to the flat band. We clearly detect both features in Fig. 2d and the relative energy distribution curves (the blue and green curves) in Fig. 2e. We also notice that the flat band is only visible with Es polarized light, while the quadratic minimum intensity disappears at the zone centre because of the photoemission matrix elements. Nonetheless, the quadratic minimum is well identified in the energy distribution curve acquired with Ep polarization (blue curve), for which, instead, the flat band is not visible. Thus, the combination of Es and Ep allows us to visualize both the quadratic minimum and the flat band, and to estimate the SOC-induced gap between them to be ~60 meV. Importantly, the presence of SOC mixes the orbital character across the opening point at Γ, and induces a finite spin Berry curvature (see again Fig. 4a and further discussion).

Fig. 2: Spectroscopy of the surface states and of the flat band.
figure 2

a, Experimental ARPES setting of linear vertical (Ep) and horizontal (Es) light polarizations (h is the Planck constant and ν the frequency of light). Es is fully in-plane and parallel to the x axis. Ep has an incidence angle of 45o, and thus has 50% of the out-of-plane contribution and 50% of the in-plane contribution parallel to the y axis. b,c, ARPES spectra recorded with Ep (b) and Es (c) lights. QM, quadratic minimum. d, Enlarged view of the red box in c highlighting the dispersion of the flat band around the Γ point at approximately −1 eV of binding energy. e, Energy distribution curve along the green line of d collected with Es polarization (green curve) and for Ep polarization (blue curve). The former shows the flat band position for kx,y = 0; the latter instead has prominent intensity corresponding to the quadratic minimum, as is also visible in b and c. f, Enlargement of the ARPES data in the proximity of the Fermi level showing the most intense Sn-derived surface states (indicated by arrows labelled (1) and (2)) and also the van Hove singularities VH1 and VH2 at the M point, with the maximum of VH1 slightly above the Fermi level. g, First-principles electronic structure of a finite slab of TbV6Sn6 on structural relaxation of the atoms at the Sn termination. As in f, arrows labelled (1) and (2) indicate the most intense Sn-derived surface states.

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Close to the Fermi level, our ARPES measurements and DFT calculations enable the investigation of the topological nature of the 166 family. We focus on the surface states originating from the multiple gapped Dirac cones described in Fig. 1b,c. In previous literature, theoretical works have attempted a topological classification of the \({{\mathbb{Z}}}_{2}\) invariant of the surface states across the BZ. According to Hu et al.16, topological surface states in GdV6Sn6 are expected to cover a significant portion of the BZ, bridging a large bulk gap across Γ. In contrast to these data, our ARPES spectra in Fig. 2f does not reveal any surface states through the Fermi level around the Γ point, probably due to a different chemical potential in TbV6Sn6,which pushes them into the unoccupied region of the electronic density of states. This observation is consistent with our slab calculation of the electronic structure in Fig. 2g, which, differently from the spectrum in Fig. 1g, fully accounts for the structural relaxation of atoms at the Sn-terminated surface, where these states are present. They are primarily unoccupied, but a portion of them bridges the gap below the Fermi energy along the \(\bar{{{\Gamma }}}\)–\(\bar{\,{{\mbox{M}}}\,}\) line. We also notice that in Fig. 2g these surface states form a small electron pocket close to the centre of the BZ. This pocket is absent in ARPES. Nonetheless, the presence of these surface states close to the K point is enforced by topology, because they originate as Fermi arcs from the Dirac cones. Therefore, in this region, we will seek the signature of the spin-polarized feature in ARPES, because these states are accessible there.

To experimentally verify this, we first calculated the expected spin texture of the Sn surface states. The calculated spin-resolved electronic structure is shown in Fig. 3a. We found that the spin component along y Sy gives the most significant and only non-zero contribution along the \(\bar{{{\Gamma }}}\)–\(\bar{\,{{\mbox{K}}}\,}\)–\(\bar{\,{{\mbox{M}}}\,}\,\)kx direction (see Supplementary Fig. 9 for the spin components along x Sx and and z Sz components). This result demonstrates the spin-momentum locking of the spin texture and its non-trivial origin. Using the ARPES analyser’s deflectors, we measured the spin-ARPES signal along \(\bar{{{\Gamma }}}\)–\(\bar{\,{{\mbox{K}}}\,}\)–\(\bar{\,{{\mbox{M}}}\,}\) direction at specific momenta for both positive and negative k values (the coloured vertical bars in Fig. 3a). In this way, not only can the theoretical predictions be proved, but the time-reversal symmetry constraint can also be verified by keeping the same matrix elements. The spin-ARPES data for Sy are shown in Fig. 3b–e (see Supplementary Information for details about data normalization). A clear spin-resolved signal in the proximity of the Fermi level confirms a non-vanishing spin polarization typical of spin-polarized states. In addition, the spin sign reverses with the momentum k, guaranteeing the time-reversal symmetry of the system. This important aspect is also compatible with the sample being well above the magnetic transition temperature (the measurements were indeed performed at 77 K). In addition, the Tb 4f levels are well separated from the near-Fermi energy region and do not hybridize significantly with the measured surface states (see also Supplementary Fig. 4).

Fig. 3: Spin texture of topological surface states.
figure 3

a, First-principles electronic structure of the Sn termination of a finite slab of TbV6Sn6 where the electronic states are coloured by their Sy character. b–e, Sy spin-resolved ARPES energy distribution curves at the specific momenta highlighted in a by the coloured vertical bars (orange (b), purple (c), green (d) and blue (e)). The grey arrows refer to the energy regions where a finite spin asymmetry for the surface states is measured.

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The energy distribution curves of Fig. 3b–e unambiguously demonstrate the spin-polarized character of the surface states in TbV6Sn6. In Supplementary Fig. 5, a thicker energy distribution curve map is also available for completeness. We notice that of the full set of states present in the DFT calculations, we are only able to resolve those that are well separated from the bulk electronic structure and that appear more prominent in intensity within the gap, which still provides sufficient evidence for the spin-momentum locking expected for these spectrocopic features. As such, this makes our finding relevant within the framework of transport experiments in kagome lattices9,31,46,47,48,49,50,51.

Standard spin-ARPES, on the other hand, cannot be used to prove the topological character of the gap between the correlated kagome flat band and the quadratic minimum because those bands are spin degenerate. Theoretically, one can access the topological character of the gap by calculating the spin Berry curvature of this system. The Chern number \({{{\mathcal{C}}}}\) of each band forming such a gap is identically zero owing to the combined action of inversion (which gives Ωn↑(k) = Ωn↑(−k) for the Berry curvature) and time-reversal symmetry (which enforces Ωn↑(k) = Ωn↓(−k)). Nonetheless, one can expect a finite spin Berry curvature \({{{\varOmega }}}_{z}^{{S}_{z}}({{{\bf{k}}}})\) due to the action of SOC. At the level of a simple first-nearest-neighbour tight-binding model with hopping amplitude t, SOC opens a gap at the Γ point between the parabolic dispersion from the Dirac band (1) and the flat band (2), as shown in Fig. 4a. The opening of a gap is in general associated with the appearance of a finite dispersion for the flat band18,52, and SOC can be thought of as a perturbation breaking the real-space topology that protects the band touching in generic frustrated hopping models53. In modern language, kagome spectra without a mathematically flat band cannot be derived as one of the two isospectral partners of a supersymmetric bipartite graph with finite Witten index54. As a result, the electronic states around the opened gap feature a finite spin Berry curvature, leading to a non-trivial spin-Chern number for the weakly dispersing flat band itself. Similar conclusions hold also for TbV6Sn6, as shown in Fig. 4b (top panel), even though the band structure is much more complex than the simplified picture shown by the tight-binding model (notice, for instance, the presence of a spectator band carrying vanishing spin Berry curvature at Γ). In addition, away from the flat band region around the Γ point, our theoretical calculations reveal enhanced spin Berry curvature at every SOC-induced gap avoided crossing between two spin-degenerate bands.

Fig. 4: Topology of the flat band region.
figure 4

a, Band structure of a first-nearest-neighbour tight-binding model on the kagome lattice with SOC. The colour highlights the finite contribution from the spin Berry curvature \({{{\varOmega }}}_{z}^{{S}_{z}}({{{\bf{k}}}})\) around the SOC-induced gaps formed by bands (1) and (2). b, Same as a but for a realistic first-principles tight-binding model of TbV6Sn6 around the flat band region (top panel) and at slightly higher binding energies (bottom panel). c,d, Circular right (CR) (c) and circular left (CL) (d) spin-ARPES energy distribution curves, collected for the spin-↑ (Sz > 0, red) and spin-↓ (Sz < 0, blue) channels. The energy distribution curves have been collected at the centre of the BZ with the light entirely in the sample’s mirror plane. Under these conditions, the geometrical contribution coming from the CD can be safely excluded. e, Extracted spin polarization for the CD (ICD for Sz<0 and Sz>0), showing a strong and finite contribution for the quadratic minimum around −1 eV as high as approximately ±90%, as well as at higher binding energies. In grey, we also report the percentage contribution of the CD (CR − CL) for spin-integrated ARPES, showing fluctuations of the maximum ± 8% in the region of interest. f–i, Same as c,d for HoV6Sn6 (f,g) and ScV6Sn6 (h,i).

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Schüler et al.55 recently proposed a methodology based on circular dichroism (CD) and spin-ARPES to directly address a signal proportional to the spin Berry curvature \({{{\varOmega }}}_{z}^{{S}_{z}}({{{\bf{k}}}})\) of quantum materials, thus allowing us to disentangle the trivial and non-trivial topologies. CD has already been used in ARPES experiments to obtain information about the Berry curvature56,57. However, the presence of inversion symmetry and SOC in TbV6Sn6 requires separating the two spin channels, that is, ±Sz, as we show in Fig. 4c,d. In addition, to ensure that any geometrical contribution will not artificially alter our measurements, we measure the CD at the BZ centre, where the geometrical contribution is exactly zero, and so it is the spin-integrated dichroic signal. Being at the BZ centre also has the advantage that this point is time-reversal symmetric; thus, we are able to detect a signal reversal for the plus and minus components of the spin. Our measurements, shown in Fig. 4e, demonstrate a substantial non-zero signal for each spin species of the quadratic minimum (approximately −1 eV) and for electronic bands at slightly higher binding energies where also a finite spin Berry curvature appears as a result of SOC (greater than approximately −1.2 eV), with a reversal between spin up and down channels (red and blue, respectively). The spin Berry curvature contribution at −1 eV comes from the quadratic minimum and this can be understood by looking at the spin-integrated CD ARPES (Supplementary Fig. 1) when compared to the measurements performed with Es and Ep light polarization: the circular light has matrix elements similar to Ep, and thus the CD ARPES results in a vanishing spectral weight for the flat band and a clear signal for the quadratic minimum, at the BZ centre. Thus, we can attribute the strong intensity peaked at −1 eV of Fig. 4e to the spin Berry curvature of the quadratic minimum. Similar conclusions can also be drawn for HoV6Sn6, as we experimentally show in Fig. 4f,g. It is worth stressing that by changing the photon energy we were not able to resolve the flat band by using circularly polarized light; by contrast, in this experimental configuration, the quadratic minimum was the only resolved feature. This result agrees with our calculations of Fig. 4b (top panel) that suggest a finite spin Berry curvature contribution around the SOC gap for the visible band (1) forming the quadratic minimum. Our analysis is proof of a topological gap in a kagome metal. In addition, our CD measurements in Fig. 4e unveil a large signal at energies below the flat band region. This result is again supported by our first-principles calculations of \({{{\varOmega }}}_{z}^{{S}_{z}}({{{\bf{k}}}})\), as we show in Fig. 4b (bottom panel), where the electronic states around −1.3 eV and −1.4 eV are characterized by an enhanced spin Berry curvature.

Kagome metals are also getting much attention since they represent the perfect playground for several intertwined many-body orders9. The unconventional charge density wave (CDW) is one of these. Its origin, whether it arises from electron–phonon coupling, electron–electron interactions or a combination thereof, is still a matter of debate. Differently from TbV6Sn6 and HoV6Sn6, ScV6Sn6 shows a CDW phase below the temperature TCDW ≈ 92 K, characterized by a distinctly different structural mode than that observed in the archetypal AV3Sb5 (A = K, Rb, Cs) compounds39 and FeGe (ref. 40). In Fig. 4h,i we present our spin-resolved CD results for ScV6Sn6 at low temperature, that is, inside the charge ordered phase, whereas in Supplementary Fig. 10 we show that, in addition to an increase in the noise level due to thermal broadening, the aforementioned CD results are identical above and below TCDW. Clearly, around −1 eV, we see a net spin asymmetry that reverses sign when the light polarization is changed. This result unambiguously demonstrates that the spin Berry curvature is robust against the onset of the ordered phase and that the SOC-induced energy scale associated with the appearance of a finite spin Berry curvature is larger than that correlated to the CDW symmetry breaking. Interestingly, our first-principles calculations, shown in Supplementary Fig. 8, reveal that, on unfolding the band structure of the distorted ScV6Sn6 onto the primitive unit cell, the CDW distortion affects only marginally the electronic properties. This is in striking contrast to the effect of the CDW order in AV3Sb5 compounds, where sizable band gaps open around the chemical potential58,59.

In conclusion, we have demonstrated the topological nature of XV6Sn6 kagome metals by exploiting the combination of spin-ARPES and DFT calculations and leveraging the sensitivity at the multiple energy scales relevant to kagome systems. As well as unveiling a net spin polarization of the surface states of TbV6Sn6, which originates from the non-trivial \({{\mathbb{Z}}}_{2}\) invariant of the SOC-induced bulk gaps close to the Fermi energy, we have crucially shown that the correlated flat band region is characterized by a finite spin Berry curvature, establishing its topological character. In addition, we have reported the resilience of the non-trivial topology against the onset of the charge ordered phase in ScV6Sn6, revealing its ubiquitous nature across the series. This will motivate the investigation of the spin Berry curvature in other kagome metals as well, such as AV3Sb5 and FeGe, where a non-trivial flat band separation has also been predicted and in which the charge order has a strong effect on the electronic properties. Within a more general perspective, our work constitutes evidence of the multidimensional topological nature, that is, from surface to bulk states, of the 166 kagome family. It ultimately establishes these systems as a new domain for correlated topological metallicity with a non-trivial spin Berry curvature of the wave function manifold.

Methods

Experimental details

Single crystals of XV6Sn6 (Tb, Sc and Ho) were grown using a flux-based growth technique as reported in ref. 60. X (chunk, 99.9%), V (pieces, 99.7%) and Sn (shot, 99.99%) were loaded inside an alumina crucible with the molar ratio of 1:6:20 and then heated at 1,125 °C for 12 h. Then, the mixture was slowly cooled to 780 oC at a rate of 2 °C h−1. Thin plate-like single crystals were separated from the excess Sn flux by centrifugation at 780 °C. The samples were cleaved in ultrahigh vacuum at a pressure of 1 × 10−10 mbar. The spin-ARPES data were acquired at the APE-LE end station (Trieste) using a VLEED-DA30 hemispherical analyser. The energy and momentum resolutions were better than 12 meV and 0.02 Å−1, respectively. The temperature of the measurements was kept constant throughout the data acquisitions (16 K and 77 K), above the magnetic transition of the system (<5 K). Both linear and circular polarized light was used to collect the data from the APE undulator of the synchrotron radiation source ELETTRA (Trieste).

Theoretical details

We employed first-principles calculations based on the DFT as implemented in the Vienna ab-initio simulation package61, within the projector-augmented plane-wave method62. The generalized gradient approximation as parametrized by the Perdew-Burke-Ernzerhof functional for the exchange-correlation potential was used63 by expanding the Kohn–Sham wave functions into plane waves up to an energy cutoff of 400 eV. We sample the BZ on an 12 × 12 × 6 regular mesh by including SOC self-consistently. For the calculation of the surface spectral function, the Kohn–Sham wave functions were projected onto a Tb d, V d and Sn s, p-type basis. The calculation of the spin Berry curvature requires a Wannier Hamiltonian where the lattice symmetries are properly enforced. For this reason, we used the full-potential local-orbital code64, v.21.00-61 (https://www.FPLO.de/). The spin Berry curvature for band n is then defined as

$${{{\varOmega }}}_{xy}^{z}({{{\bf{k}}}})=\mathop{\sum}\limits_{{E}_{n} > {E}_{m\ne n}}\frac{\left\langle n\right\vert {v}_{s,x}^{z}\left\vert m\right\rangle \left\langle m\right\vert {v}_{y}\left\vert n\right\rangle (x\leftrightarrow y)}{{({E}_{n{{{\bf{k}}}}}-{E}_{m{{{\bf{k}}}}})}^{2}},$$
(1)

with the spin operator σz and velocity operator \({v}_{i}=\frac{1}{\hslash }\partial H/\partial {k}_{i}\) (i = x, y). \(\left\vert n{{{\bf{k}}}}\right\rangle\) is the eigenvector of the Hamiltonian H with the eigenvalue Enk. Equation (1) is computed by using our in-house post-wan library (Code availability).