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Quantum spin mixing in Dirac materials


The spin of the electron is nowadays replacing the charge as basic carrier of information not only in spintronics applications, but also in the emerging field of quantum information. Topological quantum materials, where spin-momentum locking is believed to lead to particularly long spin lifetimes, are regarded as a promising platform for such applications. However, spin-orbit coupling, that is essential to all topological matter, at the same time gives rise to spin mixing and decoherence as a major obstacle for quantum computing. Here, we give experimental evidence that hot-spots of spin-mixing and spin-conserving contributions of the spin-orbit operator coexist in an archetypal topological Dirac metal, and that these hot spots can have a strongly anisotropic distribution of their respective wave vectors with respect to the spin quantization direction. Our results can be understood within a theory that takes into account the decomposition of the spin-orbit Hamiltonian into spin-conserving and spin-flip terms, contributing to a better understanding of quantum decoherence in topological materials, in general.


In spintronic devices, the spin of the electron becomes the main carrier of information, providing a link between magnetism and conventional charge-based electronics. In this approach, spin-up () and spin-down () states of the electron spin serve as a classical bit to represent and store binary information1,2. On the microscopic level, however, the electron spin by its nature has a quantum mechanical origin. As a consequence, the statistical interpretation of quantum mechanics provides possibilities to find the eigenvalue \(\frac{\hslash }{2}\) and \(-\frac{\hslash }{2}\) for spins pointing up and down along a given quantization axis. In particular, any given spin state can be represented as a superposition,

$$\left|\psi \right\rangle =a\left|\uparrow \right\rangle +b\left|\downarrow \right\rangle ,$$

of these two states. Quantum physics therefore allows information to be represented and processed in a significantly different manner, beyond the classical interpretation of spintronics. In this picture of spin-based quantum computing, the coherent superposition of two distinct spin states represents a single quantum bit, a so-called qubit2,3.

A fundamental mechanism that allows the control of the electron spin in condensed matter is provided by spin–orbit coupling (SOC). Due to SOC, the eigenstates of the Hamiltonian are no longer wave functions with a pure spin character, but a superposition of up and down states. This manifests in a manifold of phenomena. On the one side, SOC is the source of spin-momentum locking found in the electronic structure of topological quantum materials, leading to particularly long spin lifetimes. By contrast, the mixture of spin-up and spin-down states due to SOC enables that scattering events will flip the spin of an electron with a certain probability. For instance, the Elliott–Yafet spin-relaxation mechanism, being driven by SOC, represents a major source of spin decoherence2,4,5.

To understand this apparent discrepancy, the SOC operator can be divided into a spin-conserving \(\xi {({L}_{\hat{s}}{S}_{\hat{s}})}^{\uparrow \uparrow }\) and a spin-flip \(\xi {({L}_{\hat{s}}{S}_{\hat{s}})}^{\uparrow \downarrow }\) part:

$$\xi {{{{{{{\bf{L}}}}}}}}\cdot {{{{{{{\bf{S}}}}}}}}=\xi {({L}_{\hat{s}}{S}_{\hat{s}})}^{\uparrow \uparrow }+\xi {({L}_{\hat{s}}{S}_{\hat{s}})}^{\uparrow \downarrow }.$$

For a fixed direction of the spin quantization axis (SQA), these two contributions correspond to states with a pure spin character and fully spin-mixed states, respectively. Here, ξ is the strength of SOC, L and S are the orbital and spin angular momentum operators, \({L}_{\hat{s}}={{{{{{{\bf{L}}}}}}}}\cdot \hat{s}\), \({S}_{\hat{s}}={{{{{{{\bf{S}}}}}}}}\cdot \hat{s}\), and \(\hat{s}\) is a unit vector in the direction of the SQA. As the spin–orbit operator is a non-local operator, it depends on the energy and wave vector of the respective states. In crystalline solids, thus, the electronic structure can have regions in the Brillouin zone (BZ) of particularly large spin relaxation. Such regions have, for instance, direct consequences in the interpretation of ultrafast dynamics of spin-dependent properties1,5,6,7,8,9,10. A direct experimental access to the contrasting contributions of SOC in the electronic structure has so far remained elusive. Yet, such information would help to identify spots of long/short spin lifetime and might pave the way towards a precise control of spin states via on-demand tailoring of the SOC.

Here we used spin- and momentum-resolved photoelectron spectroscopy to map the SOC spin-mixing parameter in the Fermi surface (FS) of the prototypical heavy transition metal tungsten (W). Tungsten has the highest charge- to spin-current conversion efficiency among all 5d transition metals, suggesting a significant role of SOC in its FS topology11,12,13,14,15. We have investigated the twofold rotational symmetric (C2v) W(110) surface. In reminiscence of a crystalline topological insulator, the mirror symmetry of W(110) leads to several d-orbital-derived surface states with a helical spin texture16,17,18,19. The W(110) surface thus represents a paradigmatic model system to understand the non-local distribution of spin-conserving and spin-mixing parts of the SOC operator in the electronic structure. Our experimental and theoretical results give direct evidence that spin mixing in the W(110) surface becomes strongly anisotropic, with a pronounced non-uniform distribution of spin-mixing hotspots in the FS. Our results have important consequences for spintronics and understanding the efficiency of spin-torques and spin-current generation, as well as spin-coherence effects in quantum materials.


Multiple d-derived Dirac fermions in W(110)

Figure 1a presents the layout of our spin-resolved momentum microscopy experiments. Photoelectrons are collected simultaneously over the full solid angle above the sample, allowing the efficient measurement of the spin-resolved spectral function over the complete surface BZ (SBZ) (see ‘Methods’)20,21,22. The momentum image shows the electronic states at the Fermi energy EF.

Fig. 1: Multiple d-derived Dirac fermions in the heavy transition metal W(110).
figure 1

a Experimental geometry for spin- and momentum-resolved photoelectron study. The spin-resolved photoemission momentum map at the Fermi energy EF for W(110) is shown. The dashed contour indicates the outline of the surface Brillouin zone of W(110), together with the high-symmetry points (\(\overline{{{\Gamma }}}\), \(\overline{{{{{{{{\rm{H}}}}}}}}}\), \(\overline{{{{{{{{\rm{N}}}}}}}}}\), \(\overline{{{{{{{{\rm{S}}}}}}}}}\)). bd Full 3D (kx, ky, EB) spin-resolved momentum map with a photon energy of hν = 50 eV, cutting along \(\overline{{{{{{{{\rm{H}}}}}}}}}-\overline{{{\Gamma }}}-\overline{{{{{{{{\rm{H}}}}}}}}}\) (b), \(\overline{{{{{{{{\rm{S}}}}}}}}}-\overline{{{\Gamma }}}-\overline{{{{{{{{\rm{S}}}}}}}}}\) (c), and ky = 0.4 Å−1 (d) directions of W(110) as a function of binding energies EB (E − EF from 0 to −1.7 eV). Measured intensities and spin polarization are encoded using the displayed 2D colour code, where the spin polarization Py is indicated by red and blue colours, and the colour saturation encodes the intensity. e Schematic of the three observed Dirac states for W(110): one appears at the \(\overline{{{\Gamma }}}\) point and two coexist at ky = ±0.4 Å−1.

Figure 1b–d show sections through the electronic structure between EF and EF − 1.7 eV. The W(110) surface is known to host in total three gapless points in the SBZ forming Dirac cones with linear dispersion: one appears at the centre and two are away from the high-symmetry points of the SBZ17,18,19. In our measurements, we can clearly observe all three Dirac cones in the SBZ. Their positions are summarized in Fig. 1e. These three Dirac cones can be classified into two sets with distinct group velocities: one at the \(\overline{{{\Gamma }}}\) point and two at ky = ±0.4 Å−1. They are located inside a spin–orbit-induced bulk band gap that resembles the formation of the Dirac-cone surface band inside the fundamental gap of a topological insulator17. The linear bands cross at the \(\overline{{{\Gamma }}}\) point at EB = 1.25 eV (see Fig. 1b, c). At ky = ±0.4 Å−1, the Dirac point lies at EB = 0.75 eV, as shown in Fig. 1d. The presence of an odd number of gapless Dirac cones indicates a non-trivial topological character. Tungsten has been therefore classified as a d-orbital-derived topological crystalline material16 where the Dirac cones are protected by the mirror symmetry of the (110) surface.

Full spin texture

Because of the combination of the twofold surface symmetry and time-reversal symmetry, the out-of-plane component of spin polarization vanishes at all k-points in W(110)23, i.e., Pz = 0. A negligible out-of-plane spin polarization is further confirmed by our first-principle calculations. As a result, the initial-state spin polarization is purely in-plane, with components Px and Py. During the photoemission by linearly polarized light, additional contributions to the spin polarization may arise due to the lowered symmetry by the incidence direction of the light, as discussed in detail in refs. 24,25. Here we chose a geometry where the SQA lies within the optical plane, such that these effects are minimized (see ‘Methods’) and the measured spin polarization is expected to closely reflect the W(110) FS spin texture. In particular, when the W(110) surface is excited with linearly polarized light, a purely in-plane spin polarization has been previously reported, whereas a non-zero Pz component can be obtained using circularly polarized light26.

Figure 2a shows the measured spin-resolved FS contour of W(110) for a vertical Py in-plane polarization, the same spin direction as discussed in Fig. 1. The measured FS contour agrees well with previous results that have probed Py18. In particular, we find a high polarization reaching Py values up to ±0.7 in the pronounced cross-shaped contour. These states can be related to d-orbital-derived surface resonances27. The spin-resolved measurement reveals that a Rashba-like spin splitting is found around the \(\overline{{{\Gamma }}}\) point (A), whereas the vertical contour of  the FS (B) at ky = ±0.4 Å−1 merge into the off-centred Dirac cone observed in Fig. 1d. The ellipse-shaped contours around the \(\overline{S}\) points are surface states that exhibit a small Rashba-like spin splitting.

Fig. 2: Full spin texture.
figure 2

a, b Measured spin-resolved Fermi surface (FS) of W(110). Measured intensities and spin polarization are encoded using the displayed 2D colour code, where the spin polarization Py (a) or Px (b) is indicated by red and blue colours, respectively, and the colour saturation encodes the intensity. c, d Theoretical spin-resolved FS of W(110). Colours indicate the in-plane spin component along the ky (c) or kx (d) direction. Letters A–F indicate distinguished features in the FS contour with high/low spin polarization as a guide to the eye.

A significant change of the spin polarization is found for the Px component in Fig. 2b: in general, Px changes the sign upon the mirror operation ky → −ky, but is unchanged upon kx → −kx. This reflects the fact that the spin angular momentum described by pseudovectors fulfils time-reversal and mirror symmetries. The overall magnitude of the measured Px only reaches values of ±0.5 and is significantly reduced compared to Py. In particular, Px vanishes along the vertical sections of the main FS contour (C) at ky ≈ ±0.4 Å−1 and only small polarizations are observed along the horizontal sections (D). This provides already a qualitative picture of the in-plane spin texture: the spin is locally aligned tangential to the FS contour, in reminiscence of the spin warping observed in some topological insulators rather than being locked normal to the k-vector as in a regular Rashba model.

Figure 2c, d show the theoretical spin-resolved FS for Py and Px, respectively. The calculation reproduces the main details of the experiment, in particular the Px and Py spin polarizations along the cross-shaped FS contour. As in the experiment, the maximum polarization is observed in the Py component along the vertical sections in vicinity of the off-centred Dirac cone (E), whereas Px vanishes for these states (F). Furthermore, we clearly observe the Rashba-splitting of the surface states around the \(\overline{S}\) points. The calculation further confirms the experimental observation that both Px and Py, at the same time, show a strongly reduced polarization along the vertical kx = 0 mirror plane of W(110). The quantitative overall agreement between theory and experiments confirms that our measured results mainly represent the initial-state spin polarization.

Figure 3a shows the spin texture over the whole SBZ. As discussed above, in the outer contour of the cross-like feature, spin-momentum locking leads to a counterclockwise rotation around the BZ centre. At larger binding energies, these states converge into the Dirac-cone states. The two nearly degenerate ellipse-shaped surface states around \(\overline{S}\) are very close to each other. The Rashba-like opposite spin chirality explains the low observed spin polarization for Px and Py (see Fig. 2a, b).

Fig. 3: Anisotropic spin mixing.
figure 3

a Theoretical full spin texture of W(110). b, c Theoretical momentum-resolved spin-mixing parameters \({b}_{{{{{{{{\bf{b}}}}}}}}\hat{s}}^{2}\) on the Fermi surfaces of W(110) for the spin quantization axis (SQA) along y (b) and x (c) axes. Blue colours represent a pure spin character of the Bloch states along the SQA, whereas red colours indicate a fully mixed spin state, where spin-up and spin-down electrons are found with an equal probability of 0.5.

Anisotropic quantum spin mixing

Due to the helical spin texture, one would anticipate a similarly high degree of spin polarization for the Px and Py momentum maps. For instance, this would be the expected case for the spin texture of an archetypal Rashba surface state with isotropic free-electron-like bands. Such model systems are the L-gap surface states on the (111) faces of noble metals like Au(111)20. In contrast, we find a pronounced dependence of the observed spin polarization on the choice of the SQA: the overall spin polarization Px is substantially suppressed, compared to Py. To understand the underlying mechanism of how the choice of the SQA of an incoming conduction electron affects the degree of spin polarization, we carried out ab initio calculations5 to analyse the momentum distribution of the spin-mixing parameter.

Figure 3b, c show the theoretical momentum-resolved spin-mixing parameters \({b}_{{{{{{{{\bf{b}}}}}}}}\hat{s}}^{2}\) (see ‘Methods’)5 of the W(110) FS when the SQA points along the y and x axes, respectively. A fully spin-mixed state (\({b}_{{{{{{{{\bf{b}}}}}}}}\hat{s}}^{2}\) = \(\frac{1}{2}\)) is represented by a superposition of \(\left|\uparrow \right\rangle\) and \(\left|\downarrow \right\rangle\) spin states with a probability of 0.5. As a spin-resolved measurement integrates over an ensemble of many observed electrons, spin-mixing hot areas (red colours in Fig. 3) consequently appear with no effective spin polarization in the experiment. Likewise, a pure spin state corresponds to \({b}_{{{{{{{{\bf{b}}}}}}}}\hat{s}}^{2}\) = 0 and results in a high spin polarization (positive or negative) along the respective SQA. The measured spin-polarization maps in Fig. 2a, b thus provide a direct experimental access to the distribution of the spin-mixing parameter in the SBZ.

By rotating the SQA from the y- to the x-axis, the spin mixing varies considerably. With the SQAy, spin mixing is relatively uniform over the FS. A few hotspots of large spin mixing are located at places of large band curvature and along the vertical kx = 0 plane. This picture changes dramatically when the SQAx. In the latter case, full spin mixing is prominent in large areas of the SBZ. In these regions, the Bloch wave functions become a superposition of spin-up and spin-down states of the same magnitude.

This result is in perfect agreement with our experimental observations in Fig. 2b, where an overall low spin polarization is found. If an electron is scattered into such states, there is a high probability that the spin flips. They thus represent a source of decoherence, degrading the ability of qubits and spintronics to retain encoded information. As a consequence, the strong spin mixing gives rise to hot areas of enhanced spin flip when the SQA points along the x-, but not when the SQA points along the y-direction.


Due to SOC, the spin and orbital character of the Bloch states depends strongly on the spin orientation with respect to the crystal axes. This leads to drastic changes in the spin-conserving \(\xi {({L}_{\hat{s}}{S}_{\hat{s}})}^{\uparrow \uparrow }\) and a spin-mixing \(\xi {({L}_{\hat{s}}{S}_{\hat{s}})}^{\uparrow \downarrow }\) part of the electronic states with respect to the choice of the SQA. We note that the total SOC term (Eq. (2)) is a scalar product, which is independent of the SQA. This representation thus allows us to describe the anisotropic distribution of spin mixing in the entire BZ. In particular, a large \(\xi {({L}_{\hat{s}}{S}_{\hat{s}})}^{\uparrow \downarrow }\) indicates a mixing of spin states due to the presence of SOC.

The distribution of the spin mixing in the BZ has immediate consequences for spin dynamics and spin-related properties1,5,6,7,8,9,10. For instance, in regions of the BZ where the spin-conserving part of SOC is dominant, the electron spin is protected from dissipative effects, giving rise to relatively long spin lifetimes. The spin information of these states is thus preserved, being useful when a long spin lifetime is needed in spintronic devices and quantum information applications. On the other hand, regions with an enhanced spin-mixing will have a major contribution to spin-relaxation processes and ultrafast spin dynamics.

Although the eigenenergies of the Bloch states are independent of the SQA, the choice of the SQA can alter the momentum distribution of spin-conserving and spin-mixing contributions. For simplicity, spin relaxation as it is described by the Elliott–Yafet mechanism2,4 does not take into account these individual relaxation hotspots in the BZ. Nevertheless, it has been shown that the macroscopic Elliott–Yafet relaxation parameter can be obtained directly by integrating the wave-vector-dependent spin-relaxation parameter from Fig. 3b, c over the BZ5. The wave vector of the respective electron states, however, becomes of importance in single-crystalline materials and junctions, where tunnelling currents, spin-dependent excitations, and spin–orbit torque are selectively driven by small areas in the FS of the materials1,5,14,28,29. Spin-conserving and spin-mixing contributions thus are dominated by hotspots in the FS and can be exploited by appropriately designed devices.

Our findings show that the spin mixing of the Bloch states in a non-magnetic material can change drastically when the spin quantization direction varies. As a consequence, the spin-mixing parameter becomes strongly anisotropic. The Dirac surface states, marked by H, serve as the main source of anisotropy, with the spin-mixing parameter changing by >50% between SQAx and SQAy. This result can be attributed to strong spin-momentum locking perpendicular to the kx momentum direction. In contrast, the states marked by L show almost no anisotropy. A large spin mixing for both SQAs is here mediated by a strong overlap with the projected bulk bands27.

The spin-mixing parameter introduced here is a direct consequence of the superposition principle of quantum mechanics. Our observations confirm experimentally the possibility to visualize the effect of a superposition of quantum mechanical wave functions, namely the spin-up and spin-down states that form a mixed spin state due to a strong SOC. We demonstrate the potential scope of spin-momentum locking for quantum spin-mixing manipulation in Dirac materials. Our findings are widely applicable in abundant crystals with SOC and offer promising perspectives to fine-tune the quantum (de)coherence within one material by changing the spin polarization of a conduction electron in a spintronic device.


Spin-resolved momentum microscopy

Spin- and momentum-resolved photoelectron spectroscopy experiments were carried out at the NanoESCA beamline30 of the Elettra synchrotron in Trieste (Italy), using p-polarized photons at the photon energy hν = 50 eV. All measurements were performed while keeping the sample at a temperature of 130 K. Photoelectrons emitted into the complete solid angle above the sample surface were collected using a momentum microscope20,22. The momentum microscope directly forms an image of the distribution of photoelectrons as a function of the lateral crystal momentum (kx, ky) that is recorded on an imaging detector20,22.

An imaging spin filter based on the spin-dependent reflection of low-energy electrons at a W(100) single crystal31 allows the simultaneous measurement of the spin polarization of photoelectrons in the whole SBZ. Images were recorded after reflection at a scattering energy at 26.5 and 30.5 eV, which changes the spin sensitivity S of the detector between 42% and 5%, respectively. From these images, the spin polarization at every (kx, ky) point in a momentum map at certain binding energies (e.g., Figs. 1 and 2 in the main study) is derived21,31,32. As a d-orbital-derived topological metal, sizable photoemission intensities of the Dirac states are mainly observed by excitation with p-polarized light33. As verified by our first-principles calculations (see Fig. 2c, d), our measurements using p-polarized light reproduce well the ground-state spin polarization, such that pronounced effects by optical orientation, which are observed for p-orbital topological materials, only play a minor role34,35.

The spin-resolved momentum microscopy measurements were performed along the quantization axis of the spin filter, being the same direction as \(\overline{{{\Gamma }}}-\overline{{{{{{{{\rm{N}}}}}}}}}\), ky of W(110). In order to measure the transverse component of the spin polarization parallel to the kx (\(\overline{{{\Gamma }}}-\overline{{{{{{{{\rm{H}}}}}}}}}\)) direction of W(110), we rotated the sample around 90°. The kx direction of the sample then is parallel to the quantization axis of the spin filter (see Fig. 2 of the main manuscript). In this way, the SQAs for the measurements of the Px and Py components always lie within the optical plane, such that time-reversal and mirror symmetries of the spin polarization are preserved.

The W(100) crystal of the spin filter was prepared by following the same procedure as described below for W(110). This standard procedure is known to lead to clean tungsten surfaces31,32. The spin-resolved photoemission momentum maps for each constant energy contour, as shown in Fig. 1a, were measured only for 6 min. This allows to collect spin-resolved data in a wide range of binding energies, EB. The complete 3D (kx, ky, EB) information of the spin-resolved spectral function is obtained within a 2 h measurement time. Despite many attempts, it has not been possible so far to achieve a comparably detailed information by conventional spin-resolved photoemission spectroscopy36.

Cleaning procedure for a tungsten single crystal

The procedure for cleaning W(110) consists of two steps: (i) cycles of low-temperature flash heating (T ~ 1200 K) at an oxygen partial pressure of \({P}_{{{{{{{{{\rm{O}}}}}}}}}_{2}}=5\times 1{0}^{-8}\) mbar to remove the carbon from the surface and (ii) a single high-temperature flash (T ~ 2400 K) to remove the oxide layer32. The cleanliness of the W(110) surface was checked by low-energy electron diffraction and Auger electron spectroscopy.

First-principles calculations

Using the full-potential linearized augmented-plane-wave method as implemented in the FLEUR code (, we performed density functional theory calculations of 7 layers of W(110). We adopted the structural parameters given by Heide et al.37. Exchange and correlation effects were treated within the local-density approximation (LDA)38. The spin–orbit interaction was included self-consistently and the spin quantization direction was chosen with respect to \(\overline{{{\Gamma }}}-\overline{{{{{{{{\rm{N}}}}}}}}}\) and \(\overline{{{\Gamma }}}-\overline{{{{{{{{\rm{H}}}}}}}}}\) directions of W(110). The plane-wave cutoff was chosen as \({k}_{\max }=4.2\ {a}_{0}^{-1}\) with a0 as Bohr’s radius and the muffin-tin radii set to 2.5 a0.

To determine the FS with high precision, we calculated the eigenspectrum on a dense uniform mesh of 256 × 256 k-points in the full BZ. The Fermi-level crossings were then obtained by triangulation. To improve the agreement with experiment, we shifted the position of the Fermi level in our calculations downwards by 70 meV as compared to the theoretical value for the Fermi energy.

Our investigation for the momentum distribution of the spin-mixing parameter is based on density functional calculation within the LDA39. We employ the full-potential Korringa–Kohn–Rostoker Green function method40 with exact treatment of the atomic cell shapes41,42. After a self-consistent full-potential calculation performed within the scalar-relativistic approximation, SOC is added when calculating the FS properties43. The spin-mixing parameter is given by

$${b}_{{{{{{{{\bf{b}}}}}}}}\hat{s}}^{2}=1/2-{\langle {S}_{\hat{s}}\rangle }_{k}/\hslash .$$

Wave functions are chosen in a way that the spin \(\langle {S}_{\hat{s}}\rangle\) along the SQA \(\hat{s}\) is maximized. \({b}_{{{{{{{{\bf{b}}}}}}}}\hat{s}}^{2}\) can then obtain values between 0 (pure spin state) and \(\frac{1}{2}\) (fully mixed spin state). Usually, the Bloch states are of nearly pure spin character. However, at special spin flip hotspots in the BZ spin mixing may increase significantly up to 1/2, which corresponds to the case of fully spin-mixed states.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.


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Y.-J.C. and C.T. thank the staff of Elettra for their help and hospitality during their visit in Trieste, and beamline staff M. Jugovac, G. Zamborlini, V. Feyer (PGI-6, FZ-Jülich), and T. O. Menteş (Elettra) for their assistance during the experiment and providing the W(110) crystal. We are indebted to J.-P. Hanke and Y. Mokrousov for discussions. Y.-J.C., C.T., and C.M.S. gratefully acknowledge funding by the BMBF (Grant number 05K19PGA). The authors gratefully acknowledge the computing time granted through JARA on the supercomputer JURECA at Forschungszentrum Jülich.


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Y.-J.C. and C.T. carried out the experiment and analyzed the experimental data. M.H. and B.Z. carried out first-principle calculations and analyzed the theoretical data with assistance by G.B. Y.-J.C. drafted the manuscript with assistance by C.T. C.T. designed and coordinated the research together with C.M.S. and S.B. All authors discussed the results and contributed to the manuscript.

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Correspondence to Ying-Jiun Chen or Christian Tusche.

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Chen, YJ., Hoffmann, M., Zimmermann, B. et al. Quantum spin mixing in Dirac materials. Commun Phys 4, 179 (2021).

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