Altermagnetic lifting of Kramers spin degeneracy

Lifted Kramers spin degeneracy (LKSD) has been among the central topics of condensed-matter physics since the dawn of the band theory of solids1,2. It underpins established practical applications as well as current frontier research, ranging from magnetic-memory technology3–7 to topological quantum matter8–14. Traditionally, LKSD has been considered to originate from two possible internal symmetry-breaking mechanisms. The first refers to time-reversal symmetry breaking by magnetization of ferromagnets and tends to be strong because of the non-relativistic exchange origin15. The second applies to crystals with broken inversion symmetry and tends to be comparatively weaker, as it originates from the relativistic spin–orbit coupling (SOC)16–19. A recent theory work based on spin-symmetry classification has identified an unconventional magnetic phase, dubbed altermagnetic20,21, that allows for LKSD without net magnetization and inversion-symmetry breaking. Here we provide the confirmation using photoemission spectroscopy and ab initio calculations. We identify two distinct unconventional mechanisms of LKSD generated by the altermagnetic phase of centrosymmetric MnTe with vanishing net magnetization20–23. Our observation of the altermagnetic LKSD can have broad consequences in magnetism. It motivates exploration and exploitation of the unconventional nature of this magnetic phase in an extended family of materials, ranging from insulators and semiconductors to metals and superconductors20,21, that have been either identified recently or perceived for many decades as conventional antiferromagnets21,24,25.

Traditionally, lifted Kramers spin-degeneracy has been considered to originate from two possible internal symmetry-breaking mechanisms.The first one refers to time-reversal symmetry breaking by magnetization of ferromagnets, and tends to be strong due to the non-relativistic exchange-coupling origin [18].
The second mechanism applies to crystals with broken inversion symmetry, and tends to be comparatively weaker as it originates from the relativistic spin-orbit coupling [19][20][21][22].A recent theory work based on spin-symmetry classification has identified an unconventional magnetic phase, dubbed altermagnetic [23,24], that allows for lifting the Kramers spin degeneracy without net magnetization and inversion-symmetry breaking.Here we provide the confirmation using photoemission spectroscopy and ab initio calculations.We identify two distinct unconventional mechanisms of lifted Kramers spin degeneracy generated by the altermagnetic phase of centrosymmetric MnTe with vanishing net magnetization [23][24][25][26].Our observation of the altermagnetic lifting of the Kramers spin degeneracy can have broad consequences in magnetism.It motivates exploration and exploitation of the unconventional nature of this magnetic phase in an extended family of materials, ranging from insulators and semiconductors to metals and superconductors [23,24], that have been either identified recently or perceived for many decades as conventional antiferromagnets [24,27,28].
A recently developed spin-symmetry classification focusing on collinear magnets and, within the hierarchy of interactions, on the strong non-relativistic exchange, has identified a third elementary type of magnetic phases in addition to the conventional ferromagnets and antiferromagnets [23,24].The exclusively distinct spin-symmetry characteristics of this emerging third, altermagnetic class are the opposite-spin sublattices connected by a realspace rotation transformation (proper or improper and symmorphic or non-symmorphic), but not connected by a translation or inversion [23,24].In contrast, the conventional ferromagnetic (ferrimagnetic) class has one spin lattice or opposite-spin sublattices not connected by any symmetry transformation, and the conventional antiferromagnetic class has opposite-spin sublattices connected by a real-space translation or inversion.For the case of inversion, the Kramers spin degeneracy of bands in these conventional antiferromagnets is protected even in the presence of the relativistic spin-orbit coupling [29].For the translation connecting the opposite-spin sublattices, lifting of the Kramers spin degeneracy in these antiferromagnets requires both spin-orbit coupling and inversion-symmetry breaking in the crystal, in analogy to ordinary non-magnetic systems.
The unconventional nature of altermagnets is that the rotation symmetry connecting the opposite-spin sublattices protects an antiferromagnetic-like compensated magnetic order with a vanishing net magnetization while, simultanously, it enables a ferromagnetic-like lifting of the Kramers spin degeneracy without breaking the crystal inversion symmetry and without additional symmetry breaking by the relativistic spin-orbit coupling [23,24].
Here we will refer to this mechanism as "strong" altermagnetic lifting of the Kramers spin degeneracy.
Apart from the signature antiferromagnetic-like vanishing magnetization and ferromagneticlike strong spin-degeneracy lifting, whose presence have been traditionally considered as mutually exclusive in one physical system, altermagnets can host a range of novel phenomena that are unparalleled in either the conventional ferromagnets or antiferromagnets [23,24].
Within the realm of (lifted) Kramers spin-degeneracy physics, a unique property associated with the alternating sign of the spin polarization in the altermagnet's Brillouin zone is the presence of an even number of spin-degenerate nodal surfaces crossing the zone-center (Γ-point) in the non-relativistic band structure.In Fig. 1a we demonstrate that these spin degeneracies can be lifted by the relativistic spin-orbit coupling in altermagnets even without breaking the crystal inversion symmetry.We will refer to this mechanism as "weak" altermagnetic lifting of the Kramers spin degeneracy.A comparison of these unconventional weak and strong mechanisms of lifted Kramers spin degeneracy, enabled by altermagnetism, are illustrated in Figs.1a,b.
Both the strong and the weak altermagnetic lifting of the Kramers spin degeneracy can enrich fields ranging from spintronics, ultrafast magnetism, magneto-electrics and magnonics, to topological matter, dissipationless quantum nanoelectronics and superconductivity [23,24].For example, the strong altermagnetic lifting of the Kramers spin degeneracy has been theoretically shown to enable analogous spin-polarized currents to those used for reading and writing information in ferromagnetic memory devices while, simultaneously, removing the capacity and speed limitations imposed by a net magnetization [23,24,[30][31][32][33][34].
Here, using angle-resolved photoemission spectroscopy, we directly identify the weak and strong altermagnetic lifting of the Kramers spin degeneracy in the band structure of MnTe.
A schematic crystal structure of α-MnTe is shown in Fig. 1c,d.The two crystal sublattices A and B of Mn atoms, whose magnetic moments order antiparallel below the transition temperature of 310 K, are connected by a non-symmorphic six-fold screw-axis rotation, and are not connected by a translation or inversion [23,25].The resulting non-relativistic electronic structure of this altermagnet is of the g-wave type [23] with three spin-degenerate nodal planes parallel to the k z -axis and crossing Γ and K points, and one additional spin degenerate nodal plane orthogonal to the k z -axis and crossing the Γ point (k z = 0 nodal plane).These four nodal planes are highlighted in the bottom panel of Fig. 1a.
Angle-resolved photoemission spectroscopy (ARPES) measurements, shown in Fig. 2, were performed within the k z = 0 nodal plane along k x (Γ − K path) and k y (Γ − M path) using an X-ray photon energy of 667 eV.The experiments were performed on the soft X-ray ARPES beamline ADRESS at the Swiss Light Source synchrotron facility [45,46].Samples used in these measurements are thin MnTe films grown by molecular-beam epitaxy on a single-crystal InP(111)A substrate [25,47].We used a vacuum suitcase to transfer the thin-film samples from the growth to the soft X-ray ARPES chamber without breaking ultra-high-vacuum conditions (for details on the sample growth and characterization, and on the measurement techniques see Methods).
In Fig. 2a we show the measured raw data along the k x -axis (bottom panel) and compare with one-step simulation of the photoemission process (top panel), using the Korringa-Kohn-Rostoker ab-initio approach that represents the electronic structure in terms of single-particle Green's functions [48,49].The intense spectral weight around -3.5 eV binding energy, indicated by a magenta dashed line in the experimental and theoretical panels of Fig. 2a, corresponds to a resonance due to Mn d-states.For a better visualization of the bulk electronic structure of MnTe, this spectral weight is filtered out in the experimental ARPES band maps shown in Figs.2b,c.Refinements by the curvature mapping [50] extracted from the area highlighted by a white-dashed rectangle are shown in insets of top panels of Figs.2b,c.These are compared to the corresponding relativistic ab-initio electronic structure calculations plotted in the bottom panels of Figs.2b,c.The theoretical bands, with red and blue colors depicting opposite spin polarizations along the z-axis, show the weak altermagnetic lifting of the Kramers spin degeneracy within the k z = 0 nodal plane.The relativistic bandstructure calculations were performed assuming the Néel vector along the in-plane y-axis (see Fig. 1c), consistent with earlier magnetic and magneto-transport measurements of the Néel-vector easy axis in epitaxial thin films of MnTe [25,51].Altermagnetism and spinorbit coupling thus generate in this case an unconventional spin polarization of bands that is orthogonal to the direction of the magnetic-order vector.
The experimental ARPES band maps in Figs.2b,c are fully consistent with the ab-initio band structures.This includes the overall band dispersions, as well as the significantly larger splitting of the top two bands along the k x -axis (Γ − K path, Fig. 2b) than along the k y -axis (Γ − M path, Fig. 2c).The splitting is highlighted in Fig. 2b by the red double-arrow in the experimental curvature map, and the two split bands have opposite spins in the corresponding ab-initio band structure.We again emphasize that this weak altermagnetic lifting of the Kramers spin degeneracy requires relativistic spin-orbit coupling and is unconventional as it is observed in the bulk band structure of an inversion-symmetric crystal.The extraordinary spin-splitting magnitude ∼ 100 meV and the quadratic dispersion around the Γ-point (see also Fig. 3c), consistently observed in experiment and theory, further highlight the unconventional nature of this lifting of the Kramers spin degeneracy in altermagnetic MnTe.
Fig. 2d shows a k z = 0 constant-energy map measured at the X-ray photon energy of 667 eV, obtained by integrating the measured data over a 50 meV interval of binding energies from the top of the valence band.The observed 6-fold symmetry indicates that within the probing area of this ARPES measurement, there is a comparable population of three Néel-vector easy axes, corresponding to the Γ−M 1−3 axes, that are crystallographically equivalent in the ideal hexagonal lattice of MnTe.Our observation of a multi-domain state is consistent with earlier magnetotransport measurements of the MnTe epilayers [25,47].
We point out that domains with all these three Néel-vector easy axes exhibit larger spin splitting along Γ − K 1−3 paths than along Γ − M 1−3 paths, as shown in Extended Data  Fig. 1.Therefore, even when the population of the three domains is comparable within the sample probing area (X-ray spot position), a significantly larger splitting is expected for the Γ − K 1−3 paths than for the Γ − M 1−3 paths.This corroborates the excellent agreement between the experimentally observed and the calculated band splittings in Fig. 2b,c.
The top-left panel of Fig. 3a shows the refinements by the curvature mapping corresponding to Fig. 2d.Together with the one-step ARPES simulation assuming an equal population of the three easy axes, shown in the top-right panel of Fig. 3a, it confirms the 6-fold symmetry of this constant-energy cut.In the series of panels in Fig. 3a, we then systematically explore the symmetry of the constant-energy maps measured and calculated at different binding energies, indicated by symbols A-D in the band dispersion shown in Fig. 3c.
Analogous set of measurements and calculations is shown in Fig. 3b for a different probing area on the sample (different X-ray spot position).While the maps in Fig. 3a show the 6-fold symmetry for all binding energies, the maps in Fig. 3b have a lower 2-fold symmetry at energies near the top of the valence band (binding energies A-C).The 6-fold symmetry is observed in Fig. 3b only deeper in the valence band (binding energy D).The one-step ARPES simulations in Fig. 3b were performed assuming a single-domain state with the Néel vector along the easy axis corresponding to the Γ − M 1 axis.The agreement between experiment and theory for all the studied constant-energy maps confirms that in the probing area of the MnTe epilayer corresponding to Fig. 3b, there is a prevailing population of one of the three Néel-vector easy-axis domains (Γ − M 1 axis).A comparison to another probing area on the sample with an intermediate domain-population character between those of Figs.3a,b is shown in Extended Data Fig. 2. Note that, in Fig. 3b, the more prominent lowering of the symmetry from 6 to 2-fold near the top of the valence band correlates with the dominant contribution of p-orbitals of the heavy Te atoms, which significantly enhances the strength of spin-orbit coupling in this spectral range (see Extended Data Fig. 3).
As explained in the introduction and illustrated in Fig. 1, the strong altermagnetic lifting of the Kramers spin degeneracy can be identified in the electronic structure only outside the four nodal planes that are spin-degenerate in the non-relativistic limit.In Fig. 4 we compare the measured and simulated ARPES data inside and outside the nodal planes.Soft X-ray ARPES band maps for k z = 0.35 Å−1 (X-ray photon energy of 368 eV) along a path parallel to Γ − K, i.e., within one of the nodal planes, are shown in Figs.4a,b.To highlight the finite k z value, we label the path as Γ − K. Data for the same k z value and a path Γ − M, i.e., outside the nodal planes, are shown in Figs.4c,d.In both experiment and theory, we observe a significantly larger band splitting in Figs.4c,d (strong altermagnetic), reaching a half-eV scale, than Figs.4a,b (weak altermagnetic) in the part of the spectrum labeled by B 1 and B 2 .The spin-resolved one-step ARPES simulations of this part of the spectrum then suggest that a sizable spin-polarization signal should be detectable by spin-resolved ARPES (SARPES).This applies in particular to the Γ − M path featuring the strong altermagnetic lifting of the spin degeneracy.
The spin-resolved measurements were performed on the UV SARPES end-station COPHEE at the Swiss Light Source [52].Since the COPHEE sample holder was not compatible with the Omicron plate used in the vacuum suitcase, and the system did not allow for decapping the MnTe surface, we performed the SARPES measurements on in situ cleaved bulk-crystal samples.The bulk single crystals of MnTe were grown using the self-flux method.Their structural quality was confirmed by X-ray diffraction.Magnetization measurements by a superconducting quantum interference device verified the compensated magnetic ordering with the Néel temperature at 310 K and the Néel vector in the z-plane, consistent with earlier reports on bulk crystals [28], and also consistent with the magnetic characteristics of our thin MnTe films.(For more details on the characterization of the bulk MnTe crystals  see Methods and Supplementary information.) The consistency between the electronic structures of the MnTe thin-film and bulk-crystal samples is illustrated on soft X-ray ARPES band maps shown in Extended Data Figs.3a,b ory confirms the prediction [24,34] that altermagnetism can originate directly from crystal symmetries, without requiring strong electronic correlations.Specifically, it confirms that altermagnetism stems from the local crystal anisotropy that breaks translation and inversion symmetry but preserves a rotation symmetry connecting the opposite-spin sublattices.
The crystal-symmetry basis makes altermagnetism one of the elementary phases of matter which, remarkably, has been omitted for nearly a century of the band theory of solids.Our results highlight the strength of the spin-group symmetry classification in unraveling new magnetic phases and in describing the hierarchy of energy scales that underpin their rich phenomenology and potential applications [17,24,34].Combined with the free-electron dispersion of high-energy final states, the resulting precise definition of k z allows accurate determination of the 3D electronic structure.As in the case of MnTe, this advantage of SX-ARPES has been demonstrated, e.g., also on ferroelectric Rashba semiconductors [56], transition-metal dichalcogenides [55,57], high-fold chiral fermion systems [58], etc.
The SX-ARPES experiments were conducted in the photon energy range 350-700 eV at the SX-ARPES end-station [59] of the ADRESS beamline at the Swiss Light Source, Paul Scherrer Institute, Switzerland [60].All presented data were acquired with π-polarized X-rays.The photoelectrons were detected using the PHOIBOS-150 analyzer with an angular resolution of ≈ 0.1 • and using a deflector mode without changing the sample angles.
The combined (beamline and analyzer) energy resolution varied between 50 and 100 meV in the above energy range.The experiments were performed in a vacuum of better than 1×10 −10 mBar and at a sample temperature of around 15 K.The investigated MnTe thin film samples were transferred from the MBE in JKU Linz using a vacuum suitcase.In the presented data, the coherent spectral fraction was enhanced by subtracting the angle-integrated spectral intensity as seen in Fig. 2a-c of main text.The constant energy-surface maps were integrated within a range of ±50 meV.The conversion of the measured photoelectron kinetic energies and emission angles to binding energies and momenta was accomplished using the kinematic formulas which account for the photon momentum [59].
The spin-resolved ARPES (SARPES) measurements were conducted at 24 eV at COPHEE experimental station at the Swiss Light Source SIS beamline [61,62] on in situ cleaved bulk single crystals at 21 K. Combined with an angle-resolving photoelectron spectrometer it produces complete data sets consisting of photoemission intensities (Fig. 4e), as well as spin polarization curves (Fig. 4g) with the combined experimental resolution of ≈25 meV and ≈100 meV, respectively.
Calculations.The experimental results were compared with ab initio electronic structure calculations, performed for MnTe in P6 3 /mmc (Space group:194) symmetry using the lattice parameter as determined from the XRD measurements [51].
The calculations in Fig. 2, 3 and 4 were carried out using spin-polarized fully relativistic Korringa-Kohn-Rostoker (SPRKKR) Green's function method in the atomic sphere approximation, within the rotationally invariant GGA+U scheme as implemented in the SPRKKR formalism [65,66].The screened on-site Coulomb interaction U and exchange interaction J of Mn are set to 4.80 eV and 0.80 eV respectively.The angular momentum expansion of the s,p,d,f orbital wave-functions has been used for each atom on a 28×28×15 k-point grid.The energy convergence criterion has been set to 10 −5 Ry.Lloyd's formula has been employed for accurate determination of the Fermi level [66][67][68].
The photoemission calculations for a semi-infinite surface of MnTe(001) with Mn atoms as the termination layer at the surface were performed within the one-step model of photoemission in the spin-density-matrix formulation as implemented in the SPRKKR package [69].

Fig. 1 .
Fig. 1.Illustration of weak and strong altermagnetic lifting of Kramers spin degeneracy.a, Top and middle panels: ab initio band structure of MnTe at k z = 0 along the Γ − K path for relativistic spin-orbit coupling turned off and on, resp.Néel vector is along the crystal y-axis (see panels c,d), corresponding to the Γ − M axis (see bottom panel of b).Red and blue colors correspond to opposite z-components of spin.Bottom panel: Schematics of the Brillouin zone with four spin-degenerate nodal planes in the electronic structure with spin-orbit coupling turned off.b, Same as a at k z = 0.35 Å−1 along the Γ − M path illustrating the strong altermagnetic lifting of Kramers spin degeneracy.Red and blue colors correspond to opposite y-components of spin.Bottom panel highlights the Γ − M path outside the four nodal planes (the red and blue colors highlight the alternating symmetry of the spin polarization in the plane).c,d Schematic view of the crystal and magnetic structure of MnTe in the y − z and x − y plane, resp.The red and blue shadings in c,d mark Te-octahedra around the Mn sites A and B with opposite spins which are related by spin rotation combined with six fold crystal rotation and half-unit cell translation along the z-axis.

Fig. 2 .
Fig. 2. Weak altermagnetic lifting of Kramers spin degeneracy in the nodal plane.a, Bottom panel: Measured soft X-ray (667 eV) ARPES band map at k z = 0 along k x (Γ − K path) on epitaxial thin-film MnTe.Top panel: Corresponding one-step ARPES simulation.Magenta dashed line highlights an intense spectral weight around -3.5 eV binding energy corresponding to a resonance of Mn d-states.b, Measured ARPES band map along k x (Γ − K 1 path) after filtering out the intense spectral weight due to the Mn d-state resonance.Inset: Refinement of the measured data by curvature mapping.Bottom panel: Ab initio bands with red and blue colors corresponding to opposite z-components of spin.The Néel vector is aligned along the Γ − M 2 direction in the calculations.c, Same as b along k y (Γ − M 1 path).d, Constant-energy map obtained by integrating the measured data over a 50 meV binding-energy interval from the top of the valence band.

Fig. 3 .
Fig. 3. Constant-energy maps and Néel-vector easy-axis domains.a, Left column: Refinements by the curvature mapping of measured constant-energy maps for binding energies A-D indicated in panel c.Right column: Corresponding one-step ARPES simulations.b, Same as a for a different probing area on the sample (different X-ray spot position).Dashed red contours highlight the 6-fold (2-fold) symmetry in top right panels of a (b).c, Refinement by the curvature mapping of the band map from the main experimental panel of Fig. 2a at k z = 0 along the Γ − K path with the indicated binding energies A-D.d, Schematics of the 6-fold symmetry of constantenergy maps (bottom-left) for an equal (comparable) population of the three Néel-vector easy axes (top-left), and a lowered 2-fold symmetry of constant-energy maps (bottom-right) for one of the three easy-axes domains prevailing (top-right).

Fig. 4 .
Fig. 4. Weak and strong lifting of Kramers spin degeneracy at k z ̸ = 0. a, Measured soft X-ray (368 eV) ARPES band map at k z = 0.35 Å−1 along the Γ − K path (left: unrefined data, right: refined data).b, Corresponding one-step ARPES simulations.Red and blue colors show opposite y-components of spin.c,d Same as a,b along the Γ − M path.e, Experimental UV (24 eV) ARPES constant-energy maps at k z = 0.12 Å−1 measured on bulk-crystal MnTe.f, Corresponding one-step UV ARPES simulations.g, Experimental UV ARPES band map along the Γ − M path (left), corresponding total-intensity energy-distribution curve (middle) and SARPES (right).The spin polarization is detected along the Γ − M axis.In all theoretical panels, the considered Néel vector and the spin-polarization projection are along an axis corresponding to the Γ− M1 axis (also highlighted by the cyan arrow in panels e,f), and the considered paths are Γ− K1 and Γ − M1 .
Figs. 4c,d whose expected spin polarization is due to the strong altermagnetic lifting of the Kramers spin degeneracy.The spin polarization is experimentally confirmed by the UV SARPES measurements in Fig. 4g.In the middle panel of Fig. 4g we plot the measured totalintensity energy-distribution curve (EDC), and the corresponding SARPES signal is shown in the right panel of Fig. 4g.As expected, we observe the alternating sign of the spinpolarization component along the Néel vector, consistent with the presence of the strong altermagnetic lifting of the Kramers spin degeneracy for the Γ − M path.In conclusion, we have observed two types of unconventional lifting of Kramers spin degeneracy in altermagnetic MnTe.The weak altermagnetic mechanism generates extraordinary relativistic spin-splitting magnitude ∼ 100 meV and quadratic dispersion around the Γ-point.The strong altermagnetic mechanism in the magnetically compensated and centrosymmetric MnTe reaches a remarkable half-eV scale.The experimental observations are in excellent agreement with ab initio calculations.The agreement between the spin-split band structure observed in ARPES and obtained from density-functional the-

METHODS
Photoelectron spectroscopy.Angle-resolved photoemission spectroscopy (ARPES) was used for investigating the electronic structure of MnTe -including the Fermi surface, band structure, and one-electron spectral function A(ω, k) -which are resolved in electron momentum k (see Ref.[54] for more details).The extension of photon energies into the soft X-ray range (SX-ARPES) from a few hundred eV to approximately 2 keV enhances the probing depth of this technique, characterised by the photoelectron escape depth λ, by a factor of 3-5 compared to the conventional vacuum ultraviolet photon energies (UV-ARPES).This enables access to the intrinsic bulk properties, which is essential for three-dimensional (3D) materials like MnTe.The increase of λ reduces the intrinsic broadening δk z of the outof-plane momentum k z , defined by the Heisenberg uncertainty principle as δk z ≈ λ −1[55].