3D bulk-resolved <i>g</i>-wave altermagnetic order parameter in CrSb

There are two traditional types of collinear magnetic ordering: ferromagnetism^1,2, in which all spins point in the same direction, and antiferromagnetism¹⁵, in which the spin polarization alternates up and down from site to site. In an antiferromagnet, the magnetic structure can be decomposed into two magnetic sublattices, resulting in a doubling of the magnetic unit cell¹⁶ under translation. Recently, a third distinct classification was proposed, dubbed altermagnetism^{9,10,11,12,13}. Like antiferromagnets, altermagnets possess alternating up- and down-spin orientations from site to site. However, the distinction is that in an altermagnet the spin sublattices are connected by rotational symmetries rather than just translation (or inversion). This leads to the lifting of Kramers spin degeneracy¹⁷ and a momentum-dependent non-relativistic spin splitting of the electronic band structures¹⁸. Therefore, altermagnetism represents a distinct form of ordering, in which the momentum-dependent spin-split property of ferromagnets is combined with the spin-compensated zero net magnetization of antiferromagnets, opening new opportunities for magnetic memory and spintronic device applications¹⁹.

Numerous altermagnet candidates have been identified from ab initio calculations^{9,10,19,20,21}, with experiments on MnTe giving good empirical correspondence with theoretical expectations^22,23,24,25. However, although many semiconducting or insulating materials have been identified, only a handful of metallic altermagnet candidates have so far been proposed. The realization of a metallic altermagnet, with an ordering temperature far above 300 K, would be particularly desirable for efficient electronic transport in technological device settings²¹.

Of metallic candidates, surface-sensitive photoemission spectra of RuO₂ (refs. ^26,27), KV₂Se₂O (ref. ²⁸) and CrSb (refs. ^29,30,31,32) have been interpreted to show hallmarks of altermagnetic spin splitting. However, for the case of RuO₂, a variety of subsequent bulk-sensitive measurements^33,34,35 cast considerable doubt on whether this material is an altermagnet. Further photoemission studies^36,37 showed that this material seems to possess a topological surface state with Rashba-like spin splitting, which may give the appearance of altermagnetically lifted Kramers spin degeneracy although the bulk is actually nonmagnetic. A similar story has unfolded with KV₂Se₂O, for which the surface termination layer may be altermagnetic³⁸, yet neutron diffraction clearly resolves conventional antiferromagnetism throughout the bulk³⁹. These cautionary tales motivate bulk-sensitive studies of candidate altermagnets, to firmly establish their intrinsic magnetic properties.

We can frame our thinking of magnetic order parameter symmetries by analogy to unconventional superconductors. In a conventional s-wave BCS superconductor, the order parameter is the gap function \(\Delta ({\bf{k}})\), which is an isotropic energy separation corresponding to a dispersion possessing the same symmetries as the crystal. By contrast, unconventional superconductors⁴ are those in which the pairing symmetry breaks crystal point group symmetries, or where the gap function exhibits a non-trivial phase structure (like a sign change) not required by the crystalline symmetry.

These concepts have recently been applied to magnetic systems^5,6. Let us define the order parameter for an unconventional magnet as the exchange splitting (energy difference) between up- and down-spin states at a given wavevector, \(\Delta ({\bf{k}})=E({\bf{k}},{\rm{\uparrow }})-E\,({\bf{k}},{\rm{\downarrow }})\). This produces spin-split Fermi sheets separated by the (perpendicular) wavevector \({k}_{\perp }\simeq \Delta ({\bf{k}})/(\hbar {v}_{{\rm{F}}})\), where v_F = ∂E/(ħ∂k) is the Fermi velocity. In this way, a ferromagnet is analogous to a BCS superconductor—its nonzero magnetization splits the Fermi surface into unequal majority and minority spin species, characterized by an isotropic energy gap between them. It follows that for an unconventional magnet of p-, d-, f- and g-wave symmetry, the Fermi surface will possess 1, 2, 3 and 4 highly symmetric nodal planes, respectively, at which up-character swaps to down, and vice versa.

CrSb crystallizes in the hexagonal P6₃/mmc NiAs-type structure, with chromium atoms stacked along the crystallographic c-axis in octahedrally coordinated layers, whereas the antimony atoms fill the interstitial sites in a trigonal prismatic arrangement⁴⁰ (Fig. 1a,b). Below an ordering temperature of around 740 K (Extended Data Fig. 1), the chromium sites possess alternating magnetic moments oriented along the c-axis⁴¹. The trigonal arrangement of antimony necessarily requires a 6₃ screw rotation to map a spin-up chromium site to its spin-down counterpart. Consequently, although CrSb breaks both pure timereversal and primitive lattice-translation symmetries, it preserves a combined symmetry pairing this non-symmorphic spatial operation with time reversal. These symmetry properties motivate the proposal^{10,13,21,32,42,43} that CrSb is a metallic altermagnet in which \(\Delta (G{\bf{k}})=-\Delta ({\bf{k}})\), where G is this composite operation.

In this work, we map the symmetry of Δ(k) in CrSb by performing magnetic quantum oscillation measurements through the de Haas–van Alphen effect^14,44, and show that this material is a g-wave altermagnet. Quantum oscillation experiments are an especially well-suited technique for resolving the nodal planes of an unconventional magnet, and hence the order parameter symmetry. This is because Δ(k) does not just go to zero for k on a nodal plane—it is also mirror-symmetric about the plane. Aligning a magnetic field within a nodal plane therefore produces quantum oscillation orbits for up- and down-spin Fermi sheets that are different, but can be mapped onto each other by a mirror operation about the nodal plane, and which therefore enclose the same area. This leads to a single quantum oscillation frequency per Fermi pocket for fields in nodal planes. By contrast, for fields oriented away from nodal planes, the up- and down-spin Fermi sheets are not related by any symmetry operation, and therefore typically enclose different areas, leading to two distinct quantum oscillation frequencies. This thereby yields an expected smoking-gun signature of altermagnetic spin splitting.

Throughout this article, we shall refer to rotations of the orientation of an applied magnetic field H through azimuthal angles φ defined in the crystallographic ab plane, and polar angles θ between c and the ab plane (Fig. 1c,e). Note that for hexagonal CrSb, the a direction is equivalent to b. The c−a plane is a highly symmetric nodal plane in which Kramers degeneracy is enforced by symmetry, whereas c–ab is antinodal, in which spin degeneracy is only symmetry-enforced at c and ab. In CrSb, g-wave splitting should result in nodal planes every δφ = 60° for rotations in the a−ab plane, and every δθ = 90° rotating through the c−ab plane.

We present the experimentally deduced central Fermi surface pocket of CrSb, with its high-symmetry nodal planes between up and down sheets, in Fig. 1c–f. This sheet is a closed ellipsoid-like pocket of hole character, shaped like a dogbone. In Fig. 1g, we plot background-subtracted magnetic torque Δτ as a function of H, to focus on the oscillatory component, measured by cantilever beam magnetometry (Methods). The blue waveform was measured for H ∥ a at θ = 90°, φ = 0°, with the orange data collected a small rotation away in the c−a plane to θ = 88°, φ = 0°. The corresponding fast Fourier transform (FFT) spectra in Fig. 1h for both of these orientations exhibit a singular frequency peak at 4.1 kT.

Let us now contrast this with the case of fields in the c−ab (antinodal) plane. The green curve of Fig. 1g was obtained for H ∥ ab (that is, θ = 90°, φ = 90°), with a monofrequency profile of 3.6 kT. On a small θ rotation of 3° towards c, a distinct beat pattern emerges (red curve), with two peaks clearly resolved in the frequency spectra. This is characteristic of the Fermi surface becoming spin-split into two non-degenerate sheets—one spin-up and the other spin-down.

Figure 1i,j tracks this split peak structure for fields close to the ab direction. Incrementing θ in 1° steps through the c−ab plane, we find that the singular on-axis peak of 3.6 kT clearly splits into two distinct frequency branches, which by 4° of rotation are separated by 0.6 kT. This is reflected in the profile of the waveforms themselves, which exhibit a pronounced beat structure that is acutely sensitive to θ.

The full angular profiles of quantum oscillation frequency spectra for the c−a (nodal) and c−ab (antinodal) planes are shown in Fig. 2. Quantum oscillation frequencies are directly proportional to the Fermi surface cross-sectional area \({{\mathcal{A}}}_{\perp }\) normal to H (refs. ^45,46). We compare with simulated quantum oscillation frequency spectra from density functional theory (DFT) calculations (see Methods for details). The dogbone pocket exhibits quantum oscillation frequencies of 4.1 kT for H ∥ a and 3.6 kT for H ∥ ab. Two other bands of electron character also cross the Fermi level, leading to another more geometrically complex Fermi sheet with a web-like shape (Fig. 2). The web sheet produces a very complicated dependence of quantum oscillation frequency on angle in the low-frequency range ⪅2 kT. Away from nodal planes, the web is also split into up and down sheets, but its geometrical complexity makes this more difficult to resolve than for the dogbone. We therefore predominantly focus on the dogbone for H oriented close to a or to ab, which produces quantum oscillation orbits with frequencies ⪆3 kT, well above those from the web.

Symmetry-enforced spin degeneracy causes the quantum oscillation frequencies from both the spin-up and spin-down Fermi sheets to be equivalent for fields in the nodal c−a plane, as shown by our simulation (Fig. 2b). By contrast, for fields in the antinodal c−ab plane, a large frequency splitting should occur over the approximate angular interval 60° < θ < 120° (Fig. 2e,f), before the geometric profile of the dogbone leads to an accidental degeneracy. This is exactly what we observed in Fig. 1i,j and is clearly visible in the spectral intensity near θ = 90° in Fig. 2e. In Fig. 2i, we present a magnification of this region to focus on this spectral range. The overlap of frequencies at H ∥ ab, which is abruptly split on incrementing θ, is characteristic of the altermagnetic lifting of spin degeneracy, producing two spin-split Fermi sheets. Notably, whereas in the nodal plane both spin-up and spin-down frequency traces are expected to repeat every 90°, in the antinodal plane this reduces to every 180°—exactly as observed experimentally.

To trace out the spin splitting of the CrSb Fermi surface away from its nodal planes, we measured quantum oscillations in a tilted plane of rotation, chosen such that an arc of low symmetry connects two symmetry-enforced nodal orientations. Along this arc, the spin splitting should be considerable, but collapse to yield monofrequency waveforms at nodal orientations. We experimentally performed this by first rotating the sample platform 14° orthogonally to the axis of the solenoid before loading it into the magnet. Then, we rotated in situ through a polar angle (labelled α in Fig. 3), about an axis orthogonal to H, but always at a tilt of 14° relative to the sample holder. This setup means that H may still be oriented along a, but subsequent rotation of the sample holder increments θ (c−a) and φ (a−ab) such that cos(θ) = sin(14°)sin(α) and tan(φ) = cos(14°)tan(α) (see Extended Data Fig. 4 for further details). We plot the resulting data for these rotational measurements in Fig. 3, including the simulated up- and down-sheet frequency profiles computed from DFT.

The correspondence between experimental observation (Fig. 3a) and theoretical prediction (Fig. 3b) is very good. A singular frequency peak is observed for H ∥ a (θ = 90°, φ = 0°) in Fig. 3d, giving a bright spot at the nodal location of φ = 0° in Fig. 3a. On incrementing α, this then splits into two distinct branches, which, within resolution, come back together again close to φ = ±60° as the next nodal plane is crossed, as expected for Δ(k) possessing g-wave symmetry.

We determined the effective cyclotron masses of the two spin-split surfaces in a Lifshitz–Kosevich temperature-dependence study of quantum oscillation amplitude in this rotation plane at α = 22°, which in the crystal reference frame corresponds to H aligned at θ = 84.8°, φ = 21.4° (Fig. 4a). Two peaks are resolved, with f₁ = 3.41 kT and f₂ = 3.82 kT. Fitting the temperature-dependent oscillatory damping to Lifshitz–Kosevich theory⁴⁶ yields effective masses \({m}_{1}^{* }\) = 1.73(5)m_e and \({m}_{2}^{* }\) = 1.84(5)m_e, where m_e is the bare electron mass. For two quantum oscillation frequency branches that are spin-split daughters of the same mother sheet, the daughters are expected to share the same mass renormalization factors and Fermi velocity profile. The quantum oscillation mass is given by \({m}^{* }=(\partial {{\mathcal{A}}}_{\perp }/\partial E){\hbar }^{2}/(2{\rm{\pi }})\), where \(\partial {{\mathcal{A}}}_{\perp }/\partial E=(\partial {{\mathcal{A}}}_{\perp }/\partial k)/(\hbar {v}_{{\rm{F}}})\). If the orbit area scales with the square of the orbit size, \((\partial {{\mathcal{A}}}_{\perp }/\partial k)\propto \sqrt{{{\mathcal{A}}}_{\perp }}\), with \({{\mathcal{A}}}_{\perp }\) in turn proportional to the associated quantum oscillation frequency. If v_F(k) is the same around the two orbits, we would therefore expect \({m}_{1}^{* }/{m}_{2}^{* }=\sqrt{{f}_{1}/{f}_{2}}\). This is what we observe: \({m}_{1}^{* }/{m}_{2}^{* }=0.94\pm 0.04\simeq \sqrt{{f}_{1}/{f}_{2}}=0.945\). This result provides strong corroboration that these two frequencies may therefore be attributed to altermagnetic spin splitting of the same parent sheet. The maximum spin splitting can likewise be estimated from the frequency difference δf = f₂ − f₁ = 0.41 kT, which according to the Onsager relation⁴⁵ translates to a difference between the areas enclosed by spin-up and spin-down orbits as \(\delta {\mathcal{A}}=\delta f(2{\rm{\pi }}e)/\hbar =\Delta \partial {\mathcal{A}}/\partial \varepsilon ={m}^{* }\Delta (2{\rm{\pi }}/{\hbar }^{2})\), with Δ the energy splitting at the Fermi level. With the observed average effective mass m^* ≃ 2.03m_e, we thereby estimate a spin splitting at the Fermi level of Δ ≈ 25 meV for this field orientation.

Putting together our results from the three different rotation planes presented in Figs. 2 and 3, we can firmly pin down the unconventional magnetic order parameter symmetry of CrSb. By tracking how the spin-split quantum oscillation frequency profile of the primary CrSb Fermi pocket evolves under rotation, we identify four nodal planes in total: three defined for all θ at φ = 0°, 60° and 120°, and one for all φ at θ = 90°. We can therefore compare the symmetry of our unconventional magnetic order parameter, Δ(θ, φ), to that of a real spherical harmonic. We find that Δ(θ, φ) obeys the symmetries of \({{\mathcal{Y}}}_{4}^{-3}(\theta ,\varphi )\propto {P}_{4}^{3}\,(\cos \theta )\sin 3\varphi \) defined in terms of the associated Legendre polynomial \({P}_{{\ell }}^{m}(x)\) (ref. ⁴⁷) (Methods). This harmonic transforms as the B_1g irreducible representation of the D_6h point group of CrSb. \({{\mathcal{Y}}}_{4}^{-3}(\theta ,\varphi )\) can be expressed in Cartesian form as yz (3x² − y²) or equivalently in spherical polars as r⁴sin³(θ)cos(θ)sin(3φ) (Fig. 1f). This function has nodes when θ = 0°, 90° and φ = 0°, 60°, 120°, just as Δ(θ, φ) does. Our study thereby demonstrates the sensitivity of quantum oscillation measurements to spin-split band structures, positioning these measurements as an ideally suited diagnostic tool for ascertaining the order parameter symmetry of unconventional metallic magnets.

In summary, our systematic bulk-sensitive quantum oscillation measurements have mapped out the unconventional magnetic order parameter symmetry of CrSb, revealing the g-wave altermagnetic spin texture. This manifests in the k-dependence of the spin splitting per Fermi pocket, Δ(k). For magnetic fields applied in highly symmetric nodal planes, Δ(k) is always zero—whereas, rotating through arcs of low symmetry Δ(k) is generally large and well-resolvable, collapsing back to zero on crossing through a nodal plane. For fields applied in nodal planes, quasiparticle orbits are spin-degenerate and mirror-symmetric. By contrast, in antinodal planes, this mirror symmetry is lost, leading to spin-split quantum oscillation frequency traces—providing a hallmark signature of altermagnetic ordering. Furthermore, we observe excellent quantitative agreement in the ratios between quantum oscillatory frequencies and effective masses, as theoretically expected for spin-split Fermi surface sheets. In combination, these results provide strong empirical evidence supporting the designation of CrSb as a room-temperature metallic altermagnet. Furthermore, our optimized growth procedure has produced high-quality crystals with residual resistivities as low as 2 μΩ cm. Given that CrSb is relatively insensitive to air and moisture, it can be synthesized to high purity, the altermagnetic ordering temperature is very high, and both chromium and antimony are relatively abundant easily sourced elements, this combination of favourable properties positions CrSb as a leading material for pursuing applications in next-generation low-energy spintronic devices.

Crystal growth

Single-crystal CrSb specimens were grown by the chemical vapour transport technique. Stoichiometric amounts of Cr (chunks, 99.995%) and Sb (Shots, 99.9999%) were used as source material. Iodine was added as a transport agent, calculated to have a pressure of 1 bar at growth conditions. The starting materials were sealed under vacuum in a quartz ampoule and placed in a horizontal two-zone furnace. The temperature was slowly ramped up to T₁ = 925 °C and T₂ = 900 °C, left for 2 weeks, and subsequently cooled at the furnace cooling rate to room temperature. The resulting crystals were hexagonal platelets up to 1.5 mm in diameter, along with larger areas possessing intergrown crystals of CrSb, several mm in size. Only single-crystal specimens were used in this study.

Sample characterization

Several crystals were picked from a batch of single crystals and crushed into a fine powder. This powdered sample was then distributed on a microscope slide, which had a thin layer of vacuum grease. Powder X-ray diffraction was measured in the Bragg–Brentano geometry on a Bruker D8, using a Cu source, with the results plotted in Extended Data Fig. 1. The measurement was performed in a 2θ range of 10°–90°, with no peaks observed below 20°.

The obtained data display sharp, well-defined peaks, indicating a high level of crystallinity. The data were analysed using the Rietveld method, yielding an excellent fit (R_Bragg = 3.39), which describes all observed peaks, thereby indicating that the samples are phase pure. The measured crystal structure is in good agreement with previous studies⁴⁰.

We also performed electrical transport, magnetization and Laue diffractometry measurements (Extended Data Fig. 1). Samples were predominantly screened by temperature-dependent resistivity measurements, used to extract their residual resistivity ratios (RRRs). To do this, we fitted the low-temperature data to the square of the temperature and extrapolated to absolute zero to determine the residual resistivity. The 300 K resistivity was then divided by this value to yield the RRR. Higher RRR values indicate longer mean free paths and hence higher crystalline quality. Typical RRR values were in the approximate range of 10–28. High-quality specimens were then oriented by Laue diffractometry, in preparation for high magnetic field de Haas–van Alphen (dHvA) effect measurements.

dHvA effect torque magnetometry measurements

High-quality samples were selected following characterization screening and brought to the National High Magnetic Field Laboratory, Tallahassee, Florida. For torque magnetometry measurements, we largely followed the methodology outlined in ref. ⁴⁸. Samples were mounted on flexible BeCu cantilevers and affixed using multiple layers of General Electric low-temperature varnish, giving good thermal contact and strong adhesion between sample and cantilever. Cantilevers were soldered in place, such that the cantilever head was suspended above a copper baseplate by a short separation distance. As the magnetic field was swept, the change in capacitance between the cantilever and baseplate, due to the magnetic torque exerted on the sample, was measured by a General Radio analogue capacitance bridge using phase-sensitive detection. The change in torque was calibrated to units of farads using an Andeen-Hagerling digital capacitance bridge.

All dHvA measurements were performed in the 41.5 T all-resistive magnet in Tallahassee. A ³He sample environment was used, along with a probe mounting of our custom design. Rotations of the sample orientation with respect to the magnetic field were performed in situ using a brushless linear motor. Angles were calibrated by the change in sign of the torque background—identifying high-symmetry directions of the crystal—and verified using a Hall sensor.

The oscillatory component Δτ was isolated from the background magnetic torque τ by performing a locally estimated scatterplot smoothing (LOESS)⁴⁹ subtraction. In general, owing to the intricate web sheet of the CrSb Fermi surface, the dHvA waveform at a given angle could be quite complicated because of the presence of numerous frequency components. To simplify our analysis and concentrate on the dogbone Fermi sheet, we often performed combined high-pass filtering with short LOESS windows in our analysis. The dogbone frequencies are most prominent above 3 kT, and so we performed Butterworth high-pass filtering of frequencies in inverse field in this range. This was combined with a short sliding LOESS window over τ, which effectively fits any slow oscillations within the background (assumed to be quadratic in H), therefore producing a Δτ waveform dominated by higher-frequency components. For the Δτ traces presented in Fig. 1, this involved using a LOESS window of 0.7 T. In Fig. 2, we used a window of length 1.2 T to show the strong spectral weight at lower frequencies due to the web. By contrast, in Fig. 4, we used a window of only 0.6 T to focus on the >3 kT components in our temperature-dependence study.

DFT calculations

DFT calculations for CrSb were performed using the all-electron, full-potential linearized augmented plane-wave (FP-LAPW) method as implemented in the WIEN2k code⁵⁰. The electronic structure was converged on a 43 × 43 × 28 Monkhorst–Pack k-point mesh within the Brillouin zone of the primitive hexagonal unit cell. Exchange–correlation effects were treated within the generalized gradient approximation. We specified two distinct Cr sites (Cr₁ and Cr₂) within the primitive unit cell, corresponding to Cr atoms adopting up and down spin polarization. Calculations were initialized so that one Cr site has a higher spin-up density and the other has a spin-down density. The onsite spin polarization was then allowed to vary throughout the self-consistency cycles until the compensated collinear ground state was reached. Quantum oscillation frequency analysis of the resultant Fermi surface sheets was determined using SKEAF (ref. ⁵¹). Fermi surface visualization was performed using py_FS (refs. ^48,52).

We assumed that ambient-pressure CrSb in the NiAs-type structure (P6₃/mmc) adopts lattice parameters a = 4.12 Å, b = 4.12 Å and c = 5.47 Å. Within the unit cell, there are two equivalent Cr sites and two equivalent Sb sites, as specified by Extended Data Table 1. We reduce the symmetry of the crystal lattice from P6₃/mmc to P3m1 by specifying that the two Cr sites adopt opposite spins.

DFT calculations converge on a Fermi surface, in which the bands associated with the down and up ‘dogbone’ surfaces are open about the high-symmetry point, corresponding to a cylindrical topology. This is inconsistent with our quantum oscillation measurements, in which we resolve oscillations from these sheets for magnetic fields very close to the a and ab directions. No frequencies would be observed for these field orientations if the sheets were cylindrical. Therefore, we propose that these bands form closed Fermi surface sheets with dogbone-like geometry. To ‘close’ the open Fermi surface sheets of our DFT calculations, we shift our band edges relative to the Fermi energy. The dogbone-like sheets were shifted down by 0.11 eV so that the calculated frequencies along the a, ab and c directions are in good agreement with the quantum oscillation data. As the dogbone sheets are of hole character, we shifted up the ‘web’ sheets (of electron character) by 0.015 eV to keep the total carrier number constant (see Supplementary Information).

Consideration of spin–orbit coupling

Real materials always exhibit some spin–orbit coupling and many-body electronic correlations, meaning a purely non-relativistic framework is only ever an idealization. Nevertheless, our symmetry-based interpretation remains robust. Because CrSb possesses an inversion-symmetric crystal structure, the spatial symmetries protecting the orientation of the nodal planes remain intact. Furthermore, under the intense magnetic fields used in our experiments, field-assisted tunnelling (magnetic breakdown) allows quasiparticles to traverse small hybridization gaps opened by weak spin–orbit coupling, effectively restoring the pristine altermagnetic trajectories. As detailed in the Supplementary Information (in which we also explicitly account for electronic correlations), these considerations justify our simplified symmetry picture introduced here, yielding a direct mapping between the quantum oscillation frequency spectra and the underlying altermagnetic order parameter Δk.

Energy splitting from quantum oscillation frequencies

From the Onsager relation⁴⁵, we can equate a quantum oscillation frequency to a reciprocal space area as

The cyclotron mass of an orbit, m*, is related to the rate of change of the orbital area by

By taking the derivative of equation (1) with respect to E, we then determine

We can use this to determine the energy difference associated with the frequency splitting of two bands:

Spherical harmonic notation

In the text, we represent the symmetry of the altermagnetic spin splitting of CrSb in terms of the real spherical harmonic \({{\mathcal{Y}}}_{4}^{-3}\hspace{0.04pt}(\theta ,\varphi )\).

The complex spherical harmonics can be defined in terms of the associated Legendre polynomials as \({Y}_{{\ell }}^{m}(\theta ,\varphi )={N}_{{\ell }m}{{\rm{e}}}^{{\rm{i}}m\varphi }{P}_{{\ell }}^{m}\hspace{0.03pt}(\cos \theta )\), where N_ℓm is a normalization factor, \({P}_{{\ell }}^{m}(x)\) is an associated Legendre polynomial, and \({Y}_{{\ell }}^{m}(\theta ,\varphi )\) is the complex spherical harmonic for ℓ ≥ 0 and m ∈ [−ℓ, ℓ]. The complex spherical harmonics are eigenfunctions of the total angular momentum operator \({\widehat{L}}^{2}\) and of the generator of rotations about the azimuthal axis \({\widehat{L}}_{z}\), spanning a complete orthonormal basis.

The complex spherical harmonics are defined up to a phase factor e^imφ, and so their magnitude does not change as a function of φ. Therefore, it is convenient to work in the basis of the real spherical harmonics, which have explicit φ dependence, when describing the symmetry of an unconventional magnetic order parameter. We can define the real spherical harmonics \({{\mathcal{Y}}}_{{\ell }}^{m}(\theta ,\varphi )\) in terms of linear combinations of complex harmonics according to

or, equivalently, in terms of the associated Legendre polynomials

Defining the real spherical harmonics this way means they form a complete set that spans the same basis as the complex spherical harmonics; however, importantly, they have well-defined varying magnitudes as a function of φ. This allows us to map the \({{\mathcal{Y}}}_{4}^{-3}\) real spherical harmonic to the g-wave symmetry profile of the altermagnetic order parameter in CrSb.

Contactless resistivity measurements

Contactless resistivity measurements were conducted using the proximity detector oscillator⁵³ technique in pulsed magnetic fields. A selected CrSb sample was mounted on a hand-wound planar coil of 15 turns, acting as the inductive component of the oscillator. The coil diameter was customized to match the sample width for optimal filling factor. A counter-wound outer coil enclosing the same area as the inner coil was added to compensate magnetic flux induced during the field pulse, minimizing background pickup.

As the applied magnetic field is swept, changes in the resistivity ρ and susceptibility χ_s of the sample lead to changes in the inductance of the oscillator and produce a shift in the resonant frequency of the oscillator, which can be described by

where η is the filling factor, d is the sample thickness, and μ_r = 1 + χ_s is the relative magnetic permeability⁵³. For a metallic material such as CrSb, eddy currents restrict the penetration of the radiofrequency field to a characteristic skin depth \(\delta =\sqrt{2\rho /({\mu }_{{\rm{r}}}{\mu }_{0}\omega )}\), where ω is the excitation frequency, such that the frequency response is dominated by changes in the resistivity ρ.

Proximity detector oscillator measurements reported in this study were performed in a 65-T pulsed magnet at the Dresden High Magnetic Field Laboratory in Dresden, Germany, following the methodology in ref. ⁵⁴. A customized ³He cryostat was fitted to the magnet, providing a base temperature of approximately 600 mK throughout the pulses. A raw resonant frequency of 25 MHz was achieved, which was fed into a heterodyne mixing circuit to down-convert the signal to about 10.5 MHz, which was subsequently acquired using a high-definition oscilloscope.

Quantum oscillatory components were analysed over a magnetic field range of 38–63 T using a LOESS background subtraction with an 8-T window and a second-order polynomial background subtraction. A quantum oscillation of frequency 0.8 kT was clearly resolved (Extended Data Fig. 7).

We note that, for sufficiently high magnetic fields, altermagnets have been predicted to exhibit certain distinguishing quantum oscillatory features, such as a distinct frequency splitting at a field-induced Lifshitz transition separating the up and down sheets⁵⁵. However, we measured up to a maximal field strength of 64 T (Extended Data Fig. 7) and observed no such signatures. This is probably due to the very high ordering temperature (and hence energy scale) of altermagnetism in CrSb, which remains robust up to these large field strengths.

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Author notes

1. These authors contributed equally: Mengmeng Long, Theodore I. Weinberger, Zheyu Wu

Authors and Affiliations

1. Cavendish Laboratory, University of Cambridge, Cambridge, UK

   Mengmeng Long, Theodore I. Weinberger, Zheyu Wu, Mads F. Hansen, Ran Tao, Mridul Shrestha, F. Malte Grosche & Alexander G. Eaton

2. National High Magnetic Field Laboratory, Tallahassee, FL, USA

   Dave Graf

3. Hochfeld-Magnetlabor Dresden (HLD-EMFL), Helmholtz-Zentrum Dresden-Rossendorf, Dresden, Germany

   Yurii Skourski

Contributions

M.L., M.F.H. and R.T. grew single-crystal CrSb specimens. M.L. characterized the samples. M.L., T.I.W., Z.W., M.S., D.G., Y.S. and A.G.E. performed quantum oscillation experiments. M.L., T.I.W., Z.W., F.M.G. and A.G.E. analysed and interpreted the data. T.I.W. performed DFT calculations. A.G.E. conceived the project. F.M.G. and A.G.E. supervised the project. T.I.W., F.M.G. and A.G.E. wrote the paper, with input from all co-authors.

Corresponding authors

Correspondence to Theodore I. Weinberger or Alexander G. Eaton.

Competing interests

The authors declare no competing interests.

Peer review information

Nature thanks Minghu Fang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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