Experimental evidence for Zeeman spin–orbit coupling in layered antiferromagnetic conductors

Most of solid-state spin physics arising from spin–orbit coupling, from fundamental phenomena to industrial applications, relies on symmetry-protected degeneracies. So does the Zeeman spin–orbit coupling, expected to manifest itself in a wide range of antiferromagnetic conductors. Yet, experimental proof of this phenomenon has been lacking. Here we demonstrate that the Néel state of the layered organic superconductor κ-(BETS)2FeBr4 shows no spin modulation of the Shubnikov–de Haas oscillations, contrary to its paramagnetic state. This is unambiguous evidence for the spin degeneracy of Landau levels, a direct manifestation of the Zeeman spin–orbit coupling. Likewise, we show that spin modulation is absent in electron-doped Nd1.85Ce0.15CuO4, which evidences the presence of Néel order in this cuprate superconductor even at optimal doping. Obtained on two very different materials, our results demonstrate the generic character of the Zeeman spin–orbit coupling.


INTRODUCTION
Spin-orbit coupling (SOC) in solids intertwines electron orbital motion with its spin, generating a variety of fundamental effects 1,2 . Commonly, SOC originates from the Pauli term H P ¼ h 4m 2 e σ Á p ∇VðrÞ in the electron Hamiltonian 3,4 , where h is the Planck constant, m e the free-electron mass, p the electron momentum, σ its spin, and V(r) its potential energy depending on the position. Remarkably, Néel order may give rise to SOC of an entirely different nature, via the Zeeman effect 5,6 : where μ B is the Bohr magneton, k = p/ℏ the electron wave vector, B the magnetic field, while g ∥ and g ⊥ define the g-tensor components with respect to the Néel axis. In a purely transverse field B ⊥ , a hidden symmetry of a Néel antiferromagnet protects double degeneracy of Bloch eigenstates at a special set of momenta in the Brillouin zone (BZ) 5,6 : at such momenta, g ⊥ must vanish. The scale of g ⊥ is set by g ∥ , which renders g ⊥ (k) substantially momentum dependent, and turns H so Z into a veritable SOC [5][6][7][8] . This coupling was predicted to produce unusual effects, such as spin degeneracy of Landau levels in a purely transverse field B ⊥ 9,10 and spin-flip transitions, induced by an AC electric rather than magnetic field 10 . Contrary to the textbook Pauli SOC, this mechanism does not require heavy elements. Being proportional to the applied magnetic field (and thus tunable!), it is bound only by the Néel temperature of the given material. In addition to its fundamental importance, this distinct SOC mechanism opens new possibilities for spin manipulation, much sought after in the current effort [11][12][13] to harness electron spin for future spintronic applications. While the Zeeman SOC mechanism may be relevant to a vast variety of antiferromagnetic (AF) conductors such as chromium, cuprates, iron pnictides, hexaborides, borocarbides, as well as organic and heavy fermion compounds 6 , it has not received an experimental confirmation yet.
Here we present experimental evidence for the spin degeneracy of Landau levels in two very different layered conductors, using Shubnikov-de Haas (SdH) oscillations as a sensitive tool for quantifying the Zeeman effect 14 . First, the organic superconductor κ-(BETS) 2 FeBr 4 (hereafter κ-BETS) 15 is employed for testing the theoretical predictions. The key features making this material a perfect model system for our purposes are (i) a simple quasi-twodimensional (quasi-2D) Fermi surface and (ii) the possibility of tuning between the AF and paramagnetic (PM) metallic states, both showing SdH oscillations, by a moderate magnetic field 15,16 . We find that, contrary to what happens in the PM state, the angular dependence of the SdH oscillations in the AF state of this compound is not modulated by the Zeeman splitting. We show that such a behavior is a natural consequence of commensurate Néel order giving rise to the Zeeman SOC in the form of Eq. (1).
Having established the presence of the Zeeman SOC in an AF metal, we utilize this effect for probing the electronic state of Nd 2 −x Ce x CuO 4 (NCCO), a prototypical example of electron-doped high-T c cuprate superconductors 17 . In these materials, superconductivity coexists with another symmetry-breaking phenomenon manifested in a Fermi-surface reconstruction as detected by angle-resolved photoemission spectroscopy (ARPES) [18][19][20][21] and SdH experiments [22][23][24][25] . The involvement of magnetism in this Fermi-surface reconstruction has been broadly debated [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] . Here we present detailed data on the SdH amplitude in optimally doped NCCO, tracing its variation over more than two orders of magnitude with changing the field orientation. The oscillation behavior is found to be very similar to that in κ-BETS. Given the crystal symmetry and the position of the relevant Fermi-surface pockets, this result is firm evidence for antiferromagnetism in NCCO. Our finding not only settles the controversy in electrondoped cuprate superconductors but also clearly demonstrates the generality of the Zeeman SOC mechanism.
Before presenting the experimental results, we recapitulate the effect of Zeeman splitting on quantum oscillations. The superposition of the oscillations coming from the conduction subbands with opposite spins results in the well-known spin-reduction factor in the oscillation amplitude 14 : ; here m e and m are, respectively, the free-electron mass and the effective cyclotron mass of the relevant carriers. We restrict our consideration to the first-harmonic oscillations, which is fully sufficient for the description of our experimental results. In most threedimensional (3D) metals, the dependence of R s on the field orientation is governed by the anisotropy of the cyclotron mass. At some field orientations, R s may vanish, and the oscillation amplitude becomes zero. This spin-zero effect carries information about the renormalization of the product gm relative to its freeelectron value 2m e . For 3D systems, this effect is obviously not universal. For example, in the simplest case of a spherical Fermi surface, R s possesses no angular dependence whatsoever, hence no spin zeros. By contrast, in quasi-2D metals with their vanishingly weak interlayer dispersion such as in layered organic and cuprate conductors, spin zeros are 42-47 a robust consequence of the monotonic increase of the cyclotron mass, m / 1= cos θ, with tilting the field by an angle θ away from the normal to the conducting layers.
In an AF metal, the g-factor may acquire a k dependence through the Zeeman SOC mechanism. It becomes particularly pronounced in the purely transverse geometry, i.e., for a magnetic field normal to the Néel axis. In this case, R s contains the factor g ? averaged over the cyclotron orbit, see Supplementary Note I for details. As a result, the spin-reduction factor in a layered AF metal takes the form: where m 0 ≡ m(θ = 0°). Often, the Fermi surface is centered at a point k * , where the equality g ⊥ (k * ) = 0 is protected by symmetry 6as it is for κ-BETS (see Supplementary Note II). Such a k * belongs to a line node g ⊥ (k) = 0 crossing the Fermi surface. Hence, g ⊥ (k) changes sign along the Fermi surface, and g ? in Eq. (2) vanishes by symmetry of g ⊥ (k). Consequently, the quantum-oscillation amplitude is predicted to have no spin zeros 9 . For pockets with Fermi wave vector k F well below the inverse AF coherence length 1/ξ, g ⊥ (k) can be described by the leading term of its expansion in k. For such pockets, the present result was obtained in refs. 9,10,48 . According to our estimates in the Supplementary Note III, both in κ-BETS and in optimally doped NCCO (x = 0.15), the product k F ξ considerably exceeds unity. Yet, the quasi-classical consideration above shows that for k F ξ > 1 the conclusion remains the same: g ? ¼ 0, see Supplementary Note IV for the explicit theory.
We emphasize that centering of the Fermi surface at a point k * with g ⊥ (k * ) = 0 such as a high-symmetry point of the magnetic BZ boundary 6is crucial for a vanishing of g ? . Otherwise, Zeeman SOC remains inert, as it does in AF CeIn 3 , whose d Fermi surface is centered at the Γ point (see Supplementary Note V and refs. [49][50][51], and in quasi-2D AF EuMnBi 2 , with its quartet of Dirac cones centered away from the magnetic BZ boundary 52,53 . With this, we turn to the experiment.

AF organic superconductor κ-(BETS) 2 FeBr 4
This is a quasi-2D metal with conducting layers of BETS donor molecules, sandwiched between insulating FeBr À 4 -anion layers 15 . The material has a centrosymmetric orthorhombic crystal structure (space group Pnma), with the ac plane along the layers. The Fermi surface consists of a weakly warped cylinder and two open sheets, separated from the cylinder by a small gap Δ 0 at the BZ boundary, as shown in Fig. 1 15,16,54 .
The magnetic properties of the compound are mainly governed by five localized 3d-electron spins per Fe 3+ ion in the insulating layers. Below T N ≈ 2.5 K, these S = 5/2 spins are ordered antiferromagnetically, with the unit cell doubling along the c axis and the staggered magnetization pointing along the a axis 15,55 . Above a critical magnetic field B c~2 −5 T, dependent on the field orientation, antiferromagnetism gives way to a saturated PM state 56 .
The SdH oscillations in the high-field PM state and in the Néel state are markedly different (see Fig. 1b). In the former, two dominant frequencies corresponding to a classical orbit α on the Fermi cylinder and to a large magnetic breakdown (MB) orbit β are found, in agreement with the predicted Fermi surface 16,54 .  [2][3][4][5] T and [12][13][14] T in the AF and PM state, respectively. c The BZ boundaries in the AF state with the wave vector Q AF = (π/c, 0) and in the PM state are shown by solid-black and dashed-black lines, respectively. The dotted-blue and solid-orange lines show, respectively, the original and reconstructed Fermi surfaces 16 . The shaded area in the corner of the magnetic BZ, separated from the rest of the Fermi surface by gaps Δ 0 and Δ AF , is the δ pocket responsible for the SdH oscillations in the AF state. The inset shows the function g ⊥ (k).
The oscillation amplitude exhibits spin zeros as a function of the field strength and orientation, which is fairly well described by a field-dependent spin-reduction factor R s (θ, B), with the g-factor g = 2.0 ± 0.2 in the presence of an exchange field B J ≈ −13 T, imposed by PM Fe 3+ ions on the conduction electrons 46,57 . In the Supplementary Note VI, we provide further details of the SdH oscillation studies on κ-BETS. Below B c , in the AF state, new, slow oscillations at the frequency F δ ≈ 62 T emerge, indicating a Fermi-surface reconstruction 16 . The latter is associated with the folding of the original Fermi surface into the magnetic BZ, and F δ is attributed to the new orbit δ, see Fig. 1c. This orbit emerges due to the gap Δ AF at the Fermisurface points, separated by the Néel wave vector (π/c, 0) 58 . Figure 2 shows examples of the field-dependent interlayer resistance of κ-BETS, recorded at T = 0.42 K, at different tilt angles θ. The field was rotated in the plane normal to the Néel axis (crystallographic a axis). In excellent agreement with previous reports 16,59 , slow oscillations with frequency F δ ¼ 61:2 T= cos θ are observed below B c , see inset in Fig. 2b. Thanks to the high crystal quality, even in this low-field region the oscillations can be traced over a wide angular range |θ| ≤ 70°.
The angular dependence of the δ-oscillation amplitude A δ is shown in Fig. 3. The amplitude was determined by fast Fourier transform (FFT) of the zero-mean oscillating magnetoresistance component normalized to the monotonic B-dependent background, in the field window between 3.0 and 4.2 T, so as to stay below B c (θ) for all field orientations. The lines in Fig. 3 are fits using the Lifshitz-Kosevich formula for the SdH amplitude 14 : where A 0 is a field-independent prefactor, B = 3.5 T (the midpoint of the FFT window in 1/B scale), m the effective cyclotron mass (m = 1.1m e at θ = 0°1 6 , growing as 1= cos θ with tilting the field as in other quasi-2D metals 60,61 ), K = 2π 2 k B / he, T = 0.42 K, T D the Dingle temperature, and R MB the MB factor. For κ-BETS, R MB takes the form , with two characteristic MB fields B 0 and B AF associated with the gaps Δ 0 and Δ AF , respectively. The Zeeman splitting effect is encapsulated in the spin factor R s (θ). In Eq. (1), the geometry of our experiment implies B ∥ = 0, thus in the Néel state R s (θ) takes the form of Eq. (2).
Excluding R s (θ), the other factors in Eq. (3) decrease monotonically with increasing θ. By contrast, R s (θ) in Eq. (2), generally, has an oscillating angular dependence. For g ? ¼ g ¼ 2:0 found in the PM state 46 , Eq. (2) yields two spin zeros, at θ ≈ 43°and 64°. Contrary to this, we observe no spin zeros but rather a monotonic decrease of A δ by over two orders of magnitude as the field is tilted away from θ = 0°to ±70°, i.e., in the entire angular range where we observe the oscillations. The different curves in Fig. 3 are our fits using Eq. (3) with A 0 and T D as fit parameters and different values of the g-factor. We used the MB field values B 0 = 20 T and B AF = 5 T. While the exact values of B 0 and B AF are unknown, they have virtually no effect on the fit quality, as we demonstrate in Supplementary Note VII. The best fit is achieved with g = 0, i.e., with an angle-independent spin factor R s = 1. The excellent agreement between the fit and the experimental data confirms the quasi-2D character of the electron conduction, with the 1= cos θ dependence of the cyclotron mass.
Comparison of the curves in Fig. 3 with the data rules out g ? > 0:2. Given the experimental error bars, we cannot exclude a nonzero g ? t 0:2, yet even such a small finite value would be in stark contrast with the textbook g = 2.0, found from the SdH oscillations in the high-field, PM state 46 . Below we argue that, in fact, g ? in the Néel state is exactly zero.

Optimally doped NCCO
This material has a body-centered tetragonal crystal structure (space group I4/mmm), where (001) conducting CuO 2 layers alternate with their insulating (Nd,Ce)O 2 counterparts 17 . Bandstructure calculations 62,63 predict a hole-like cylindrical Fermi surface, centered at the corner of the BZ. However, ARPES 18-21,64 reveals a reconstruction of this Fermi surface by a (π/a, π/a) order. Moreover, magnetic quantum oscillations [23][24][25] show that the Fermi surface remains reconstructed even in the overdoped regime, up to the critical doping x c (≈ 0.175 for NCCO), where the superconductivity vanishes 65 . The origin of this reconstruction remains unclear: while the (π/a, π/a) periodicity is compatible with the Néel order observed in strongly underdoped NCCO, coexistence of antiferromagnetism and superconductivity in electron-doped cuprates remains controversial. A number of neutron-scattering and muon-spin rotation studies [32][33][34][35] have detected short-range Néel fluctuations but no static order within the superconducting doping range. However, other neutron scattering 36,37 and magnetotransport [38][39][40] experiments have produced evidence of static or quasi-static AF order in superconducting samples at least up to optimal doping x opt . Alternative mechanisms of the Fermi-surface reconstruction have been proposed, including a d-density wave 28 , a charge-density wave 29 , or coexistent topological and fluctuating short-range AF orders 30,31 .
To shed light on the relevance of antiferromagnetism to the electronic ground state of superconducting NCCO, we have studied the field-orientation dependence of the SdH oscillations of the interlayer resistance in an optimally doped, x opt = 0.15, NCCO crystal. The overall magnetoresistance behavior is illustrated in Fig.  4a. At low fields, the sample is superconducting. Immediately above the θ-dependent superconducting critical field, the magnetoresistance displays a non-monotonic feature, which has already been reported for optimally doped NCCO in a magnetic field normal to the layers 22,66 . This anomaly correlates with an anomaly in the Hall resistance and has been associated with MB through the energy gap, created by the (π/a, π/a)-superlattice potential 65 . With increasing θ, the anomaly shifts to higher fields, consistently with the expected increase of the breakdown gap with tilting the field.
SdH oscillations develop above about 30 T. Figure 4b shows examples of the oscillatory component of the magnetoresistance, normalized to the field-dependent non-oscillatory background resistance R backg , determined by a low-order polynomial fit to the as-measured R(B) dependence. In our conditions, B ≲ 65 T, T = 2.5 K, the only discernible contribution to the oscillations comes from the hole-like pocket α of the reconstructed Fermi surface 22 . This pocket is centered at the reduced BZ boundary, as shown in the inset of Fig. 5. While MB creates large cyclotron orbits β with the area equal to that of the unreconstructed Fermi surface, even in fields of 60-65 T the fast β oscillations are more than two orders of magnitude weaker than the α oscillations 24,65 .
The oscillatory signal is plotted in Fig. 4b as a function of the out-of-plane field component B ? ¼ B cos θ. In these coordinates, the oscillation frequency remains constant, indicating that FðθÞ ¼ Fð0 Þ= cos θ and thus confirming the quasi-2D character of the conduction. In the inset, we show the respective FFTs plotted against the cos θ-scaled frequency. They exhibit a peak at F cos θ ¼ 294 T, in line with previous reports. The relatively large width of the FFT peaks is caused by the small number of oscillations in the field window [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64] T. This restrictive choice is dictated by the requirement that the SdH oscillations be resolved over the whole field window at all tilt angles up to θ ≈ 72°. In Supplementary Note VIII, we provide an additional analysis of the amplitude at fixed field values, confirming the FFT results.  Inset: The first quadrant of the BZ with the Fermi surface reconstructed by a superlattice potential with wave vector Q = (π/a, π/a). If this potential involves Néel order, the function g ⊥ (k) (red line in the inset) vanishes at the reduced BZ boundary (dashed line). The SdH oscillations are associated with the oval hole pocket α centered at (π/2a, π/2a) 22 . The error bars are defined as described in Supplementary Note VIII.
Furthermore, in Fig. 4b one can see that the phase of the oscillations is not inverted and stays constant in the studied angular range. This is fully in line with the absence of spin zeros, see Eq.
(2). The main panel of Fig. 5 presents the angular dependence of the oscillation amplitude (symbols), in a field rotated in the (ac) plane. The amplitude was determined by FFT of the data taken at T = 2.5 K in the field window 45 T ≤ B ≤ 64 T. The lines in the figure are fits using Eq. (3), for different g-factors. The fits were performed using the MB factor R MB ¼ ½1 À expðÀB 0 =BÞ 24,65 , the reported values for the MB field B 0 = 12.5 T, and the effective cyclotron mass m(θ = 0°) = 1.05m 0 65 , while taking into account the 1= cos θ angular dependence of both B 0 and m. The prefactor A 0 and Dingle temperature T D were used as fit parameters, yielding T D = (12.6 ± 1) K, close to the value found in the earlier experiment 65 . Note that, contrary to the hole-doped cuprate YBa 2 Cu 3 O 7−x , where the analysis of earlier experiments 47,67 was complicated by the bilayer splitting of the Fermi cylinder 47 , the single-layer structure of NCCO poses no such difficulty.
Similar to κ-BETS, the oscillation amplitude in NCCO decreases by a factor of about 300, with no sign of spin zeros as the field is tilted from θ = 0°to 72. 5°. Again, this behavior is incompatible with the textbook value g = 2, which would have produced two spin zeros in the interval 0°≤ θ ≤ 70°, see the green dash-dotted line in Fig. 5. A reduction of the g-factor to 1.0 would shift the first spin zero to about 72°, near the edge of our range (blue dotted line in Fig. 5). However, this would simultaneously suppress the amplitude at small θ by a factor of ten, contrary to our observations. All in all, our data rule out a constant g > 0.2.

DISCUSSION
In both materials, our data impose on the effective g-factor an upper bound of 0.2. At first sight, one could simply view this as a suppression of the effective g to a small nonzero value. However, below we argue that, in fact, our findings imply g ? ¼ 0 and point to the importance of the Zeeman SOC in both materials. The quasi-2D character of electron transport is crucial for this conclusion: as mentioned above, in three dimensions, the mere absence of spin zeros imposes no bounds on the g-factor.
In κ-BETS, the interplay between the crystal symmetry and the periodicity of the Néel state 5,6,48 guarantees that g ⊥ (k) vanishes on the entire line k c = π/2c and is an odd function of k c − π/2c, see the inset of Fig. 1c and Supplementary Figure 2 in Supplementary Note II. The δ orbit is centered on the line k c = π/2c; hence g ? in Eq. (2) vanishes, implying the absence of spin zeros, in agreement with our data. At the same time, quantum oscillations in the PM phase clearly reveal the Zeeman splitting of Landau levels with g = 2.0 46 . Therefore, we conclude that g ? ¼ 0 is an intrinsic property of the Néel state.
In optimally doped NCCO, as already mentioned, the presence of a (quasi)static Néel order has been a subject of debate. However, if indeed present, such an order leads to g ⊥ (k) = 0 at the entire magnetic BZ boundary (see Supplementary Note II). For the hole pockets, producing the observed F α ≃ 300 T oscillations, g ? ¼ 0 by symmetry of g ⊥ (k) (see inset of Fig. 5 and Supplementary Fig.  3). Such an interpretation requires that the relevant AF fluctuations have frequencies below the cyclotron frequency in our experiment, ν c~1 0 12 Hz at 50 T.
Finally, we address mechanismsother than Zeeman SOC of Eq. (1)that may also lead to the absence of spin zeros. While such mechanisms do exist, we will show that none of them is relevant to the materials of our interest.
When looking for alternative explanations to our experimental findings, let us recall that, generally, the effective g-factor may depend on the field orientation. This dependence may happen to compensate that of the quasi-2D cyclotron mass, m 0 = cos θ, in the expression (2) for the spin-reduction factor R s , and render the latter nearly isotropic, with no spin zeros. Obviously, such a compensation requires a strong Ising anisotropy [g(θ = 0°) ≫ g(θ = 90°)]as found, for instance, in the heavy fermion compound URu 2 Si 2 , with the values g c = 2.65 ± 0.05 and g ab = 0.0 ± 0.1 for the field along and normal to the c axis, respectively 68,69 . However, this scenario is irrelevant to both materials of our interest: In κ-BETS, a nearly isotropic g-factor, close to the free-electron value 2.0, was revealed by a study of spin zeros in the PM state 46 . In NCCO, the conduction electron g-factor may acquire anisotropy via an exchange coupling to Nd 3+ local moments. However, the low-temperature magnetic susceptibility of Nd 3+ in the basal plane is some five times larger than along the c axis 70,71 . Therefore, the coupling to Nd 3+ may only increase g ab relative to g c , and thereby only enhance the angular dependence of R s rather than cancel it out. Thus we are led to rule out a g-factor anisotropy of crystal-field origin as a possible reason behind the absence of spin zeros in our experiments.
As follows from Eq. (2), another possible reason for the absence of spin zeros is a strong reduction of the ratio gm/2m e . However, while some renormalization of this ratio in metals is commonplace, its dramatic suppression (let alone nullification) is, in fact, exceptional. First, a vanishing mass would contradict m/m e ≥1, experimentally found in both materials at hand. On the other hand, a Landau Fermi-liquid renormalization 14 g → g/(1 + G 0 ) would require a colossal Fermi-liquid parameter G 0 ≥ 10, for which there is no evidence in NCCO, let alone κ-BETS with its already mentioned g ≈ 2 in the PM state 46 .
A sufficient difference of the quantum-oscillation amplitudes and/or cyclotron masses for spin-up and spin-down Fermi surfaces might also lead to the absence of spin zeros. Some heavy fermion compounds show strong spin polarization in magnetic field, concomitant with a substantial field-induced difference of the cyclotron masses of the two spin-split subbands 72,73 . As a result, for quantum oscillations in such materials, one spin amplitude considerably exceeds the other, and no spin zeros are expected. Note that this physics requires the presence of a very narrow conduction band, in addition to a broad one. In heavy fermion compounds, such a band arises from the f electrons but is absent in both materials of our interest.
Another extreme example is given by the single fully polarized band in a ferromagnetic metal, where only one spin orientation is present, and spin zeros are obviously absent. Yet, no sign of ferromagnetism or metamagnetism has been seen in either NCCO or κ-BETS. Moreover, in κ-BETS, the spin-zero effect has been observed in the PM state 46 , indicating that the quantumoscillation amplitudes of the two spin-split subbands are comparable. However, for NCCO one may inquire whether spin polarization could render interlayer tunneling amplitudes for spinup and spin-down different enough to lose spin zeros, especially in view of an extra contribution of Nd 3+ spins in the insulating layers to spin polarization. In Supplementary Note IX, we show that this is not the case.
Thus we are led to conclude that the absence of spin zeros in the AF κ-BETS and in optimally doped NCCO is indeed a manifestation of the Zeeman SOC. Our explanation relies only on the symmetry of the Néel state and the location of the carrier pockets, while being insensitive to the mechanism of the antiferromagnetism or to the orbital makeup of the relevant bands.