Interplay of Dirac electrons and magnetism in CaMnBi2 and SrMnBi2

Dirac materials exhibit intriguing low-energy carrier dynamics that offer a fertile ground for novel physics discovery. Of particular interest is the interplay of Dirac carriers with other quantum phenomena such as magnetism. Here we report on a two-magnon Raman scattering study of AMnBi2 (A=Ca, Sr), a prototypical magnetic Dirac system comprising alternating Dirac carrier and magnetic layers. We present the first accurate determination of the exchange energies in these compounds and, by comparison with the reference compound BaMn2Bi2, we show that the Dirac carrier layers in AMnBi2 significantly enhance the exchange coupling between the magnetic layers, which in turn drives a charge-gap opening along the Dirac locus. Our findings break new grounds in unveiling the fundamental physics of magnetic Dirac materials, which offer a novel platform for probing a distinct type of spin–Fermion interaction. The results also hold great promise for applications in magnetic Dirac devices.

The standard theory of magnetic Raman scattering is based on the Elliott-Fleury-London theory 1 . The Raman scattering operator is given by Here is the spin exchange interaction between the local moments at site i and site j, � and � are unit polarization vectors of the incoming and scattering light, respectively, and ̂ is the vector connecting site i and site j. The Raman cross section at zero temperature is given by the imaginary part of the correlation function ( ) = − ∫ d 〈T � + ( ) � (0)〉 0 , where ⟨...⟩0 represents the quantum mechanical average over the ground state, and Tt is the time ordering operator.
We use the spin-wave approach within the framework of perturbative theory to calculate Raman spectra 2 . By introducing the Holstein-Primakoff transformation, we express spins in A (spin up) and B (spin down) sublattices in terms of H-P boson operators and . The system Hamiltonian H = is expanded in powers of 1/S as: Here 0 is a constant classical energy, � 0 contains quadratic terms of magnons and represents the linear spin-wave (LSW) correction to the classical energy, and � 1 contains magnon quartic terms, representing two-body magnon-magnon interactions. The higher order terms are ignored in the present treatment.
To calculate the correlation function ( ), we first apply Fourier and Bogoliubov transformations to diagonalize the quadratic LSW part � 0 in terms of Bogoliubov magnons α and β − . The � 1 part then contains Bogoliubov magnons in quartic order, and the magnon-pair-scattering term α + β − + β − ′ α ′ in � 1 is treated within the ladder approximation. The interaction vertex is reserved to the lowest (1/S) 0 order, and the ladder diagrams are summed up exactly.  Figure 1). In the linear spin-wave theory, these frequencies take the following analytical form: which holds when Jc<4J2 and < , and these constraints for both cases are valid in the cases studied in the present work. From the experimentally determined frequencies, we can extract the three exchange couplings J1, J2 and Jc using the above formulas.

Supplementary Note 3: Error estimation of the exchange energies
The characteristic frequencies can be expressed as = ( ) (11) The experimentally determined frequencies have errors around their averages = � + (12) The corresponding exchange energies can then be written as = ̅ + , where ̅ is determined via the equation � = ( ̅ ), and is obtained through the expansion where � � | = ̅ is the Jacobi's determinant, which leads to Supplementary Note 4: Effects of possible spin anisotropy on the exchange energies in SrMnBi2 and CaMnBi2 In Sr(Ca)MnBi2 materials Dirac carriers in Ca(Sr)Bi layers are subjected to SOC effect which may introduce spin anisotropy in the exchange couplings between Mn ions. To take into account the spin anisotropy effect we consider the following model where η is the parameter describing the anisotropy, and < 1 ensures that spins are aligned along the z direction.
To determine the parameters J1, J2, Jc and η, we make use of an additional characteristic point in the measured Raman spectra, namely the absorption edge in the 1 channel. Following the same procedure described above, we find that compared to the isotropic model the difference in the exchanges energies caused by the anisotropy is less than 10%, which indicates that the results obtained using the isotropic model are quite reasonable.

Supplementary Note 5: Effects of single-ion anisotropy in CaMnBi2, SrMnBi2 and BaMn2Bi2
It has been reported in inelastic neutron scattering experiments that in BaMn2Bi2 materials there is a spin gap of 16 meV 4 . The magnetic excitations can be well fit to a Heisenberg model plus single-ion anisotropy terms 4 where > 0 ensures that spins are aligned along the z direction.
We find that all characteristic frequencies will be shifted in presence of finite D terms. However, the relation for G-type AFM and We can see that in presence of D terms, the exchange interactions are slightly changed. The interchange coupling SJc becomes slightly smaller in presence of D, but the difference is less than 10%. It is not surprising that D terms can generate a relatively big spin gap but do not strongly affect the exchange interaction results extracted from Raman experiments, since D terms mainly affect low energy magnetic excitations, but Raman techniques mainly probe high energy physics.

Supplementary Note 6: Determination of characteristic frequencies and their errors
The spectra for BaMn2Bi2 and SrMnBi2 have a good signal-to-noise level, and the frequencies can be directly read out from the raw spectra (ω2 in BaMn2Bi2 and ω1 in SrMnBi2) or obtained from taking the first derivative (ω1 in BaMn2Bi2 and ω2 in SrMnBi2). For CaMnBi2, one must be more careful as the original polarized spectra have a higher noise level, making it hard to accurately determine the characteristic spectral points. On the other hand, spin-wave calculations indicate that both polarized spectra (A1g and B1g) have exactly the same characteristic points. This means that the combined A1g and B1g spectra, i.e., the unpolarized data, possess the same characteristic spectral points. This enables an alternative scenario, where we have collected many unpolarized spectra at 10 K with a much better signal-to-noise ratio, as shown in Supplementary Figure 2. Using the unpolarized spectra (A1g+B1g), one can easily determine the characteristic frequencies and their errors, in the same way as done above for BaMn2Bi2 and SrMnBi2. We have examined the validity of the method using SrMnBi2, in which there is little difference between the parameters derived from the polarized spectra and the unpolarized ones, respectively (Supplementary Note 7).
Then we can extract the frequency parameters and quantitatively estimate the associated errors in the following procedure (see Supplementary Figure 2). 1) A general linear fitting was made for a linear region selected from the raw spectra or the first-derived data. This gives the standard deviations in intensity (or its derivative); 2) For ω1 and ω2 associated with the local maximums/minimums of the raw spectra or the first-derivatives, we can identify a region which starts from the maximums/minimums and vertically extends by double standard deviations (the heights of the red dashed error boxes). All the data points in the region are possible maximums/minimums. This simply fixes the corresponding deviations in frequency (the widths of the error boxes) and provides the standard errors; 3) For ω3 not associated with a local extreme, we first determined the baselines (black lines) by fitting the background at the high-frequency end. The left bound of the error box is reached when the positive intensity deviations from a baseline begin to exceed the standard deviations estimated in the first step. Similarly the right bound is defined at the position where the intensity deviations approach the standard ones. This gives the frequency parameter ω3 and the associated errors. The characteristic frequencies obtained from the experimental spectra strictly following the above procedure and the extracted exchange parameters with the error bars, are listed in Supplementary Table 1.

Supplementary Note 7: Comparison of the parameters extracted from the polarized and unpolarized spectra in SrMnBi2
Following the procedure described in Supplementary Note 6, we can also obtain the frequency parameters in SrMnBi2 with the unpolarized spectra (see Supplementary Figure 3). And the obtained characteristic frequencies and the extracted exchange parameters from the polarized and unpolarized spectra, are listed in Supplementary Table 2 for comparison. There is little difference between both cases. This demonstrates that the unpolarized spectra work well as the polarized ones in obtaining the characteristic frequency points.

Supplementary Note 8: Electronic band structures and Dirac points in SrMnBi2 and CaMnBi2
We consider the spin-fermion Hamiltonian introduced in the main text The first term is a two-orbital tight-binding model for the itinerant electrons in Bi 6px and 6py orbits (in the Sr(Ca)Bi layer). The second term contains the atomic spin-orbit coupling. i±z � refers to the local moment of Mn in layers above or below the Bi site i. For simplicity, we do not consider the influence of the Bi 6pz orbit. Since the observed magnetic moments in these materials are about 4 μB per Mn, we treat Si' as classical spins. We take the i ′ j ′ H values obtained from the Raman measurements, so that the ground state of the model has either a G-AFM (SrMnBi2) or a C-AFM (CaMnBi2) order. We then treat the effects of the AFM order on the band structure of the itinerant electrons at the mean-field level. Within above approximations, the Mn local moments serve as local magnetic fields that couple to the itinerant electrons and modify their dispersion. The Hamiltonian is then written as H ≈HTB+ HS, and = 〈 〉 is the sublattice magnetic moment for n = A, B sublattice. For SrMnBi2, the magnetic order is G-AFM, where the Mn ions in upper and lower layers belong to the A and B sublattices, respectively, and at the mean-field level, the effect from the two layers cancels out exactly, hence H ≈HTB. But for CaMnBi2, the C-AFM order induces an uncompensated magnetic field, which acts as a mass term since it has different signs on the two sublattices.
We rewrite HTB into a matrix form 3 : where Hso and Hnn' are 2×2 matrices. The spin-orbit coupling takes the form where is the coupling constant. In our calculations, we consider two cases, = 0, and = 0.6 eV. 3 For the hopping integrals, we assume that the dominant terms are the intraorbital ones, and we neglect the interorbital hopping. We then obtain the following hopping matrices for SrMnBi2 and CaMnBi2: Here the hopping parameters and the chemical potential µ are determined by fitting to the DFT band structure, and their values are summarized in Supplementary Table 4. Note that due to the buckling of the Sr cations, ≠ for general k. The band structure can be obtained by diagonalizing the mean-field Hamiltonian. Without the spin-orbit coupling, for SrMnBi2, the energy in each band reads 1(2),± = 2 2 cos cos ± � 4 2 sin 2 sin 2 + 4� 1 ( ) cos + 1 ( ) cos � 2 (27) By requiring � 4 2 sin 2 sin 2 + 4� 1 ( ) cos + 1 ( ) cos � 2 = 0, we obtain four Dirac points. For 1 > 1 , they are located at = 0, = ±arccos (− 1 1 ⁄ ), and = 0, = ±arccos (− 1 1 ⁄ ). Focusing on one Dirac point = 0, = ±arccos (− 1 1 ⁄ ), the dispersion is very anisotropic, as shown in Fig. 4(a) in the main text. Turning on the spin-orbit coupling pushes apart the Dirac points as shown in Fig. 4(b).
Following the same procedure, we obtain the electronic band structure for CaMnBi2. Without the spin-orbit coupling, 1(2),± = 2 2 cos cos ± �̃2 +4� 1 ( ) cos + 1 ( ) cos � 2 (28) where ̃= | | is the effective magnetic field to itinerant electrons due to the C-AFM order. If we neglect the magnetic coupling, the band structure has continuous Dirac points along the lines = ±arccos(− 1 1 ⁄ cos ) and = ±arccos(− 1 1 ⁄ cos ). The magnetic field term ̃ acts as a mass term and opens a gap between the upper and lower branches of the Dirac bands. Note that a finite spin-orbit coupling may also opens a gap, but it is much smaller compared to the one associated with the AFM order (see Fig.4(c)-(d), Supplementary Figure 4).