Magnetic crystalline-symmetry-protected axion electrodynamics and field-tunable unpinned Dirac cones in EuIn2As2

Knowledge of magnetic symmetry is vital for exploiting nontrivial surface states of magnetic topological materials. EuIn2As2 is an excellent example, as it is predicted to have collinear antiferromagnetic order where the magnetic moment direction determines either a topological-crystalline-insulator phase supporting axion electrodynamics or a higher-order-topological-insulator phase with chiral hinge states. Here, we use neutron diffraction, symmetry analysis, and density functional theory results to demonstrate that EuIn2As2 actually exhibits low-symmetry helical antiferromagnetic order which makes it a stoichiometric magnetic topological-crystalline axion insulator protected by the combination of a 180∘ rotation and time-reversal symmetries: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C}_{2}\times {\mathcal{T}}={2}^{\prime}$$\end{document}C2×T=2′. Surfaces protected by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2}^{\prime}$$\end{document}2′ are expected to have an exotic gapless Dirac cone which is unpinned to specific crystal momenta. All other surfaces have gapped Dirac cones and exhibit half-integer quantum anomalous Hall conductivity. We predict that the direction of a modest applied magnetic field of μ0H ≈ 1 to 2 T can tune between gapless and gapped surface states.

. The Eu crystallographic sites in magnetic space group C2 2 2 1 (No. 20.33). Positions are given in the setting of the parent space group P 6 3 /mmc (No. 194) with lattice parameters (a, b, 3c). m i are components of the ordered magnetic moment µ. This table was created using the Bilbao Crystallography Server [1].

Atoms
Coordinates the known chemical structure of the material [3,4] was performed allowing the scale factor and atomic positions to vary. The site occupations and thermal parameters could not be refined. The magnetic structure was then refined using the 6 K absorption-corrected data corresponding to the magnetic order. This refinement used the magnetic symmetry files created on the Bilbao crystallography server [1] for C2 2 2 1 and the parameters found from the refinement to the 30 K data. To account for the localized nature of the Eu 2+ magnetic moments, a refinement constraint of equal values of the total ordered magnetic moment µ for both sites was used. Supplementary Figure 4 illustrates the refinement which returned a goodness-of-fit value of R F = 7.50. The determined magnetic structure at 6 K is the brokenhelix order diagrammed in Fig. 1d as well as Supplementary Figure 5 with turn angles of φ rr = −80(2) • and φ rb = 130(1) • , and µ = 5.9(2) µ B /Eu lying within the ab plane.
This result is confirmed by a refinement to T = 6 K data collected on the CORELLI instrument for a different sample. Data corresponding to a reduced incident neutron energy bandwidth centered at E = 50 meV (1.54Å) and spanning 45-55 meV were used in order to accurately perform a correction for neutron absorption using mag2pol [2]. The absorption correction and refinement procedures employed were similar to these previously described for the TRIAX data analysis. Sets of 18 nuclear (22 K) and 108 magnetic (6 K) independent reflections were used. We obtained turn angles of φ rr = −68(2) • and φ rb = 124(1) • , µ =   6.1(2) µ B /Eu and a goodness-of-fit value of R F = 17.2. Figure 6 illustrates the quality of this refinement.
In the following, we detail our determination of the T N2 <T≤T N1 magnetic order using TRIAX and CORELLI data.
A reliable refinement of the TRIAX data for T N2 < T ≤ T N1 could not be completed.
This is due to the rather weak magnetic intensities at these temperatures because of the smaller value of µ and the strong neutron absorption. Nevertheless, data in     The line indicates the refinement, which has a goodness of fit of R F = 7.5.
Supplementary Table III. Integrated intensities for Bragg peaks recorded on TRIAX at T = 6 K. The correction for neutron absorption was applied using mag2pol [2] as described in Ref. [5]. When present, nuclear contributions to the Bragg peaks were subtracted using the 30 K data given in Table. II.
(h, 0, l) Integrated Intensity Absorption Corr. Intensity To complement these results, we made a refinement for the T N2 < T ≤ T N1 magnetic phase using MSG P 6 1 2 2 and data for 20 independent Bragg peaks corresponding to τ 1 taken with CORELLI at 16.3 K. Similar to above, data corresponding to a reduced incident neutron energy bandwidth centered at E = 50 meV (1.54Å) and spanning 45-55 meV were used in order to accurately perform a correction for neutron absorption using mag2pol [2].
In agreement with our TRIAX estimation, the refinement returns µ = 2.5(5) µ B /Eu with In the following, we detail our determination of the magnetic phases' temperature evolution using CORELLI data.
The CORELLI data also allowed us to precisely and efficiently visualize the appearance of magnetic Bragg peaks and their positions upon cooling down to T = 6 K. Figure 2b displays data from magnetic order parameter measurements and Next, Supplementary Fig. 10d shows that the Fourier components of the hyperfine field found by 151 Eu Mössbauer spectroscopy measurements also show evidence for the two mag- A Curie-Weiss fit to high-temperature χ(T ) data is shown in Supplementary Figure 11a and yields an effective magnetic moment of µ eff = 8.05 (7)  In order to determine the hyperfine field B hf , the spectra in Supplementary Figure 12 are fit as follows. Spectra for T 10 K and T 18 K are fit to a sum of Lorentzian lineshapes with the positions and intensities derived from a full solution to the nuclear Hamiltonian [9]. However, spectra taken between 10 K and 18 K are fit using a model that derives a distribution of hyperfine fields from an (assumed) incommensurate magnetic structure with a sinusoidally modulated value for the ordered magnetic moment µ [10,11]. We next describe this distribution model.
If we denote the antiferromagnetic (AF) propagation vector as k (instead of τ ), assume that the modulation in µ along the direction of the propagation vector can be written in terms of its Fourier components, and that the hyperfine field is a linear function of µ at any given site, then the variation of B hf with distance x along k can be written as: [10] B hf (kx) = Bk 0 + n l=0 Bk 2l+1 sin[(2l + 1)kx] . (1) Bk n are the odd Fourier coefficients of the field modulation and kx is a position in reciprocal  Supplementary Figure 12 shows that increasing the temperature not only leads to a gradual reduction in B hf , but also to a clear increase in the linewidth. This broadening reflects a distribution of environments for the Eu, and may arise from either dynamic effects (e.g. slow paramagnetic relaxation) or from a static distribution of hyperfine fields. The former is inconsistent with both the observed evolution of the spectral shapes and with the continued observation of well-defined magnetic Bragg peaks in the neutron diffraction data.
This leaves a static distribution of hyperfine fields as the source of the line broadening.
As Eu 2+ is the only magnetic species present, a distribution in B hf necessarily reflects a distribution of moment magnitudes µ. We therefore turn to the distribution model given by Supplementary Equation (1)  To summarize, our Mössbauer measurements indicate that the initial AF order that develops on cooling through 17.5 K is dominated by an incommensurate, sine-like, modulation of µ along the direction of the AF propagation vector, but a significant uniform contribution is likely also present. On further cooling, both contributions grow, but at 16 K the modulation starts to decline and the uniform term quickly dominates, until by 9 K only order with a uniform (fixed-size) value of µ remains. The two helical magnetically ordered phases are described by a magnetic unit cell that is three times larger alongĉ than the crystallographic unit cell, c mag = 3c. As a result, the two additional translationsĉ and 2ĉ, which areĉ mag /3 and 2ĉ mag /3 with respect to the magnetic unit cell, need to be considered. Combined with the crystallographic SG, this leads to additional symmetry operations which are non-symmorphic with respect to the magnetic unit cell. Moreover, while T by itself is no longer a symmetry element of the magnetically ordered phase, T combined with some spatial operations of the crystallographic SG are symmetry elements of the magnetic phase, and the MSG is of black-white type. The MSG is defined by the subset of operations generated by the generators of the paramagnetic MSG P 6 3 /mmc1 combined with the translations {ĉ =ĉ mag /3, 2ĉ =ĉ mag /3} which leave the magnetic order invariant.
We first analyze the symmetries of the pure 60 • -helix phase that appears upon cooling through T N1 = 17.6(2) K as shown in Fig. 2b in the main text. The ordered magnetic moments µ are ferromagnetically aligned in each Eu layer. As shown in Supplementary where 1 is the identity element. Note that a helix with opposite helicity (i.e. clockwise rotation aroundĉ) is described by MSG P , which is a symmetry element for the 60 • -helix order, is lost in the broken-helix state. The broken-helix phase is described by MSG C2 2 2 1 (No. 20.33), which is generated by the element set Here, we have set the origin to lie in Eu layer 3 (blue). Since the MSG contains the element We find that gapless surface Dirac cones, which can be shifted away from the center of the surface Brillouin zone, occur on the (110) and (110)   Several magnetic phase transitions occur in EuIn 2 As 2 upon cooling, which in Landau theory for second-order phase transitions may be related to transitions from a symmetry group to one of its subgroups. Starting from the high-temperature paramagnetic phase, the compound undergoes a transition to pure 60 • -helix order below T N1 = 17.6(2) K, followed by a second transition to broken-helix order below T N2 = 16. The other two paths in Supplementary Figure 18 correspond  Also, note that the broken-helix state also exhibits a handedness, yet both right-and lefthanded chiralities are described by MSG C2 2 2 1 .
Polarizing the Eu moments along a specific direction via applying an external magnetic field offers tuning of the topological properties of the material. As shown in Fig. 4b and Supplementary Figure 11b, at T = 5 K a magnetic field of µ 0 H ≈ 1 to 2 T aligns the Eu magnetic moments parallel to the field, which corresponds to a field-polarized state. Below, we study the symmetry and topological consequences of a field-polarized state by considering three orientations for the external field.
We first consider a magnetic field along the high-symmetry direction [110], which is the direction that the blue Eu ordered magnetic moments lie along.
The combinations of the generators in Supplementary Equation (6)   Note that the surfaces hosting exotic states are the same as those for pure 60 • -helix order. The difference here is that the Dirac cones are no longer completely unpinned, but are constrained to occur alongΓ-X because of the additional mirror symmetry m [001] 00 c 2 resulting from the combination of inversion I and the screw operation C 2,[001] 00 c 2 . It becomes evident from the previous analysis that we can use an external magnetic field to either induce or destroy topological states on different surfaces of EuIn 2 As 2 . For instance, applying a magnetic field along [110] inside the pure 60 • -helix phase gaps out Dirac cones on (100), (010), (210) and (120) surfaces and pins the gapless Dirac cones on (110) and (110) to theΓ-Z direction in the surface Brillouin zone. If, on the other hand, a field parallel to [001] is applied in the pure 60 • -helix phase, the Dirac cones remain gapless on all six surfaces but are now all constrained to lie along theΓ-X direction.
In Supplementary Table IV, we show the symmetry-based indicators X BS of topology for the magnetic helical phases and the field-polarized phases. While symmetry indicators are absent for the MSGs describing the two helical phases, since X BS = 1 means that there exist