Magnetic structure and excitations of the topological semimetal YbMnBi 2
Jian-Rui Soh, Henrik Jacobsen, Bachir Ouladdiaf, Alexandre Ivanov, Andrea Piovano, Tim Tejsner, Zili Feng, Hongyuan Wang, Hao Su, Yanfeng Guo, Youguo Shi, Andrew T. Boothroyd
MMagnetic structure and excitations of the topological semimetal YbMnBi Jian-Rui Soh, Henrik Jacobsen, Bachir Ouladdiaf, Alexandre Ivanov, Andrea Piovano, Tim Tejsner,
2, 3
Zili Feng, Hongyuan Wang,
5, 6
Hao Su, Yanfeng Guo, Youguo Shi, and Andrew T. Boothroyd ∗ Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK Institut Laue-Langevin, 6 rue Jules Horowitz, 38042 Grenoble Cedex 9, France Nanoscience Center, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China University of Chinese Academy of Sciences, Beijing 100049, China (Dated: August 15, 2019)We investigated the magnetic structure and dynamics of YbMnBi , with elastic and inelastic neu-tron scattering, to shed light on the topological nature of the charge carriers in the antiferromagneticphase. We confirm C-type antiferromagnetic ordering of the Mn spins below T N = 290 K, and de-termine that the spins point along the c -axis to within about 3 ◦ . The observed magnon spectrumcan be described very well by the same effective spin Hamiltonian as was used previously to modelthe magnon spectrum of CaMnBi . Our results show conclusively that the creation of Weyl nodesin YbMnBi by the time-reversal-symmetry breaking mechanism can be excluded in the bulk. PACS numbers: 75.25.-j, 75.30.Ds, 75.30.Gw, 74.70.Xa
I. INTRODUCTION
Dirac and Weyl materials are semimetals whose va-lence and conduction bands have a linear dispersion inthe vicinity of the Fermi energy . These gapless bandcrossings, which are protected by topology or crystallinesymmetries, can give rise to massless quasi–particle exci-tations which can be described by the relativistic Dirac orWeyl equations. Materials that host such fermions pos-sess a range of desirable physical properties: exception-ally high electrical and thermal conductivities, immunityto disorder and ballistic electronic transport .Weyl semimetals (WSMs) can occur in crystals withbroken spatial inversion symmetry (IS), broken time-reversal symmetry (TRS), or both. Examples of the firsttype (with broken IS only) were found in 2015 , butrealizations of WSMs with broken TRS are still rare .Recently, the layered AFM YbMnBi was proposed as apotential candidate . The evidence from angle-resolvedphotoemission spectroscopy (ARPES) is quite convinc-ing , and there is also some support from optics .The tetragonal unit cell of YbMnBi , which can be de-scribed by the P /nmm space group (No. 129), includesalternating Bi square layers that host the possible Weylfermions, and MnBi tetrahedral layers which con-tain magnetic moments on the Mn atoms [See Fig. 1(a)].In the antiferromagnetically (AFM) ordered phase, be-low T N = 290 K, neighbouring Mn spins are reported tobe antiparallel within the ab plane, but crucially, they areferromagnetically stacked along the c –axis . Thismeans that magnetic coupling to the Bi conduction statesis allowed at the mean–field level, which can lead to bandsplitting.In Ref. 10, it was argued that creation of Weyl pointsby TRS breaking in YbMnBi requires a ∼ ◦ cantingof the Mn moments away from the c -axis. If present, this canting would generate a net ferromagnetic componentin the ab -plane of YbMnBi , and would account for theWeyl nodes and arcs observed in the ARPES data. Sucha small deviation in the moment direction from the c -axiswould not have been discernible in the (100) magneticpeak studied in the previous neutron diffraction measure-ments , so the possibility that YbMnBi might bea WSM by this mechanism remains to tested.Moreover, if the AFM order of manganese createsWeyl fermions, which then dominate the electronic trans-port , then these quasiparticle excitations could playbe expected to play some role in the exchange couplingbetween Mn moments which could in turn influence themagnon spectrum. As the magnetic order is key to thebehavior of YbMnBi as a topological material, measure-ments of the magnon spectrum, and the exchange param-eters derived from it, could provide additional informa-tion on the presence of Weyl fermions near the Fermienergy.In light of this, we set out in this study, (i) to searchfor evidence of a canted magnetic structure by neutrondiffraction, and (ii) to investigate the magnon spectrumin the AFM phase of YbMnBi through inelastic neu-tron scattering. To achieve the required sensitivity tothe predicted ferromagnetic component of the proposedcanted magnetic structure, we performed careful mea-surements of the weak (00 l ) nuclear reflections. Further-more, to identify any anomalies in the magnetic exchangebetween Mn moments associated with the presence ofWeyl fermions, we compare the observed magnon spec-trum with that of Dirac semimetal CaMnBi , which isisostructural to YbMnBi . We demonstrate that the Mnsublattice in YbMnBi has C-type AFM ordering below T N = 290 K, with the moments aligned along the c -axis towithin 3 ◦ (at 95% confidence level). Moreover, we find noevidence from the magnon spectrum for anomalous mag- a r X i v : . [ c ond - m a t . s t r- e l ] A ug FIG. 1. (a) The unit cell of YbMnBi for the space group P /nmm (No. 129). The proposed Weyl fermions are con-tained in the Bi square net in the center of the unit cell.The magnetic exchange between the ab -plane nearest neigh-bor ( J ), ab -plane next-nearest neighbor ( J ), and c -axis near-est neighbor ( J c ) Mn ions were used in the linear spin-wave model to describe the magnon spectrum. (b) The def-inition of high symmetry lines and planes in the first Bril-louin zone of the tetragonal lattice. The spin-wave spectrumin the ( h l ) and ( hk
0) reciprocal lattice planes was mappedin this work. Here, the reciprocal lattice vector is definedas, G = h b + k b + l b , where | b | = | b | = 2 π/a and | b | = 2 π/c . netic coupling between the Mn spins. Our results ruleout the existence of magnetically-induced Weyl fermionsin the bulk of YbMnBi , but leave open the possibilitythat the ∼ ◦ canting of the Mn moments needed to formthe Weyl nodes might occur at the surface. II. EXPERIMENTAL DETAILS
Single crystalline YbMnBi was grown by the self-fluxmethod. The starting materials were mixed together ina molar ratio of Yb:Mn:Bi = 1:1:8. The mixture wasplaced into an alumina crucible, sealed in a quartz tube,then slowly heated to 900 ◦ C and kept at this tempera-ture for 10 hours. The assembly was subsequently cooleddown to 400 ◦ C at a rate of 3 ◦ C/hour. It was finallytaken out of the furnace at 400 ◦ C and was put into acentrifuge immediately to remove the excess Bi. Thestructure and quality of the single crystals was checkedwith laboratory x–rays on a 6–circle diffractometer (Ox-ford Diffraction) and Laue diffractometer (Photonic Sci-ence). A superconducting quantum interference device(SQUID) magnetometer (Quantum Design) was used tostudy the magnetization of YbMnBi as a function oftemperature. These zero-field-cooled (ZFC) magnetom-etry measurements were performed in the temperaturerange 10 to 370 K in a field of 1 T applied parallel to the a - and c -axes of YbMnBi .Elastic neutron scattering of a YbMnBi single crystal with a mass of 76 mg was performed on a 4–circle diffrac-tometer (D10) at the Institut Laue-Langevin (ILL) reac-tor source. The intensities of the reflections were studiedover the temperature range of 20 to 400 K. A pyrolyticgraphite (PG) monochromator was used to select the in-cident neutron wavelength of λ = 2 .
36 ˚A. The rockingcurve of each peak was obtained by measuring the num-ber of scattered neutrons at each rocking angle ( ω ) witha 80 ×
80 mm area detector.Inelastic neutron scattering measurements were per-formed on the triple-axis neutron spectrometer IN8 with the FlatCone detector at the ILL. A YbMnBi single crystal (mass 1 g) was initially oriented with the a and c crystal axes horizontal to map the spin-wave spec-trum in the ( h l ) scattering plane (see Fig. 1). Thecrystal was subsequently rotated by 90 ◦ (such that thecrystalline a and b axes were in the scattering plane)to access the ( h k
0) plane. Constant-energy maps weremeasured at various energies, ∆ E = E i − E f . Theoutgoing neutron wavevector was fixed at k f = 3 ˚A − ( E f = 18 . k i , with an incident beammonochromator. For energy transfers ∆ E ≥
40 meV, aPG (002) double-focusing monochromator was used, andfor ∆
E <
40 meV an elastically-bent, perfect Si (111)double-focusing monochromator was used.The array of 31 detectors on the FlatCone device al-lows for the simultaneous acquisition of scattered inten-sity along arcs in reciprocal space. By rotating the singlecrystal about the scattering plane normal, these arcs cansweep out areas in k -space to give reciprocal space maps. III. RESULTS AND ANALYSIS
The x-ray diffraction patterns of single crystallineYbMnBi obtained from the 6-circle and Laue diffrac-tometers are fully consistent with the P /nmm spacegroup, with cell parameters a = 4 . c =10 . < . ◦ ) points to a high crys-talline quality of the flux-grown crystals.The temperature dependence of the magnetic suscep-tibility of YbMnBi , with the field applied parallel tothe a and c crystal axes, is shown in Fig. 2(a). Theanomaly in the χ c data at T N (cid:39)
290 K is associated withthe onset of AFM order in the Mn sublattice. Thisvalue for the N´eel temperature is consistent with thosereported in earlier studies of YbMnBi , as well asthe neutron diffraction data presented in this work (seelater). Below T N , the magnetic susceptibility becomesstrongly anisotropic with respect to applied field, where χ a > χ c . This bifurcation of χ ( T ) at T N suggests thatthe manganese moments, in the ordered phase, are moresusceptible to an in-plane field than a field applied alongthe c -axis, in agreement with earlier reports . At lowtemperatures (below 50 K), the susceptibility grows in ac (001)(002)(100) FIG. 2. (a) Temperature dependence of the magnetic suscep-tibility of YbMnBi measured with the field applied along the a and c axes ( χ a and χ c , respectively). The single crystal wascooled in zero field and measured in an applied field strengthof 1 T. (b) Temperature dependence of the integrated inten-sity of the (001), (002) and (100) peaks. The red line is apower law fit to the temperature dependence of the (100) re-flection which gives a transition temperature of T N =290(1) K.(c) Measured intensity of the (001) peak, together with linescalculated for tilt angles of 0 ◦ , 5 ◦ and 10 ◦ . The inset showsthe variation of the χ with tilt angle. both field directions. This upturn is likely due to a small concentration of a Mn-containing paramagnetic impurityphase, and is observed in other members of the A MnBi family ( A = Sr, Ca, Ba) . A. Elastic Neutron Scattering
Neutron diffraction data in the temperature range 20to 400 K are presented in Fig. 2(b). As the sample wascooled below T = 290 K, the (100) peak, which is oth-erwise forbidden in the P /nmm space group, was ob-served. This reflection is consistent with a magneticpropagation vector of k = . The onset of this purelymagnetic peak at T N reveals the incipient AFM order ofthe Mn sublattice. The temperature dependence of theintegrated peak intensity fits very well to a power law, I obs ∝ | T N − T | β , with critical exponent β = 0.38(2),consistent with the 3D Heisenberg universality class.The predicted canting of the Mn moments awayfrom the c –axis should produce a small ab –planeferromagnetic component. Given that magnetic neutronscattering is sensitive to the component of the orderedmoment perpendicular to the scattering vector Q , wecan isolate this small in-plane component by studyingthe intensity of reflections with Q (cid:107) c . If there were anin-plane ferromagnetic component then the intensity of(00 l ) peaks should increase on cooling below T N , as wasobserved in a sister compound SrMnSb , where a smallin-plane ferromagnetic contribution to the nuclear peakwas reported .To minimize the reduction of the scattered intensitydue to the magnetic form factor of Mn , we studied thereflections with the smallest Q , namely the (001) and(002) peaks, as shown in Fig. 2(b). We observe no dis-cernible change in the integrated intensity of these peaksapart from the gradual increase with decreasing tempera-ture which can be attributed to the Debye–Waller factor.In Fig. 2(c) we show the intensity of the (001) peak ona magnified scale, together with lines calculated assumingtilt angles of 0 ◦ , 5 ◦ and 10 ◦ . The 0 ◦ curve is a quadraticfit to the data, and the other two curves are obtainedby adding the calculated magnetic intensity of the (001)peak to the 0 ◦ curve based on the measured intensity ofthe (100) peak. We also calculated the variation of the χ goodness-of-fit statistic as a continuous function of tiltangle, see inset to Fig. 2(c). From the χ distribution,we find that the probability of a tilt angle greater than3 ◦ is only 5%.These results imply that the ordered moments inYbMnBi are collinear and aligned along the c -axis towithin 3 ◦ at a 95% confidence level. Hence, a 10 ◦ cant-ing of Mn moments away from the c -axis, as requiredto create the Weyl nodes, can be excluded. FIG. 3. Constant-energy maps in the ( h l ) plane in recip-rocal space, illustrated in Fig. 1(b), at various ∆ E , plottedin reduced lattice units (r.l.u.). In each panel, the top andbottom half correspond to the data and model, respectively. B. Inelastic Neutron Scattering
Constant-energy maps of the scattering intensityrecorded in the ( h l ) and ( hk
0) reciprocal lattice planesat various energy transfers, ∆ E , are shown in Figs. 3and 4, respectively. We discuss the data from the differ-ent scattering planes in turn, starting with the ( h l ) data,which appears in the top half of each panel in Fig. 3.We find the lowest energy spin-wave mode at the Γpoint, with an energy gap of ∆ E (cid:39)
10 meV. This gapis caused by the magnetic anisotropy which favors spin
FIG. 4. Constant-energy maps in the ( hk
0) plane in re-ciprocal space, illustrated in Fig. 1(b), at various ∆ E . Ineach panel, the left and right half correspond to the data andmodel, respectively. alignment along the c axis. At ∆ E = 20 meV, we findpinch points in the magnon spectrum at the high symme-try point Z , that is, halfway between Γ points in adjacentBrillouin zones along l . These pinch points form as a re-sult of the dispersion along the c -axis. For ∆ E ≥
30 meV,the magnon dispersion along l goes away, and the inten-sity becomes independent of l . In other words, the Mnspin dynamics becomes two-dimensional. The spectrumreaches a maximum along the R − X − R high symmetryline at ∆ E = 60 meV.We now turn to the reciprocal space maps in the ( hk FIG. 5. The observed and calculated spin-wave spectrum ofthe Mn spins in YbMnBi along high symmetry directions,as defined in Fig. 1(b). The calculated magnon spectrum isin good agreement with the measured spin-wave dispersion(red markers), which was obtained from constant-energy cutsthrough the intensity maps in the ( h l ) and ( hk
0) planes. spond to the left half of each panel in Fig. 4. Just as inthe ( h l ) plane, we observe the lowest energy excitationsat the Γ point in the Brillouin zone at ∆ E = 10 meV. For10 ≤ ∆ E ≤ . ab plane. At ∆ E = 26.5 meV,we observe a saddle in the spin-wave spectrum appear-ing at the high symmetry point M . The maximum inthe dispersion is once again found at the X point, at∆ E ≤
60 meV.To obtain the spin-wave dispersion, cuts were madealong the Z − Γ − X and M − Γ − X high symmetrylines [see Fig. 1(b)] through the measured intensity mapsin the ( h l ) and ( hk
0) planes, respectively. The inten-sity in cuts at various ∆ E was fitted with peak func-tions to identify the magnon wavevectors for each ∆ E .In Fig. 5 we present the measured spin-wave dispersiondetermined this way.In order to model the observed magnon spectrum weemployed the effective spin Hamiltonian H = (cid:88) i,j J ij S i · S j − (cid:88) i D ( S zi ) , (1)where J ij is the (isotropic) exchange between Mn spins S i and S j on sites i and j , and D is a single-ion anisotropyparameter making the c axis an easy axis. In the firstsummation, we include first and second nearest neigh-bors in the ab plane ( J and J ), and nearest neighborsalong the c axis ( J c ). We used linear spin-wave theoryas implemented in the SpinW software to calculate themagnon spectrum.By fitting the linear spin-wave model to the mea-sured dispersion we find values for the parameters SJ = 22.6(5) meV, SJ = 7.8(5) meV, SJ c = − . SD = 0.37(4) meV (see Supplemental Material fordetails ), where S is the spin quantum number, whichfor Mn is S = 5 /
2. Based on these parameters, wepresent the calculated constant-energy intensity maps inthe ( h l ) and ( hk
0) planes on the lower and right halvesof the panels in Figs. 3 and 4, respectively, and we plotthe calculated magnon spectrum along high symmetrydirections in Fig. 5. Overall, we find that the calculatedspectrum agrees very well with the data.
IV. DISCUSSION
As neutron diffraction probes the entire volumeof the sample, our results rule out the possibilityof magnetically-induced Weyl nodes in the bulk ofYbMnBi . On the other hand, neutron diffraction wouldnot be sensitive to a canting of the magnetic momentsat the surface of the sample. Such a canting, if present,would reconcile the results of the present study with thework by Borisenko et al. .In YbMnBi , the spontaneous magnetic order in theMn sublattice coexists with massless quasiparticle exci-tations arising from the Bi square net. Armed with thebest-fit parameters of the linear spin-wave model, we arenow in the position to address whether the magnon spec-trum in YbMnBi differs in any detectable way comparedwith other related systems. For instance, one might ex-pect to see differences in the inter-layer exchange cou-pling parameter J c if the conducting states on the Bilayers were very unusual in YbMnBi .To elucidate this, we compare the fitted spin-wavemodel parameters obtained in this work with those ofCaMnBi , which is isostructural to YbMnBi . CaMnBi possesses a near identical N´eel temperature to YbMnBi of T N = 290 K, and is predicted to be a Diracsemimetal. Using the same Hamiltonian (1), thethree magnetic exchange parameters in CaMnBi werefound to be SJ = 23.4(6) meV, SJ = 7.9(5) meV and SJ c = − . which are the same as those ofYbMnBi to within experimental error.The anisotropyparameter for CaMnBi , SD = 0 . , which reflects that the energygap at Γ is slightly smaller in CaMnBi than in YbMnBi .These results demonstrate that the magnon spectrum ofYbMnBi does not show any anomalous behavior rela-tive to that of CaMnBi .More broadly, this suggests that replacing the divalentalkali-earth metal Ca on the A site of A MnBi with therare-earth Yb ion does not significantly enhance thecoupling between the magnetism in the octahedral MnBi layers and the charge carriers in the Bi square net. Thisis despite the fact that the A atom is situated along thedirect exchange path between the Mn and Bi atoms. In arecent review of the wider A Mn P n family of compounds,Klemenz et al. suggested another route to enhance thecoupling between magnetism and the topological chargecarriers, namely to have a magnetic ion on the A site(like Eu ) rather than non–magnetic ions such as Ca ,Sr , Ba and Yb . This was prompted by the factthat the A site atom is in closer proximity to the squareBi compared to the Mn ion and might lead to a greaterorbital overlap and thus magnetic exchange interaction.In fact, this was considered in Refs. 10 and 12, where theelectronic structure and optical properties of EuMnBi and YbMnBi were compared. The divalent rare-earthions on the A site of both A MnBi compounds have com-parable ionic radius and very similar relative positions tothe Bi square layer, but with the difference that Eu hashalf-filled 4 f orbitals compared to the fully-filled case forYb . This leads to a large pure-spin magnetic momentof 7 µ B on the A site of EuMnBi , and a non-magnetic ionon the A site of YbMnBi . These studies demonstrate amarked increase in coupling between magnetism and thetopological charge carriers in EuMnBi compared to thatin YbMnBi , which is consistent with magnetotransportstudies . This suggests that in EuMnBi , com-pared to YbMnBi , a greater coupling of magnetism tothe pnictide square net can be achieved with magneticspecies on the A site, which for the extended A Mn P n (or 112-pnictide) family, is closer to the pnictide layercompared to Mn.Finally, it is instructive to compare the physical prop-erties of YbMnBi with that of YbMnSb , which isisostructural to YbMnBi and also exhibits Mn AFMorder with a similar magnetic ordering temperature of T N = 345 K. A comparison of the band structures ofthe two 112 pnictides reveal a greater extent of inver-sion in the conduction and valence bands in YbMnBi ,with several band crossings at E F as shown Refs. 10and 11, compared to that in YbMnSb . Moreover, theShubnikov–de Haas (SdH) oscillation of the magneto–transport in both compounds reveals that the effec-tive mass of the charge carriers in YbMnBi ( m ∗ c ∼ . m e ) is approximately twice that of YbMnSb asreported in Refs. 35 and 36.These features can be understood from the relativesizes of the spin–orbit coupling (SOC) in the pnictidesquare conducting layers, which is significantly larger inYbMnBi as Bi is ∼ . M high sym-metry line is not protected by symmetry, the doubly–degenerate pnictide (Sb 5 p or Bi 6 p ) bands hybridizeand give rise to an avoided Dirac crossing. As such,the stronger SOC in YbMnBi produces a larger energygap in the electronic bands, resulting in a heavier ef-fective mass of the charge carriers compared to that inYbMnSb . This is consistent with the work in Ref. 37,which explored the effect of the masses of pnictides onthe physical properties of BaMn P n ( P n = Sb, Bi). Inthat work, Liu et al. also proposed that a more suitableplatform to realize massless Dirac fermions is in replac- ing Bi with lighter elements in the same group. Thisdemonstrates that the 112 pnictide family of compoundsoffers strong tunability of the effective mass of the chargecarriers from the size of the SOC.
V. CONCLUSION
We have presented the magnetic structure and magnonspectrum of the candidate Weyl semimetal YbMnBi .The (0 0 l ) family of nuclear reflections does not displayany additional magnetic contribution below T N , and thisrules out the mechanism for creation of Weyl nodes viaTRS-breaking through canting of the Mn spins. Hence,we demonstrate that bulk YbMnBi is a Dirac semimetalrather than a host for the WSM state. We have not ruledout the possibility of spin canting at the surface, whichcould reconcile the present results with those of Ref. 10.The lack of any anomalous features in the magnon spec-trum implies a weak coupling between magnetism andthe topological charge carriers. YbMnBi belongs to thewider A Mn P n family of compounds which are currentlyattracting strong interest owing to its strong potential forspintronic applications. We hope that the understandingof YbMnBi achieved here will contribute to the devel-opment of strategies for enhancing the exchange couplingbetween charge transport and magnetism, and for reduc-ing the effective mass of the quasiparticles. ACKNOWLEDGMENTS
The authors wish to thank D. Prabhakaran and F.Charpenay for technical assistance, and M. Newportfor fabricating the Al mount used in the INS experi-ment. We are also grateful to M. C. Rahn and P. Stef-fens for the data analysis software, P. Manuel and D.D. Khalyavin for help with preliminary neutron stud-ies on WISH, ISIS (beamtime RB1720113), M. Gut-mann for checking the single crystal quality of YbMnBi on SXD, ISIS and N. Qureshi for orienting the crystalfor the INS experiment on OrientExpress , ILL (beam-time EASY-365). The D10 and IN8 experiment numberswere DIR-159 and 4-01-1572 respectively. This workwas supported by the U.K. Engineering and PhysicalSciences Research Council, Grant Nos. EP/N034872/1and EP/M020517/1, the Natural Science Foundationof Shanghai (Grant No. 17ZR1443300), the ShanghaiPujiang Program (Grant No. 17PJ1406200), the Na-tional Key Research and Development Program of China(Grant No. 2017YFA0302901), the Beijing Natural Sci-ence Foundation (Grant No. Z180008) and the K. C.Wong Education Foundation (Grant No. GJTD-2018-01). J.-R. Soh acknowledges support from the SingaporeNational Science Scholarship, Agency for Science Tech-nology and Research. ∗ [email protected] A. A. Burkov, Nat. Mat. , 1145 (2016). N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev.Mod. Phys. , 015001 (2018). J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Ann. Rev. Con.Mat. Phys. , 195 (2016). D. Pesin and L. Balents, Nat. Phys. , 376 (2010). M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010). B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao,J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen,Z. Fang, X. Dai, T. Qian, and H. Ding, Phys. Rev. X ,031013 (2015). S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang,B. Wang, N. Alidoust, G. Bian, M. Neupane, C. Zhang,S. Jia, A. Bansil, H. Lin, and M. Z. Hasan, Nat. Comms. , 7373 (2015). S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian,C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang,A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, andM. Z. Hasan, Sci. , 613 (2015). L. X. Yang, Z. K. Liu, Y. Sun, H. Peng, H. F. Yang,T. Zhang, B. Zhou, Y. Zhang, Y. F. Guo, M. Rahn,D. Prabhakaran, Z. Hussain, S.-K. Mo, C. Felser, B. Yan,and Y. L. Chen, Nat. Phys. , 728 (2015). S. Borisenko, D. Evtushinsky, Q. Gibson, A. Yaresko,K. Koepernik, T. Kim, M. Ali, J. van den Brink,M. Hoesch, A. Fedorov, E. Haubold, Y. Kushnirenko,I. Soldatov, R. Sch¨afer, and R. J. Cava, Nat. Comms ,3424 (2019). D. Chaudhuri, B. Cheng, A. Yaresko, Q. D. Gibson, R. J.Cava, and N. P. Armitage, Phys. Rev. B , 075151(2017). M. Chinotti, A. Pal, W. J. Ren, C. Petrovic, and L. De-giorgi, Phys. Rev. B , 245101 (2016). S. Klemenz, S. Lei, and L. M. Schoop, Annu. Rev. Mater.Res. , 185 (2019). A. Wang, I. Zaliznyak, W. Ren, L. Wu, D. Graf, V. O.Garlea, J. B. Warren, E. Bozin, Y. Zhu, and C. Petrovic,Phys. Rev. B , 165161 (2016). J. Y. Liu, J. Hu, D. Graf, T. Zou, M. Zhu, Y. Shi, S. Che,S. M. A. Radmanesh, C. N. Lau, L. Spinu, H. B. Cao,X. Ke, and Z. Q. Mao, Nat. Comms. , 646 (2017). A. Pal, M. Chinotti, L. Degiorgi, W. Ren, and C. Petrovic,Physica B , 64 (2018). I. A. Zaliznyak, A. T. Savici, V. O. Garlea, B. Winn,U. Filges, J. Schneeloch, J. M. Tranquada, G. Gu,A. Wang, and C. Petrovic, J. Phys.: Conf. Ser. ,012030 (2017). M. C. Rahn, A. J. Princep, A. Piovano, J. Kulda, Y. F.Guo, Y. G. Shi, and A. T. Boothroyd, Phys. Rev. B ,134405 (2017). A. Hiess, M. Jim´enez-Ruiz, P. Courtois, R. Currat,J. Kulda, and F. Bermejo, Physica B , 1077(2006). M. Kempa, B. Janousova, J. Saroun, P. Flores, M. Boehm,F. Demmel, and J. Kulda, Physica B , 1080(2006). m. m. see supplemental material athttp://link.aps.org/supplemental/10.1103/PhysRevB.00.000000 for laboratory x-ray diffraction patterns anddata analysis methods,. Y. F. Guo, A. J. Princep, X. Zhang, P. Manuel,D. Khalyavin, I. I. Mazin, Y. G. Shi, and A. T. Boothroyd,Phys. Rev. B , 075120 (2014). L. Li, K. Wang, D. Graf, L. Wang, A. Wang, and C. Petro-vic, Phys. Rev. B , 115141 (2016). Y.-Y. Wang, Q.-H. Yu, and T.-L. Xia, Chin. Phys. B ,107503 (2016). G. L. Squires,
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