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Spontaneous Chirality Flipping in an Orthogonal Spin-Charge Ordered Topological Magnet
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Spontaneous Chirality Flipping in an Orthogonal Spin-Charge Ordered Topological Magnet
H. Miao ,1,,?J. Bouaziz ,2,,?G. Fabbris ,3,W. R. Meier ,4F. Z. Yang,1H. X. Li,1,5C. Nelson,6E. Vescovo,6
S. Zhang,7A. D. Christianson ,1H. N. Lee,1Y. Zhang,8,9C. D. Batista ,8,10and S. Blügel2
1Materials Science and Technology Division, Oak Ridge National Laboratory,
Oak Ridge, Tennessee 37831, USA
2Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA,
D-52425 Jülich, Germany
3Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA
4Department of Materials Science and Engineering, University of Tennessee, Knoxville, Tennessee, USA
5Advanced Materials Thrust, The Hong Kong University of Science and Technology (Guangzhou),
Guangzhou, China
6National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA
7Max-Planck-Institut fur Physik komplexer Systeme, Nothnitzer Stra?e 38, 01187 Dresden, Germany
8Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA
9Min H. Kao Department of Electrical Engineering and Computer Science, University of Tennessee,
Knoxville, Tennessee 37996, USA
10Quantum Condensed Matter Division and Shull-Wollan Center, Oak Ridge National Laboratory,
Oak Ridge, Tennessee 37831, USA
(Received 4 December 2023; revised 24 January 2024; accepted 13 February 2024; published 21 March 2024)
The asymmetric distribution of chiral objects with opposite chirality is of great fundamental interest
ranging from molecular biology to particle physics. In quantum materials, chiral states can build on inversion-
symmetry-breaking lattice structures or emerge from spontaneous magnetic ordering induced by competing
interactions. Although the handedness of a chiral state can be changed through external fields, a spontaneouschirality flipping has yet to be discovered. We present experimental evidence of chirality flipping via
changing temperature in a topological magnet EuAl
4, which features orthogonal spin density waves (SDW)
and charge density waves (CDW). Using circular dichroism of Bragg peaks in the resonant magnetic x-rayscattering, we find that the chirality of the helical SDW flips through a first-order phase transition with
modified SDW wavelength. Intriguingly, we observe that the CDW couples strongly with the SDW and
displays a rare commensurate-to-incommensurate transition at the chirality flipping temperature. Combiningwith first-principles calculations and angle-resolved photoemission spectroscopy, our results support a Fermi
surface origin of the helical SDW with intertwined spin, charge, and lattice degrees of freedom in EuAl
4.O u r
results reveal an unprecedented spontaneous chirality flipping and lay the groundwork for a new functionalmanipulation of chirality through momentum-dependent spin-charge-lattice interactions.
DOI: 10.1103/PhysRevX.14.011053 Subject Areas: Condensed Matter Physics, Magnetism
Chirality, a geometrical concept that distinguishes an
object from its mirror image, has been proposed for over
three decades as a potential mechanism for novel quantum
states including spontaneous quantum Hall liquids [1],
chiral spin liquids [2], and magnetic skyrmions [3,4].Recently, chirality has experienced a revival in the context
of correlated and geometrically frustrated electronic sys-tems [5–12]. In these settings, chiral spin, charge, orbital,
and pairing fields become strongly coupled, giving rise tointertwined orders [10] and long-range entangled quasipar-
ticles [11,12] . In magnetic systems, the 1D and 2D chiral
spin textures, as respectively shown in Figs. 1(b) and1(c),
have been widely observed in noncentrosymmetric lattices[13,14] . In these systems, the chirality or handedness,
χ?/C61, is usually transmitted to the spin system via
the relativistic Dzyaloshinskii-Moriya (DM) interactioninduced by spin-orbit coupling. Chiral spin orders canalso spontaneously emerge in centrosymmetric materials[13–17], where competing magnetic interactions, such as theThese authors are contributed equally to this work.
?Corresponding author: miaoh@ornl.gov
?Corresponding author: j.bouaziz@fz-juelich.de
Published by the American Physical Society under the terms of
theCreative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article ’s title, journal citation,
and DOI.PHYSICAL REVIEW X 14,011053 (2024)
2160-3308 =24=14(1) =011053(8) 011053-1 Published by the American Physical SocietyRuderman-Kittel-Kasuya-Yosida (RKKY), can yield equally
populated χ?1and?1states. Experimentally, chirality
manipulation has been achieved by applying external fields
[13,18] . An outstanding question that remains is if the sign
of chirality can be controlled by other means.
In this paper, we uncover an unprecedented spontaneous
chirality flipping in EuAl 4. EuAl 4hosts nanometric sky-
rmions and the topological Hall effect under a magnetic
field along the caxis [20–22]. At zero external magnetic
field, EuAl 4exhibits a tetragonal structure with I4=mmm
symmetry (centrosymmetric space group no. 139) at roomtemperature. Figure 1(d)summarizes the zero magnetic field
symmetry breaking orders of EuAl
4(see also Supplemental
Material Fig. S1 and Table S1 [19]). Below TCDW ?140K,
an incommensurate charge density wave (ICDW) develops
along the crystalline caxis with QCDW ??0;0;0.183?in
reciprocal lattice unit (r.l.u.). The QCDW gradually shifts to
the commensurate position at ?0;0;1=6?via decreasing
temperature, which is typical for ICDW systems due to the
lattice commensurate energy [23].A tT?1?
SDW?15.4K, the
system breaks the time-reversal symmetry Tby forming a
double- Qspin density wave (SDW) along the [110] and
?1?10/C138directions. Below T?2?
SDW?13.3K, the spin
moment of the double- QSDW gains a finite c-axis compo-
nent, leading to new interference magnetic peaks alongthe [100] and [010] direction. The fourfold rotational sym-
metry C4in the abplane is broken at T?3?
SDW?12.3K,
resulting in a stripe helical SDW [20,21] . Interestingly,
although the helical SDW persists down to the lowest
temperature at zero magnetic field, an additional first-order
phase transition sets in at Tχ?10.1K.
Here we use the circular dichroism (CD) of Bragg peaks
in the resonant magnetic x-ray scattering (XRMS) todemonstrate that the first-order phase transition at T
χleads
to a spontaneous chirality flipping. CD XRMS is a direct
experimental probe of chiral electronic orders [24–28].
Under the resonance condition, where the incident photonenergy ωmatches the energy differences between occupied
and unoccupied atomic energy levels, the x-ray scattering
amplitude from site ncan be written as [24] (see the
Appendix)
f
n?ω??? ??0·???f0?ω??i???0×???·cMnf1?ω?;?1?
where ??and ??0are the polarization vectors of the incident
and scattering x rays, respectively, and cMnis the magnetic
moment of site n.f0?ω?is the anomalous charge scattering
form factor that can be added to the Thomson scattering.
f1?ω?is the linear magnetic scattering form factor. For the
experimental geometry shown in Fig. 2(a), the CD of the
(a) (b) (d)
(c)
FIG. 1. Emergent chiral magnetic orders and the zero-field phase diagram of EuAl 4. (a) Spontaneous chiral symmetry breaking yields
degenerated χ?1andχ??1states. External field or intertwined orders can lift the degeneracy. (b) Schematics of 1D and 2D chiral
spin textures, helical SDW (left) and Bloch-type skyrmion (right). (c) Possible microscopic mechanisms that drive chiral magneticstates. Left: relativistic DM interaction Din noncentrosymmetric lattice determines the sign of χ(see Supplemental Material
Note 1 [19]). The wavelength of the spin order λis proportional to the relative energy scales of atomic exchange energy JandD. Right:
quasi-nested Fermi surface in hexagonal and tetragonal structures give rise to frustrated RKKY interactions along Q
1,Q2,Q3that satisfy
Q1?Q2?Q3?0. The inset depicts possible Fermi surface topologies in the hexagonal and tetragonal lattices that can yield nearly
degenerated Q1,Q2,Q3orders with Q1?Q2?Q3?0. The Fermi surface topology of EuAl 4is similar to the schematic Fermi surface
in the tetragonal lattice, where yellow and cyan represent electron and hole band, respectively (Supplemental Material Fig. S4). (d) Phase
diagram of EuAl 4without external magnetic field. Orthogonal CDW and SDW superlattice peaks are marked in the 3D and 2D Brillouin
zone, respectively. Chiral SDW emerges below T3
SDW and breaks the C4rotational symmetry.H. MIAO et al. PHYS. REV . X 14,011053 (2024)
011053-2helical SDW can be formulated as [28]
I?Q?CR?I?Q?CL??τχ?Dyz; ?2?
where τ?sgn??QSDW· ?x?=jQSDW· ?xj/C138, ?xis the unit vector
along the xdirection as shown in Fig. 2(a).Dyzis a function
of magnetic moment in the ( y-z) plane [Fig. 2(a)] and is
independent of τandχ.I?Q?CRandI?Q?CLrepresent x-ray
intensity obtained under circular right (CR) and circular left(CL) incident photon energy, respectively. FollowingEq.(2), the CD of the χ?1helical SDW [Fig. 1(b)]
is positive for a propagation vector Q
SDW and negative
for?QSDW. An achiral SDW, such as the double- QSDW
above T?3?
SDW will, therefore, yield zero CD. To probe the
magnetic chirality of EuAl 4, the photon energy is tuned to
the Eu L3edge ( 2p-5d). Figure 2(a) shows the x-ray
fluorescence scan at T?5K. The single peak at ωres?
6.977keV confirms the Eu2?electronic configuration in
EuAl 4[21]. The energy scan at fixed QSDW??0.19;0;4?
at 5 K show giant magnetic resonance at ωres, confirming its
magnetic origin.
In Figs. 2(b)and2(c), we first show the CD of a structural
B r a g gp e a k( 0 ,0 ,4 )a t T?5K. Yellow and cyan curves
represent I?Q?CRandI?Q?CL, respectively. The asymmetry
of the CD, F?Q??I?Q?CR?I?Q?CL=I?Q?CR?I?Q?CL,i s
s h o w ni nF i g . 2(c). As expected, the (0, 0, 4) structural Bragg
peak shows F?Q??0. Note the large noise away from the
Bragg condition is due to the nearly zero scattering intensity,
proving the high sample quality. We then move to the helicalSDW at T ?9Kχ. As shown in Figs. 2(d) and2(e),we observe giant CD at both QSDW and?QSDW with
F?QSDW??40%a n d F??QSDW???90%. This observa-
tion proves that the helical SDW below Tχis chiral with
χ?1.T h el a r g e Findicates that the entire photon illumi-
nated SDW (on the order of 50×50μm2)h a st h es a m e χ
and hence macroscopically breaks the symmetric chiral
distribution. We then move to TχSDW.
Remarkably, as shown in Figs. 2(f)and2(g), the CD changes
sign with F?QSDW???40%a n d F??QSDW??90%. This
observation establishes a spontaneous chirality flipping from
χ?1toχ??1. The comparably large F?Q?below and
above T?4?
SDWfurther suggests that the chirality flipping is also
realized on a macroscopic length scale. The chiral density is
back to nearly zero upon warming up above T?3?
SDW.T h eg i a n t
asymmetric chiral distribution and spontaneous chirality
flipping between the helical SDW states constitute the mainexperimental results of this work.
The spontaneous chirality flipping raises questions con-
cerning its microscopic origin. Because of the coexistence
of CDW and SDW, we first determine the complex spin-
charge correlations by tracing the temperature-dependentevolution of CDW and SDW wave vectors below 20 K. The
scanning trajectories in the reciprocal space are shown in
the inset of Figs. 3(a)–3(c). Figure 3(d) summarizes the
extracted Q
DW(in r.l.u.). As shown in Fig. 3(a), the double-
QSDW first emerges below T?1?
SDW along the [110] and
?1?10/C138directions and is smoothly connected with the
spin canted double- Qphase. In the chiral SDW phase
below T?3?
SDW,QSDW increases monotonically along the
(a) (arb. units)(b)
(c)(d)
(e)(f)
(g)(h)
(i)
FIG. 2. Discovery of spontaneous chirality flipping in EuAl 4. (a) Experimental geometry of CD XRMS. The photon energy was tuned
to Eu L3edge to probe the magnet order parameter. For a 1D chiral SDW, the CD is given by Eq. (2), where the sign of F?Q?changes
fromQ?QSDW toQ??QSDW in the rocking scattering geometry. Here we define χ??1ifF?QSDW?>0. Fluorescence scan
(bottom left) at T?5K shows single peak at ωres?6.977keV, confirming Eu2?configuration. Magnetic resonance scan (bottom
right) at QSDW??0.169;0;4?. The strong resonant enhancement confirms its magnetic origin. (b) –(i) CD of the structural and magnetic
Bragg peaks. Yellow and cyan curves in (b), (d), (f), and (h) represent CR and CL incident photon polarization, respectively. Red, green,and blue curves in (c), (e), (g), and (i) represent positive, zero, and negative F?Q?. We note that due to the finite Hcomponent and
narrow width of magnetic peaks, horizontal axis from H??0.17to 0.17 was not shown in (e), (g), and (i). Giant CD is observed below
T
3
SDW?12.3K. The sign change of the CD shown in (e) and (g) establishes the chirality flipping across Tχ.SPONTANEOUS CHIRALITY FLIPPING IN AN ORTHOGONAL … PHYS. REV . X 14,011053 (2024)
011053-3[100] and [010] direction and displays a discontinuous leap
at the chirality flipping transition. For the CDW, the QCDW
first shows a rare commensurate-incommensurate transition
in the temperature range ?T?3?
SDW;20K/C138?TCDW. The
QCDW then jumps back to the commensurate value
and remains Tindependent in the χ??1SDW phase
(?Tχ;T?3?
SDW/C138). Finally, in the χ?1SDW phase ( TtheQCDW once again becomes incommensurate. This
complex temperature-dependent evolution of the CDW
and the SDW is characteristic of intertwined spin, charge,
and lattice degrees of freedom.
The incommensurability of both QSDW andQCDW and
the presence of large itinerant carriers in EuAl 4indicate a
Fermi surface effect. Figures 3(e) and 3(f) show the
calculated 3D Fermi surface and Eu-Eu magnetic inter-action J?q?of EuAl
4in the tetragonal phase (see
Supplemental Material Fig. S2 for the electronic structure
determined by angle-resolved photoemission spectroscopy
[19]). The highest value of J?q?determines the N? eel
temperature and the wave vector qpof the helical SDW
state. As shown in Fig. 3(f),J?qp?along the [100] direction
features a typical paramagnetic spin susceptibility of ametal with a sharp and significant finite- qpeak at
qp?0.19r.l.u., consistent with experimental data at
5 K. The estimated magnetic transition temperature,
Tcal
SDW∝J?qp?=3kB?14.8K(kBbeing the Boltzmann
constant), is also in agreement with experimental obser-
vation [see Supplemental Material Figs. S3 –S6 for the
effects of Coulomb interactions, electron temperature and
magnetoelastic coupling on J?q?]. These findings provide
strong numerical evidence for a Fermi surface driven
helical SDW in EuAl 4. Interestingly, the QCDW matches
the calculated charge susceptivity peak along the Γ-Z
direction [29]. Although the primary driving force of the
CDW in EuAl 4remains to be determined, the presence of
nested Fermi surface is usually helpful to select the QCDW
by forming a CDW gap near the Fermi level [30,31] .
While the intertwined spin, charge, and lattice degrees of
freedom are established in EuAl 4, the microscopic origin of
the giant asymmetric chiral distribution and spontaneous
chirality flipping calls for further studies. The CD of the
SDW is robust under temperature cycling above both
the achiral double- Qphase and C4symmetry breaking
(see Supplemental Material Figs. S7 and S8 [19]), suggesting
(a)
(d) (f)(b) (c) (e)
FIG. 3. Intertwined SDW and CDW with orthogonal wave vectors. (a),(b) T-dependent Hscan and HKscan near the SDW wave
vectors. (c) T-dependent Lscan near the CDW wave vector. The scanning trajectories are shown in each panel. Dashed lines indicate the
magnetic transition temperatures. (d) Extracted T-dependent CDW and SDW wave vectors. Blue and purple squares are corresponding
tojQSDWjin the unit of 2π=a0and?2π=a0????
2p
, respectively. In the spin canted double- Qphase between T?2?
SDWandT?3?
SDW, thec-axis spin
component yields superlattice peaks along [100] and [010] directions whose values are???
2p
times of the principal peak values along
?1?10/C138and [110] directions. Yellow circles represent jQCDWjin the unit of 2π=c0.a0andc0are lattice constants in the tetragonal unit
cell. (e) DFT calculated electronic structure in the tetragonal phase. Green arrows indicate the in-plane “nesting vector ”that favors
helical SDW. (f) Calculated RKKY interactions along [100] (black line) and [110] (red line) directions. The prominent peak at qp?0.19
r.l.u. is consistent with experimentally observed SDW at 5 K.H. MIAO et al. PHYS. REV . X 14,011053 (2024)
011053-4hidden chiral interactions above 20 K. Because of
the intertwined nature of SDW and CDW, it is tempting toassociate the chiral interaction with a chiral CDW.
Encouragingly, the CDW in EuAl
4is found to be transverse
[32], where the CDW driven lattice distortions and soft
phonon modes are perpendicular to the CDW propagation
vector [29]. Assuming a linear transverse CDW along the
propagation direction, the C4rotational symmetry is expected
to be broken below TCDW.H o w e v e r ,t h e C4symmetry
breaking in the abplane is observed only at T?3?
SDW?12.3K?
TCDW?140K[20,21] . These observations, therefore, sup-
port a chiral CDW in EuAl 4. It is highly interesting to point
out that the CDW related “nesting ”vector connects the
topological semi-Dirac bands [33]. Similar type of band
structure has been studied in 3D-quantum Hall systems,where electron-electron scattering between the Dirac bands
involves chiral lattice excitations [34,35] .T h er o l eo fc h i r a l
phonons in EuAl
4is therefore an interesting open question.
Finally, we discuss the origin of chirality flipping. Since
chirality is a discontinuous physical quantity, the chirality
flipping requires a first-order phase transition that is con-
sistent with the jump of QSDWatTχ. The absence of hysteresis
observed in this study and previous works [20,21] suggests
that the transition is weak first order [31,36] .S i n c et h e
itinerant electrons play a pivotal role for both CDWand SDW
in EuAl 4, the chiral magnetic interaction is also likely
momentum dependent (see Supplemental Material Notes 1and 2 for a simplified model [19]). Depending on Q,t h ec h i r a l
interaction energetically favors χ?1or?1SDW. Indeed, as
we show in Figs. 3(a)and3(d), we observe a significant jump
ofQ
SDWatTχwith minor lattice and CDW modifications.
In summary, we discovered a spontaneous chirality flip-
ping in an orthogonal spin-charge ordered itinerant magnet.Our results highlight EuAl
4and associated materials as a rare
platform for emergent chiral interactions and open a new
avenue for chiral manipulations through intertwined orders.
We thank Matthew Brahlek, Miao-Fang Chi, Xi Dai,
Satoshi Okamoto, Andrew May, Brian Sales, Jiaqiang Yan,and Gabriel Kotliar for stimulating discussions. This
research was supported by the U.S. Department of
Energy, Office of Science, Basic Energy Sciences,
Materials Sciences and Engineering Division (x-ray and
ARPES measurement). CD XRMS used resources (beamline 4ID) of the Advanced Photon Source, a U.S. DOE
Office of Science User Facility operated for the DOE
Office of Science by Argonne National Laboratory underContract No. DE-AC02-06CH11357. ARPES and XRMS
measurements used resources at 21-ID-1 and 4-ID beam
lines of the National Synchrotron Light Source II, a U.S.
Department of Energy Office of Science User Facility
operated for the DOE Office of Science by BrookhavenNational Laboratory under Contract No. DE-SC0012704.
W. R. M. acknowledges support from the Gordon and Betty
Moore Foundation ’s EPiQS Initiative, Grant GBMF9069 toD.M. Y. Z. is partly supported by the National Science
Foundation Materials Research Science and EngineeringCenter program through the UT Knoxville Center forAdvanced Materials and Manufacturing (Grant No. DMR-2309083). J. B. and S. B. acknowledge financial support from
the European Research Council (ERC) under the European
Union's Horizon 2020 research and innovation program (GrantNo. 856538, project “3D MAGiC ”) and computing time
granted by the JARA-CSD and VSR Resource AllocationBoard provided on the supercomputers CLAIX at RWTH
Aachen University and JUREC A at Juelich Supercomputer
Centre under Grants No jara0219 and No jiff13. S. B. acknowl-edges financial support from Deutsche Forschungsge-meinschaft (DFG) through CRC 1238 (Project No. C01)and SPP 2137 (Project No. BL 444/16-2).
APPENDIX: METHODS
1. Sample growth
EuAl
4crystals were grown from a high-temperature
aluminum-rich melt [21,33] . Eu pieces (Ames Laboratory,
Materials Preparation Center 99.99?%) and Al shot (Alfa
Aesar 99.999%) totaling 2.5 g were loaded into one side of
a 2-mL alumina Canfield crucible set. The crucible set wassealed under 1=3atm argon in a fused silica ampoule.
The ampoule assembly was placed in a box furnace
and heated to 900°C over 6 h ( 150°C=h) and held for 12 h
to melt and homogenize the metals. Crystals were pre-cipitated from the melt during a slow cool to 700°C over
100 h ( ?2°C=h). To liberate the crystals from the remain-
ing liquid, the hot ampoule was removed from the furnace,inverted into a centrifuge, and spun.
2. XRMS
Resonant magnetic x-ray scattering measurements were
performed at the integrated in situ and resonant hard x-ray
studies (4-ID) beam line of National Synchrotron LightSource II (NSLS-II). The photon energy, which is selected
by a cryogenically cooled Si(111) double-crystal mono-
chromator, is 6.977 keV. The sample is mounted in aclosed-cycle displex cryostat in a vertical scatteringgeometry, and the magnetic σ-πscattering channel is
measured using an Al(222) polarization analyzer and
silicon drift detector.
3. CD XRMS
The CD XRMS were performed at the 4-ID-D beam
line of the Advanced Photon Source (APS), ArgonneNational Laboratory (ANL). The photon energy was tunedto the Eu L
3resonance (6.977 keV) using a double-crystal
Si (111) monochromator. Circularly polarized x rays were
generated using a 180-μm-thick diamond (111) phase
plate [34], focused to 200×100μm2full width at half
maximum (FWHM) using a toroidal mirror, and further
reduced to 50×50μm2FWHM with slits. TemperatureSPONTANEOUS CHIRALITY FLIPPING IN AN ORTHOGONAL … PHYS. REV . X 14,011053 (2024)
011053-5was controlled using a He closed-cycle cryostat.
Diffraction was measured in reflection from the sample[001] surface using vertical scattering geometry and anenergy dispersive silicon drift detector (approximately0.15 keV energy resolution).
4. Angle-resolved photoemission spectroscopy (ARPES)
The angle-resolved photoemission spectroscopy (ARPES)
experiments were performed on single crystals EuAl
4.T h e
samples were cleaved in situ in a vacuum better than
3×10?11torr. The experiment was performed at beam line
21-ID-1 at the NSLS-II. The measurements were takenwith synchrotron light source and a Scienta-OmicronDA30 electron analyzer. The total energy resolution of theARPES measurement is approximately 15 meV. The samplestage is maintained at T?20K throughout the experiment.
5. Density functional theory (DFT)
The first-principles simulations are performed using the
all-electron full-potential Korringa-Kohn-Rostoker (KKR)Green function method [37] in the scalar relativistic
approximation. The inclusion of the spin-orbit couplingself-consistently does not alter the magnetic interactions or
Fermi surface. The Eu
2?and its 4felectrons are treated
using the DFT ?Uapproach [38]with value of U?2eV.
The calculations are carried out using an angular momen-tum cutoff of lmax ?4in the orbital expansion of the
Green function. Accurate self-consistent results wereobtained using 58 integration points along the complex
energy contour and a 30×30×30kmesh for the Brillouin
zone integration. The magnetic interactions between theEu magnetic atoms are computed in real space using theinfinitesimal rotation method [39]. These interactions are
then Fourier transformed into reciprocal space with a cutoff
of 10 lattice constants to accurately account for the long-
range RKKY interactions. The Fermi surface is imaged bycomputing q-dependent density of states [40].
6. XRMS cross section
Under the electric dipole approximation, the resonant
scattering process involves transitions between the corestate jζ
υiwith energy Eυand an unoccupied state jψηiwith
energy Eηin both absorption and emission channel. The
scattering amplitude can be written as
fres?ω??X
ij??0
i??jX
ηhζυjRijψηihψηjRjjζυi
ω??Eη?Eυ??iΓ?X
ij??0
i??jTij;
?A1?
where ??and ??0are the polarization vectors of the incident
and scattering x rays, respectively. Ris the position
operator. When the resonant atom has a parity-evenmagnetic moment cMnat site n, and assuming cylindrical
symmetry, the magnetic scattering takes the form [24]:
fn?ω??? ??0·???f0?ω??i???0×???×cMnf1?ω?
?? ??0·cMn????·cMn?f2?ω?: ?A2?
Since f2?ω?is usually much smaller than f1?ω?, the
magnetic scattering is dominated by the second term ofEq.(A2). For the 1D helical SDW, the CD has been
derived, respectively, for the tilting and rocking geometries
shown in Fig. 2(a) [28] . For the chiral Bloch-type SDW,
I?Q?
CR?I?Q?CL??τχ?Dyzin the rocking geometry :
?A3?
For the achiral N? eel-type SDW,
I?Q?CR?I?Q?CL?χByzin the tilting geometry :?A4?
Note that for the N? eel-type SDW, the CD does not change
sign from the positive to negative propagation vector. Our
experiment was performed in the rocking geometry that
shows sign change for the chiral Bloch-type SDW.
7. Nonresonant XRMS cross section
For nonresonant scattering, the x-ray scattering ampli-
tude can be written as
fnon?ω?∝X
jh0jei ?q·rj!
ji? ??0/C3· ???
?i?ω
mc2/C20mc
e?h0j?q×? ?ML? ?q?×?q/C138j0i· ?PL
?mc
e?h0j? ?MS? ?q?/C138j0i· ?PS/C21
; ?A5?
where ?MLand ?MSare Fourier transform of orbital and spin
moment density, respectively. Here,
?PL?4sin2θ? ??0/C3× ???; ?A6?
?PS? ??× ??0?? ?kf× ??0/C3???kf· ??????ki× ?????ki· ??0/C3?
???kf× ??0/C3?×??ki× ???: ?A7?
Equations (A5)–(A7) show that the nonresonant x-ray
scattering can also probe spin and orbital magnetic moment
and display CD. However, the factor ?ω=mc2in Eq. (A5)
at?ω?10keV is on the order of ?10?4. Therefore, for
the nonresonant x-ray scattering, the cross section related
to Eqs. (A6) and (A7) is extremely small, typically
10–30counts =s for magnetic materials.H. MIAO et al. PHYS. REV . X 14,011053 (2024)
011053-6In our study, we observed both CD and asymmetry of
the CD that is described in Eq. (A4). Furthermore,
F?Q??I?Q?CR?I?Q?CL=I?Q?CR?I?Q?CL?90%i sv e r y
large, excluding contributions from Eqs. (A6) and(A7) as the
origin of chirality flipping.
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