Spontaneous Chirality Flipping in an Orthogonal Spin-Charge Ordered Topological Magnet

The asymmetric distribution of chiral objects with opposite chirality is of great fundamental interests 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 spontaneous chirality flipping has yet to be discovered. In this letter, we present experimental evidence of chirality flipping via changing temperature in a topological magnet EuAl$_4$, which features orthogonal spin and charge density waves (SDW/CDW). Using circular dichroism of Bragg peaks in the resonant magnetic x-ray scattering, 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. Combining with first principles calculations and angle resolved photoemission spectroscopy, we establish the Fermi surface origin of the helical SDW with intertwined spin, charge, and lattice degrees of freedom in EuAl$_4$. Our results reveal an unprecedented spontaneous chirality flipping and lays the groundwork for a new functional manipulation of chirality through momentum dependent spin-charge-lattice interactions.


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 EuAl4.Our results reveal an unprecedented spontaneous chirality flipping and lays the groundwork for a new functional manipulation of chirality through momentum dependent spin-charge-lattice interactions.
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 4 , chiral spin liquids 5 , and magnetic skyrmions 6,7 .Recently, chirality has experienced a revival in the context of correlated and geometrically frustrated electronic systems [8][9][10][11][12][13][14][15] .In these settings, chiral spin, charge, orbital, and pairing fields become strongly coupled, giving rise to intertwined orders 13 and long-range entangled quasiparticles 14,15 .
Experimentally, chirality manipulation has been achieved by applying external fields 2,3 .An outstanding question that remains is if the sign of chirality can be controlled by other means.
In this letter, we uncover an unprecedented spontaneous chirality flipping in EuAl4.EuAl4 hosts nanometric skyrmions and the topological Hall effect under a magnetic field along the c-axis [20][21][22] .
The fourfold rotational symmetry, C4, in the ab-plane is broken at  !"# (() =12.3 K, 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  ) =10.1 K.
Here we use the circular dichroism (CD) of Bragg peaks in the resonant magnetic x-ray scattering (XRMS) to demonstrate that the first order phase transition at  ) leads to a spontaneous chirality flipping.CD-XRMS is a direct experimental probe of chiral electronic orders [24][25][26][27][28] .Under the resonance condition, where the incident photon energy, w, matches the energy differences between occupied and unoccupied atomic energy levels, the x-ray scattering amplitude from site n can be written as 24 (see Methods): where ̂ and ′ are the polarization vectors of the incident and scattering x-rays, respectively, and  * 5 is the magnetic moment of site n. , () is the anomalous charge scattering form factor that can be added to the Thomson scattering. % () is the linear magnetic scattering form factor.For the experimental geometry shown in Fig. 2a, the CD of the helical SDW can be formulated as 28 : where =sign((QSDW•  ⃗)/|QSDW•  ⃗|),  ⃗ is the unit vector along the x-direction as shown in Fig. 2a.
01 is a function of magnetic moment in the (y-z) plane (Fig. 2a) and is independent of  and .
() -. and () -/ represent x-ray intensity obtained under circular right (CR) and circular left (CL) incident photon energy, respectively.Following Eq. ( 2), the CD of the c=1 helical SDW (Fig. 1b) is positive for a propagation vector QSDW and negative for -QSDW.An achiral SDW, such as the double-Q SDW above  !"# (() will, therefore, yield zero CD.To probe the magnetic chirality of EuAl4, the photon energy is tuned to the Eu L3-edge (2p-5d).Figure 2a shows the x-ray fluorescence scan at T=5 K.The single peak at wres=6.977keV confirms the Eu 2+ electronic configuration in EuAl4 21 .The energy-scan at fixed QSDW=(0.19, 0, 4) at 5 K show giant magnetic resonance at wres, confirming its magnetic origin.
The large F indicates that the entire photon illuminated SDW (on the order of 50 µm × 50 µm) has the same c and hence macroscopically breaks the symmetric chiral distribution.We then move to  ) <T=11 K< !"# (() .Remarkably, as shown in Fig. 2f and 2g, the CD changes sign with F(QSDW) =-40% and F(-QSDW) =90%.This observation establishes a spontaneous chirality flipping from c=1 to c=-1.The comparably large () below and above  !"# (2) further 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  !"# (() .The giant asymmetric chiral distribution and spontaneous chirality flipping between the helical SDW states constitute the main experimental results of this work.
The spontaneous chirality flipping raises questions concerning its microscopic origin.Due to the coexistence of CDW and SDW, we first determine the complex spin-charge correlations by tracing the temperature dependent evolution of CDW and SDW wavevectors below 20 K.The scanning trajectories in the reciprocal space are shown in the inset of Fig. 3a-c.Figure 3d  ).Finally, in the c=1 SDW phase (T< ) ), the QCDW 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 and QCDW and the presence of large itinerant carriers in EuAl4 indicate a Fermi surface effect.Figure 3e and f show the calculated 3D Fermi surface and Eu-Eu magnetic interaction, J(q), of EuAl4 in the tetragonal phase (see Supplementary Figure S2 for the electronic structure determined by angle-resolved photoemission spectroscopy).The highest value of J(q) determines the Néel temperature and the wavevector qp of the helical SDW state.As shown in Fig. 3f, J(qp) along the [100] direction features a typical paramagnetic spin susceptibility of a metal with a sharp and significant finite-q peak at qp=0.19 r.l.u., consistent with experimental data at 5 K.The estimated magnetic transition temperature,  !"# 345 ∝ J(qp)/3kB~14.8K (kB being the Boltzman constant), is also in agreement with experimental observation (see Supplementary Figure 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 EuAl4.Interestingly, the QCDW matches the calculated charge susceptivity peak along the G-Z direction 29 .Although the primary driving force of the CDW in EuAl4 remains 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 EuAl4, 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-Q phase and C4 symmetry-breaking (see Supplementary Figures S7 and S8), suggesting hidden chiral interactions above 20 K. Due to the intertwined nature of SDW and CDW, it is tempting to associate the chiral interaction with a chiral CDW.Encouragingly, the CDW in EuAl4 is 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 C4 rotational symmetry is expected to be broken below TCDW.However, the C4 symmetry-breaking in the ab-plane is observed only at  !"# (() =12.3 K<< TCDW=140 K 20,21 .
These observations, therefore, support a chiral CDW in EuAl4.It is highly interesting to point out that the CDW related "nesting" vector connects the topological semi-Dirac bands .

Methods:
Sample Growth: EuAl4 crystals 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 2.5g were loaded into one side of a 2-mL alumina Canfield Crucible Set.The crucible set was sealed under 1/3 atm 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 precipitated from the melt during a slow cool to 700 • C over 100 h (−2 • C/h).To liberate the crystals from the remaining liquid, the hot ampoule was removed from the furnace, inverted into a centrifuge, and spun.

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 Light Source II (NSLS-II).The photon energy, which is selected by a cryogenically cooled Si(111) double-crystal monochromator is 6.977 keV.The sample is mounted in a closed-cycle displex cryostat in a vertical scattering geometry, and the magnetic s-p scattering channel is measured using an Al(222) polarization analyzer and silicon drift detector.

CD-XRMS:
The CD-XRMS were performed at the 4-ID-D beamline of the Advanced Photon Source (APS), Argonne National Laboratory (ANL).The photon energy was tuned to the Eu L3 resonance (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 200x100 µm 2 full-width-half-maximum (FWHM) using a toroidal mirror, and further reduced to 50x50 µm 2 FWHM with slits.Temperature was controlled using a He closed-cycle cryostat.Diffraction was measured in reflection from the sample [001] surface using vertical scattering geometry and an energy dispersive silicon drift detector (approximately 0.15 keV energy resolution).

ARPES:
The ARPES experiments were performed on single crystals EuAl4.The samples are cleaved in situ in a vaccum better than 3×10 −11 torr.The experiment was performed at beam line 21-ID-1 at the NSLS-II.The measurements are taken resonant x-ray scattering, the cross-section related to Eq. (M6) and (M7) is extremely small, typically 10~30 counts/s for magnetic materials.
In our study, we not simply observed CD, but the asymmetry of the CD that is described in Eq. (M4).Furthermore, () = () <, − () <= () <, + () <= ⁄ ~90%, is very large, excluding contributions from Eqs. (M6) and (M7) as the origin of chirality flipping.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 H component and narrow width of magnetic peaks, horizontal axis from H=-0.17 to 0.17 summarizes the extracted QDW (in r.l.u.).As shown in Fig. 3a, the double-Q SDW first emerges below  !"# (%) along the [110] and [1-10] directions and is smoothly connected with the spin canted double-Q phase.In the chiral SDW phase below  !"# (() , QSDW increases monotonically along the [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 [ !"# (() , 20K]≪TCDW.The QCDW then jumps back to the commensurate value and remains T-independent in the c=-1 SDW phase ([ ) ,  !"# (() ]

Figure 1 :
Figure 1: Emergent chiral magnetic orders and the zero-field phase diagram of EuAl4.a, Spontaneous chiral symmetry breaking yields degenerated c=1 and c =-1 states.External field or intertwined orders can lift the degeneracy.b, schematics of 1-dimensional (1D) and 2D chiral spin textures, helical SDW (left) and Bloch-type Skyrmion (right).c, possible microscopic mechanisms that drive chiral magnetic states.Left: relativistic DM-interaction, D, in non-centrosymmetric lattice determines the sign of c (See Supplementary Note 1).The wavelength of the spin order, l, is proportional to the relative energy scales of atomic exchange energy, J, and D. Right: quasinested Fermi surface in hexagonal and tetragonal structures give rise to frustrated RKKY interactions along Q1, Q2, Q3 that 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, Q3 orders with Q1+Q2+Q3 =0.The Fermi surface topology of EuAl4 is similar to the one in the tetragonal lattice, where yellow and cyan represent electron and hole band, respectively (Supplementary Figure S4).d, phase diagram of EuAl4 without external magnetic field.Orthogonal CDW and SDW superlattice peaks are marked in the 3D and 2D Brillouin zone, respectively.Chiral SDW emerges below  !"# (

Figure 2 :
Figure 2: Discovery of spontaneous chirality flipping in EuAl4.a, experimental geometry of CD-XRMS.The photon energy was tuned to Eu L3-edge 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 from Q=QSDW to Q=-QSDW in the rocking scattering geometry.Here we define c=+1 if F(QSDW)>0.Fluorescence scan (bottom left) at T=5 K shows single peak at wres=6.977keV, confirming Eu 2+ configuration.
was not shown in e, g, and i.Giant CD is observed below  !"# ( =12.3 K.The sign change of the CD shown in e and g establishes the chirality flipping across  ) .

Figure 3 :
Figure 3: Intertwined SDW and CDW with orthogonal wavevectors.a and b, T-dependent Hscan and HK-scan near the SDW wave vectors.c, T-dependent L-scan near the CDW wave vector.