Signature of spin-phonon coupling driven charge density wave in a kagome magnet

The intertwining between spin, charge, and lattice degrees of freedom can give rise to unusual macroscopic quantum states, including high-temperature superconductivity and quantum anomalous Hall effects. Recently, a charge density wave (CDW) has been observed in the kagome antiferromagnet FeGe, indicative of possible intertwining physics. An outstanding question is that whether magnetic correlation is fundamental for the spontaneous spatial symmetry breaking orders. Here, utilizing elastic and high-resolution inelastic x-ray scattering, we observe a c-axis superlattice vector that coexists with the 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document}×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document}×1 CDW vectors in the kagome plane. Most interestingly, between the magnetic and CDW transition temperatures, the phonon dynamical structure factor shows a giant phonon-energy hardening and a substantial phonon linewidth broadening near the c-axis wavevectors, both signaling the spin-phonon coupling. By first principles and model calculations, we show that both the static spin polarization and dynamic spin excitations intertwine with the phonon to drive the spatial symmetry breaking in FeGe.

Hove singularity for CDW in FeGe.
Here,  4 = 2Y(1 + ) 9 − cos 9  is the magnon dispersion, with  the exchange constant, which will be taken as the unit of energy below, and  represents the single-ion anisotropy along the z axis.The two magnon modes,  = 1 and  = 2, have opposite helicities.
Introducing the Fourier transform of the lattice displacement fields as  6,* = T  (;) are corresponding annihilation (creation) operators of longitudinal phonons, the phonon-magnon coupling is written as: sin}N 3 ; +  -3 Oj  %,4 ;  9,-4-3 ; + .. (5) Here,  is the mass of a magnetic ion,  ),3 and  2,3 are the dispersion of acoustic and optical phonons given by  ),3 = T where the subscript of the phonon dispersion is removed for simplicity as  3 =  ),3, .The phonon self-energy is computed by the second-order perturbation using the Matsubara formalism.The self-energy is schematically shown in Fig. S6, and its analytic form is given by: where  # and  0 are bosonic Matsubara frequencies given by  # = 2 and  0 = 2, respectively, with  and  begin integers.After the summation over the Matsubara frequency  0 and the analytic continuation to real frequency as  # →  + , where  is a small imaginary number, one obtains the phonon retarded self-energy at zero temperature as: As mentioned previously, phonon-magnon coupling should disappear at  → 0 for longitudinal phonons.One can confirm this behavior by noticing While magnon dispersions in FeGe have been reported yet, based on the current theoretical analysis, we speculate that longitudinal phonons decay into two magnons at near the A point, experiencing the sudden broadening of the spectral function.We shall note that our analysis is based on a simplified 1D spin-chain and hence serves as a proof-of-principle analysis.In FeGe, as we have shown in the main text, the complicated lattice distortion is expected to give rise more complicated spin-phonon coupling and phonon self-energy effects.
surface topology and ARPES intensity plots along highsymmetry directions.a and b are Fermi surface mappings of FeGe measured by circular left (CL) and circular right (CR) incident photon polarization, respectively.The data were taken at photon energy hn=100 eV.The surface high symmetry points are defined in Fig.1 of the main text.c, DFT+DMFT calculated Fermi surface at kz=0 in the A-type AFM phase at U=4.2 eV and JH=0.88 eV shows good consistence with experimental data.d and e, ARPES intensity plots along the high symmetry - and - direction, respectively.In agreement with previous study, the van Hove singularity at the  point is slightly above the Fermi level 24 .The overall Fermi surface topology and band structure are very similar to FeSn 3 , challenging a weak-coupling picture based on van absence of Kohn-anomaly at  !// and QL.a, experimental S(Q, w) along the G(0, 0, 4)-M(0.5 ,0, 4)-L(0.5, 0, 4.5)-A(0, 0, 4.5)-G(0, 0, 4) direction determined by inelastic xray scattering at 200 K. b-e, IXS spectra at Q=(0.5, 0, 4), (0.5, 0, 4.5), (0, 0, 4.5) and (0, 0, 4.45).These curves were taken at 120 K, slightly above TCDW to avoid strong elastic intensity from CDW and charge dimer.Dashed black curves are fittings of the experimental data (see Methods).The red squares shown in b-e are the extracted phonon peak positions at 200 K from the same Qs and the same phonon modes.The lateral size of the red squares represents the instrumental energy resolution.The vertical error bars in panel b-e represent 1-standard deviation assuming Poisson counting statistics.vibrations of the B1u phonon modes at the three equivalent L-points.a,b,c, for L points (0.5, 0, 0.5), (0, 0.5, 0.5), (-0.5, 0.5, 0.5), respectively.The arrows indicate movements of Fe and Ge atoms.In addition to the 2 × 2 × 2 superstructure, we have also employed other lattice distortion ansätze, including (i)1 × 2 × 2 superstructure, corresponding to any one of the three B1u mode at L-point; (ii) √3 × √3 × 2 superstructure corresponding to the B1u phonon mode at H-point; and (iii) a √5 × √5 × 2 superstructure.All these ansätze yield ground state energies higher than the 2 × 2 × 2 superstructure and the original ideal Kagome structure.This indicates that the 2 × 2 × 2 superstructure optimized by DFT is indeed the true ground state even though DE for small U is close to zero.This small energy difference at small U may be consistent with experimental observation where both the CDW transition temperature and correlation length can be reduced significantly by post annealing process 51 .Supplementary Figure 6: Calculated electronic band structures of FeGe in AFM phase.a, Electronic band structures calculated by DFT.b, c, d, Electronic spectra functions calculated by DFT+DMFT at  = 3 ,  $ = 0.75 ;  = 4.2 ,  $ = 0.88  and  = 5 ,  $ = 0.95 , respectively.The van-Hove singularities are located at M point near the Fermi level.
eV, J H =0.95 eV U=4.2 eV, J H =0.88 eV d DFT+DMFT U=3.0 eV, J H =0.75 eV Supplementary Figure7: Spin-polarization-induced 2×2×2 superstructure in FeSn.As we show in Fig.S2, the Fermi surface topology is very similar between the isostructure FeSn and FeGe.Indeed, FeSn also features A-type AFM with a TN~365 K 36 .Therefore, it is important to understand why FeSn fails to stabilize the 2×2×2 superstructure.a,b,c, Similar to FeGe, the calculated phonon spectra of FeSn also show dramatic change of the  %& phonon at L-point as increasing U, as marked by red circles.d, Left y-axis shows the energy difference between the locally "charge-dimerized" 2 × 2 × 2 superstructure and the ideal Kagome phase, Δ =  '()*+,-./0,*−  1)+20, .In contrast to FeGe, the charge dimerized superstructure has lower energy only at very large U (>2.4 eV), suggesting that FeSn favors a ground state with ideal Kagome structure.Possible reason for the difference between these compounds is that the c-axis lattice parameter of FeSn is about 10% larger than that of FeGe, which makes the dimerization along c-axis unfavorable in FeSn.Indeed, the spin polarization enhancement in FeGe is about twice larger than FeSn for U<1 eV.

2|𝑞|𝑐𝑜𝑠 9
at small .Thus, (, ) indeed becomes zero at  = 0 .With this self-energy, the phonon retarded Green's function is given by: summarizes the results of the present analysis using two sets of magnon dispersions, one without single-site anisotropy  = 0 (a-c) and one with anisotropy  = 0.05 (df).Here, the intensity of the phonon self-energy is scaled by  9 8 ⁄ √.To compute the Green's function (, ), phonon-magnon coupling  9 8 ⁄ √ = 0.2 and the bare band width of phonon dispersions 2Y  ⁄ = 3 are used.When  = 0, magnons have gapless excitations at  = 0, i.e.,  T = 0. Therefore, the continuum of the imaginary part of the phonon self-energy has a lower bound set by magnon excitation energy  4 because the energy and momentum conservation  3 =  3 is always satisfied [see Fig. 11 (a)].With nonzero ,  T = 2Y(2 + ), therefore  3 >  3 and there is a gap  T between the continuum and magnon dispersion  3 [Fig.11 (d)].The real part of the self-energy is related to the imaginary part by the Kramers-Kronig relation.Therefore, it has a negative peak near the lower bound of the continuum of the imaginary part and a positive peak slightly above the lower bound [Figs.11 (b,e)].Obviously, the continuum of the imaginary part of the self-energy has upper bound 2 3UV 9 ⁄ at  = .The phonon Green's function reflects these behaviors of the self-energy.When phonon dispersion  3 is below the continuum, the phonon spectral function has a well-defined peak at  =  3 + ℜN,  3 O , and when  3 hits the continuum, the phonon spectral function rapidly loses its intensity [Figs.11 (c,f)].

Figure 11 :
Figure 11: Numerical results of the phonon self-energy (, )and Green's function (, ).a-c,  = 0 and d-f,  = 0.05.(a,d) The Imaginary part of (, ), (b,e) the real part of (, ), and (c,f) the imaginary part of (, ).The intensity of the phonon self-energy is scaled by is the phonon-magnon coupling constant, representing the modulation of exchange by atomic displacements,  6,* is a longitudinal displacement of the lattice site at sublattice  at position .is the lattice constant, which will be taken to be unity.6,* (;) is a magnon annihilation (creation)operator at sublattice  at position .For simplicity, we only consider the Fourier transform of magnon annihilation and creation operators as  6,* = T