# Observation of Unconventional Charge Density Wave without Acoustic Phonon
Anomaly in Kagome Superconductors AV3Sb5 (A=Rb,Cs)

H. X. Li  Material Science and Technology Division, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA  T. T. Zhang  Department of
Physics, Tokyo Institute of Technology, Okayama, Meguro-ku, Tokyo 152-8551,
Japan  Tokodai Institute for Element Strategy, Tokyo Institute of Technology,
Nagatsuta, Midori-ku, Yokohama, Kanagawa 226-8503, Japan  T. Yilmaz  National
Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York
11973, USA  Y. Y. Pai  Material Science and Technology Division, Oak Ridge
National Laboratory, Oak Ridge, Tennessee 37831, USA  C. Marvinney  Material
Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831, USA  A. Said  Advanced Photon Source, Argonne National
Laboratory, Argonne, Illinois 60439, USA  Q. Yin  Department of Physics and
Beijing Key Laboratory of Opto-Electronic Functional Materials and Micro-
devices, Renmin University of China, Beijing, China  C. Gong  Department of
Physics and Beijing Key Laboratory of Opto-Electronic Functional Materials and
Micro-devices, Renmin University of China, Beijing, China  Z. Tu  Department
of Physics and Beijing Key Laboratory of Opto-Electronic Functional Materials
and Micro-devices, Renmin University of China, Beijing, China  E. Vescovo
National Synchrotron Light Source II, Brookhaven National Laboratory, Upton,
New York 11973, USA  R. G. Moore  Material Science and Technology Division,
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  S. Murakami
Department of Physics, Tokyo Institute of Technology, Okayama, Meguro-ku,
Tokyo 152-8551, Japan  Tokodai Institute for Element Strategy, Tokyo Institute
of Technology, Nagatsuta, Midori-ku, Yokohama, Kanagawa 226-8503, Japan  H. C.
Lei<EMAIL_ADDRESS>Department of Physics and Beijing Key Laboratory of Opto-
Electronic Functional Materials and Micro-devices, Renmin University of China,
Beijing, China  H. N. Lee  Material Science and Technology Division, Oak Ridge
National Laboratory, Oak Ridge, Tennessee 37831, USA  B. Lawrie  Material
Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831, USA  H. Miao<EMAIL_ADDRESS>Material Science and Technology
Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

(August 27, 2024)

###### Abstract

The combination of non-trivial band topology and symmetry breaking phases
gives rise to novel quantum states and phenomena such as topological
superconductivity, quantum anomalous Hall effect and axion electrodynamics.
Evidence of intertwined charge density wave (CDW) and superconducting order
parameters has recently been observed in a novel kagome material AV3Sb5
(A=K,Rb,Cs) that features a $\mathbb{Z}_{2}$ topological invariant in the
electronic structure. However, the origin of the CDW and its intricate
interplay with topological state has yet to be determined. Here, using hard
x-ray scattering, we demonstrate a three-dimensional (3D) CDW with $2\times
2\times 2$ superstructure in (Rb,Cs)V3Sb5. Unexpectedly, we find that the CDW
fails to induce acoustic phonon anomalies at the CDW wavevector but yields a
novel Raman mode that quickly damps into a broad continuum below the CDW
transition temperature. Our observations exclude strong electron-phonon
coupling driven CDW in AV3Sb5 and point to an unconventional particle-hole
condensation mechanism that couples CDW, superconductivity and topological
band structure.

The charge density wave (CDW), a translational symmetry-breaking electronic
fluid, plays a crucial role in unconventional superconductors and intertwined
electronic orders [1, 2, 3, 4, 5]. While CDWs have been isolated from
topological excitations, recently an experimental evidence of topological CDW
with chiral flux is observed in a new kagome metal, AV3Sb5 (A=K, Rb, Cs) [6],
whose crystal structure and 3D Brillouin zone are shown in Fig. 1(a) and (b),
respectively.

Figure 1: The crystal structure of AV3Sb5 and CDW induced phonon anomalies.
(a) shows the crystal structure of AV3Sb5 (space group P6/mmm, No. 191), which
consists of V-Sb slabs that are separated by alkali elements. The V-Sb slab
contains a V-kagome lattice and two Sb sites laying in the kagome plane and
above the V-triangle. (b) shows the 3D BZ and high symmetry points of AV3Sb5.
(c) and (d) show the valence and core electron density in the normal and CDW
phase, respectively. Due to EPC, the core electrons will follow the CDW period
and yield acoustic phonon anomalies. (e) Nesting induced Kohn anomalies. (f)
Strong coupling induced phonon softening in an extended momentum space near
$\mathbf{Q}_{\text{CDW}}$. Red, yellow, green, blue and purple colors
represent temperature from $T\gg$ $T_{\text{CDW}}$ to $T<$$T_{\text{CDW}}$.
The phonon hardening for $T<$$T_{\text{CDW}}$ is due to the CDW amplitude
mode.

Density functional theory (DFT) calculations of the electronic structure find
a $\mathbb{Z}_{2}$ topological invariant and saddle points at the M point that
may lead to a CDW instability at low temperature [7, 8, 6, 9, 10].
Interestingly, scanning tunneling microscopy (STM) studies of CsV3Sb5 observed
evidence of roton pair-density wave and Majorana zero mode, suggesting novel
topological superconductivity intertwines with CDW [11, 12]. Despite the
intimate correlations between the CDW, superconductivity (SC) and topological
band structure [7, 8, 6, 13, 9, 10, 14, 11, 12, 15, 16], the nature of the
CDW, in particular the role of electron-phonon coupling (EPC), remains
unresolved, hindering the proper interpretations of SC in AV3Sb5.

Theoretically, the CDW mechanisms are broadly separated into two categories:
weak coupling scenarios based on Fermi-surface instabilities [17, 18, 19] and
strong coupling theories derived from local electron-electron or electron-
phonon interactions [20, 21, 22, 23]. As we show in Fig. 1(c) and (d), due to
the finite EPC, the formation of CDWs distorts the underlying lattice and
typically results in acoustic phonon anomalies. For a weak coupling mechanism,
the phonon softening is sharply confined near the CDW wavevector,
$\mathbf{Q}_{\text{CDW}}$, and is known as Kohn anomaly[19] [illustrated in
Fig. 1(e)]. In contrast, strong coupling mechanisms usually give rise to
phonon anomalies in an extended Q-range near $\mathbf{Q}_{\text{CDW}}$ [24,
25, 26, 27], as shown in Fig. 1(f). Here we combine inelastic x-ray scattering
(IXS), Raman spectroscopy and angle-resolved photoemission spectroscopy
(ARPES) to uncover a novel particle-hole condensation in AV3Sb5 (A=Cs, Rb).
Unlike conventional CDW materials, we demonstrate that neither longitudinal
(LA) nor transverse (TA) acoustic phonon modes show CDW induced phonon
anomalies in AV3Sb5 (A=Cs, Rb). Instead, we discover a novel Raman mode that
emerges at the CDW temperature, $T_{\text{CDW}}$, and is quickly damped below
$T_{\text{CDW}}$ into a continuum that broadly peaked at 20 meV. Intriguingly,
the three-dimensional (3D) $2\times 2\times 2$ superstructure, determined by
hard x-ray diffraction (XRD), connects particle-hole scattering channels in
the electronic band structure and yields a band-inversion near the Fermi
level, $E_{\text{F}}$, below $T_{\text{CDW}}$. Our results uncover the
intricate interplay between non-trivial band topology and intertwined symmetry
breaking phases and shed light on the unusual SC in CsV3Sb5.

We first determine the bulk CDW superlattice peaks using meV resolution XRD.
Figure 2(a) and (b) show the temperature dependent $L$-scan of RbV3Sb5 at
$\mathbf{Q}=(3.5,0,0)$ and $(3.5,0,0.5)$ in reciprocal lattice units (r.l.u.).

Figure 2: 3D-CDW determined by meV-resolution XRD. (a,b) The temperature
dependent CDW peaks of RbV3Sb5 at $\mathbf{Q}=(3.5,0,0)$ and $(3.5,0,0.5)$.
The CDW peak at half-integer $L$ demonstrates a 3D CDW with $2\times 2\times
2$ superstructure. The CDW peaks are extremely sharp along $L$, with HWHM of
$\sim 0.0025$ r.l.u., at T=98 K. (c) and (d) show the extracted temperature
dependent CDW peak intensity at $\mathbf{Q}=(3.5,0,0)$ and $(3.5,0,0.5)$,
respectively. The peak intensity takes a sharp upturn at the CDW onset
temperature of 103 K for RbV3Sb5, which is consistent with the specific heat
measurement (see Appendix A). The absence of hysteresis is consistent with a
second order phase transition. CDW peaks in CsV3Sb5 are shown in (e) and (f).
The peak intensity at $\mathbf{Q}=(3,0.5,0)$ is over 10 times larger than the
one at $\mathbf{Q}=(3.5,0,0)$ [panel (e)], consistent with larger structure
factors at $\mathbf{Q}=(H,K+0.5,0)$, $H=$odd integer [8]. The absence of peaks
at $\mathbf{Q}=(3,0,0.5)$, $(3.5,0.5,0)$, $(3.5,0,0.25)$ confirms the same
$2\times 2\times 2$ superstructure in CsV3Sb5. The error bars in panel
(a),(b),(e),(f) represent one standard deviation assuming Poisson counting
statistics. The error bars in panel (c) and (d) denote the $2\sigma$ returned
from the pseudo-voigt fittings that extract the peak intensity.

The onset temperature at $T=103$ K is consistent with the CDW transition
temperature in RbV3Sb5 (see Appendix A). The absence of hysteresis, as shown
in Fig. 2(c) and (d), suggests that the CDW order is a second order phase
transition. The observation of a CDW peak at half integer $L$ in Fig. 2(b)
proves that the CDW in RbV3Sb5 forms a 3D $2\times 2\times 2$ superstructure
rather than a 2D $2\times 2$ superstructure as suggested by previous STM and
XRD measurements [8, 6, 14, 11, 12]. Indeed, the CDW correlation length,
determined by the half-width-at-half-maximum (HWHM), is larger than 1000 Å
along the $L$-direction, establishing a long range ordered 3D-CDW. As shown in
Fig. 2(e) and (f), the same 3D-CDW superlattice structure is also observed in
CsV3Sb5, suggesting a ubiquitous CDW mechanism for all AV3Sb5. Interestingly,
we find that while the CDW is long-range ordered, the integrated CDW intensity
that is proportional to the CDW order parameter is extremely small. Comparing
with fundamental Bragg peaks, the CDW peak intensity is $3\sim 5$ orders of
magnitude weaker, demonstrating small lattice distortions.

With the 3D-CDW established, we explore the origin of the CDW in AV3Sb5 using
IXS and Raman spectroscopy. As we have shown in Fig. 1, the formation of CDWs
is always accompanied by acoustic phonon anomalies because of EPC. Indeed,
even in the cuprates, where CDWs are short ranged and possibly dynamical, a
$15\%$ LA phonon softening has been observed [24, 25], demonstrating acoustic
phonon is a sensitive probe of CDW correlations. Recent DFT calculations of
the phonon spectrum of AV3Sb5 show large negative frequency of the LA near the
M and L point, suggesting EPC induced CDW [13] (see supplementary materials in
ref.[28]). To test this scenario, we performed IXS measurement along
$(3,0,0)-(3.5,0,0)$ and $(3,3,0)-(3.5,2.5,0)$ to selectively enhance LA and TA
phonon modes in our experimental geometry (see [28]. Unexpectedly, as we show
in Fig. 3(a)-(e), CDWs in AV3Sb5 failed to induce phonon anomalies on both TA
and LA modes from $T=300$ K $>$ $T_{\text{CDW}}$ to 50 K $<$ $T_{\text{CDW}}$.

Figure 3: Evidence of unconventional electronically driven CDW. (a) compares
the IXS determined LA of RbV3Sb5 along $\Gamma(3,0,0)$ to M$(3.5,0,0)$ at
temperatures of $T=300$ K and 50 K ($T_{\text{CDW}}$=103 K). (b) Extracted
temperature dependent phonon dispersions from (a). (c) TA in RbV3Sb5 measured
along $\Gamma(3,3,0)$ to M$(3.5,2.5,0)$ at room temperature (300 K). (d) and
(e) compare temperature dependent IXS spectra at $\mathbf{Q}=(3.47,0,0)$ and
$(3.5,2.5,0)$. The dashed lines in (d) and (e) highlight LA and TA,
respectively. All IXS data are Bose-factor corrected. The raw IXS data is
shown in ref.[28]. (f),(g) Temperature dependent Raman spectra on CsV3Sb5 from
150 K to 3 K. Beside the sharp optical phonon peak at 11, 15 and 17 meV, there
is a continuum that is broadly centered near 19 meV (The 11 meV peak may
actually be at lower energy but appeared to be at 11 meV due to the cutoff of
the longpass filter at 90 cm-1). The black dots in (f) mark the peak positions
of the continuum near 19 meV. When cooling towards $T_{\text{CDW}}$, the
center of this continuum gradually shifts from 19.5 to 19 meV and starts to
build a strong, sharp and asymmetric peak at $T_{\text{CDW}}$ = 94 K. Moving
to lower temperature, this peak is quickly damped and hardened to 20 meV at 30
K. Below 30 K, two new phonon modes at 25.4 and 27.5 meV ared observed, which
may correspond to the stripe phase observed by a recent STM study [14]. The
error bars in panel (d), (e) represent one standard deviation assuming Poisson
counting statistics. The error bars in panel (b) denote the $2\sigma$ returned
from the fitting that extract the spectral peak position.

The absence of phonon softening excludes strong EPC as the driven force of the
CDW in AV3Sb5. Instead, our inelastic and elastic results indicate an
electronic-driven CDW mechanism with extremely weak electron acoustic-phonon
interactions.

To confirm this scenario, we performed temperature dependent Raman
spectroscopy on CsV3Sb5 with $T_{\text{CDW}}$=94 K. The results are shown in
Fig. 3(f) and (g). According to DFT calculations (see supplementary materials
in [28]), four Raman active modes at 8.6, 15.1, 16.2 and 17.4 meV,
corresponding to $E_{1g}$, $E_{2g}$, $A_{1g}$ and $B_{2g}$ modes,
respectively, are allowed for the normal state structure. At 150 K, we observe
three optical phonon modes at 11, 15 and 17 meV, which we attribute to
$E_{1g}$, $E_{2g}$ and $B_{2g}$ modes (The 11 meV peak may actually be at
lower energy but appeared to be at 11 meV due to the cutoff of the longpass
filter at 11 meV). Interestingly, we observe a continuum that is broadly
centered around 19.5 meV. As we cool down towards the $T_{\text{CDW}}$, the
center of this continuum gradually shifts to lower energy and eventually
develops into a strong, sharp and asymmetric peak at 19 meV, demonstrating its
intimate correlation with the CDW. Furthermore, 19 meV is nearly two times of
the STM determined single-particle gap at $E_{\text{F}}$ [6, 14, 11, 12],
consistent with the two-particle process of Raman scattering. Unexpectedly,
moving to lower temperature, this peak is suppressed and hardened to 20 meV at
30 K. The novel temperature dependence of the broad Raman mode strongly
supports an electronic origin of the CDW. Remarkably, although Raman
spectroscopy measures excitations at the Brillouin zone center, this
unconventional mode shows clear softening above $T_{\text{CDW}}$ and hardening
below $T_{\text{CDW}}$, reminiscent of the CDW phason and amplitudon
excitations near the $\mathbf{Q}_{\text{CDW}}$ [26, 27]. A similar effect has
been observed in CDW material TaSe2 [27], which also has a 3D $2\times 2\times
2$ superstructure. We suspect that CDWs in AV3Sb5 might be a novel realization
of particle-hole condensation in a metallic system, where the particle-hole
excitation is highly damped by itinerant electrons. In this scenario, the
absence of phonon softening might be a signature of large CDW phason gap that
is induced by commensurability effect [29]. Indeed, As we discuss in more
details in Appendix B, the combination of CDW commensurate period and the
unique electronic structure of AV3Sb5 allows a large gap in the CDW phase
mode. We note, below 30 K, we observe additional Raman active modes at 25.4
and 27.5 meV, demonstrating additional symmetry breaking phase at low
temperature. Comparing with previous STM study [14], this new low-temperature
phase is possibly the unidirectional CDW. Interestingly, this new symmetry
breaking phase correlates strongly with the CDW-related continuum, where the
intensity of the continuum significantly suppressed when the new Raman modes
emerge.

To disclose the particle-hole scattering channel that is connected by the
3D-CDW, in Fig. 4, we show the low-temperature ($T=15$ K) electronic structure
of RbV3Sb5 determined by ARPES measurements and DFT calculations.

Figure 4: 3D-CDW induced band-inversion. (a) and (b) show Fermi surface maps
measured at photon energies of $h\nu=100$ and 90 eV, corresponding to
$k_{z}=\pi$ and $0.15\pi$ around the $\overline{\text{M}}$ point,
respectively. Data was taken at $T=15$ K. The blue lines in (a) and (b) mark
the high symmetry cuts shown in (c) and (d), respectively. The inner potential
is extracted to be around 8.2 eV (see supplementary materials [28]). (c) and
(d) selectively enhance the electron band near the unfolded L point and the
hole band near the unfolded M point. The yellow dots in (c) and (d) are
extracted from the MDC peaks shown in (e). The blue dashed line in (d) is a
guide to the eye. The blurry spectral weight near the Fermi level in (d)
likely comes from the bulk projected surface state [8].(e) and (f) show
stacking plots of MDCs to further reveal the electron and hole bands in panel
(c) and (d). (g) shows the electronic band structure of RbV3Sb5 calculated by
VASP [30, 31, 32] after a full relaxation on atomic positions with an atom
pair-wise correction method (DFT-D3) [33] since the van der Waals forces play
an important role along the c-layer stacking direction. The Fermi energy is
shifted by 100 meV in order to match the experimental results. There is a hole
pocket in the vicinity of M point ($\Gamma$-M direction) carrying a parity of
“+” and an electron pocket in the vicinity of L point carrying a parity of
“-”, contributed by different bands. Our DFT result confirm the parity
distribution of the band structure that is reported to possess a non-trivial
$\mathbb{Z}_{2}$ topological invariant and induce topological surface states
around M point [8].

As shown in Fig. 4(a) and (b), the overall Fermi surface topology is similar
to previous ARPES studies of KV3Sb5 and CsV3Sb5 [34, 8]. Photon energies of
$h\nu=100$ and 90 eV are corresponding to $k_{z}=\pi$ and $0.15\pi$ around the
$\overline{\mathrm{M}}$ point, respectively (see supplementary material [28]
for the $k_{z}$ map). The most important observations are shown in Fig. 4(c)
and (d), corresponding to the blue-cut in Fig. 4(a) and (b), respectively. At
$k_{z}=\pi$, we observe a shallow electron-band with a band bottom of $\sim
50$ meV, while at $k_{z}=0.15\pi$, we observe a steep hole-like band
dispersing towards the Fermi level. The shallow electron and hole band can be
further revealed by the stacking momentum distribution curves (MDCs) displayed
in Fig. 4(e) and (f). Similar electronic structure is also observed in CsV3Sb5
(see supplementary materials [28]). Comparing with the DFT calculations shown
in Fig. 4(g), we find that the electron and hole bands correspond to band
#49,50 at the L point [Fig. 4(g)] and band #47,48 at the M point,
respectively. The parity distribution confirms the opposite parity between
these two bands, which is reported to hold a non-trivial $\mathbb{Z}_{2}$
topology [8]. We note that at $k_{z}=0.15\pi$, a weak electron-like spectral
weight is observed near the Fermi level. It likely comes from the bulk
projected surface state as suggested in a previous study [8]. Our observations
establish two particle-hole channels that are relevant to the CDW: (i) the
quasi-nested electron and hole pockets at the M and L point [see Fig. 1(b)]
and (ii) the saddle points at the M point [4, 5, 9, 10]. The former is likely
responsible for the $L$-component of the 3D-CDW , while the latter may play
crucial role for the in-plane component as suggested by previous studies [35,
5, 10]. Interestingly, in the CDW phase, the shallow electron and hole-like
bands will fold to the Brillouin zone center and open a band gap between bands
with opposite parity. This CDW induced band inversion near the Fermi level may
possibly responsible for the novel SC at low temperature [14, 11, 12, 16].

In summary, we demonstrated a novel electronic driven 3D-CDW in the
$\mathbb{Z}_{2}$ kagome superconductor AV3Sb5, where the formation of CDW
failed to induce phonon anomalies near $\mathbf{Q}_{\text{CDW}}$. Our
observations exclude strong EPC driven CDW in AV3Sb5 and point to an
unconventional particle-hole condensation mechanism that couples the CDW and
the topological band structure.

Acknowledgements: We thank J. G. Cheng, R. Thomale, Z. Q. Wang, S. F. Wu, B.
H. Yan and J. X. Yin for stimulating discussions. This research was sponsored
by the U.S. Department of Energy, Office of Science, Basic Energy Sciences,
Materials Sciences and Engineering Division (IXS, ARPES data analysis, Raman
spectroscopy) and by the Laboratory Directed Research and Development Program
of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U. S.
Department of Energy (IXS and ARPES experiment). This research used resources
(IXS experiment at beamline 30ID) 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 under Contract No. DE-AC02-06CH11357. ARPES
measurements used resources at 21-ID-1 beamlines of the National Synchrotron
Light Source II, a US Department of Energy Office of Science User Facility
operated for the DOE Office of Science by Brookhaven National Laboratory under
contract no. DE-SC0012704. Extraordinary facility operations were supported in
part by the DOE Office of Science through the National Virtual Biotechnology
Laboratory, a consortium of DOE national laboratories focused on the response
to COVID-19, with funding provided by the Coronavirus CARES Act. Raman
spectroscopy was conducted at the Center for Nanophase Materials Sciences,
which is a DOE Office of Science User Facility. T.T. Z. and S. M. acknowledge
support from Tokodai Institute for Element Strategy (TIES) funded by MEXT
Elements Strategy Initiative to Form Core Research Center. S. M. also
acknowledges support by JSPS KAKENHI Grant Number JP18H03678. H.C.L
acknowledges the supports from the National Key R&D Program of China (Grant
Nos. 2016YFA0300504, and 2018YFE0202600), the National Natural Science
Foundation of China (Grant Nos. 11774423 and 11822412), the Fundamental
Research Funds for the Central Universities, and the Research Funds of Renmin
University of China (Grant Nos. 18XNLG14 and 19XNLG17). C.M. was supported by
the Intelligence Community Postdoctoral Research Fellowship Program at the Oak
Ridge National Laboratory, administered by Oak Ridge Institute for Science and
Education through an interagency agreement between the U.S. Department of
Energy and the Office of the Director of National Intelligence.

##  Appendix A Methods

Sample: Single crystals of AV3Sb5 (A = Rb, Cs) were grown from Rb ingot
(purity 99.9%), Cs ingot (purity 99.9%), V powder (purity 99.9%) and Sb grains
(purity 99.999%) using the self-flux method, similar to the growth of RbV3Sb5
[13]. The mixture was put into an alumina crucible and sealed in a quartz
ampoule under partial argon atmosphere. The sealed quartz ampoule was heated
to 1273 K for 12 h and soaked there for 24 h. Then it was cooled down to 1173
K at 50 K/h and further to 923 K at a slowly rate. Finally, the ampoule was
taken out from the furnace and decanted with a centrifuge to separate CsV3Sb5
single crystals from the flux. The obtained crystals have a typical size of
$2\times 2\times 0.02$ mm3. CDW transition is clearly observed in transport
and specific heat measurement shown in Fig. A1(a) and (b), respectively.

Figure A1: (a) and (b) show transport and specific heat measurement of RbV3Sb5
(red) and CsV3Sb5(blue).

Inelastic x-ray scattering: The experiments were conducted at beam line
30-ID-C (HERIX) at the Advanced Photon Source (APS). The highly monochromatic
x-ray beam of incident energy $E_{i}$ = 23.7 keV (l = 0.5226 Å) was focused on
the sample with a beam cross section of 35$\times$15 $\mu$m2 (horizontal ×
vertical). The total energy resolution of the monochromatic x-ray beam and
analyzer crystals was $\Delta E\sim$1.5 meV (full width at half maximum). The
measurements were performed in transmission geometry. Typical counting times
were in the range of 30 to 360 seconds per point in the energy scans at
constant momentum transfer $\mathbf{Q}$. $H,\ K,\ L$ are defined in the
hexagonal structure with a=b=5.472 Å, c=9.073 Å at the room temperature for
RbV3Sb5, and a=b=5.495 Å, c=9.309 Å at the room temperature for CsV3Sb5.

ARPES experiment: The ARPES experiments were performed on single crystals of
RbV3Sb5 and CsV3Sb5. The samples were cleaved $in$-$situ$ in a vaccum better
than 5$\times$10${}^{-11}\ torr$. The experiment was performed at beamline
21-ID-1 at National Synchrotron Light Source II, Brookhaven. The measurements
were taken with synchrotron light source and a Scienta-Omicron DA30 electron
analyzer. The total energy resolution of the ARPES measurement is $\sim$15
meV. The sample stage was maintained at low temperature ($T=$15 K) throughout
the experiment.

Raman experiment: Raman spectroscopy was performed in a Montana Instruments
closed cycle Cryostation s100 and utilized a Hubner Photonics 532 nm diode
pumped laser excitation and an Isoplane SCT-320 imaging spectrograph with a
400B-eXcelon CCD camera and a 2400 groove/mm visible-holographic grating.
Semrock dichroic and long pass filters were integrated in the optics train.
For all Raman spectra reported here, a 2 mW laser excitation power was used,
and Raman spectra were acquired for 300 s each. The sample was initially
cooled to 3 K then heated to 150 K. All of the reported Raman spectra were
then acquired while monotonically recooling the sample from 150 K to 3 K. The
optics train was refocused at each temperature after waiting for the sample
temperature to stabilize in order to correct for changes due to thermal
expansion and to ensure that the sample was in thermal equilibrium before
acquiring a spectrum.

DFT calculations: The electronic band structure of RbV3Sb5 calculated by VASP
[30, 31, 32] after a full relaxation on atomic positions with an atom pair-
wise correction method (DFT-D3) [33]. The phonon band structure of RbV3Sb5 is
calculated using VASP within the Perdew-Burke-Ernzerhof exchange correlation
based on density functional perturbation theory. An equivalent k-point mesh of
$9\times 9\times 6$ is used in the self-consistent calculation, and the cutoff
energy for the plane-wave basis is 400 eV. Prior to the phonon spectra
calculation, crystal structure is relaxed with the residual force on each atom
is less than 0.001 eV/Å.

##  Appendix B Commensurability effect

For incommensurate CDW (IC-CDW), the condensation energy, $E_{cond}$ is
independent of the phase of CDW, $\phi$. For commensurate CDW (C-CDW),
however, the condensation energy becomes phase dependent:

$E_{\text{cond}}(\phi)=-\frac{n(\epsilon_{\text{F}})\Delta_{\text{CDW}}^{2}}{\lambda}(\frac{\Delta}{D})^{M-2}cos(M\phi)$
(1)

where $\lambda$ is the dimensionless EPC constant, $n(\epsilon_{\text{F}})$
and $\Delta$ are the density-of-state at $E_{F}$ and the CDW gap,
respectively. $M$ is the CDW period relative to the lattice constant $a_{0}$.
$D$ is the electronic band width. This phase dependent $E_{\text{cond}}(\phi)$
means that gliding the CDW requires finite energy, corresponding to a finite
gap in the CDW phase mode:

$\omega_{\phi}(q)=(\frac{\omega_{F}^{2}}{M}+v_{F}^{2}\frac{m}{m^{*}}q^{2})^{1/2}$
(2)

where
$\omega_{F}^{2}=\frac{4M^{2}}{\lambda}\frac{m\Delta^{2}}{m^{*}}(\frac{\Delta}{D})^{M-2}$.
$m^{*}$ is the effective mass. At $q=q_{CDW}$, Eq. 2 gives:

$\omega_{\phi}=\sqrt{\frac{4M}{\lambda}\frac{m}{m^{*}}}\Delta(\frac{\Delta}{D})^{\frac{M-2}{2}}$
(3)

Usually $D\gg\Delta$ and therefore, $\omega_{\phi}\sim 0$. However, for $M=2$,
which is the case of AV3Sb5,
$\omega_{\phi}=\sqrt{\frac{4M}{\lambda}\frac{m}{m^{*}}}\Delta$. Most
interestingly, a simple parabolic fitting of the ARPES data shown in Fig. 4
gives $\frac{m}{m^{*}}\sim$3 for the shallow electron and hole pockets,
suggesting large $\omega_{\phi}$ in AV3Sb5. As we shown in Fig. A2c, due to
the large phason gap, the particle-hole condensation will not interacting with
acoustic phonon modes, as these modes are laying inside the phason gap.

Figure A2: (a) schematically shows commensurate (yellow) and incommensurate
(cyan) CDW periods with respect to the lattice (red). (b) Yellow and cyan
curves are the calculated phase modes for C-CDW and IC-CDW, respectively. (c)
shows the CDW mechanism without electron-acoustic phonon interaction. The
temperature dependent particle-hole excitations are shown as red, yellow and
blue curves. The dashed line show LA mode, whose energy at $q_{CDW}$ is less
than $\omega_{\phi}$. Shaded pink area, indicates the momentum position of
Raman measurement.

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