# Charge density waves and electronic properties of superconducting kagome
metals

Hengxin Tan  Yizhou Liu  Department of Condensed Matter Physics, Weizmann
Institute of Science, Rehovot 7610001, Israel  Ziqiang Wang  Department of
Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA  Binghai Yan
Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot
7610001, Israel

(August 28, 2024)

###### Abstract

Kagome metals $A$V3Sb5 ($A=$ K, Rb, and Cs) exhibit intriguing
superconductivity below $0.9\sim 2.5$ K with possibly $\mathbb{Z}_{2}$
topological surface states, after a charge density wave (CDW) transition
around $80\sim 100$ K. The nature of the CDW phase and its relation to the
superconductivity remain elusive. In this work, we investigate the electronic
and structural properties of the CDW phase by first-principles simulations. We
reveal an inverse star of David structure as the $2\times 2$ CDW ground state
of the kagome lattice. The CDW transition is driven by the Peierls instability
characterized by the softening of a breathing phonon mode in the kagome
lattice. The simulated scanning tunneling microscopy (STM) images of the
inverse star of David structure agree with recent experiments on the KV3Sb5
surface. The nontrivial $\mathbb{Z}_{2}$-type topological band structure
remains the same after the CDW transition. Further, we evaluate the electron-
phonon coupling in the CDW phase and find it is too weak to rationalize the
superconductivity in all three materials. It implies the existence of
unconventional pairing of these kagome metals. Our results provide essential
knowledge toward understanding the superconductivity and topology in kagome
metals.

##  I Introduction

The kagome lattice constitutes uniformly tiled triangles and hexagons in a
plane. Related insulating materials are actively studied for intriguing
phenomena like spin liquidsShores  _et al._ (2005), geometrical
frustrationRamirez (1994), charge density waves (CDW)Isakov  _et al._ (2006),
and unconventional superconductivity of doped Mott insulatorsKo  _et al._
(2009). For the more recent interest of topology, several magnetic kagome
metals were found to be magnetic Weyl semimetalsYang  _et al._ (2017); Morali
_et al._ (2019); Liu  _et al._ (2019) and exhibit a large anomalous Hall
effectNakatsuji  _et al._ (2015); Nayak  _et al._ (2016); Liu  _et al._
(2018); Ye  _et al._ (2018). Very recently, nonmagnetic kagome metals $A$V3Sb5
($A=$ K, Rb, and Cs)Ortiz  _et al._ (2019) were reported to exhibit both
charge density waves and superconductivity, and $\mathbb{Z}_{2}$-type
nontrivial band topologyOrtiz  _et al._ (2020); Ortiz and Sarte (2020), which
sparked immediate interest in these materialsYang  _et al._ (2020); Yu  _et
al._ (2021); Wang  _et al._ (2020).

The pristine phases of $A$V3Sb5 exhibit a uniform kagome lattice of V atoms at
room temperature. The $\mathbb{Z}_{2}$ topological metal was found by
calculations and angle-resolved photoemission spectroscopyOrtiz  _et al._
(2020). As cooling down to about $80\sim 100$ K, $A$V3Sb5 goes through a CDW
transition in experiment. The superconductivity emerges from the CDW phase
with T${}_{c}=0.9\sim 2.5$ K (see Table 1 for summary). For CsV3Sb5, very
recent experiments reveal possible nodal superconductivityZhao  _et al._
(2021a) and the competition between superconductivity and CDWChen  _et al._
(2021). For the CDW phase, X-ray diffractionOrtiz  _et al._ (2020); Ortiz and
Sarte (2020) and scanning tunneling microscopy (STM)Jiang  _et al._ (2020);
Zhao  _et al._ (2021b); Liang  _et al._ (2021); et al. (2021) revealed the
formation of a $2\times 2$ superlattice, for which Refs.Jiang  _et al._ ,
2020; Zhao  _et al._ , 2021b; Liang  _et al._ , 2021 assumed a Star of David
(SD) distortion in the kagome plane and the proximity of the Fermi surface to
the van Hove singularity. However, solid knowledge on the lattice and
electronic properties for the origin of the CDW state are missing in order to
the understand of the superconductivity and topology.

In this work, we have investigated the structural and electronic properties of
the CDW phase by first principles calculations. We find that the CDW
transition is driven by a breathing phonon mode of the kagome lattice. The
resultant $2\times 2$ superlattice energetically favors an inverse Star of
David (ISD) pattern rather than the SD, for all three compounds. The STM
images simulated for the ISD phase agree with the recent experiment. The CDW
distortion modulates the Fermi surface but preserves the $\mathbb{Z}_{2}$
topology of the pristine phase. We evaluate the electron-phonon coupling in
the CDW phase and find it is too weak to interpret the superconductivity in
all three materials. It may imply the existence of unconventional
superconductivity pairing.

Figure 1:  Crystal structures of KV3Sb5 in the pristine phase, the Star of
David 2$\times$2 CDW phase, and the inverse Star of David phase. Corresponding
phonon dispersions are shown in the lower panels. A 2$\times$2 supercell is
shown in (a). Imaginary (negative) phonon frequency in (d) corresponds to the
breathing mode of the kagome lattice. Such a breathing-instability leads to
CDW distortions. The breathing-out and breathing-in lead to two different
structures in (b) and (c), where the red arrows indicate the lattice
distortion due to the breathing mode. In CDW phases, V1 and V2 represent
inequivalent vanadium sites of the kagome lattice, and Sb1 and Sb2 are
inequivalent sites of the honeycomb lattice.

##  II Methods

Density Functional Theory calculations are performed within the Perdew-Burke-
Ernzerhof-type generalized gradient approximation Perdew  _et al._ (1996),
which is implemented in the Vienna $ab$-$initio$ Simulation Package (vasp)
Kresse and Furthmüller (1996a, b). The projected augmented wave Blöchl (1994)
potentials with nine valence electrons for the $A$ atom. The cutoff energy for
the plane wave basis set is 300 eV. The zero damping DFT-D3 van der Waals
correction Grimme  _et al._ (2010) is employed throughout the calculations.
Spin-orbital coupling (SOC) is considered in the electronic structure
calculations but not for the structural relaxation, phonon dispersion and STM
simulation. The phonon dispersion is calculated by using the finite
displacement method as implemented in the phonopy code Togo and Tanaka (2015).
The superconducting temperatures $T_{c}$ are estimated under the electron-
phonon coupling scenario, where the deformation potential and the electron-
phonon coupling constants are estimated by the frozen phonon scheme.

##  III Results

###  III.1 Charge density waves

Figure 2:  Total energy profiles for the pristine phase, Star of David (SD)
and inverse Star of David (ISD) phases. The distortion stands for the
amplitude of the breathing phonon mode where the negative values are used for
SD in order to distinguish from ISD. The vertical axis stands for the total
energy per supercell (36 atoms). Spin-orbit coupling is considered.  Figure 3:
Electronic structures. (a-c) show the band structure for the pristine phase.
(d-f) show the Fermi surface for the pristine phase, where the color stands
for the Fermi velocity. (g-i) show the unfolded band structure of the inverse
Star of David phase. (j-l) show the total DOS of the pristine (blue) and
inverse Star of David (red) phases. The red color represents the $1\times 1$
spectrum weights. Spin-orbit coupling is included. Figure 4:  Surface orbital-
resolved DOS and simulated STM images for the $2\times 2$ CDW phases of
KV3Sb5: (a) and (b) for the Star of David structure, (c) and (d) for the
inverse Star of David structure. The V1, V2, Sb1 and Sb2 in (a) and (d) are
labeled in Fig. 1 (b) and (c). The DOS of V-$d$ and Sb-$p$ states are
projected. The V1, V2, Sb1 and Sb2 are superposed on the images for reference.

The pristine phase of $A$V3Sb5 crystallizes in the hexagonal structure with
the space group of $P6/mmm$ (No.191), as shown in Fig. 1(a). The calculated
lattice constants are 5.41 Å ($a$) and 8.89 Å ($c$) for KV3Sb5, 5.42 Å and
9.09 Å for RbV3Sb5 and 5.44 Å and 9.33 Å for CsV3Sb5. Within this phase, there
are four independent Wyckoff sites: (0, 0, 0) for alkali atom, (0.5, 0, 0.5)
for V, (0, 0, 0.5) for Sb1 and (1/3, 2/3, $z$) for Sb2, where $z$ = 0.7566 for
K, 0.7501 for Rb and 0.7425 for Cs. The V atoms form the kagome layer and Sb1
occupies the centers of the V hexagons. This kagome layer is sandwiched by the
honeycomb lattices of Sb2. Neighboring kagome layers are separated by alkali
ions. Three materials are nonmagnetic in our calculations, consistent with
experimentsOrtiz  _et al._ (2019); Kenney  _et al._ (2021).

The pristine phases exhibit a structural instability. Let’s take KV3Sb5 as an
example. Its phonon band structure in Fig. 1(d) shows a softening accounstic
phonon modes at the Brillouin Zone boundary near $M$ and $L$ points,
indicating the strong instability. Extended to the $2\times 2$ supercell, the
soft mode corresponds to a breathing phonon ($A_{1g}$ symmetry) of V atoms in
the kagome lattice [Figs. 1(b) and (c)]. Breathing-in and out lead to two
different structures. The SD structure forms when V1 (V2) atoms moves away
(toward to) the center, which is similar to the known CDW of 1T TaS2Wilson
_et al._ (1975), while the ISD structure forms by an inverse deformation to
the SD. Both SD and ISD exhibit an in-plane 2$\times$2 modulation of the
structure. The 2$\times$2 superlattice have the same space group of $P6/mmm$
as the pristine phase but two inequivalent sites (V1 and V2).

For all three compounds, the optimized SD and ISD structures are locally
stable. The imaginary frequencies disappear in their phonon spectra except a
faint soft feature in the SD case (see more details in the Supplementary
Material (SM)SM ). We find that the SD phase is higher in energy than the ISD
phase for all the three materials. As shown in Fig. 2, the ISD phase is more
and more energetically favored from K, to Rb and Cs. Besides, the distortion
also becomes larger from K, to Rb and Cs. Therefore, the ISD is more likely
the ground state structure, compared to the SD, in the $2\times 2$ CDW phase.
This observation will be verified by the STM images in Section III.3.

The CDW is induced by the Peierls instability related to the Fermi surface
nesting and phonon softeningPeierls (1996); Kohn (1959). The $q$ vector of the
soft phonon at $M$ coincides with the Fermi nesting vector between neighboring
van Hove singularities in the Fermi surface of $A$V3Sb5 (see Fig.3). As a
consequence, the CDW deformation strongly reduces the density of states (DOS)
at the Fermi energy by suppressing the van Hove singularities. Related Fermi
surface nesting, CDW and their competition with superconductivity are
extensively discussed for the kagome lattice model in literatureWen  _et al._
(2010); Yu and Li (2012); Kiesel  _et al._ (2013); Wang  _et al._ (2013).

Above energetics and structures of two CDW phases are studied for bulk
materials. When forming a surface with different charging or atomic-
terminations, the ISD or SD structure may deviate on the surface. To examine
the CDW stability on the surface, we have performed surface calculations using
a slab model that is four-unitcell thick. Recent experimentsJiang  _et al._
(2020); Zhao  _et al._ (2021b); et al. (2021) observed both $A$ and Sb
terminations on cleavage surfaces. We find that the ISD phase is still as
stable as the bulk phase for the $A$-termination. However, the ISD deformation
tends to be weaker (closer to the pristine case) on the Sb-terminated surface
than in the bulk. The SD phase exhibits a similar trend. But the ISD structure
is still more energetically favored than the SD one. The surface atomic
relaxation indicates the instability of the $2\times 2$ CDW on the Sb-
terminated surface. We note that recent experimentset al. (2021); Zhao  _et
al._ (2021b) reveals the existence of a 4$\times$1 pattern when cooling down
further after the 2$\times$2 CDW transition. The emerging 4$\times$1 pattern
may be due to electron correlations related to above surface instability and
electron-phonon interaction.

Further, the very recent STM measurementLiang  _et al._ (2021) revealed a
three-dimensional CDW, i.e., the $2\times 2\times 2$ CDW, for CsV3Sb5. This
coincides with the inter-layer instability characterized by the soft phonon
mode at the $L$ (0.5 0.5 0.5) point in the phonon spectrum [Fig.1(d)]. Our
calculations confirm that the $\pi$ CDW-phase-shift between neighboring layers
reduces slightly the binding energy by $\sim$ 10 meV per $2\times 2$ layer for
CsV3Sb5. In the $2\times 2\times 2$ CDW, the SD phase is no longer meta-stable
and relaxes to the ISD phase spontaneously. Therefore, the ISD deformation is
still preferred in energy. Additionally, the $2\times 2\times 2$ CDW exhibits
nearly the same electronic structure as the $2\times 2\times 1$ one, because
of the weak inter-layer interaction. For simple, we focus on the $2\times
2\times 1$ (i.e., $2\times 2$) CDW phase in the following discussions.

###  III.2 Band structures

The band structures of the pristine phase are shown in Fig. 3(a)-(c) for three
materials, which are similar to the previous report. We should note that the
Fermi energy (charge neutral point) in our work is about 0.1 eV higher than
that of Ref.Ortiz  _et al._ (2019). Such a minor derivation lifts the Fermi
energy above the van Hove singularity point at $M$ and leads to a slightly
different Fermi surface (FS) near the Brillouin zone boundary [Fig. 3(d)-(f)].
The FSs exhibt a strong 2D feature, which is characterized by the cylinder-
like FS (Sb-$p$ states) centered around $\Gamma$, and the large hexagonal FS
(V-$d$ states). There are also much smaller FSs (V-$d$ states) near $M/L$ and
$K/H$. Additionally, one can also observe the 2D feature in the weak
$\Gamma-A$ dispersion in the phonon spectrum [Fig.1(d)].

The 2$\times$2 ISD CDW distortion modifies the FSs. We unfold the band
structure of the ISD phase to the pristine Brillouin zone in Figs. 3(g)-(i).
These bands near the Brillouin zone boundary (e.g., $M$ and $K$) are strongly
modified, including the large hexagonal FS and small FSs, although the
cylinder-like FS is nearly unchanged. It is not surprising that the breathing
distortion changes mainly the V-$d$ driven states. The van Hove singularities
slightly below the Fermi energy at $M$ are strongly suppressed by the CDW
distortion. Consequently, CDW reduces the total density of states (DOS) near
the Fermi energy [Fig. 3(j)-(l)], which is also consistent with recent STM
spectraJiang  _et al._ (2020); et al. (2021); Liang  _et al._ (2021). The Cs
compound exhibits stronger modifications in FSs and DOS than the K and Rb
compounds. This is mainly because the V atom displacement becomes larger from
K, to Rb and Cs compounds (see Fig.2).

The band structure of the pristine phase was reported to exhibit nontrivial
band inversion Ortiz  _et al._ (2020); Ortiz and Sarte (2020). For the CDW
phases, we calculate the $\mathbb{Z}_{2}$ topological invariants by the parity
egeinvalues Fu and Kane (2007) . We find that both SD and ISD phases are still
$\mathbb{Z}_{2}$ nontrivial topological metals (see SMSM  for details).

###  III.3 STM images

The existence of SD/ISD CDW can be examined via the STM images in experiments.
Therefore, we have simulated STM images for two CDW phases for all three
materials. We also take the K compound for example here. We calculate the
local density of states (LDOS) in the plane that is $\sim$3 Å above the Sb-
terminated surface (in a slab modelSM ), to simulate the STM dI/dV mapping.
For a given bias $V_{b}$, the LDOS is summed in the energy range of $V_{b}\pm$
25 meV. Figures 4(a)&(d) show the surface DOS contributed by V1, V2, Sb1 and
Sb2 atoms [labeled in Fig. 1(b) and (c)], which rationalizes the differences
of STM images between SD and ISD phases

The most characteristic feature near the Fermi energy is that the ISD
structure exhibits a large, bright $2\times 2$ spots by the V1-hexagon centers
while the SD structure exhibits bright circles around the V1-hexagon with the
hexagon center being dark. The V1-hexagon centers turn quickly dark as moving
away from the zero energy, which agrees with recent STM results for KV3Sb5 in
Ref.Jiang  _et al._ (2020). Also supported by the lower total energy, we
conclude that the ground state of the $2\times 2$ CDW phase exhibits more
likely the ISD structure for these three compounds. The high LDOS at
V1-hexagon centers is caused by the large peak of V1-$d_{z^{2}}$ and small
V1-$d_{xz,yz}$ in DOS [Fig.4(d)]. In contrast, no such a peak appears for the
V1-$d_{z^{2}}$ on SD surface [Fig.4(a)].

At $V_{b}=0.05$ eV, the V1-hexagon brightness greatly decreases due to the
decreasing DOS of V-$d_{z^{2}}$ on the ISD surface. At the same time, the
V-$d_{xz,yz}$ DOS increases, where the V2-$d_{xz,yz}$ contribution is slightly
larger than the V1-$d_{xz,yz}$ one. Thus, small V2-triangles turn brighter.
For the SD surface, the STM image at $V_{b}=0.05$ eV remains almost the same
as that at zero, because the relative amplitudes of DOS between V1 and V2 are
nearly unchanged. For $V_{b}=0.2$ eV, the ISD surface exhibits bright circles,
because V1 is larger in DOS than V2. In contrast, the SD surface shows small,
bright spots at V2-triangles, caused by the relative larger DOS of V2 than
that of V1.

At $V_{b}=-0.1$ eV, the ISD surface looks less-contrast in the STM image
compared to the zero bias case, due to the decreasing V-$d_{z^{2}}$ and
increasing V-$d_{xz,yz}$ in DOS. For the SD surface, all the atomic sites show
similar brightness due to the small DOS peaks of V-$d$ and Sb-$p$ near $-$0.1
eV. At $V_{b}=-0.4$ eV, both ISD and SD surfaces show a 1$\times$1 honeycomb
pattern. The honeycomb feature reflects the top Sb layer (Sb1 and Sb2),
although V is comparable to Sb in DOS.

We note that STM images of RbV3Sb5 and CsV3Sb5 (see SM SM ) share similar
features as those of KV3Sb5.

###  III.4 Superconductivity

$A$V3Sb5 exhibits superconductivity from the CDW normal state. The most recent
STM experiment et al. (2021) on CsV3Sb5 reveals a superconducting gap $\sim
0.5$ meV, which puts the superconductor in the strong-coupling regime with
2$\Delta/k_{B}T_{c}$ around 4.6. A natural question is whether the
superconductivity is driven by the electron-phonon coupling in Bardeen-Cooper-
Schrieffer (BCS) scenario Bardeen  _et al._ (1957). Therefore, we have
calculated the electron-phonon coupling constant ($\lambda$) for the ISD phase
and estimated the $T_{c}$ based on McMillan’s formula which applies in the
strong-coupled regime McMillan (1968),

$T_{c}=\frac{\theta_{D}}{1.45}\exp\left[-\frac{1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}\right],$
(1)

with $\theta_{D}$ being the Debye temperature which can be extracted through a
quadratic fitting of the phonon density of states of the low frequency
acoustic bands: $g(\omega)=\alpha\omega^{2}$. Using the fitted value of
$\alpha$, the Debye temperature is determined by
$\theta_{D}=(\hbar/k_{B})(9N/\alpha)^{1/3}$ where $N$ and $k_{B}$ are the
total number of atoms within a unit cell and the Boltzmann constant,
respectively. This relation is determined by the total number of phonon modes
$3N=\int^{\omega_{D}}_{0}g(\omega)\leavevmode\nobreak\ \text{d}\omega$ where
$\omega_{D}=k_{B}\theta_{D}/\hbar$ is the Debye angular frequency.

$\lambda$ ($\mu^{*}$) is a dimensionless parameter describing the strength of
effective electron-electron attractive (screened-repulsive) interaction Poole
_et al._ (1999) and the superconductivity can happen only when
$\lambda>\mu^{*}$. The value of $\mu^{*}$ is assumed in the range 0.10–0.15 An
and Pickett (2001); Si  _et al._ (2013). And $\lambda$ is calculated based on
deformation potential approximation Khan and Allen (1984); An and Pickett
(2001),

$\begin{split}\lambda=&N_{F}\langle
V_{\text{ph}}\rangle_{E_{F}}=N_{F}\sum_{\omega}\Big{\langle}\frac{|\mathcal{M}_{\text{e-ph}}|^{2}}{\hbar\omega}\Big{\rangle}_{E_{F}}\\\
=&N_{F}\sum_{\omega}\frac{|D|^{2}}{2M\omega^{2}}\end{split}$ (2)

where $N_{F}$ is the electron DOS per spin per unit cell, $V_{\text{ph}}$ is
the phonon-mediated effective electron-electron interaction energy,
$\mathcal{M}_{\text{e-ph}}$ is the electron-phonon matrix element, $\omega$ is
the phonon frequency, $M$ is atomic mass, and $D$ is the electronic
deformation potential. $\langle\leavevmode\nobreak\ \rangle_{E_{F}}$ refers to
the averaging at Fermi energy $E_{F}$. According to Eq. (2) phonon modes with
lower frequency and stronger deformation potential contributes most to
$\lambda$. We approximate $\lambda$ by considering only the breathing mode and
the deformation potential $D\approx\Delta E_{k}/\Delta u_{\text{breathing}}$
with $\Delta E_{k}$ being the average shift of band energy at Fermi energy and
$\Delta u_{\text{breathing}}$ is the deformation amplitude. We further
simplify the expression of $\lambda$ by
$M\omega^{2}_{\text{breathing}}=\langle\Phi\rangle_{\text{breathing}}$ with
$\langle\Phi\rangle_{\text{breathing}}$ being the average force constant of
the breathing mode. More details of the calculation of $\lambda$ and
$\theta_{D}$ are shown in SM SM .

Table 1:  Estimation of the $T_{c}$ due to electron-phonon coupling. We list the CDW transition temperature $T^{\text{exp.}}_{\text{CDW}}$, experimental superconducting temperature $T_{c}^{\text{exp.}}$ and experimental Debye temperature $\theta^{\text{exp.}}_{D}$ in unit of K Ortiz and Sarte (2020); Ortiz  _et al._ (2020); Yin  _et al._ (2021). We show calculation parameters and results: DOS per superlattice at the Fermi energy $N_{F}$ (eV-1), deformation potential $D$ (eV/Å), phonon mediated electron-electron attractive interaction strength $\lambda$, calculated Debye temperature $\theta_{D}$, and calculated $T_{c}$ for the ISD phase.  | $T^{\text{exp.}}_{\text{CDW}}$ | $N_{F}$ | $D$ | $\lambda$ | $\theta_{D}$ | $\theta^{\text{exp.}}_{D}$ | $T_{c}$ | $T^{\text{exp.}}_{c}$
---|---|---|---|---|---|---|---|---
KV3Sb5 | 78 | 11.6 | 0.52 | 0.46 | 158 | 68 | 0.38–1.13 | 0.93
RbV3Sb5 | 102 | 9.3 | 0.56 | 0.39 | 191 | 169 | 0.10–0.54 | 0.92
CsV3Sb5 | 94 | 5.2 | 0.76 | 0.30 | 121 | – | 0.0017–0.055 | 2.5

Table 1 lists the calculated parameters $N_{F}$, $D$, $\lambda$, $\theta_{D}$,
and $T_{c}$ for ISD phase of three materials. The $D$ values are one order of
magnitude smaller than the known BCS superconductor MgB2 An and Pickett (2001)
but the $\lambda$ fall into a moderate region as compensated by large $N_{F}$.
All the $\lambda$ are smaller than 1, indicating that they are in the weakly-
coupled regime. The calculated $\theta_{D}$ is larger than experimental values
for RbV3Sb5 Yin  _et al._ (2021) and KV3Sb5Ortiz and Sarte (2020). The
smallness of $\lambda$ of CsV3Sb5 is mainly due to the low DOS at the Fermi
energy. The upper limit of calculated $T_{c}$, which is already overestimated
due to $\theta_{D}$, is much smaller than corresponding experimental value.
This indicates the unconventional mechanism of the superconductivity observed
in these materials.

##  IV Summary

In summary, we have studied the crystal and electronic structures of the CDW
phases of three kagome metals $A$V3Sb5. The 2$\times$2 CDW (also the $2\times
2\times 2$ one) phase exhibits an inverse Star of David deformation.
Corresponding STM images simulated agree with recent experiments. The
nontrivial $\mathbb{Z}_{2}$ nature remains in the CDW band structure. We have
evaluated the superconducting temperature induced by electron-phonon coupling
and found it is much smaller than corresponding experimental value, implying
possible unconventional superconductivity in these materials.

##  V Acknowledgements

We thank H. Chen for inspiring discussions and A. Perara for help in
calculations. B.Y. acknowledges the financial support by the Willner Family
Leadership Institute for the Weizmann Institute of Science, the Benoziyo
Endowment Fund for the Advancement of Science, Ruth and Herman Albert Scholars
Program for New Scientists, the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (Grant No.
815869). Z.W.is supported by the U.S. Department of Energy, Basic Energy
Sciences Grant No.DE-FG02-99ER45747.

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