††thanks: Equal contribution††thanks: Equal contribution

# Two types of charge order in the superconducting kagome material CsV3Sb5

Ritu Gupta44footnotemark: 4<EMAIL_ADDRESS>Laboratory for Muon Spin
Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
Debarchan Das44footnotemark: 4<EMAIL_ADDRESS>Laboratory for Muon
Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
Charles Mielke III  Laboratory for Muon Spin Spectroscopy, Paul Scherrer
Institute, CH-5232 Villigen PSI, Switzerland  Ethan Ritz  Department of
Chemical Engineering and Materials Science, University of Minnesota, MN 55455,
USA  Fabian Hotz  Laboratory for Muon Spin Spectroscopy, Paul Scherrer
Institute, CH-5232 Villigen PSI, Switzerland  Qiangwei Yin  Department of
Physics and Beijing Key Laboratory of Opto-electronic Functional Materials &
Micro-nano Devices, Renmin University of China, Beijing 100872, China  Zhijun
Tu  Department of Physics and Beijing Key Laboratory of Opto-electronic
Functional Materials & Micro-nano Devices, Renmin University of China, Beijing
100872, China  Chunsheng Gong  Department of Physics and Beijing Key
Laboratory of Opto-electronic Functional Materials & Micro-nano Devices,
Renmin University of China, Beijing 100872, China  Hechang Lei
<EMAIL_ADDRESS>Department of Physics and Beijing Key Laboratory of Opto-
electronic Functional Materials & Micro-nano Devices, Renmin University of
China, Beijing 100872, China  Turan Birol  Department of Chemical Engineering
and Materials Science, University of Minnesota, MN 55455, USA  Rafael M.
Fernandes  School of Physics and Astronomy, University of Minnesota,
Minneapolis, MN 55455, USA  Zurab Guguchia  Laboratory for Muon Spin
Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
Hubertus Luetkens<EMAIL_ADDRESS>Laboratory for Muon Spin
Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
Rustem Khasanov<EMAIL_ADDRESS>Laboratory for Muon Spin Spectroscopy,
Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland

The kagome metals of the family $A$V3Sb5, featuring a unique structural motif,
harbor an array of intriguing phenomena such as chiral charge order and
superconductivity. CsV3Sb5 is of particular interest because it displays a
double superconducting dome in the region of the temperature-pressure phase
diagram where charge order is still present. However, the microscopic origin
of such an unusual behavior remains an unsolved issue. Here, to address it, we
combine high-pressure, low-temperature muon spin relaxation with first-
principles calculations. We observe a pressure-induced threefold enhancement
of the superfluid density, which also displays a double peak feature, similar
to the superconducting critical temperature. This leads to three distinct
regions in the phase diagram, each of which features distinct slopes of the
linear relation between superfluid density and the critical temperature. These
results are attributed to a possible evolution of the charge order pattern
from the superimposed tri-hexagonal Star-of-David phase at low pressures
(within the first dome) to the staggered tri-hexagonal phase at intermediate
pressures (between the first and second domes). Our findings suggest a change
in the nature of the charge ordered state across the phase diagram of CsV3Sb5,
with varying degrees of competition with superconductivity.

Among the series $A$V3Sb5 ($A$ = Rb, K, Cs) Neupert et al. (2022); Jiang et
al. (2021); Mielke III et al. (2022), the Cs compound manifests the highest
superconducting critical temperature $T_{c}$ $\simeq$ 2.5 K. CsV3Sb5 also
features multi-gap superconductivity and more importantly, a time reversal
symmetry breaking (TRSB) chiral charge order below $T_{co}$ = 94 K Yu et al.
(2021a); Wu et al. (2021); Khasanov and et al. (2022); Wang et al. (2021a), as
reported by scanning tunneling microscopy Zhao et al. (2021), polar Kerr
rotation Wu et al. (2021), and $\mu$SR experimentsYu et al. (2021a); Khasanov
and et al. (2022). A comprehensive understanding of the interdependence
between charge order (CO) and superconductivity (SC) is thus essential, and
can be studied by using an appropriate external perturbation. In the quest to
obtain an efficient tuning knob, hydrostatic pressure was found to be optimal.
Indeed, pressure suppresses the charge order and results into an unusual but
well-pronounced double superconducting dome in the temperature-pressure phase
diagram. Compared to the Rb and K counterparts Wang et al. (2021b); Zhu et al.
(2021); Du et al. (2021), the double peak behavior is most distinguishable in
the case of CsV3Sb5 Yu et al. (2021b); Wang et al. (2021c); Chen et al.
(2021); Zhang et al. (2021a) where $T_{c}$ is roughly tripled to 8 K around
the optimal pressure of 2 GPa from its value $T_{c}$ = 2.5 K at ambient
pressure. Thus, pressure tuned CsV3Sb5 offers a rich framework for studying
the nature of the interplay between superconductivity and charge order.

To systematically characterize and to obtain a microscopic understanding of
the complex temperature-pressure phase diagram of CsV3Sb5, here we report
high-pressure zero-field and transverse-field ${\mu}$SR as well as AC
susceptibility (up to 1.75 GPa) measurements combined with first principles
calculations. Transverse field (TF) ${\mu}$SR experiments serve as an
extremely sensitive local probe technique to measure the magnetic penetration
depth $\lambda$ in the vortex state of bulk type II superconductors. This
quantity is directly related to the superfluid density $n_{s}$ via
1/${\lambda}^{2}$ = $\mu_{0}$$e^{2}$$n_{s}/m^{*}$ (where $m^{*}$ is the
effective mass). Zero-field ${\mu}$SR has the ability to detect internal
magnetic fields as small as 0.1 G without applying external magnetic fields,
making it a highly valuable tool for probing spontaneous magnetic fields due
to TRSB in exotic superconductors. The techniques of ${\mu}$SR and DFT
complement each other ideally, as we are able to sensitively probe the
fundamental microscopic properties of CsV3Sb5 with ${\mu}$SR experiments and
correlate them with the pressure-evolution of the charge ordered state
calculated by DFT.

Figure 1: (Color online) TF and ZF-$\mu$SR time spectra. a.) Selected
TF-$\mu$SR time spectra at 0.27 K (in the superconducting state) and 10 K (in
the normal state) collected for ambient and 1.74 GPa pressure. Solid lines are
the fits of the spectra considering both the sample and background parts, as
described in the Supplementary section. b.) The fast Fourier transform of the
spectra shows an additional broadening of the peak due to an inhomogeneous
distribution of internal fields, resulting from the appearance of a vortex
lattice. c.)-d.) Representative ZF-$\mu$SR time spectra collected in zero
field well below $T_{c}$ (0.27 K) and above $T_{c}$ (10 K) for 1.18 GPa [panel
(c)] and 1.74 GPa [panel (d)]. The solid lines are fits considering two
relaxation channels, namely Gaussian-Kubo-Toyabe and a Lorentzian decay
function.  Figure 2: (Color online) Pressure dependence of the superconducting
parameters and temperature-pressure phase diagram. a.) Temperature variation
of the inverse squared magnetic penetration depth $\lambda^{-2}(T)$ for the
applied hydrostatic pressures shown in the plot legend. The solid line is a
two gap $s+s$-wave model fitting using the phenomenological $\alpha$-model.
b.) The temperature-pressure superconducting phase diagram of CsV3Sb5. The
left axis shows $T_{c}$ values obtained through TF-$\mu$SR and ac magnetic
susceptibility measurements carried out under hydrostatic pressure conditions.
The right axis presents the inverse squared London penetration depth at base
temperature, i.e., $\lambda^{-2}(0)\propto n_{s}$ as a function of pressure.
c.) Superconducting transition temperature $T_{c}$ plotted as a function of
the base-temperature superfluid density $\lambda^{-2}(0)$ for the three
different regions of the phase diagram of panel (b).

##  I results

###  I.1 Superfluid density as a function of pressure

Figure 1(a) shows the TF-$\mu$SR time spectra of CsV3Sb5 well below (0.25 K)
and above (10 K) $T_{c}$ in the presence of an applied magnetic field of 10 mT
for ambient as well as the highest applied pressure, 1.74 GPa. The weakly
damped oscillations above $T_{c}$ are essentially due to the randomly
distributed local fields created by nuclear magnetic moments. The spectra
below $T_{c}$ show a sizable increase in relaxation because of the
inhomogeneous distribution of internal fields arising due to the formation of
the flux line lattice in the vortex state. The Fourier transform of the
spectra, shown in Fig. 1b, gives the internal field distribution, which
showcases a clear broadening in the superconducting state compared to the
normal state. From the TF-$\mu$SR data, the superconducting relaxation rate
$\sigma_{\rm sc}(T)$ is extracted. The latter is related to the magnetic
penetration depth via $\frac{\sigma_{\rm
sc}(T)}{\gamma_{\mu}}=0.06091\frac{\phi_{0}}{\lambda^{2}(T)}$, where
$\gamma_{\mu}$ = 2$\pi\times$135.5 MHz/T is the muon gyromagnetic ratio and
$\phi_{0}$ is the flux quanta Brandt (1988, 2003) (details of the analysis of
the TF-$\mu$SR data are given in the Supplementary section). The resulting
temperature dependence of the inverse squared magnetic penetration depth
$\lambda^{-2}(T)$, measured under various hydrostatic pressures, are shown in
Fig. 2(a).

The $\lambda^{-2}(T)$ curves in Fig. 2(a) are well described by a two-gap
model across the entire pressure range probed (see details in the Methods
section). This indicates that the multi-gap nature of the superconducting
state of CsV3Sb5, which was previously reported by various techniques for
ambient conditions Xu et al. (2021); Duan et al. (2021); Gupta et al. (2021);
Nakayama et al. (2021), is robust against the application of hydrostatic
pressure. As we show below, a sharp increase of both $T_{c}$ and
$\lambda^{-2}(0)$ occurs at pressures in the range 1.5 – 1.7 GPa, suggesting
the complete suppression of the CO and thus a transition between the SC+CO
state to the pure SC state. It is quite remarkable, therefore, that at the
intermediate pressure 1.68 GPa, which is at the border between the SC+CO and
pure SC phases, the temperature evolution of the penetration depth shows a
prominent two-step like feature [purple curve in Fig. 2(a)]. This is a
signature of inhomogeneity, whose origin can be either extrinsic, e.g., due to
pressure inhomogeneity inside the cell, or intrinsic, due to phase separation
caused by a first-order SC+CO to SC transition.

Figure 2(b) shows the pressure dependence of $T_{c}$ and of the zero-
temperature value of $\lambda^{-2}(0)$, which is directly proportional to the
superfluid density $n_{s}$. Interestingly, both $\lambda^{-2}(0)$ and $T_{c}$
show a similar non-monotonic pressure dependence, featuring three distinct
regions marked as I, II and III in Fig 2(b). Initially, as pressure increases,
$T_{c}$ shows an increase from 2.85(9) K at ambient pressure to 6.9(3) K at a
critical pressure $p_{\rm c1}$ $\simeq$ 0.63 GPa. Upon further increase in
pressure, $T_{c}$ shows a sharp and substantial decrease down to 2.87(7) K at
0.8 GPa. This marks the onset of Region II, where $T_{c}$ increases only
slightly with pressure. Remarkably, a sudden jump in $T_{c}$ is seen around a
second critical pressure $p_{\rm c2}$ $\simeq$ 1.74 GPa, reaching a maximum
value of $T_{c}^{\rm max}$ = 8.0(1) K – which is almost three times larger
than the ambient pressure value – and nearly saturating beyond this pressure.
This regime is denoted as Region III in the $T-p$ phase diagram.

The maximum $T_{c}$ value of 8 K extracted from $\mu$SR is fairly close to the
value obtained through electrical transport, AC and DC magnetization
measurements Chen et al. (2021). Moreover, $\lambda^{-2}(0)$ qualitatively
shows a similar trend to $T_{c}$ with regard to pressure, also featuring a
double-peak behavior. Indeed, there is a nearly threefold increase in the
value of $\lambda^{-2}(0)$ from 6.5(1) $\mu\rm{m}^{-2}$ at ambient pressure to
17.3(9) $\mu\rm{m}^{-2}$ at 1.74 GPa, followed by saturation upon further
increase in pressure. The relative variation of the $\lambda^{-2}(0)$ value at
the highest applied pressure compared to its ambient pressure value is
$\delta\lambda^{-2}(0)/\lambda^{-2}(0)$ = 63 %. This is rather unusual in
comparison to BCS superconductors, where the superfluid density either depends
weakly on pressure or remains pressure independentDas et al. (2021). The
sudden enhancement of $T_{c}$ and $\lambda^{-2}(0)$ has an intimate connection
with the collapse of CO at the critical pressure $p_{\rm c2}$, and is
consistent with a first-order transition from the SC+CO phase to the pure SC
phase.

Figure 3: (Color online) Indication of time-reversal symmetry-breaking in the
superconducting state of CsV3Sb5. a.) Electronic relaxation rate $\Lambda(T)$
for CsV3Sb5 as a function of temperature obtained from ZF-$\mu$SR experiments
carried out in the presence of three different applied pressures: (a) 0, (b)
1.18 and (c) 1.74 Gpa. Each pressure corresponds to a different Region (I, II,
and III) of the $T-p$ phase diagram of Fig. 2 (c). Solid lines through the
data points are fits as described in the Methods section. The values for the
superconducting transition temperature $T_{c}$ are denoted by the vertical
dashed black line.

###  I.2 Superconducting transition temperature vs. superfluid density

Figure 2(c) displays the correlation between the superconducting transition
temperature $T_{c}$ and the superfluid density $\lambda^{-2}(0)$ obtained from
$\mu$SR experiments under different pressures. There are three distinct types
of $T_{c}(\lambda^{-2}(0))$ lines, each associated with one of the three
regions of the phase diagram in Fig 2(b). For the data points belonging to
Region I, we obtain a slope of 2.72 K$\mu$m2, which is 3.5 times larger than
the slope obtained for the data points from Region II (0.78 K$\mu$m2). For
Region III, $\lambda^{-2}(0)$ seems to be almost independent on $T_{c}$.

These linear relationships between $T_{c}$ and $\lambda^{-2}(0)$ are
characteristic of unconventional superconductors. In the present case, for
Regions I and II, the $T_{c}/\lambda^{-2}(0)$ ratio lies far away from the
values typically seen in BCS superconductors, where
$T_{c}/\lambda^{-2}~{}\approx$ 0.00025–0.015 K$\mu$m2. The large values of
$T_{c}$, despite the small values of the superfluid density in Regions I and
II, are also signatures of unconventional SC, where
$T_{c}/\lambda^{-2}(0)\sim$ 1–20 Uemura et al. (1989); Guguchia et al. (2022);
Mielke et al. (2021). Moreover, the change in the slope
$T_{c}/\lambda^{-2}(0)$ from Region I to Region II, both of which display
coexisting CO and SC, suggests a subtle modification in the CO state. Combined
with the sudden suppression in $T_{c}$ and $\lambda^{-2}(0)$ upon crossing the
Region I to Region II boundary in Fig. 2b, this provides strong evidence for a
distinct competition between SC and CO in these two regions. Finally, the fact
that $\lambda^{-2}(0)$ is nearly independent of $T_{c}$ in Region III can be
attributed to the absence of a competing CO state in this region.

###  I.3 Spontaneous fields in the superconducting state

Zero-field $\mu$SR experiments have been successfully implemented to probe
TRSB in well-known unconventional superconductors, e.g., Sr2RuO4Luke et al.
(1998); Grinenko et al. (2021a, b), UPt3Luke et al. (1993), LaNiC2Hillier et
al. (2009), LaPt3PBiswas et al. (2021). We have conducted ZF-$\mu$SR
experiments for three different applied pressures: 0, 1.18, and 1.74 GPa.
Representative ZF-$\mu$SR spectra above and below $T_{c}$ for the pressures of
1.18 and 1.74 GPa are shown in Figs. 1(c) and (d). Both display an increase in
relaxation within the superconducting state. The data were fitted using the
Gaussian-Kubo-Toyabe (GKT) depolarization function Kubo and Toyabe (1967)
multiplied by an exponential decay function exp$(-\Lambda t)$. The Gaussian-
Kubo-Toyabe function accounts for the random magnetic fields created by the
nuclear moments. As was discussed previously for KV3Sb5 Mielke III et al.
(2022), the exponential relaxation rate $(-\Lambda t)$ is sensitive to the
temperature dependence of the electronic contribution to the muon spin
relaxation. The temperature dependences of the electronic relaxation rate
$\Lambda(T)$ for the applied pressures of 0, 1.18 and 1.74 GPa are shown in
Fig. 3.

At ambient pressure, we observe an enhancement of the exponential relaxation
rate, with an onset slightly above $T_{\rm c}$, by a small amount of
$\Delta\Lambda_{\rm 0}$ = 0.024(8) $\mu\rm{s}^{-1}$. Since time-reversal
symmetry is already broken by charge order at $T_{\rm CO}$ $\simeq$ 95 K Yu et
al. (2021a); Khasanov and et al. (2022); Wu et al. (2021), we speculate that
the effect in the superconducting state is caused by the Meissner screening of
the small fields induced by the CO loop currents created at $T_{\rm CO}$. A
similar argument holds for 1.18 GPa (Region II), where both CO and SC coexist
and the associated increase in electronic relaxation is $\Delta\Lambda_{\rm
1.18}~{}=~{}0.011(2)$ $\mu\rm{s}^{-1}$.

However, at 1.74 GPa (Region III), where the CO is completely suppressed, we
still observe an increase in $\Lambda(T)$ with the onset of SC at $T_{\rm c}$
$\simeq$ 8 K. The increase of $\Lambda(T)$ with decreasing temperature in
Region III is smooth, contrasting with the sharp increase observed in Regions
I and II. As there is no CO at this pressure, a plausible origin for the
enhancement of the internal field width is the spontaneous breaking of time
reversal symmetry in the superconducting state. The increase in $\Lambda$ by a
magnitude of $\Delta\Lambda_{\rm 1.74~{}GPa}~{}=~{}0.031(3)~{}\mu\rm{s}^{-1}$
is associated with a characteristic field strength of $\Delta B$ =
$\Delta\Lambda_{\rm 1.74~{}GPa}$/$\gamma_{\mu}$ = 0.04 mT. This value is
comparable to what is observed in well-known TRSB superconductors, such as
0.05 mT for the chiral superconductor Sr2RuO4Grinenko et al. (2021b, a), 0.01
mT for the heavy-fermion superconductor UPt3Luke et al. (1993), and 0.01 mT
for the non-centrosymmetric superconductor LaNiC2 Hillier et al. (2009). These
results indicate that the SC state without charge order breaks time-reversal
symmetry. Interestingly, a recent $\mu$SR investigation of the other kagome
compounds, RbV3Sb5 and KV3Sb5, also reported evidence for TRSB in the pure SC
state Guguchia et al. (2022).

##  II Discussion

One of the intriguing findings of this paper is the observation of a double
superconducting dome inside the region where SC and CO coexist, associated
with a more than threefold enhancement of $T_{\rm c}$ and of the superfluid
density, $\sigma_{\rm sc}(0)~{}\propto~{}n_{s}$. Indeed, Fig. 2(b) shows that
with increasing pressure, $T_{\rm c}$ and the superfluid density first
increase up to 0.63 GPa, followed by a sudden drop in these parameters with
further increase in pressure, defining the boundary between Regions I and II.
Within Region II, $T_{\rm c}$ and $n_{s}$ increase slowly with increasing
pressure up to 1.54 GPa, beyond which a dramatic enhancement is seen in these
two quantities, marking the onset of Region III. This behavior is suggestive
of a state of coexistence between competing SC and CO orders, which undergoes
a transition to a pure SC state at $p_{\rm c2}$. Such a scenario would
naturally explain why not only $T_{c}$ but also $n_{s}$ displays a sudden
change from Region II to Region III, as $n_{s}$ is suppressed in the
coexisting state due to the partial gapping of the Fermi surface promoted by
CO order.

Analogously, the strong variation of $T_{c}$ and $n_{s}$ between $p_{\rm c1}$
and $p_{\rm c2}$ is indicative of a subtle modification in the nature of
either the CO or the SC state from Region I to Region II. The SC state seems
qualitatively the same in both regions: first, the $\lambda^{-2}(T)$ curves
are indicative of a fully-gapped SC state in both regions. Second, the
temperature dependence of the electronic relaxation $\Lambda(T)$ is very
similar in both Regions I and II (see Figs. 3(a) and 3(b)). Consequently, it
is plausible that the changes in $T_{c}$ and $n_{s}$ are a consequence of a
change in the CO state, in such a way that the CO of Region II competes more
strongly with SC than the CO of Region I. Experimentally, it has been proposed
that more than one CO state can be realized in these kagome metalsWang et al.
(2021c, a); Zhao et al. (2021). Different theoretical calculations also
suggest multiple nearby CO instabilities Ratcliff et al. (2021); Denner et al.
(2021); Feng et al. (2021); Lin and Nandkishore (2021); Christensen et al.
(2021); Subedi (2022); Park et al. (2021). To investigate the possibility of a
change in the nature of the CO state from Region I to Region II in CsV3Sb5, we
performed DFT calculations at different hydrostatic pressures.

Figure 4: (Color online) Theoretical modeling and DFT results. a.) Pressure
dependence of the difference in enthalpies between a charge-ordered state and
that of the undistorted kagome lattice for different possible CO phases
discussed in the text and shown in panels (e)-(g). b.) Density of states (DOS)
per formula unit calculated at ambient pressure for the different CO phases
considered. c.) Variation of the DOS per formula unit at the Fermi level with
pressure. The distinct types of CO states considered here display different
bond order configurations and are named: d.) parent; e.) planar tri-hexagonal;
f.) superimposed tri-hexagonal Star-of-David; and g.) staggered tri-hexagonal.
Black, blue, green and red points in panels (a)-(c) correspond to these CO
configurations respectively.

The $A$V3Sb5 compounds are predicted to have phonon instabilities at the $M$
and $L$ points of their hexagonal Brillouin zones Cho et al. (2021); Qian et
al. (2021). Both of the unstable phonons transform as three-dimensional
irreducible representations of the space group, which means that the high-
temperature undistorted kagome lattice may lower its energy via lattice
distortions along six independent atomic displacement patterns. Superpositions
of these patterns give rise to a large family of candidate structures and bond
order patterns, a subset of which are shown in Fig. 4 Christensen et al.
(2021).

While there is growing evidence of time-reversal symmetry breaking in the CO
phase, the predominant effect of pressure is expected to be on the lattice,
and not directly on the properties of the CO state affected by TRSB. This is
because interatomic force constants depend strongly on the bond lengths in
ionic compounds, where there is a fine balance between the repulsive and the
attractive components of the interatomic interactions. Moreover, the TRSB CO
state is generally expected to induce a regular (i.e. time-reversal symmetry
preserving) CO pattern of bond distortions due to nonlinear terms in the free
energy Lin and Nandkishore (2021); Park et al. (2021). For these reasons,
despite the fact that our first-principles approach cannot directly account
for TRSB, understanding how the crystal structure of CsV3Sb5 evolves under
pressure using non-spin-polarized DFT calculations is a viable approach to
elucidate the behavior of the CO in this material.

In Fig. 4(a), we present the enthalpy per formula unit of the three lowest
energy CO structures of CsV3Sb5 calculated from DFT as a function of pressure.
The parent structure, without CO, is shown in Fig. 4(d). Two of these CO
phases, dubbed the $(M00)+(0LL)$ “staggered tri-hexagonal” state (red, pane
(g)) and the $(MMM)+(LLL)$ “superimposed tri-hexagonal Star-of-David” phase
(green, panel (f)), host a superposition of both $M$ and $L$ point lattice
instabilities, and therefore have the $2\times 2\times 2$ periodicity
consistent with the available X-ray data. The third low energy phase, the so-
called $(MMM)$ “planar tri-hexagonal” phase (blue, panel (e)), is only
associated with the $M$ point instability. As a result, it has a $2\times
2\times 1$ periodicity that seems inconsistent with the currently available
experimental structure data.

Fig. 4(a) shows that at pressures above 5 GPa, CO is completely suppressed,
and we predict the undistorted kagome lattice to be the equilibrium structure.
This is consistent with previous results that predicted the phonon
instabilities at both $M$ and $L$ to vanish at pressures between 3 GPa and 5
GPa Zhang et al. (2021b). The threefold difference with respect to the
experimental $p_{c,2}$ value is not unexpected. Indeed, an accurate prediction
of the pressure at which the structural instabilities disappear is hard to
achieve in DFT because of the large unit cell, the sensitivity of the
electronic structure of the kagome layers on pressure, and lattice parameters
errors due to the exchange correlation approximations. Nevertheless, the
qualitative trends with pressure extracted from DFT are expected to be
reliable.

While the staggered tri-hexagonal phase (red) has lower enthalpy than the
other CO phases throughout the pressure range considered, the enthalpy
differences between the distinct CO phases are on the order of a few meV’s per
formula unit. This implies that entropic effects not captured by DFT are large
enough to switch the ordering of these phases and cause the equilibrium
structure to be a different phase from the one that minimizes the Kohn-Sham
enthalpy. The two most likely candidates consistent with the experimentally
observed $2\times~{}2~{}\times 2$ periodicity of the CO state are the
staggered tri-hexagonal (red) and the superimposed tri-hexagonal Star-of-David
phases (green). Since the energy difference between these phases become
smaller with increasing pressure, it is plausible that there is a transition
from one to the other under pressure.

To elucidate which of these two CO phases is realized in Region I, we compute
their densities of states (DOS). As shown in Fig. 4(b), there are important
differences between the DOS curves of the staggered tri-hexagonal phase (red),
the superimposed tri-hexagonal Star-of-David phase (green), and the
undistorted kagome lattice (black). In particular, the latter has a larger DOS
at the Fermi level, as well as van Hove singularities (signaled by the central
“satellite” peaks in the DOS) closer to the Fermi level, when compared to the
red and green curves (the blue curve, referring to the planar tri-hexagonal
phase, is also shown for completeness). In contrast, the red curve has the
smallest Fermi level DOS and farthest separated van Hove singularity peaks
among the three curves. While the mechanism for superconductivity in CsV3Sb5
remains under debate, one generally expects that a higher Fermi-level DOS
correlates with a higher $T_{c}$. Moreover, if van Hove singularities are
important for the SC mechanism, it is reasonable to assume that the closer
their peaks are to the Fermi level, the higher the $T_{c}$ will be. If these
assumptions hold for CsV3Sb5, then a possible explanation for the unusual
double-peak SC dome revealed by our $\mu$SR data is a transition from the
superimposed tri-hexagonal Star-of-David phase (green) in Region I to the
staggered tri-hexagonal phase (red) in Region II. This is because the latter
should compete more strongly with SC due to the larger reduction in the DOS
and further separation between the van Hove peaks. Similarly, because the
undistorted lattice (i.e. without CO) has the highest DOS, this would also
account for $T_{c}$ being the highest when all types of CO are suppressed at
$p_{c,2}$.

##  III Conclusion

Our results uncover two different regions in the part of the phase diagram of
the kagome metal CsV3Sb5 where both charge order and superconductivity are
present. Based on our DFT calculations, we propose that two distinct types of
CO order are realized: a superimposed tri-hexagonal Star-of-David phase at low
pressures and a staggered tri-hexagonal phase at intermediate pressures,
before CO is fully suppressed. These charge orders display different degrees
of competition with superconductivity, which leads to the complex phase
diagram featuring a double-peak in the pressure dependence of both the
superconducting transition temperature and the superfluid density. Throughout
the pressure range investigated, the nodeless multigap nature of the
superconducting state remains persistent, in contrast to KV3Sb5 and RbV3Sb5,
where a nodal gap emerges at higher pressures Guguchia et al. (2022).
Furthermore, we show that once charge order is fully suppressed, the
superconducting state breaks time-reversal symmetry, which makes this
compound, together with its K- and Rb-counterparts, one of the rare cases of
unconventional superconductors with spontaneously broken time reversal
symmetry.

##  IV Methods

Sample growth: Single crystals of CsV3Sb5 used for for this study were grown
by using self-flux method from Cs ingots (purity 99.9%), V 3-N powder (purity
99.9%) and Sb grains (purity 99.999%)Yin et al. (2021). The detailed
description about the growth and characterization of crystals is provided in
detail in Refs. Gupta et al. (2021); Ortiz et al. (2020, 2019)

$\mu$SR experiment: Muon spin relaxation/rotation ($\mu$SR) measurements under
pressure were performed at $\mu$E1 beam line using GPD spectrometer at Paul
Scherrer Institute, Switzerland. This spectrometer is a dedicated worldwide
unique instrument for carrying out high pressure $\mu$SR experiments. For
$\mu$SR experiments, 100 % spin polarized muons $\mu^{+}$ are implanted inside
the sample one by one, where the spin of muon will Larmor precess around the
local magnetic field $B_{\rm int}$ at the muon site. Detailed description
about the $\mu$SR technique can be found in the Refs.Yaouanc and De Reotier
(2011); Hillier et al. (2022) In case of a superconductor, the characteristic
length scale namely penetration depth can be determined by probing the
inhomogeneous field distribution of vortex lattice. Heliox cryostat with a
He-3 insert was used to access temperature down to 250 mK. Pressure upto 1.9
GPa were achieved using low and pre-defined background double wall pressure
cell made from MP35N/CuBe alloy, specifically designed for high pressure
experiments Shermadini et al. (2017); Khasanov et al. (2016). Single crystals
of CsV3Sb5 were filled with random orientations inside the pressure cell in a
compact cylindrical area of height 12 mm and diameter 6 mm. Daphne 7373 was
used as a pressure medium to ensure the hydrostatic pressure conditions as it
solidifies at much higher pressure $\sim$ 2.5 GPa. Low temperature values of
pressures were determined by tracking the superconducting transition of
Indium, measured using ac susceptibility experiment. The fraction of muon
stopping inside the sample was maximized to $\sim$ 40 %.

We performed transverse-field (TF) and zero-field ${\mu}$SR experiments at
different pressures. Pressure cell is surrounded by four detectors: Forward,
backward, left, and right with muon initial spin parallel to its momentum. The
TF-${\mu}$SR experiments were performed in field cooled condition to get
homogeneous vortex lattice formation. Approximately 106 positrons were
recorded for each data point. The asymmetry vs. time spectra were analyzed
using the open software package Musrfit Suter and Wojek (2012).

Analysis of $\lambda(T)$: The temperature variation of magnetic penetration
depth $\lambda^{-2}(T)$ is analyzed within the local (London) approach using
following expressionServenti et al. (2004):

$\frac{\lambda^{-2}(T,\Delta_{0})}{\lambda^{-2}(0,\Delta_{0})}=1+\frac{1}{\pi}\int_{0}^{2\pi}\\!\\!\int_{\Delta(T,\phi)}^{\infty}\frac{\partial
f}{\partial
E}\frac{E\,\mathrm{d}E\,\mathrm{d}\phi}{\sqrt{E^{2}-\Delta(T,\phi)^{2}}},$ (1)

where $f=(1+E/k_{\mathrm{B}}T)^{-1}$ is the Fermi distribution function. The
temperature and angular dependence of gap is given by
$\Delta(T,\phi)=\Delta_{0}\delta(T/T_{c})g(\phi)$, where $\Delta_{0}$ is the
gap value at 0 K, $\delta(T/T_{c})=\tanh\\{1.821[1.018(T_{c}/T-1)^{0.51}]\\}$
represents the temperature dependence of gap. The angular dependence of the
gap is represented by $g(\phi)$, which adapts a value 1 for isotropic $s$\-
wave model. We have fitted the $\lambda^{-2}(T)$ data at various pressures by
separating it into two gaps:

$\frac{\lambda\textsuperscript{-2}(T)}{\lambda\textsuperscript{-2}(0)}=x\frac{\lambda\textsuperscript{-2}(T,\Delta_{0,1})}{\lambda\textsuperscript{-2}(0,\Delta_{0,1})}+(1-x)\frac{\lambda\textsuperscript{-2}(T,\Delta_{0,2})}{\lambda\textsuperscript{-2}(0,\Delta_{0,2})}.$
(2)

In the above equation, $x$ is the weight factor of the bigger gap
$\Delta_{0,1}$. Fig. S3 shows the pressure dependence of the superconducting
gaps ($\Delta_{0,1}$ and $\Delta_{0,2}$). In the regions where SC and CO
coexists, i.e. Region I and II, fitting was done by considering the weight
factor $x$ as a global parameter for all pressures in the mentioned regimes.
Similarly, for Region III, the fitting was done by assuming $x$ as a global
parameter in that region. The data at intermediate pressure 1.68 GPa is fitted
by considering a linear combination of Eq. 2 with parameter values fixed from
that obtained for 1.54 GPa (corresponding to the SC+CO state) and 1.74 GPa
(corresponding to the SC state).

Analysis of ZF relaxation rate: $\Lambda(T)$ at various pressures are fitted
with following empirical relation:

$\lambda(T)=\begin{cases}\Lambda_{0},&T>T^{*}\\\
\Lambda_{0}+\Delta\Lambda\left[1-\left(\frac{T}{T^{*}}\right)^{n}\right],&T<T^{*}\end{cases}.$
(3)

where $T^{*}$ is a characteristic temperature below which the increase in
electronic relaxation starts. $\Lambda_{0}$ is the relaxation value above
$T_{c}$, and $\Delta\Lambda$ is the change in relaxation after entering into
the superconducting ground state. In Region I and II, the value of $n$ is
considerably high (14 and 15 respectively) indicating the step like increase
in the relaxation rate. On the other hand, in Region III, where CO suppress
completely, the value of $n$ is 1 suggesting a linear increase in $\lambda(T)$
below $T^{*}$ which coincides with $T_{c}$. As discussed in the main text, we
conclude that in region III, $T^{*}$ corresponds to the time reversal symmetry
breaking transition $T_{\rm TRSB}$.

DFT-calculations: All DFT calculations were performed using Projector
Augmented Waves (PAW) as implemented in the Vienna Ab initio simulation
package (VASP) version 5.4.4 Kresse and Hafner (1993); Kresse and Furthmüller
(1996a, b). We used the PBEsol exchange correlation functional, with valence
configurations of 5 _s_ 25 _p_ 66 _s_ 1, 3 _s_ 23 _p_ 63 _d_ 44 _s_ 1, and 5
_s_ 25 _p_ 3 corresponding to Cs, V, ans Sb, respectively. Lattice parameters
were found to be converged to within 0.001 Å using a plane wave cutoff energy
of 450 eV, a Monkhorst-Pack k-point mesh of 20$\times$20$\times$10 in the
primitive cell, and a 2nd order Methfessel-Paxton smearing parameter of 10 meV
Methfessel and Paxton (1989). Different phases have different specific
volumes, and hence at finite pressure, the phase that minimizes the _enthalpy_
, rather than the Kohn-Sham energy only, is the equilibrium structure at that
volume.

Bilbao Crystallographic Server and tools therein were used for symmetry
analysis of various theoretically obtained crystal structures Aroyo et al.
(2011).

Acknowledgments
We thank B. Andersen and M. Christensen for fruitful discussions. ${\mu}$SR
experiments were performed at the Swiss Muon Source (S$\mu$S), Paul Scherrer
Institute (PSI), Switzerland. H.C.L. was supported by National Key R&D Program
of China (Grant No. 2018YFE0202600) and the Beijing Natural Science Foundation
(Grant No. Z200005). T.B. and E.R. were supported by the NSF CAREER grant
DMR-2046020. RMF (theory) was supported by the Air Force Office of Scientific
Research under award number FA9550-21-1-0423.

Author contributions
R.G., D.D., C.M., Z.G., F. H., and R.K. performed ${\mu}$SR experiments. R.G.,
D.D. and R.K. analyzed the ${\mu}$SR data. Q.Y., Z.T., C.G., and H.C.L.
synthesized and characterized samples. R. M. F., T. B., and E.R. performed the
theoretical analysis. R. K. and H.L. supervised the work at PSI. R.G., D.D.,
Z.G., and R. M. F. prepared the manuscript with notable inputs from all
authors.

Competing interests
The authors declare no competing interests.

Data availability:
The data supporting the findings of this study are available within the paper
and in the Supplementary Information. The raw data are available from the
corresponding authors upon reasonable request.

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