###### Abstract

Recently, a superconducting (SC) transition from low-field (LF) to high-field
(HF) SC states was reported in CeRh2As2, indicating the existence of multiple
SC states. It has been theoretically noted that the existence of two Ce sites
in the unit cell, the so-called sub-lattice degrees of freedom owing to the
local inversion symmetry breaking at the Ce sites, can lead to the appearance
of multiple SC phases even under an interaction inducing spin-singlet
superconductivity. CeRh2As2 is considered as the first example of multiple SC
phases owing to this sub-lattice degree of freedom. However, microscopic
information about the SC states has not yet been reported. In this study, we
measured the SC spin susceptibility at two crystallographically inequivalent
As sites using nuclear magnetic resonance for various magnetic fields. Our
experimental results strongly indicate a spin-singlet state in both SC phases.
In addition, the antiferromagnetic phase, which appears within the SC phase,
only coexists with the LF SC phase; there is no sign of magnetic ordering in
the HF SC phase. The present work reveals unique SC properties originating
from the locally noncentrosymmetric characteristics.

Figure 1:  Schematics of a pair-density-wave (PDW) state in a two-dimensional
system with locally broken inversion symmetry. (a) Schematic image of crystal
structure. The unit cell contains two kinds of superconducting layers. The
position of the inversion center is indicated by the arrows. (b) Theoretically
predicted $H$ –$T$ phase diagram. The position dependence of superconducting
order parameters in each superconducting phase is shown by the schematic
images. (c) Crystal structure of CeRh2As2.

In conventional superconductors, a superconducting (SC) order parameter can be
classified as the multiplication of spin and orbital states, that is, even-
parity spin-singlet and odd-parity spin-triplet states. One possible expansion
of the framework in SC symmetry is to introduce additional degrees of freedom,
such as frequency and atomic orbital[1, 2]. As shown in Fig. 1(a), in a two-
dimensional system with locally broken inversion symmetry (centrosymmetric but
no inversion center at the SC layers), there are sub-lattice degrees of
freedom owing to the inversion symmetry operation between the two layers. In
such systems, a parity transition from a low-field (LF) even-parity SC state
to a high-field (HF) odd-parity pair-density-wave (PDW) SC state (the SC phase
is inverted layer by layer) [Fig. 1(b)] is theoretically proposed, even when
only a pairing interaction in the spin-singlet channel exists[3]. The PDW
state coexisting with the charge-density-wave state has been discussed in
high-SC transition temperature $T_{\rm SC}$ cuprates[4] and kagomé
superconductors[5]; however, the PDW phase originating from crystal symmetry
is quite rare.

CeRh2As2, where the local inversion symmetry is broken at the Ce sites, is a
recently discovered superconductor with $T_{\rm SC}\sim 0.3$ K[6]. In
CeRh2As2, an SC transition from the SC1 to SC2 phase occurring at
approximately 4 T was reported for $H\parallel c$. This is the first
experimental indication of the SC parity transition originating from the sub-
lattice degrees of freedom and has promoted many theoretical studies[7, 8, 9,
10, 11, 12, 13]. However, to date, there have been few experimental
investigations on the SC properties[14, 15, 16]. Therefore, further
experimental confirmation is required.

The crystal structure of CeRh2As2 is of the tetragonal CaBe2Ge2-type with
space group $P4/nmm$ (No.129, $D_{4h}^{7}$)[17]. There are two
crystallographically inequivalent As and Rh sites. As(1) [Rh(1)] is
tetrahedrally coordinated by Rh(2) [As(2)], as shown in Fig. 1 (c). Heavy-
fermion superconductivity is characterized by a broad maximum in resistivity
at $T_{\rm coh}\sim$ 40 K and a large specific heat jump at $T_{\rm SC}$[6].
In addition to the multiple SC phases, CeRh2As2 exhibits nonmagnetic and
magnetic phase transitions just above and below $T_{\rm SC}$, respectively[6,
14]. The specific heat shows a large anomaly at $T_{\rm SC}$ and a rather weak
anomaly at $T_{0}\sim$ 0.4 K. The anomaly at $T_{0}$ is considered as a phase
transition to an electric quadrupole density-wave state[18]. Moreover, nuclear
quadrupole resonance (NQR) measurements revealed an antiferromagnetic (AFM)
order inside the SC phase[14]. Hence, CeRh2As2 is a promising system for
studying the role of sub-lattice degrees of freedom for unconventional
nonmagnetic, AFM, and SC states as well as their interactions.

Figure 2:  NMR measurements in CeRh2As2. (a) NMR spectra at 1.2 T for
$H\parallel c$ measured at various temperatures. The dashed curves indicate
the results of a two-peak fitting. Temperature dependence of full width at
half maximum(FWHM) (b), Knight shift (c) at the As(1) site, $K^{\rm As(1)}$,
and As(2) site, $K^{\rm As(2)}$, determined by NMR spectrum at 1.2 T. $T_{\rm
SC}$ and $T_{\rm N}$ are indicated by the dashed lines. The dashed curves are
guides for the eye. (d) Temperature dependence of ac magnetic susceptibility
at 1.2 T for $H\parallel c$. (e) NMR spectra at 4.5 T for $H\parallel c$
measured at various temperatures. Temperature dependence of FWHM (f), Knight
shift (g), and ac magnetic susceptibility (h) at 4.5 T for $H\parallel c$.

Nuclear magnetic resonance (NMR) can measure spin susceptibility in the SC
state, whereas bulk magnetic susceptibility is dominated by SC diamagnetic
shielding effect. In addition, NMR is sensitive to the appearance of internal
magnetic fields at observed nuclear sites. Therefore, NMR is one of the most
powerful techniques for studying SC and magnetic properties. Furthermore, NMR
is also useful for detecting spatially modulated SC states, such as the Fulde-
Ferrell-Larkin-Ovchinnikov and PDW phases, because NMR measures the local spin
susceptibility at the nuclear positions[19].

In this paper, we report the 75As-NMR results for CeRh2As2. Up to 5 T, the
spin susceptibility decreased in the SC state, indicating spin-singlet
superconductivity in both the SC1 and SC2 phases. The decrease in the Knight
shift in the SC1 phase was in good agreement with the critical field of the
SC1 – SC2 transition. In contrast, the decrease in the Knight shift in the SC2
phase is seemingly inconsistent with the large $H_{\rm c2}$, suggesting the
spatial modulation of the SC spin susceptibility. This is the first
microscopic information on the unique SC states in CeRh2As2.

Single crystals of CeRh2As2 were grown using the Bi flux method[6]. The
magnetic field dependence of $T_{\rm SC}$ and the temperature dependence of
the critical field of the SC1 – SC2 transition were determined from the ac
susceptibility measurements using an NMR coil as shown in Supplemental
Materials (SM)[20]. The details in NMR measurements are described in SM[20].
We experimentally confirmed the superconductivity immediately after the NMR
pulses using a technique reported in a previous study[21, 22]. Reflecting two
crystallographically inequivalent As sites, two NMR peaks were observed in all
measurement ranges, as shown in Figs. 2 (a) and 2 (e). The site assignment of
two NMR peaks has been described in a previous paper[23]. The reason why the
NMR spectrum of the As(1) site is broad at 4.5 T is a slight misalignment of
the magnetic field direction from the $c$-axis ($<$2o in this experiment).

Figure 3:  Magnetic field variation of the magnetic properties in CeRh2As2.
The $c$-axis magnetic field dependence of full width at half maximum (FWHM) at
0.06 K (a) and $\Delta K~{}=$ $K$(0.06 K) - $K$(0.6 K) (b). The dashed curve
indicates the contribution of SC diamagnetism $K_{\rm dia}$. The dashed lines
are guides for the eye. (c) The $H$ – $T$ phase diagram in the present sample
of CeRh2As2 determined from the NMR and ac magnetic susceptibility
measurements. Circles denote $T_{\rm SC}$ determined by the onset temperature
of the SC diamagnetic signal from ac magnetic susceptibility. Crosses denote
the critical magnetic field of the SC1 – SC2 transition determined by the
anomaly of ac magnetic susceptibility. Diamonds indicate $T_{\rm N}$
determined by the increase in NMR linewidth. (Inset) Schematic image of the
distribution of spin susceptibility and SC gap.

First, we show the temperature evolution of the NMR spectrum for each SC
phase. As shown in Fig 2 (a), at 1.2 T (low-field SC1 phase), two sharp peaks
were observed above $T_{\rm SC}\sim 0.35$ K. Below $T_{\rm SC}$, the NMR
spectrum broadens, owing to the SC diamagnetic field, and significant site-
dependent broadening is observed below 0.25 K. The site-dependent broadening
was also observed at zero field (NQR measurements)[14], which was attributed
to the AFM order. The internal field at the As(2) site is estimated to be
$\sim$ 20 mT. In contrast, at 4.5 T in the SC2 phase, the NMR lines broaden
just below $T_{\rm SC}$ for both sites, as in the SC1 phase, but there is no
additional, site-dependent broadening at lower temperatures, down to 0.06 K.
This indicates the absence of AFM ordering below $T_{\rm SC}$ in the SC2
phase. To investigate the spin susceptibility at the two As sites, NMR Knight
shifts are determined by two-peak Gaussian fitting. The Knight shifts for the
As(1) and As(2) sites are denoted by $K^{\rm As(1)}$ and $K^{\rm As(2)}$,
respectively. $K^{\rm As(1)}$ decreases below $T_{\rm SC}$ in both SC phases.
In the SC2 phase, $K^{\rm As(2)}$ also decreases, while in the SC1 phase
$K^{\rm As(2)}$ is not measurable owing to the extremely broad spectrum below
the AFM transition temperature $T_{\rm N}$.

To clarify the magnetic field evolution of the spin susceptibility and the AFM
order, the magnetic field dependence of the full width at half maximum (FWHM)
at the lowest temperature (0.06 K) and $\Delta K~{}=$ $K$(0.06 K)-$K$(0.6 K)
are shown in Figs. 3 (a) and 3 (b), respectively. In the SC1 phase, the
linewidth of the As(2) site decreases as the magnetic field increases but
remains significantly broader than that of As(1) until 3.5 T. Between 3.5 T
and 4 T, the width of the As(2) lines drops to values only slightly higher
than that of As(1). In contrast, in the SC2 phase, the linewidth of the As(2)
site is almost the same as that of the As(1) site. The absence of an internal
field at the As(2) site indicates that the AFM phase only coexists with the
SC1 phase, not with the SC2 phase. In contrast to the magnetic-field
dependence of the linewidth, $|\Delta K|$ at both As sites does not show any
anomaly at the phase boundary and gradually decreases as the magnetic field
increases, up to 5 T. The magnetic field dependence of $|\Delta K^{\rm
As(1)}|$ can be extrapolated to zero at 14 T ($H_{\rm c2}$), suggesting that
the magnetic field dependence of $|\Delta K|$ originates from the field-
induced quasiparticle density of states (Volovik effect). $|\Delta K^{\rm
As(2)}|$ is smaller than $|\Delta K^{\rm As(1)}|$ owing to the difference of
the hyperfine coupling constant $A_{\rm hf}$[23].

From the AC magnetic susceptibility and NMR measurements, we constructed the
$H$ – $T$ phase diagram in the present sample of CeRh2As2, as shown in Fig. 3
(c). The previously reported peculiar shape of $T_{\rm SC}$ and the SC1 – SC2
transition were reproduced by our measurements, whereas $T_{\rm SC}=0.37$ K
was higher than that of the previous samples[6, 16]. This is due to the
differences in sample quality and experimental methods. Our NMR results
indicate that the AFM phase disappears in the SC2 phase, which is strong
evidence of the presence of the SC1 – SC2 transition and the different SC
properties between the two phases. Furthermore, the smooth connection of the
Knight shift between two SC states indicates that the SC2 state is a non-
polarized one. This is in complete contrast to the usual field-induced
destruction of an AFM state resulting in a polarized ferromagnetic-like state.
Therefore, the observed field-induced transition from an AFM state to a state
with no sizable polarization at high field is a highly unusual one, indicating
that this AFM state is strongly linked to the SC1 state.

The SC spin state can be deduced from the temperature variation in the NMR
Knight shift. In a conventional spin-singlet superconductor, the spin
component of the Knight shift decreases in the SC state and becomes almost
zero at $T\rightarrow 0$ K. However, when the SC spins of spin-triplet
superconductors are aligned along the magnetic field, the spin component of
the Knight shift does not change across $T_{\rm SC}$. As the Knight shift
includes various contributions, such as the temperature-independent orbital
component arising from the Van Vleck susceptibility and SC diamagnetic effect,
the absolute amount of the spin component of the Knight shift must be
evaluated. $K$ at the lowest temperature can be divided into the following
three contributions:

$\displaystyle K=K_{\rm normal}+\delta K_{\rm spin}+K_{\rm dia}.$ (1)

where $K_{\rm normal}$ is the normal-state Knight shift, $\delta K_{\rm spin}$
is the change in a spin component of the Knight shift in the SC state, and
$K_{\rm dia}$ is the SC diamagnetic component. In the SC state, the Knight
shift decreases owing to the SC diamagnetic shielding effect, and the value of
$K_{\rm dia}$ at 0 K is approximately expressed as[24]

$\displaystyle K_{\rm
dia}=-\frac{H_{c1}}{H}\frac{\ln\left(\frac{\beta\lambda_{d}}{\sqrt{e}\xi}\right)}{\ln\kappa}.$
(2)

Here, $H_{\rm c1}$ is the SC lower critical field; $\xi$ is the Ginzburg-
Landau (GL) coherence length; $\beta$ is a factor that depends on the vortex
structure and is 0.38 for the triangular vortex lattice; $\lambda_{d}$ is the
distance between the vortices and is calculated using the relation
$\phi_{0}=\frac{\sqrt{3}}{2}\lambda_{d}^{2}(\mu_{0}H_{\rm ext})$; $e$ is
Euler’s number, and $\kappa$ is the GL parameter. From the SC upper critical
field $\mu_{0}H_{\rm c2}~{}=~{}14$ T and thermodynamic critical field
$\mu_{0}H_{\rm c}~{}=~{}31$ mT[6], $\mu_{0}H_{\rm c1}~{}=~{}0.40$ mT,
$\xi~{}=~{}4.85$ nm, and $\kappa~{}=~{}319$ were obtained. As shown in Fig 3
(b), the magnetic field dependence of $K_{\rm dia}$ is negligibly small
compared to the experimental $\delta K$. In addition, the temperature
dependence of $K_{\rm normal}$ shows almost constant at low temperatures[23],
as shown in Figs. 2 (c) and 2 (g). Therefore, $\Delta K$ was dominated by
$\delta K_{\rm spin}$. As shown in Figs. 2 (c) and 2 (g), a clear decrease in
$K$ was observed in the SC state, indicating the realization of spin-singlet
superconductivity in both the SC1 and SC2 phases.

The spin-singlet superconductivity in the SC1 phase can also be confirmed by
the quantitative agreement between $H_{\rm c2}$ and the Pauli limiting field
$H_{\rm P}$ estimated from the reduction in the Knight shift. In strongly
correlated electron systems, spin-singlet superconductivity is destroyed when
the Zeeman-splitting energy is as high as the SC condensation energy, called
the Pauli limiting effect. It is well known that in a spin-singlet
superconductor, a relation holds between $H_{\rm P}$ and the decrease in the
spin susceptibility $\delta\chi$ ascribed to singlet-pair formation. This is
expressed as

$\displaystyle\frac{1}{2}\delta\chi\mu_{0}H_{\rm
P}(0)^{2}~{}=~{}\frac{1}{2}\mu_{0}H_{\rm c}^{2}.$ (3)

This equation yields $\mu_{0}H_{\rm P}(0)=\mu_{0}H_{\rm
c}/\sqrt{|\delta\chi|}$, where $\delta\chi$ can be determined from $\Delta K$
as $\delta\chi=(\mu_{\rm B}N_{\rm A}/A_{\rm hf})\Delta K$. In the SC1 phase,
$\Delta K$ at the As(1) site is approximately - 0.2 %; thus, $\mu_{0}H_{\rm
P}(0)$ is estimated to be 3.4 T with $\mu_{0}H_{\rm c}$ = 31 mT and the
hyperfine coupling constant $A_{\rm hf}^{\rm As(1)}$ = 1.55 T/$\mu_{\rm
B}$[23]. The estimated $\mu_{0}H_{\rm P}(0)$ agrees to the SC1–SC2 transition
field ($\sim 4$ T), indicating that the SC1 phase exhibits homogeneous spin-
singlet superconductivity. Such a correspondence between the critical field
and $\Delta K$ is observed in various spin-singlet heavy-fermion
superconductors[25, 26, 27, 28].

In contrast, $\Delta K$ in the SC2 phase is unusual. $K$ in the SC2 phase also
clearly decreases, and $\mu_{0}H_{\rm P}(0)$ is estimated to be 4.8 T from
$\Delta K^{\rm As(1)}~{}=~{}-0.1$ %, which is much smaller than $\mu_{0}H_{\rm
c2}(0)=14$ T. Here, we assume the same $H_{\rm c}$ as in the SC1 state. If we
consider that $H_{\rm P}(0)$ increases owing to an increase in $H_{\rm c}$ in
the SC2 phase, the $H_{\rm c}$ should be increased by a factor of three, which
is an unrealistic value in this case. It is noteworthy that $\Delta K^{\rm
As(2)}~{}=~{}-0.05$ % and $A_{\rm hf}^{\rm As(2)}$ = 0.27 T/$\mu_{\rm B}$ also
lead to a small Pauli-limiting field, $\mu_{0}H_{\rm P}(0)~{}=~{}2.8$ T. There
are several superconductors without the Pauli limiting effect owing to the
absence of the spin susceptibility reduction[29, 30, 31], but the case where
the Pauli limiting effect does not work despite a decrease in the spin
susceptibility is quite rare. The discrepancy between the decrease in the spin
susceptibility and the absence of the Pauli limiting field can be reconciled
by the presence of a spatially inhomogeneous SC state.

Recently, SC states realized in the presence of sub-lattice degrees of freedom
have been intensively studied, and spin susceptibility has been calculated in
various SC states[32]. The average spin susceptibility in the PDW state is
suggested to be the same as that in the normal state. The present Knight-shift
results are inconsistent with these theoretical suggestions. To interpret this
discrepancy, we propose a position-dependent modulation of spin susceptibility
in the PDW state. In this SC state, the SC order parameter oscillates along
the $c$ axis with a period corresponding to the $c$ lattice parameter; thus,
the spin susceptibility also oscillates with the same periodicity. NMR Knight
shift probes the local spin susceptibility at the observed nuclear site.
Therefore, the present results imply that the block layers become the ”spin-
singlet” dominant SC state and the spin susceptibility at the block layers
decreases in the SC2 states as shown in the inset of Fig. 3 (c). In contrast,
a sizable Rashba effect at the Ce site would lead to locally spin-polarized
bands. The Pauli susceptibility would then transform into a Van Vleck
susceptibility, which is insensitive to superconductivity[32, 33]. As a
result, the spin susceptibility in the SC state at the Ce sites would be of
similar size as that in the normal state.

In CeRh2As2, the SC properties are determined by the Ce layer, resulting in
the spin susceptibility of the Ce layers being crucially modified by the SC
character. We suggest the possibility that the spin susceptibility at the Ce
layers largely changes between the LF and HF SC states; that is, the spin
susceptibility at the Ce site decreases in the SC1 state but remains unchanged
in the SC2 state, although the spin susceptibility at the block layers does
not change significantly. This might explain the large difference in the Pauli
limiting field between the two phases. However, because all NMR-active Ce
isotopes are unstable, measuring the spin susceptibility at the Ce site is
impossible. Thus, we need to seek other methods to determine the spatial
modulation of the spin susceptibility in CeRh2As2. In addition, theoretical
studies on spin-susceptibility modulation in the PDW state are required.

In conclusion, we measured the SC spin susceptibility in the LF and HF SC
states. The spin susceptibility decreases in both SC states, indicating a
spin-singlet state. The discrepancy between the decrease in the spin
susceptibility and the absence of the Pauli limit might be explained by the
spatially modulated spin susceptibility in the SC2 state. In addition, NMR
measurements revealed that the AFM phase only coexists with the LF SC phase,
indicating a strong link to the SC1 state. Our findings help to understand the
unique multi-SC state in CeRh2As2.

## acknowledgments

The authors would like to thank K. Nogaki, Y. Yanase, Y. Maeno, and S.
Yonezawa for their valuable discussions. This work was partially supported by
the Kyoto University LTM Center and Grants-in-Aid for Scientific Research
(KAKENHI) (Grants No. JP19K14657, No. JP19H04696, No. JP20H00130, No.
JP21K18600, No. JP22H04933, and No. JP23H01124). C. G. and E. H. acknowledge
support from the DFG program Fermi-NESt through Grant No. GE 602/4-1.
Additionally, E. H. acknowledges funding by the DFG through CRC1143 (project
number 247310070) and the Würzburg-Dresden Cluster of Excellence on Complexity
and Topology in Quantum Matter—ct.qmat (EXC 2147, project ID 390858490).

S. O. and S. K. contributed equally to this work.

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Supplementary Materials for Parity transition of
spin-singlet superconductivity by sub-lattice degrees of freedom

Materials and Methods

Sample preparation Single crystals of CeRh2As2 were grown by the Bi flux
method. The present sample is the same as that used in the previous
measurements[1]. The sample is in a plate shape with a size of about 3.0
$\times$2.0 $\times$0.75 mm3. The superconducting transition temperature
$T_{\mathrm{sc}}=0.37$ K is higher than that in the previous report[2].

AC susceptibility measurements The magnetic field dependence of $T_{\rm SC}$
was determined by the onset temperature of the SC diamagnetic signal from the
radio-frequency ac susceptibility measurement using an NMR tank circuit. In
the superconducting state, the impedance of the circuit changes due to the
superconducting diamagnetic effect, and thus, the tuning frequency of the
circuit drastically changes just below $T_{\rm SC}$. The frequency of the AC
magnetic field is $\sim$ 10 MHz. Figures S1 and S2 show the temperature and
magnetic field scan of $\chi_{\rm ac}$, respectively. The obtained phase
diagram shown in Fig. 3 (c) is quite consistent with the previous report[2].

Field alignment For NMR measurements, we used a split SC magnet, that
generated a horizontal field and combined it with a single-axis rotator to
apply a magnetic field parallel to the $c$ axis. With this setup, we can
rotate the magnetic field direction within the $ac$ plane of the sample.

FIG. S 1:  The temperature dependence of $\chi_{\rm ac}$ at several magnetic
fields up to 7 T in CeRh2As2. A magnetic field is applied parallel to the
$c$-axis. The arrows indicate $T_{\rm SC}$ determined by the onset of a
superconducting diamagnetic signal.  FIG. S 2:  The magnetic field dependence
of $\chi_{\rm AC}$ at several temperatures in CeRh2As2. A magnetic field is
applied parallel to the $c$-axis. The solid arrows indicate $H_{\rm c2}$ and
the broken arrows indicate $H^{*}$ where $\chi_{\rm ac}$ shows a kink.  FIG. S
3:  NMR measurements in CeRh2As2. (a) NMR spectra at 1.4 T for $H\parallel c$
measured at various temperatures. The dashed curves show Gaussian curves for
guidance to the eye. Temperature dependence of full width at half
maximum(FWHM) (b), Knight shift (c) at the As(1) site, $K^{\rm As(1)}$, and
As(2) site, $K^{\rm As(2)}$, determined by NMR spectrum at 1.4 T. $T_{\rm SC}$
and $T_{\rm N}$ are indicated by the dashed lines. The dashed curves are
guides for the eye. (d) Temperature dependence of ac magnetic susceptibility
at 1.4 T for $H\parallel c$. (e) NMR spectra at 3.5 T for $H\parallel c$
measured at various temperatures. Temperature dependence of FWHM (f), Knight
shift (g), and ac magnetic susceptibility (h) at 3.5 T for $H\parallel c$.

NMR measurements An NMR spectrometer with a 100 W (at 0 dB input) power
amplifier (Thamway Product: N146-5049A) was used for the measurements. A
conventional spin-echo technique was used for NMR measurements. Low-
temperature NMR measurements down to 0.06 K were carried out with a 3He–4He
dilution refrigerator, in which the sample was immersed into the 3He–4He
mixture to avoid radio-frequency heating during measurements. We
experimentally confirmed the superconductivity just after the NMR pulses using
a technique reported in a previous study[3, 4]. The 75As-NMR spectra (nuclear
spin $I~{}=~{}3/2$, nuclear gyromagnetic ratio $\gamma/2\pi~{}=~{}7.29$ MHz/T,
and natural abundance 100%) were obtained as a function of the frequency at
fixed magnetic fields. For $H\parallel c$, because the contribution of the
nuclear quadrupole interaction can be ignored at the center peak (1/2
$\leftrightarrow$ -1/2 transition) of the NMR spectrum, the observed NMR
spectrum is shown against $K=(f-f_{0})/f_{0}$. Here,
$f_{0}=(\gamma_{n}/2\pi)\mu_{0}H$ is the reference frequency. The magnetic
field was calibrated using 63Cu (${}^{63}\gamma_{n}/2\pi=11.285$ MHz/T) and
65Cu (${}^{65}\gamma_{n}/2\pi=12.089$ MHz/T) NMR signals from the NMR coil.
The Knight shift is simply determined at the peak of the spectrum. The
anomalies in $T_{1}$ below $T_{\rm SC}$ were observed in NQR measurements[1]
and $K$ decreases below $T_{\rm SC}$. Therefore, we can claim that we are
really measuring the superconducting state.

Experimental results

FIG. S 4:  Magnetic field evolution of the NMR spectrum at 0.06 K
(superconducting state) and 0.6 K (normal state).

NMR spectra in the SC1 phase

To determine the antiferromagnetic transition temperature $T_{\rm N}$ in Fig.
3(c), in addition to 1.2 T and 4.5 T shown in Fig. 2, we also measured the
temperature variation of the NMR spectrum at 1.4 T and 3.5 T (both in the SC1
phase) as shown in Fig. S3. At 1.4 T and 3.5 T, the clear site-dependent
linewidth broadening was observed at low temperatures, indicating the
existence of antiferromagnetic transition. The NMR spectra at 1.4 T overlap
even in the normal state, making it difficult to separate the As(1) and As(2)
sites at low temperatures.

Magnetic field evolution of the NMR spectrum

Figure S4 shows the magnetic field evolution of the NMR spectrum at 0.6 K
(normal state) and 0.06 K (superconducting state). The magnetic field
dependence of the NMR spectrum is affected by a slight misalignment of the
magnetic field direction from the $c$-axis ($<$2 degrees in this experiment).
In situations where the frequency corresponding to the Zeeman energy due to
the external magnetic field (7.29 $\times\mu_{0}H$ MHz for As) and the NQR
frequency ($\sim$ 31 MHz and $\sim$ 10.8 MHz for the As(1) and As(2),
respectively) are almost the same, a slight misalignment of the magnetic field
direction from the $c$ axis causes the line-width broadening and the change in
peak position. From this figure, we estimated the magnetic field dependence of
the full width at half maximum at the lowest temperature and $\Delta K~{}=$
$K$(0.06 K)-$K$(0.6 K), as shown in Figs. 3(a) and 3(b). NMR spectra clearly
show a decrease in the Knight shift in the superconducting state and the site-
dependent linewidth broadening above 4 T (SC2 phase).

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