Science - USA (2022-04-22)

To obtain further information, we inves-
tigated the temperature variation of the
O(2)-site NMR spectrum as well as the bulk
shielding signal to determineTc. Shown in
Fig. 3A are the temperature variation of the
ac susceptibility (cAC); the square root of the
second moment (

ffiffiffiffiffi

s^2

p
), characterizing the NMR
spectral linewidth ( 27 ) and reflecting the
distribution of spin susceptibility; and the
Knight-shift change from the normal-state
value, measured at various fields. These NMR
quantities were determined from the spectra
shown in fig. S4. At 1.1 T, the decreasingK(DK)
and increasing

ffiffiffiffiffi

s^2

p
upon cooling belowTcis
the above-mentioned conventional spin-singlet
SC behavior. At 1.25 T, an additional peak with
a larger Knight shift than the normal-state
shift appears only below 0.3 K, which is far
belowTc~ 0.7 K. Moreover, the field depen-
dence of

ffiffiffiffiffi

s^2

p
shown in Fig. 3B exhibits an
anomalous enhancement nearHc2. A compar-
ison between this behavior and that reported
in organic superconductors ( 12 ) is discussed in
( 27 ). Such an anomalous

ffiffiffiffiffi

s^2

p
at low tempera-
tures indicates that an inhomogeneous super-
conductivity emerges in the high-field region.
To determine the boundary of this inho-
mogeneous state in theH-Tphase diagram
systematically, we focused on

ffiffiffiffiffi

s^2

p
, which
characterizes the distribution of spin density.
Because

ffiffiffiffiffi

s^2

p
exhibits additional rapid increase
below a temperature substantially lower than
Tc, we defineT* below which

ffiffiffiffiffi

s^2

p
exceeds the
value expected from the ordinary behavior
stemming from vortex penetration (Fig. 3B,
dashed curve) ( 27 ).T characterizes the onset
of spectral splitting. As shown in Fig. 3C,T
increases with increasingH, indicating that
the inhomogeneous state becomes more stable
whenHc2(0) is approached. Thermodynamic
measurements on Sr 2 RuO 4 have not revealed
this phase boundary. However, an anomaly in
fourfold in-plane field anisotropy of the spe-
cific heat above about 1.2 T, close toH* in our
phase diagram, has been reported ( 28 , 29 );
this anomaly was interpreted as the occurrence
of an extra phase, such as the FFLO phase ( 29 ).
The double-horn two-peak spectra with one
of the peaks having a larger value ofKthan
that of the normal state were observed only in
the high-Hand low-Tregion (Fig. 3C). This
peak splitting cannot be explained by ordinary
phase separation. Instead, such a double-horn
structure is naturally explained by the real-
space sinusoidal spin modulation with a single
wave number, as shown schematically in Fig. 1.
Such double-horn NMR spectra have often
been observed in spin-density wave states
with a sinusoidal-modulated ordered moment
( 30 ). The crucial difference in the present case
is that the spin modulation is formed with
superfluid electron pairs. Thus, the present
observation provides evidence for SC spin
smecticity. For comparison, the Q-phase of

CeCoIn 5 exhibits a spin modulation much

larger than the inhomogeneous phase in

Sr 2 RuO 4 .Thisdifferenceisconsistentwiththe

spin modulation in the CeCoIn 5 Q-phase being

dominated by magnetic order, whereas that in

Sr 2 RuO 4 is a result of the SC spin smecticity ( 27 ).

The observed spin smecticity provides un-

ambiguous evidence for the FFLO state. The

distribution of the spin density in an FFLO

state has been theoretically calculated ( 31 , 32 ).

The simulated NMR spectra ( 31 ) for the FFLO

state atH= 0.9Hc2is in good agreement with

our double-horn spectrum at 1.3 T (fig. S7).

Additionally, the field and temperature de-

pendence of the spectral shape (Fig. 2B and

fig. S4) can be understood within the FFLO

scenario. The intensity of the peak arising

from the nodal region (the right-side peak) in

the FFLO state increases abruptly with increas-

ingHand decreasingTowing to the forma-

tion of the nodal plane. Moreover, the intensity

ratio of the two peaks depends on the FFLO

wavelengthqand thus on temperature and

the magnitude of the appliedH. The real-space

nodes in the FFLO state can be viewed as

topologically protected bound states formed

inside the SC gap ( 33 , 34 ). Thep-phase shift of

the pair potential across the nodal plane leads

to zero-energy bound states with an enhanced

density of states. This enhancement makes the

spin density at the FFLO nodal plane higher

than the value in the normal state.

In the FFLO state, Cooper pairs (k↑,–k+q↓)

with a finite total momentumjjqegmBH=ℏvF

are formed between the electrons in the Zeeman-

split parts of the Fermi surfaces. Here,grep-

resents the electron-sping-factor,mBis the

Bohr magneton,ħis Planck’s constanthdi-

vided by 2p, andvFis the Fermi velocity. Conse-

quently, the SC order parameter is modulated

in the real space asD(r)=Dcos(q·r), with the

wavelengthlFFLO¼ 2 p=jjq. From the compar-

ison between the NMR spectrum at 1.3 T and

the simulation (fig. S7) ( 27 , 31 ),lFFLOis esti-

mated to be 30 times longer than the in-plane

coherence lengthx, and thuslFFLO~ 30x~

2.0mmbyusingx~ 0.066mmofSr 2 RuO 4.

This is in good agreement with the crude esti-

mationlFFLO=2pħnF/gmBH=2p^2 Dx/gmBH=

29.8x= 1.97mm, assumingg=2andD= 1.76kBTc,

wherekBis the Boltzmann constant.

As the next step to obtain the definitive

evidence of the FFLO state, it is quite im-

portant to detect the FFLO phase boundary

from other thermodynamic measurements

and to image the spatial modulation of the

SC order parameter directly, which should be

possible with scanning tunneling microscopy.

Our study reveals that Sr 2 RuO 4 is a most suit-

able superconductor for studying the FFLO

state because the FFLO state is observable

under magnetic fields one order of magni-

tude smaller than those for the other FFLO

candidates. This makes the inhomogeneous

SC state we observed in Sr 2 RuO 4 accessible to

various experimental probes.

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ACKNOWLEDGMENTS

The authors thank Y. Yanase, S. Kasahara, Y. Matsuda, S. Kittaka,

K. Machida, M. Ichioka, A. P. Mackenzie, and S. E. Brown for

valuable discussions.Funding:This work was partially supported

by the Kyoto University LTM center, Grant-in-Aid for Scientific

Research on Innovative Areas from the Ministry of Education,

Culture, Sports, Science, and Technology (MEXT) of Japan, and

JSPS Core-to-Core program (JPJSCCA20170002). We are also

supported by JSPS KAKENHI JP15H05852, JP15K21717, JP17H06136,

JP20H00130, JP15K21732, JP15H05745, JP20KK0061, JP19H04696,

JP19K14657, and JP20H05158.Author contributions:K.I. and

Y.M. implemented this project. Z.Q.M., S.Y., and Y.M. synthesized

and characterized the samples. K.K., M.M., S.K., and K.I. performed

the NMR measurements and analysis. K.K., S.K., K.I., S.Y., and

Y.M. drafted the manuscript. All authors contributed to interpret the

experimental results and to finalize the manuscript.Competing

interests:The authors declare that they have no competing

interests.Data and materials availability:Data supporting this

study’s findings are available in ( 35 ).

SUPPLEMENTARY MATERIALS

science.org/doi/10.1126/science.abb0332

Materials and Methods

Supplementary Text

Figs. S1 to S8

Reference ( 36 )

8 March 2021; accepted 22 March 2022

10.1126/science.abb0332

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