Reduction of the spin susceptibility in the superconducting state of Sr2RuO4 observed by polarized neutron scattering
A. N. Petsch, M. Zhu, Mechthild Enderle, Z. Q. Mao, Y. Maeno, I. I. Mazin, S. M. Hayden
RReduction of the spin susceptibility in the superconducting state of Sr RuO observedby polarized neutron scattering A. N. Petsch, ∗ M. Zhu, Mechthild Enderle, Z. Q. Mao,
3, 4
Y. Maeno, I. I. Mazin, and S. M. Hayden † H.H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom Institut Laue-Langevin, CS 20156, 38042 Grenoble Cedex 9, France Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan Department of Physics, Pennsylvania State University, University Park, Pennsylvania 16802, USA Department of Physics and Astronomy, George Mason University andQuantum Science and Engineering Center, Fairfax, VA 22030, USA (Dated: August 27, 2020)Recent observations [A. Pustogow et al . Nature , 72 (2019)] of a drop of the O nuclearmagnetic resonance (NMR) Knight shift in the superconducting state of Sr RuO challenged thepopular picture of a chiral odd-parity paired state in this compound. Here we use polarized neutronscattering (PNS) to show that there is a 34 ± H ∼ H c than aprevious PNS study allowing the suppression to be observed. The PNS measurements show a smallersusceptibility suppression than NMR measurements performed at similar field and temperature. Ourresults rule out the chiral odd-parity d = ˆ z ( k x ± ik y ) state and are consistent with several recentproposals for the order parameter including even-parity B g and odd-parity helical states. Introduction. —Sr RuO is a moderately correlated ox-ide metal which, which forms a good Fermi liquid andsuperconducts [1] below 1.5 K. It has been initially pro-posed as a solid state analogue [2, 3] of superfluid He–A, driven by proximity to ferromagnetism. The super-conducting state was widely assumed to possess chiralodd-parity order [4, 5] with broken time-reversal symme-try [6]. An important property of odd-parity (triplet-paired) superconductors is that for some magnetic fielddirections the spin susceptibility may show no changeupon entering the superconducting state. This propertymay be investigated by probes not sensitive to the su-perconducting diamagnetic screening currents, such asnuclear magnetic resonance (NMR) or polarized neutronscattering (PNS). Early studies of the susceptibility usingthe NMR Knight shift with O (NMR) [7] and PNS [8]detected no change while crossing the superconductingtransition when magnetic fields were applied parallel tothe RuO -planes or ab -planes. These observations sup-ported the picture of triplet-paring with an out-of-plane d -vector or an unpinned in-plane d -vector ⊥ H . A muon-spin rotation ( µ SR) study [6] observed that magneticmoments appeared below the superconducting transitiontemperature T c . This was interpreted as evidence fortime-reversal symmetry breaking (TRSB) in the super-conducting phase. Further evidence for TRSB came fromthe detection of a Kerr effect [9] associated with the su-perconducting transition.More recently, it has become clear that it is difficultto consistently describe the physical properties of the su-perconducting state with a simple odd-parity represen-tation [10]. For example, the favored d = ˆ z ( k x ± ik y )state implies the existence of edge currents which are notdetected experimentally [11–14] and H c is much lowerthan expected [10]. Also an NMR experiment failed to detect any changes in susceptibility for a c -axis field [15],even though it would have to be reduced below T c in theˆ z ( k x ± ik y ) state. It was shown that rotation of the orderparameter vector would be forbidden by the strong spin-orbit coupling [16, 17], which led one of us to concludethat “the Knight shift in Sr RuO remains a challengefor theorists; until this puzzle is resolved, we cannot usethe Knight shift argument”[16].Further progress has been made recently with a O-NMR study by Pustogow et al. [18] which detected asignificant reduction in the Knight shift on entering thesuperconducting state for in-plane fields for the first time.This result has now been reproduced by Ishida et al. [19].The observed reduction in the Knight shift on enter-ing the superconducting state of Sr RuO shows thatthe spin susceptibility is reduced. This should also beobserved in polarized neutron scattering measurements.However, a PNS study [8] with relatively poor statisticsand at a field µ H = 1 T was unable to observe a reduc-tion.In this paper, we report PNS measurements at a lowerfield ( µ H = 0 . ± ± µ H = 0 .
48 T. We discusspossible reasons for the difference between the two ob-servations and the constraints our results place on theallowed order superconducting order parameter.
The spin susceptibility as a probe of the superconduct-ing state. —The transition to a superconducting state in-volves the pairing of electrons and hence a change to thespin wavefunction. In the case of singlet pairing wherethe spin susceptibility is suppressed everywhere (barring a r X i v : . [ c ond - m a t . s up r- c on ] A ug state basis line TRSB χ ( H, T → /χ ( n ) state basis line TRSB χ ( H, T → /χ ( n )function nodes with H k function nodes with H k ∆( k ) [100] [110] [001] d ( k ) [100] [110] [001] A g k x + k y X A u ˆ x k x + ˆ y k y X z k z (h) X A g k x k y ( k x − k y ) (v) X A u ˆ x k y − ˆ y k x X z k x k y k z ( k x − k y ) (v,h) X B g k x − k y (v) X B u ˆ x k x − ˆ y k y X z k x k y k z (v,h) X B g k x k y (v) X B u ˆ x k y + ˆ y k x X z k z ( k x − k y ) (v,h) X E g ( a ) k x k z ; k y k z (v,h) X E u ( a ) ˆ x k z ; ˆ y k z (h) X
0; 1 1 / z k x ; ˆ z k y (v) X E g ( b ) ( k x ± k y ) k z (v,h) X E u ( b ) (ˆ x ± ˆ y ) k z (h) X / z ( k x ± k y ) (v) X E g ( c ) ( k x ± ik y ) k z (h) X E u ( c ) (ˆ x ± i ˆ y ) k z (h) X z ( k x ± ik y ) X D h withstrong spin-orbit coupling [20]. The left and right panels are even-parity (singlet) and odd-parity (triplet) states, respectively.Column three and ten show whether states have vertical line nodes, k k z , (v), or horizontal line nodes, ⊥ k z (h) on a 2D Fermisurface. States with k µ transform like sin k µ while states with k µ transform like cos k µ . Ticks indicate whether TRSB is present. χ is calculated from Eqn. 2 in the H → , T → exotic cases as Ising superconductivity) its temperaturedependence is described by the Yosida function [21, 22].A triplet superconductor is described by a spinor orderparameter, ∆ = (cid:20) ∆ ↑↑ ∆ ↑↓ ∆ ↓↑ ∆ ↓↓ (cid:21) = ∆ (cid:20) − d x + id y d z d z d x + id y (cid:21) , (1)where the d -vector is d = ( d x , d y , d z ). The non-interacting spin susceptibility tensor is given by [23] χ ,αβ χ ( n ) = δ αβ + Z d Ω4 π [ Y (ˆ k , T ) − < (cid:26) d ∗ α ( k ) d β ( k ) d ∗ ( k ) · d ( k ) (cid:27) , (2)where the integral is over the Fermi surface, Y (ˆ k , T ) isthe Yosida function and χ ( n ) the normal-state spin sus-ceptibility. Table I shows χ ( T → , H →
0) = χ (0)evaluated using Eqn. 2 for selected irreducible represen-tations and applied field directions.The measurements of the spin susceptibility in thesuperconducting state are complicated by diamagneticscreening due to supercurrents which precludes quanti-tative measurements with standard techniques such asSQUID magnetometry. The NMR Knight shift and PNShave been used successfully to measure the spin suscep-tibility in the superconducting state. The Knight shift, K , originates from the hyperfine interaction between thenuclear moment and the magnetic field at the nuclear siteproduced by the electrons surrounding that site, whichis only indirectly related to the total magnetization. Forinstance, core polarization and spin-dipole interaction donot contribute to the latter, but affect the Knight shift,while spin and orbital moments have different spatial dis-tributions and therefore produce different contributions to K with opposite signs in some cases [24, 25]. In con-trast, PNS probes the total magnetization density M ( r )in absolute units induced by an external magnetic field µ H . The orbital and spin magnetization are equallyweighted and M ( r ) is averaged in space. In the nor-mal state of Sr RuO , three bands cross the Fermi sur-face [26]. The partially filled Ru t g orbitals account forthe majority of the density of states at the Fermi energy.Hence the majority of the spin susceptibility is associatedwith the Ru site probed by PNS.In PNS, the spatially varying density M ( r ) is mea-sured by diffraction. This technique was first applied toV Si by Shull and Wedgwood [27]. It has also been usedto probe cuprate [28] and iron-based [29, 30] supercon-ductors where singlet pairing has been observed. EarlyPNS measurements by Shull and Wedgwood [27] of thetemperature dependence of the induced magnetizationin V Si showed that the susceptibility only dropped toabout of its normal state value for applied fields of ≈ . H c . This residual susceptibility is due to the or-bital susceptibility present in transition metals [31, 32].Importantly, such cancellation effects work very differ-ently in NMR and PNS, for instance, in V metal NMRshows no or a very small change [33, 34] in the Knightshift across T c , because of nearly exact cancellation ofthe core polarization (an effect specific to NMR) and theFermi-contact term.NMR and PNS probe susceptibility of superconduct-ing in the mixed state and typically in relatively highmagnetic fields ∼ N ?F ( H ) ∝ N ?F ( n ) H/H c comes from low-energy localized states in the vortexcores [35], while in superconductors with lines of gapnodes N ?F ( H ) ∝ N ?F ( n ) p H/H c comes from the vicin-ity of the gap nodes in the momentum space, and par-tially from outside the vortex cores [36]. The same statesgive rise to a linear heat capacity and also contribute tothe spin susceptibility. For example, a linear field de-pendence of χ = M/H has been observed [29] in thesuperconducting state of Ba(Fe − x Co x ) As . Experimental method. —The experimental set up andcrystal were the same as described in Duffy et al. [8].A single crystal (C117) of Sr RuO with dimensions of1.5 mm × × T c of this sample is1.47 K and µ H c (100 mK) = 1 .
43 T for H || [110] [8, 41].We used the three-axis spectrometer IN20 at the Insti-tut Laue-Langevin, Grenoble [42]. A vertical magneticfield was applied perpendicular to the scattering planealong the [100] direction. Measurements of the nuclearBragg reflections (002) and (011) verified that the fieldwas within 0 . ◦ of the [100] direction. The PNS exper-iments were performed in the superconducting state atthe (011) Bragg reflection with µ H = 0 . E = 63 meV. The spectrome-ter beam polarization was 93.2 ± µ H = 0 . M ( r ) in theunit cell, induced by a large magnetic field µ H . Furtherdetails of the method and theory are given in our previousstudies [8, 29]. Due to the periodic crystal structure, theapplied magnetic field induces a magnetization densitywith spatial Fourier components M ( G ), where G are thereciprocal lattice vectors, and M ( G ) = Z unit cell M ( r ) exp( i G · r )d r . (3)We measure the flipping ratio R , defined as the ra-tio of the cross-sections, I + , I − , of polarized incidentbeams with neutrons parallel or anti-parallel to the applied magnetic field. A detector insensitive to thescattered spin polarization and summing over the fi-nal spin states was used. Because the induced mo-ment is small, the experiment is carried out in the limit( γr / µ B ) M ( G ) /F N ( G ) (cid:28)
1. In this limit [8], the flip-ping ratio is, R = 1 − γr µ B M ( G ) F N ( G ) , (4)where the nuclear structure factor F N ( G ) is known fromthe crystal structure and γr = 5 . × − m. For the(011) reflection we used F N = 4 . × − m f.u. − . Results. —In the present experiment we apply mag-netic fields along the [100] direction to allow compari-son with NMR measurements [18, 19]. We first estab-lished the normal state susceptibility χ ( n ) by makingmeasurements at T = 1 . µ H = 2 . − R , is proportional to the in-duced moment (Eq. 4) and our error bars are determinedby the number of counts in I ± and hence the countingtime. Thus, the 2.5 T measurement (closed diamond)provides the most accurate estimate of the normal statesusceptibility, shown by the horizontal solid line. Thedata has been converted to χ using Eqn. 4 [43]. A fur-ther measurement (closed circle) was performed in thesuperconducting state at T = 0 .
06 K and µ H =0.5 Twith a total counting time of 52 hours in order to ob-tain good statistics and a small error bar. The differencebetween χ ( n ) and the χ ( T =0.06 K, µ H =0.5 T) pointdemonstrates a clear drop in χ of 34 ± O Knight shift [19] K at comparable field and temperature. Specifically, at µ H ’ . T and T ’
60 mT, we have χ (PNS) /χ ( n ) =0 . ± .
06 and K (NMR) /K ( n ) = 0 . ± .
08. In the Sup-plementary Material [39], we show how the measured sus-ceptibility χ is corrected for Fermi liquid effects (Stonerenhancement) and the orbital contribution. The cor-rected non-interacting spin susceptibility χ , which canbe compared with theory, is shown in Fig. 1(c).As discussed above, PNS and NMR probe the mag-netization in different ways so there are reasons whythe results may be different. For the (011) Bragg re-flection, the PNS method measures M [ G = (011)] ≈ M Ru × f Ru ( G ) + 1 . × M O (2) × f O ( G ) ≈ M Ru × f Ru ( G ),where f is the magnetic form factor and M is the mag-netic moment on a given site. Note that the O(1) oxy-gen sites [see Fig. 1(c)] do not contribute to the mag-netic signal observed at this reflection. Further, the mo-ment on the oxygen O(2) sites is known to be small fromNMR Knight shift measurements [19] and DFT calcu-lations [44, 45]. Thus, our measurement is essentiallysensitive only to the Ru sites where most of the moment T c (0.5T) T c (1T) (a) = M ( G ) / H ( - B T - f. u . - ) s -wavenodes0.5T H H H H c2 (c) H (T)00.20.40.60.811.2 / ( n ) , K / K ( n ) , / ( n ) H c2 (b) H H (T)NMR, O(1),20mK, PustogowNMR, O(1),66mK, Ishida,60mK, Kittaka O(1)b aO(2)Ru c FIG. 1. PNS measurements of the susceptibility χ = M ( G ) /µ H at G = (011). (a) T -dependence of χ shows a drop below T c for µ H k [100]. Results of Duffy et al. [8] with µ H = 1T k [110] (open symbols) are shown scaled by the Ru magnetic formfactor [37]. T -dependencies (dotted and solid lines) are calculated using the Yosida functions [21, 22, 38] and χ ( H ) from (b).(b) H -dependence of χ at T = 60 mK. χ ( H ) is fitted to simple s -wave and nodal models (see text). (c) H -dependence of thescaled non-interacting spin susceptibility χ ( H ) /χ ( n ) determined from (b) by correction for Stoner enhancement and orbitalcontribution [39]. Also shown are O NMR Knight shift, K , data at similar temperatures from Pustogow et al.[18] and Ishidaet al.[19], and the measured linear coefficient of the specific heat, γ = C e /T from Kittaka et al.[40]. Both quantities have beenscaled to their normal state values. resides. In contrast, recent NMR experiments [18, 19]probed the oxygen O(1) sites.As mentioned, χ ( T ) for an isotropic s -superconductoris expected to follow the Yosida function [21]. This func-tion can be easily modified [22] to account for a super-conductor with vertical line nodes. At low temperaturesthe drop in χ ≡ M/H will be field-dependent due tothe introduction of vortices by the magnetic field. Mod-eling the effect of vortices on the spin susceptibility istheoretically difficult; in addition, Sr RuO shows a firstorder phase transition at H c with a step in the spinmagnetization [46–48]. Nevertheless, in Fig. 1(c) we fittwo simple illustrative low-temperature field dependen-cies of χ ( H ) with χ ,s ( H ) = χ (0) + ∆ χ H/H c and χ , nodal ( H ) = χ (0) + ∆ χ p H/H c for the s -wave andnodal cases using the actual H c [35, 36]. Our s -wave andnodal field-dependent fits yield zero-field residual valuesof χ (0) /χ ( n ) of 0 . ± .
09 and 0 . ± .
15 respectively.Both fits give χ ≈ × − µ B T − f.u. − for µ H = 1 T.Our data is consistent with two interesting scenariosas H →
0: (i) a rapid reduction of χ below H c (solidline in Fig. 1(c)); (ii) a large residual contribution to χ in the H → χ ( H ) and the linear coefficient of specific heat γ ( H ) = C e /T can detect the low-energy states intro- duced by vortices. For a singlet superconductor, theyare expected to show similar behavior [48]. In Fig. 1(b,c)we reproduce the measured [40] γ ( H ) for H k [100] andthe recent NMR Knight shift [18, 19] which both yieldsmaller values when normalized to the normal state [39].It is instructive to compare the present PNS data withthat of Duffy et al. [8] measured with H k [110]. From Ta-ble I one can see that the effect of the field is expected tobe the same for the [100] and [110] directions for all orderparameter symmetries, except for two of the E u states.In Fig. 1(a), we also show the PNS results of Duffy etal. [8] (open squares) measured at a field of 1 T with H k [110]. These were probed at the (002) Bragg peak andhave therefore been scaled by the ratio of the Ru formfactors [37, 49] at (011) and (002) for comparison. Thesolid and dotted lines show the expected T -dependencebased on the fitted χ ( H ) in Fig. 1(b,c) and the Yosidafunction [21, 22]. It is likely that Duffy et al. [8] were un-able to resolve a change because of the lower statisticalaccuracy and use of a higher field, where the suppressioneffect is smaller due to the field induced density of states.However, the E u state d = (ˆ x − ˆ y ) k z (which shows nochange in χ for H k [110] and a change for H k [100]) can-not currently be ruled out by the PNS experiments. Discussion. —Many superconducting states have beenproposed for Sr RuO . Some of those are shown inTable I and, following the observations of Pustogow etal. [18], there have also been new theoretical proposals[50–52]. The PNS measurements reported here yield anon-interacting spin susceptibility in the superconduct-ing state χ ( µ H = 0 . /χ ( n ) = 0 . ± .
06 whichis larger than the NMR Knight shift [18, 19]. Thus thePNS data better match different states. We concludedabove that the residual χ (0) /χ ( n ) = 0 . ± .
09 or0 . ± .
15 for non-nodal [e.g. ˆ x k x + ˆ y k y , ˆ z ( k x ± ik y )]or (near) nodal [e.g. s , d x − y , ( k x ± ik y ) k z ] gaps re-spectively. Thus, our PNS measurements do not rule outall odd-parity states, but they do rule out those with χ (0) /χ ( n ) = 1 in Table I. This includes the previouslywidely considered chiral p -wave d = ˆ z ( k x ± ik y ) state[2, 4, 5, 53]. Odd-parity states with in-plane d vectorssuch as the helical triplet (e.g. d = ˆ x k x + ˆ y k y ) statesproposed by Rømer et al. [50] have a partial ( ≈ χ s and therefore are not ruled out. Otherstates that are qualitatively compatible with our obser-vations include the TRSB s + id x − y and non-unitaryˆ x k x ± i ˆ y k y states [50] proposed by Rømer et al . [50],states resulting from the 3D model of Røising et al. [51],the TRSB d xz ± iyz of Zutic and Mazin[16, 54], or thenewest d + ig proposal by Kivelson et al. [52].The smaller χ (0) /χ ( n ) = 0 . ± .
15 for nodal statesis compatible with even-parity order parameters withdeep minima or nodes in the gap, listed above. Par-ticularly, if other (impurity or orbital) contributions areincluded in our estimated χ which are not due to quasi-particles excitations [39, 55, 56]. The nodal s -waveand d x − y -wave states are supported by thermal con-ductivity [57], angle-dependent specific heat [40], pene-tration depth [58] and quasiparticle interference exper-iments [59], albeit not by the evidence of TRSB, orby the recently observed discontinuity in the c shearmodulus[60]. Future polarized neutron scattering mea-surements at lower fields and other field directions willplace further constraints on the allowed paired states.The authors are grateful for helpful discussions withJ. F. Annett, S. E. Brown, K. Ishida, A. T. Rømer, andS. Yonezawa. 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Rev. , 961 (1969). upplementary Material:Orbital contributions to the susceptibility, Fermi liquid corrections and a two fluidmodel Our polarized neutron scattering measurement probes the total magnetic moment (or associ-ated susceptibility) including spin and orbital contributions. In addition, the spin susceptibility inSr RuO is subject to an exchange or Stoner enhancement. The effect of the Stoner enhancement istemperature and field dependent. Here we describe how these two effects can be taken into accountwhen comparison is made with theory. BACKGROUND
Theoretical models describing the spin susceptibilityin superconductors such as Sr RuO typically aim tocalculate the non-interacting susceptibility. In the nor-mal state this would be µ B N F and would be obtainedfrom a density functional theory (DFT) band structurecalculation, where the non-interacting unrenormalizedelectron density of states at the Fermi energy is N F .In Landau Fermi Liquid theory [1], the H → χ and spe-cific heat coefficient γ = C e /T are given by χ = µ B N ?F and γ = ( π / k B N ?F , where N ?F is the quasiparticledensity of states.Polarized neutron scattering measurements probe thetotal magnetic moment (or associated susceptibility) in-cluding spin and orbital contributions. In addition, χ in Sr RuO is subject to a Stoner enhancement wherethe Stoner factor is I . Our experiments are performedin a magnetic field, where we define the susceptibility χ ( H ) = M/µ H and χ ( n ) denotes the normal statevalue (say at H c ).The susceptibility due to the orbital and spin compo-nents is then given by χ ( H ) = χ orb + χ spin ( H ) = χ orb + χ ( H )1 − Iχ ( H ) . (1)DFT calculations estimate χ orb ≈ . × − µ B T − f . u . − , although, DMFT suggests anenhancement by fluctuations of the orbital contributionup to a factor of two [2, 3] to ∼ × − µ B T − f.u. − . Itwill be convenient to define the ratio of the orbital andspin parts of the susceptibility in the normal state f as f = χ orb χ spin ( n ) = χ orb − Iχ ( n ) χ ( n ) . (2)Based on the DFT calculations, we estimate f = 0 . γ to estimate Stoner enhancement. The mea-sured susceptibility for fields parallel to the a -axis is χ a = 0 .
91 emu mole − = 1 . × − µ B T − f . u . − [4].From the measured susceptibility and the estimate of χ orb we obtain χ spin ( n ) = 1 . × − µ B T − f . u . − = 1 . × − mole − . The linear specific heat is linear spe-cific heat γ ( n ) = 40 mJ mole − K − , yielding a Wilsonratio W = ( χ/γ ) × ( π k B ) / (3 µ µ B ) = 1 .
5. Assuming aStoner enhancement factor equal to the Wilson ratio, thelast term of Eq. 1 is W = 11 − Iχ ( n ) . We can estimate Iχ ( n ) = S = 0 .
33. The enhancementof the spin susceptibility can also be understood throughFermi liquid theory ( F a parameter) [5]. In the case of He, which has W ≈
3, Fermi liquid effects are requiredto understand the temperature dependence of the spinsusceptibility of the superfluid phases [6].
CORRECTION FOR ORBITAL SUSCEPTIBILITYAND STONER ENHANCEMENT
In Fig. 1 of the main paper we plot susceptibilitiesnormalized to their normal state values as a functionof field and temperature. We should remember that inthe case of the polarized neutron scattering we measure M ( G ) /µ H in absolute units. This susceptibility con-tains a form factor due to the electron orbital hence thevalues are lower those quoted above by a factor of 0.68[7].We define relative susceptibilities X ( H ) = χ ( H ) /χ ( n ) , and X ( H ) = χ ( H ) /χ ( n ) , where χ ( H ) refers to the measured susceptibility in Eq. 1and χ ( H ) refers to the corresponding non-interactingspin susceptibility. Fig. 1(a,b) shows the measured X ( B, T ). X ( H ) and X ( H ) are related through theequations X ( H ) = χ orb + χ ( H )1 − Iχ ( H ) χ orb + χ ( n )1 − Iχ ( n ) (3)= (cid:18) X ( H ) + f [1 − X ( H )]( f + 1)(1 − S ) (cid:19) − S − SX ( H ) (4) X ( H ) = f ( X ( H ) −
1) + X ( H )1 + (1 + f ) S ( X −
1) (5) a r X i v : . [ c ond - m a t . s up r- c on ] A ug Thus we use Eqn. 5 with f = 0 .
11 and S = 0 .
33 to correctfor orbital susceptibility and Stoner enhancement of thesusceptibility and relate the data and curves in Fig. 1(b)and (c). In particular, as discussed in the main text,theory makes predictions for the field dependencies ofthe non-interacting spin susceptibility χ ,s ( H ) = χ (0) +∆ χ H/H c and χ , nodal ( H ) = χ (0) + ∆ χ p H/H c . X ( H ) = χ ( H ) /χ ( n ) X ( H ) X ( H ) χ ( µ H = 0 . χ (0) , χ (0) ( s -wave) 0.47(9) 0.55(9) χ (0) , χ (0) (nodal) 0.16(14) 0.29(15)TABLE I. Measured and corrected susceptibilities. The tableshows the effect of the correction for the orbital susceptibilityand Stoner enhancement (Fermi liquid effects) on the suscep-tibility in the superconducting state. TWO-FLUID MODEL FOR SPINSUSCEPTIBILITY AND HEAT CAPACITY
As mentioned above, the linear coefficient of the elec-tronic specific heat γ = C e /T and the non-interactingspin susceptibility χ should be proportional to the quasi-particle density of states N ?F . This would imply that χ and γ are also proportional to each other and hence, therelative values γ ( H ) /γ ( n ) and X ( H ) = χ ( H ) /χ ( n )should be equal. In Fig. 1(c) of the main paper the rela-tive values γ ( H ) /γ ( n ) and χ ( H ) /χ ( n ) are shown, how-ever, they display different field dependencies. This couldbe due to various causes (i-iv). (i) It could be that the or-bital susceptibility in Sr RuO is larger than the appliedvalue of 1 . × − µ B T − f . u . − calculated by DFT and isinstead closer to the fluctuation enhanced doubled valuepredicted by DMFT [2, 3]. (ii) The sample we probed byPNS has a larger residual γ than the sample used in thespecific heat measurements reproduced in Fig. 1(b,c) inthe main paper[8]. Sample dependence of the linear coef-ficient γ has been reported in Sr RuO in the past[8, 9].(iii) There is a nuclear polarization of some of the Sr andRu nuclei due to the applied field and low temperature,which would give an additional term in the PNS flippingratio [10, 11]. (iv) The two-fluid model, which describesa supercondcutor as a superposition of a superconduct-ing condensate and a normal Fermi-liquid with no inter-ference, is not a sufficiently accurate description for thesuperconducting state in Sr RuO . The measured O NMR Knight shifts reproduced inFig. 1(b,c)[12, 13] in the main paper are as well as the spinsusceptibility subject to Stoner enhancement and shouldalso contain orbital contributions. If we assume the sameStoner factor (S=0.33), that we observe that the relativenon-interacting Knight shifts K ( H ) /K ( n ) analogue to X in Eq. 5 agrees with the reproduced relative linear co-efficient of the electronic specific heat γ ( H ) /γ ( n ) withoutany orbital contribution ( f = 0). [1] P. Coleman, Introduction to Many-Body Physics (Cam-bridge University Press, 2015).[2] M. Kim, J. Mravlje, M. Ferrero, O. Parcollet, andA. Georges, Phys. Rev. Lett. , 126401 (2018).[3] A. Tamai, M. Zingl, E. Rozbicki, E. Cappelli, S. Ricc`o,A. de la Torre, S. M. Walker, F. Bruno, P. King,W. Meevasana, M. Shi, M. Radovi´c, N. Plumb,A. Gibbs, A. Mackenzie, C. Berthod, H. Strand, M. Kim,A. Georges, and F. Baumberger, Phys. Rev. X (2019).[4] K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z. Q.Mao, Y. Mori, and Y. Maeno, Nature , 658 (1998).[5] I. I. Mazin and R. E. Cohen, Ferroelectrics , 263(1997).[6] A. J. Leggett, Phys. Rev. Lett. , 536 (1965).[7] P. J. Brown, International Tables for CrystallographyVol. C (Kluwer, Dordrecht, 1992) p. 391.[8] S. Kittaka, S. Nakamura, T. Sakakibara, N. Kikugawa,T. Terashima, S. Uji, D. A. Sokolov, A. P. Mackenzie,K. Irie, Y. Tsutsumi, K. Suzuki, and K. Machida, J.Phys. Soc. Japan , 093703 (2018).[9] S. NishiZaki, Y. Maeno, and Z. Mao, J. Phys. Soc. Jpn. , 572 (2000).[10] C. G. Shull and R. P. Ferrier, Phys. Rev. Lett. , 295(1963).[11] Y. Ito and C. G. Shull, Phys. Rev. , 961 (1969).[12] A. Pustogow, Y. Luo, A. Chronister, Y.-S. Su, D. A.Sokolov, F. Jerzembeck, A. P. Mackenzie, C. W. Hicks,N. Kikugawa, S. Raghu, E. D. Bauer, and S. E. Brown,Nature , 72 (2019).[13] K. Ishida, M. Manago, K. Kinjo, and Y. Maeno, J. Phys.Soc. Japan89