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[J. F. Landaeta](https://orcid.org/0000-0002-4229-2115), [K. Semeniuk](https://orcid.org/0000-0003-2680-128X), J. Aretz, [K. R. Shirer](https://orcid.org/0000-0003-4570-0361), D. A. Sokolov, [N. Kikugawa](https://orcid.org/0000-0003-3975-4478), Y. Maeno, [I. Bonalde](https://orcid.org/0000-0003-2478-5618), J. Schmalian, A. P. Mackenzie, [E. Hassinger](https://orcid.org/0000-0003-2911-5277)

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[Evidence for vertical line nodes in <math>  <msub>    <mi>Sr</mi>    <mn>2</mn>  </msub>  <msub>    <mi>RuO</mi>    <mn>4</mn>  </msub></math> from nonlocal electrodynamics](https://mdr.nims.go.jp/datasets/fb6687f7-1e8a-49d0-9f2a-33f73c8b941b)

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Evidence for vertical line nodes in Sr2RuO4 from nonlocal electrodynamicsJ. F. Landaeta,1, 2 K. Semeniuk,2 J. Aretz,2 K. Shirer,2 D. A. Sokolov,2 N. Kikugawa,3Y. Maeno,4 I. Bonalde,5 J. Schmalian,6, 7 A. P. Mackenzie,2, 8 and E. Hassinger1, 2, ∗1Institute of Solid State and Materials Physics, TU Dresden, 01069, Dresden, Germany2Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany3National Institute for Materials Science, Tsukuba, 305-0003, Japan45 Toyota Riken - Kyoto University Research Center (TRiKUC), Kyoto 606-8501, Japan5Centro de Física, Instituto Venezolano de Investigaciones Científicas, Apartado 20632, Caracas 1020-A, Venezuela6Institute for Theoretical Condensed Matter Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany7Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany8Scottish Universities Physics Alliance, School of Physics and Astronomy,University of St. Andrews, St. Andrews KY16 9SS, United Kingdom(Dated: December 13, 2023)By determining the superconducting lower and upper critical fields Hc1(T ) and Hc2(T ), respectively, in ahigh-purity spherical Sr2RuO4 sample via ac-susceptibility measurements, we obtain the temperature depen-dence of the coherence length ξ and the penetration depth λ down to 0.04 Tc. Given the high sample quality, theobserved T 2 dependence of λ at low temperatures cannot be explained in terms of impurity effects. Instead, weargue that the weak type-II superconductor Sr2RuO4 has to be treated in the non-local limit. In that limit, thepenetration depth data agree with a gap structure having vertical line nodes, while horizontal line nodes cannotaccount for the observation.INTRODUCTIONUnderstanding the superconductivity of Sr2RuO4 is a chal-lenge that has now spanned nearly three decades [1]. The highpurity of the crystals available for study, combined with therelatively simple and well-understood normal state [2] meansthat this should be a soluble problem, and it has become amilestone for the whole field of unconventional superconduc-tivity [3–9]. Progress is hindered by the lack of a completeunderstanding of its superconducting order parameter. Re-cent studies of the spin susceptibility in the superconductingstate have called into question the long-held paradigm of aspin-triplet, odd-parity order parameter, and provided strongevidence for a spin singlet, even-parity state [10–12]. Thequestion of whether the order parameter breaks time-reversalsymmetry or not is also the subject of ongoing investigations[13–16].To further inform the rejuvenated theoretical effort thatthese new results have stimulated, it is important to find newways to address one of the other issues about which thereis apparently conflicting information: the nodal structure ofthe superconducting gap. It is widely agreed, on the basis of,for example, ultrasound [17], penetration depth [18], heat ca-pacity [19–21] and thermal conductivity [22–24] that the gapin Sr2RuO4 has nodes, but different conclusions have beenreached about whether these are horizontal [20, 25] or vertical[19, 22, 24]. Any information on this issue is important, be-cause horizontal line nodes would imply mechanisms incorpo-rating inter-plane or inter-orbital pairing [26–29], while verti-cal line nodes would be consistent with pairing states formedfrom in-plane electronic states [29–32], the latter being natu-ral for a quasi-2D material.In this paper, we approach the problem using a techniquethat has not so far been employed in the study of Sr2RuO4.We perform magnetic susceptibility measurements on a nearlyspherical sample sculpted from an ultra-high purity singlecrystal with Tc = 1.5K for µ0H ∥ c. The measurements en-able a quantitative determination of the temperature depen-dence of both the lower and upper critical fields, Hc1 and Hc2.From this information, we derive the temperature dependenceof the in-plane penetration depth λ and the superconductingcoherence length ξ . From the former, we show that non-localelectrodynamics, proposed in Ref. [33] for nodal supercon-ductors, must be used to analyse the penetration depth resultsin Sr2RuO4 with this level of purity. Calculations in this non-local limit distinguish the responses from vertical and hori-zontal line nodes, and our results are shown to be consistentwith the prediction for vertical nodes.EXPERIMENTAL RESULTSTo study the critical fields of Sr2RuO4, we measured themagnetic field dependence of the ac-susceptibility at differ-ent temperatures with the external magnetic field µ0H appliedparallel to the crystallographic c axis. We used a high-puritysample cut into a sphere with a diameter of 470 µm using fo-cussed ion beam (FIB) milling as shown in Fig. 1(a) (for de-tails, see Ref. [34]). The spherical shape gives a well-defineddemagnetization factor independent of the magnetic field di-rection and removes uncertainties from other shapes that canstrongly influence the measured field values, particularly thatof Hc1.The temperature dependence of the real part of the magneticac-susceptibility χ ′(T ) at H = 0 is shown in Fig. 1(a). Thesuperconducting critical temperature Tc = 1.5 K reveals a highsample purity, comparable to samples having the highest Tcvalues [35]. Figure 1(b) shows the field dependence of the acsusceptibility χ ′(H) at 100mK. These are the up sweep dataarXiv:2312.05129v2  [cond-mat.supr-con]  11 Dec 202321 . 0 1 . 5 2 . 0- 1 . 0- 0 . 50 . 00 1 0 2 0 3 0 4 0 5 0 6 0 7 0- 1 . 0- 0 . 50 . 00 1 0 2 0( a )S r 2 R u O 4� 0 H  =  0  TT c�' (arb. u.)T  ( K )H vH pH c 2( b )�' (arb. u.)� 0 H  ( m T ) T  =  1 0 0  m K� 0 H  | |  cH *c 1�' (arb. u.) H *c 1� 0 H  ( m T )Figure 1. (a) Temperature dependence of ac susceptibility χ ′(T ) ofthe spherical sample of Sr2RuO4. The Tc is defined at the onset of thesuperconducting transition, highlighted with an arrow. (b) Magneticfield dependence (up sweep) of χ ′(H) at 100 mK. The upper criticalfield Hc2 is defined at the onset of the transition, and the featuresrelated to the vortex physics, Hp and Hv, are defined at the peakand valley as indicated by arrows. The inset shows a zoom near H∗c1,defined as the field where the susceptibility departs from the constantminimum value.that have been corrected for the remanent field of the magnet[34]. We identify four features: the lower critical field H∗c1,which is uncorrected for demagnetisation, the upper criticalfield Hc2, a peak Hp, and a valley Hv, the latter two beinglikely associated with the superconducting vortex physics [36,37]. Hc2 is defined as the onset of the normal state transitionand H∗c1 as the first deviation of the susceptibility from thefull screening in the Meissner state, as defined in the insetto Fig.1(b) (see also the Supplementary information for moredetails [34]). To obtain the actual lower critical fieldHc1 = H∗c1(1−N)−1, (1)knowledge of the demagnetization factor N is required. Cru-cial for our analysis is that for a spherical sample N = 1/3[38]. It is important to note that the feature in the ac suscepti-bility due to H∗c1 remains the same if we sweep the magneticfield up, starting in zero-field cooled or under field-cooledconditions (see the supplementary material [34] ). These re-sults show that we can clearly detect a sharp signature at H∗c1.(a)(b)Figure 2. (a) Magnetic field dependence (up sweep) of χ ′(H) at dif-ferent selected temperatures of the spherical sample with the fieldapplied along the c axis. The H∗c1, Hp, Hv and Hc2 are indicated withsymbols. (b) Superconducting phase diagram extracted from (a).Figure 2 (a) exhibits the magnetic field dependence of thesusceptibility at different temperatures. The four features de-scribed in Figure 1(b) are indicated with arrows on the suscep-tibility curves. From this study, we identified the signature ofH∗c1 up to 1.25 K and the peak-valley (Hp-Hv) vortex featuresup to 1.1 K. The Hp and Hv features approach each other aswe increase the temperature, making them indistinguishableand undetectable for temperatures above 1.1 K.From the data in Fig. 2(b)), we extrapolated to T =0 K H∗c1(0) = 9.27mT, Hp(0) = 49 mT, Hv(0) = 60 mT and3Hc2(0) = 67 mT. Using the demagnetizing factor of a sphere,we obtain from Eq. (1) for the T → 0 limit of the lower criticalfield Hc1(0) = 13.9mT.Accurate knowledge of Hc1 is important for the rest of ouranalysis, so we have checked our results against others fromthe literature: H∗c1 = 7mT was obtained in specific heat mea-surements at 60mK in samples with a slab geometry [19]. Avalue of H∗c1 = 7mT was determined using SQUID magne-tometry at T = 20mK [39], while thermal-conductivity mea-surements find H∗c1 = 8mT at T = 320mK [22] and 12mTfor a long plate-shaped sample parallel oriented to the field atT = 300mK [23]. Those values reveal a strong influence ofthe geometry of the sample on the value of H∗c1. For compar-ison, we use an estimated typical demagnetizing factor N ≈ 0for a plate-shaped sample oriented parallel to the field whilefor a slab with proportions of a× b× c = 0.5× 0.5× 0.33,N ≈ 0.5 (see the supplementary material [34]). This leadsto estimated Hc1 values between 13 mT and 16 mT for thesemeasurements. Our value of Hc1(0) = 13.9mT is hence invery good agreement with previous measurements using dif-ferent techniques.0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 001 0 02 0 03 0 04 0 05 0 0��, � (nm)T  /  T c�Figure 3. Temperature dependence of the superconducting coher-ence length (orange) and the penetration depth (blue) as determinedfrom the upper and lower critical fields via Eq. (2).We can perform a still more rigorous check of the accuracyof our critical field values by estimating some fundamentalsuperconducting parameters, calculating the thermodynamiccritical field, and comparing it to that deduced from specificheat data. To this end, we useHc2 =Φ02πξ 2 ,Hc1 =Φ04πλ 2 C (κ) , (2)that relate the two critical fields with the penetration depthλ and the coherence length ξ of a type II superconductor.Here, Φ0 is the flux quantum and κ = λ/ξ > κc =1√2theGinzburg-Landau parameter. The function C(κ) was deter-mined numerically from the solution of the Ginzburg-Landauequations by Brandt [40], who also gave a simple analyticinterpolation formula that is highly accurate (< 10−3) forall values κ > κc and reproduces the limits C (κc) = 1 andC (κ ≫ 1) = logκ + 0.49693 [41] (see supplementary mate-rial [34]).Using Eq. (2) and our result for Hc2(0) and Hc1(0) yieldsξ0 = 70nm and λ0 = 134nm for the zero-temperature valuesof coherence length and penetration depth, respectively. Thisis consistent with the measurement by Muon Spin Rotation(µSR) λ0 = 126 nm and the calculation of the contribution ofeach band to the magnetic penetration depth based on Angle-Resolved Photo-electron Spectroscopy (ARPES) λ0 = 130 nm[42]. From λ0 and ξ0 we obtain κ0 ≡ κ(T = 0) = 1.92, i.e.Sr2RuO4 is not a strong type II superconductor as the valueof the Ginzburg-Landau parameter is not far from the limitto type I superconductivity (κc ≈ 0.71). These results yieldthe thermodynamic critical field Hc(0) =Hc2(0)√2κ0= 24.67mT,in very good agreement with Hc = (23±2)mT deduced fromspecific heat data [34]. In summary, our measurements giveprecise values of Hc1 and superconducting parameters in fullagreement with previous results.TEMPERATURE-DEPENDENT PENETRATION DEPTHAND COHERENCE LENGTHThrough Eq. (2), the temperature dependencies of Hc1(T )and Hc2(T ) are directly related to the T -variation of the in-plane penetration depth λ (T ) and coherence length ξ (T ), re-spectively. In Fig. 2 we show both length scales as they varywith temperature. As expected, λ (T ) and ξ (T ) grow withtemperature. Focussing on the temperature variation of thepenetration depth, we show in Figure 4(a))∆λ (T )λ0=λ (T )−λ0λ0, (3)which exhibits a T 2 dependence for temperatures below ≈0.5Tc .One possible explanation of the T 2 dependence, that is ap-propriate for systems such as the cuprates is the effect of im-purities in superconductors with line nodes. Hirschfeld andGoldenfeld showed that for an unconventional superconductorwith vertical line nodes, the scattering due to impurities wouldlead to a change in the temperature dependence of λ (T ) ∝ Tto T 2 below a crossover temperature T ∗imp ≈ 0.83(Γ∆0)1/2,where Γ is the scattering rate and ∆0 is the magnitude of thesuperconducting gap[43]. In our extremely clean sample withTc ≈ 1.5K, we obtain T ∗imp ≤ 0.05Tc, such that the effect of im-purities cannot explain our experiment across the wide rangeof temperatures where we see the T 2 dependence.There have been several previous reports of T 2 behaviourof ∆λ/λ0 in Sr2RuO4 within the Meissner state [18, 44, 45].However, these were on crystals with lower Tc and higher T ∗imp.40 . 0 0 . 2 0 . 4 0 . 6 0 . 80 . 00 . 51 . 01 . 52 . 00 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 50 . 00 . 10 . 20 . 0 0 . 1 0 . 20 . 00 . 10 . 2� �  / � � 0  +  0 . 5B o n a l d e  e t  a l .B o n a l d e  T *i m p�� /��0 ( T  /  T c ) 2S p h e r e  T *i m p( c )( b )H o r i z o n t a l  l i n e s n o n - l o c a lV e r t i c a l  l i n e sn o n - l o c a l�� /��0T  /  T c( a )S p h e r e  T *i m pV e r t i c a l  l i n e sn o n - l o c a lH o r i z o n t a l  l i n e s n o n - l o c a l�� /��0( T  /  T c ) 2Figure 4. (a) Temperature dependence of ∆λ/λ0 vs. (T/Tc)2. The data from Bonalde et al.[18] is shifted for clarity. The black and bluedashed lines represent the T ∗imp from Bonalde et al. and the spherical sample.(b) Shows the expected behavior of ∆λ/λ0 for vertical (solid line)and horizontal (dashed) line nodes in the nonlocal electrodynamics limit. (c) Shows ∆λ/λ0 vs. (T/Tc)2; in this scale, the difference betweenthe vertical and horizontal nodes becomes clearer in this scale.Measurements of the penetration depth with tunnel diode os-cillator (TDO) technique [18] are shown in Fig. 4(b) [18],while Refs. [44, 45] used microwave surface impedance mea-surements. For the samples used for these measurements, weestimate, using the relationship between the scattering rate Γand Tc [46], T ∗imp ≈ 0.2Tc. It was therefore uncertain whetherthe observed behaviour should be attributed to impurity scat-tering or not. In our data, there is no such ambiguity: thedata shown in Fig.4 are in the clean limit. At first sight, thispresents a puzzle. In the local electrodynamics, applicable tomost unconventional superconductors, only point nodes cangive a T 2 dependence of the penetration depth [47]. In con-trast, all the even parity states under discussion as potentialorder parameters for Sr2RuO4 either have line nodes or arefully gapped, giving only T -linear or exponential dependen-cies of ∆λ/λ0 at low temperatures.The resolution to this apparent paradox lies in consideringnon-local electrodynamics. Nonlocal effects in the electro-magnetic response below Tc go back to the analysis of Pip-pard for type I superconductors [48], where spatial modes ofthe magnetic field with wavelengths smaller than the coher-ence length give rise to reduced screening currents that shieldthe external field, affecting the length scale up to which thefield can penetrate.It was pointed out by Kosztin and Leggett [33] that a non-local response can also play a role in type II superconductorsif they possess nodes of the gap function. Qualitatively, thisis due to the effective coherence length ξk ∼ vF/∆k that di-verges along the nodal directions where the gap function ∆kvanishes. However, one must keep in mind that in the groundstate, the entire Fermi surface contributes to the phase stiffnessand the relative importance of the nodal points is negligible.On the other hand, the temperature dependence of the pene-tration depth is dominated by thermal quasiparticle excitationsnear the nodes. The effective coherence length of those exci-tations is the thermal de Broglie length ξT ∼ vF/T . The resultby Kosztin and Leggett for a system with vertical line nodescan then be written in the form:∆λ (T )λ0∼ ∆λ (T )λ0∣∣∣∣locλ0ξT, (4)where ∆λ (T )/λ0|loc = log2 T∆0is the well-established re-sult for the local electromagnetic response of a nodal su-perconductor. Hence, it follows that in the non-local limit∆λ (T )/λ0 ∝ κ0T 2/∆20. Here ∆0 is the gap amplitude thatalso enters in the low-temperature density of states ρ (ω) =ρF |ω|/∆0 of the nodel superconductor. This non-locality istied to the condition λ0 ≪ ξT , which translates to T ≪ T ∗ =∆0/κ0. The change in the electromagnetic response due tothermally excited nodal quasiparticles is reduced for wave-length of the magnetic field smaller than ξT . In strongly typeII superconductors with high κ , the range of temperaturesover which non-local effects are relevant becomes vanishinglysmall. However, the low κ of Sr2RuO4 leads to a much largerrange of temperatures for which the non-local physics is ap-plicable; in the Supplementary Material, we estimate T ∗ to beas high as 0.8Tc.Intriguingly, the non-local regime offers a qualitative dis-tinction between the effects of vertical and horizontal linenodes on ∆λ/λ0, because horizontal nodes lie in the planeof the screening super-currents. This regime was analyzed byKusunose and Sigrist in Ref.[49] and their result for horizontalline nodes can be formulated as∆λ (T )λ0∼ ∆λ (T )λ0∣∣∣∣locλ0ξTlogξTλ0, (5)Hence, the suppression of the electromagnetic response byquasiparticles with horizontal nodes is less strong and, with∆λ (T )/λ0 ∝ T 2 log(T ∗/T ), in principle distinguishable fromthose of vertical line nodes where ∆λ (T )/λ0 ∝ T 2. Whilethese qualitative arguments are limited to the regime of lowesttemperatures, in the supplementary material [34], we demon-strate the full analysis of the electromagnetic response forhorizontal and vertical line nodes. The theoretical curves,5shown in Figure 4(b) alongside the experimental data, de-pend on the two dimensionless numbers κ0 that we deter-mined earlier and 2∆0/(kBTc). For the theoretical curvesshown in the Figure 4(c), we used 2∆0/(kBTc) = 3.53 andκ0 = 1.92. In the supplementary material, we also show datafor 2∆0/(kBTc) = 3.16, deduced from the heat capacity de-pendence Cs(T ) = αT 2 in the superconducting state [50], thatallows for very similar conclusions.In Fig. 4(c), we replot the same theoretical results and ex-perimental data as a function of (T/Tc)2, which highlightsthe qualitative difference in the predictions for horizontal andvertical line nodes at low temperatures. In both Figs. 4 (b)and (c), the predictions for vertical line nodes are a system-atically better match to the data than those for horizontal linenodes. This remains true if we allow 2∆0/(kBTc) as an openfit parameter. Our conclusions are robust as long as thereare no strong variations of the gap amplitude among the vari-ous Fermi surface sheets. Then, the data are compatible withthe existence of purely vertical line nodes but not compatiblewith order parameters containing solely horizontal nodes. Formixed order parameters with both types of node, explicit cal-culations of ∆λ/λ0 would be required to determine whetheror not the predictions are compatible within experimental er-ror with our data.In conclusion, we have used measurements of Hc1 and Hc2on an extremely high purity single crystal of Sr2RuO4 to showthat its in-plane low-temperature coherence length ξ0 = 70nmand penetration depth λ0 = 134nm, and that ∆λ/λ varies asT 2 in the clean limit. Analysis of our results using non-localelectrodynamics confirms that the observations are compati-ble with vertical line nodes in its superconducting order pa-rameter. Our measurements and analysis are of relevance tothe ongoing quest to understand the order parameter symme-try of Sr2RuO4, and invite careful measurement and analysisof ∆λ/λ in other unconventional superconductors in whichthe non-local regime is experimentally accessible.Note added. We note that Ref. [51] reports measurementsof ∆λ as a function of uniaxial pressure Sr2RuO4 that alsohighlight the importance of non-local effects.ACKNOWLEDGEMENTSWe thank D. Bonn, J.-P. Brison, S. Brown, P. Hirschfeld,G. Palle, R. Prozorov, A. Ramires, and H. Suderow forstimulating exchange and helpful discussions. E.H. ac-knowledges funding from Deutsche Forschungsgemeinschaft(DFG) via the CRC 1143-247310070 (project C10) and theWuerzburg-Dresden cluster of excellence EXC 2147 ct.qmatComplexity and Topology in Quantum Matter - project num-ber 390858490. J.S. was supported by the DFG throughTRR 288-422213477 Elasto-Q-Mat (project A07). This workis supported by JSPS KAKENHI (No. JP18K04715, No.JP21H01033, and No. JP22K19093). J.S. also acknowledgesthe hospitality of KITP, where part of the work was done.KITP is supported in part by the National Science Foundationunder Grant No. NSF PHY1748958 and NSF PHY-2309135.SUPPLEMENTARY MATERIALExperimental methodsThe magnetic AC susceptibility was measured using acustom-made pair of compensated pickup coils, each hav-ing a length of 2 mm and 4500 turns [52]. As a drive coil,we used a superconducting modulation coil with an excita-tion field of 175 µT at a frequency of 5 Hz. The output sig-nal from the pickup coils was amplified by a low-temperaturetransformer with a 1:100 amplification ratio, followed by alow-noise amplifier SR560 from Stanford Research Systems.Our experimental setup incorporated a National Instruments24-bit PXIe-4463 signal generator and 24-bit PXIe-4492 os-cilloscope for data acquisition, utilizing digital lock-in ampli-fication. The noise level of our measurements was approxi-mately 70 pV/√Hz.The ac-susceptibility measurements were conducted withinan MX400 Oxford dilution refrigerator down to 40 mK. Thetemperature was measured with a thermometer directly cou-pled with a conductive high-purity annealed silver wire to thesample, but situated in the compensated region of the magnet.We only show the real part of the magnetic susceptibility sincethe imaginary part is not relevant here. We corrected for theremnant field of the superconducting magnet, which remainedat a constant value within a range of ±2 mT while sweepingthe magnetic field between -120 and 120 mT, as carried out inour study. This is explained in detail below. The earth mag-netic field is not shielded in our measurements.Sample preparationThe spherical sample of Sr2RuO4 was produced via fo-cused ion beam (FIB) milling. The procedure is illustrated inFig. S1. The initial cube-shaped crystal with a side length of0.5 mm was mounted on the tip of a rotor of a piezo-actuatedstepper motor, which was then mounted onto the specimenplatform of a Helios G4 Xe Plasma FIB microscope (man-ufactured by FEI). The crystal was oriented with its c axisparallel to the motor rotation axis, which in turn was perpen-dicular to the ion beam. Using the motor, the sample wasexposed to the ion beam from different directions, and excessmaterial was ablated by Xe plasma with a voltage of 30 kV,eventually giving the sample a spherical shape. This energygives an amorphous surface layer of a thickness of typically100 nm. The sample was kept stationary during the milling.The motor was actuated using a controller located outside themicroscope chamber. Currents of the order of 1µA were usedfor the initial pass. These were eventually reduced to the levelof several nA as the final shape was approached. To keep trackof the a axis direction, a shallow mark was etched on the sideof the sphere using the ion beam.6Figure S1. Various stages of fabrication of the spherical Sr2RuO4sample. The panels are arranged chronologically. The images weretaken inside the FIB microscope chamber, using an electron columnpositioned at 52◦ with respect to the ion column.DETERMINATION OF CRITICAL FIELD VALUESThis paragraph describes the data-analysis procedure usedto extract the critical field values necessary for determiningthe values of penetration depth and coherence length. Wecarried out measurements of the magnetic susceptibility as afunction of temperature for H = 0 to determine Tc, and as afunction of the magnetic field at different temperatures. Here,we only show the real part of the magnetic susceptibility sincethe imaginary part is not relevant. For each temperature, westart measurements at a magnetic field with an absolute valueµ0H > 100 mT in the normal state, sweep to the same field ofopposite sign and back to the starting field. As will be shownbelow, these measurements give the same result for H∗c1 asstarting in the zero-field-cooled condition. Fig. S2(a) showsthe raw data of the real part of the magnetic susceptibility foran example temperature. The curves are shifted along the fieldaxis in a way that there is a shift of both up and down-sweepcurves with respect to each other, and that the Hc2 values (de-fined here at the maxima indicated by an arrow) are not sym-metric with respect to the H = 0 reading of the magnet. Weascribe this to two effects:i) The up-sweep and down-sweep curves are shifted rela-tive to each other along the field-axis. This shift is of theorder of 1 mT and likely due to a lag during the sweep.This typically occurs in sweep-mode measurements dueto the data acquisition loop in which the magnetic fieldat the time of susceptibility measurement is differentfrom the magnetic field at the time of the field reading,the latter being the recorded and plotted field value.ii) The asymmetry with respect to the recorded H = 0value is of the order 2 mT and it originates in a remanentfield in the superconducting magnet, i.e. trapped fluxlines from previous magnetization of the magnet, that- 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 8 0- 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 8 0- 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0� 0 HT = 5 6  m K6 . 6  m T- 1 0 . 9  m T6 5  m T6 4  m T- 6 8 . 7  m Tχ' (arb. u.)  u p  s w e e p d o w n  s w e e p- 6 9 . 6  m T9 . 2 2  m T( c )6 6 . 7  m T-  6 6 . 7  m Tχ' (arb. u.)� 0 H u p  s w e e p d o w n  s w e e p9 . 2 5  m TT = 5 6  m K( a )( b )9 . 2 2  m Tχ' (arb. u.)� 0 H u p  s w e e p d o w n  s w e e pT = 5 6  m K9 . 2 5  m TFigure S2. The magnetic field dependence of χ ′(H) of the sphericalsample at 56 mK. (a) Raw data of χ ′(H). (b) χ ′(H) with correctionin µ0H and (c) is a zoom-in of the panel (b) around the magneticfield range of Hc1. The interception of the lines in (c) corresponds toour definition of H∗c17leads to a finite remanent field in the center of the mag-net even for zero applied current through the supercon-ducting magnet coil. In the example curve, this meansthat a current corresponding to a field of ≈ −2 mT hasto be applied so that this remanent field is compensatedand that the real magnetic field at the sample is zero.In order to correct for both effects, we use the fact that Hc2is expected to be symmetric in field. We read off the value ofHc2 as the maximum in χ ′ for each sweep at positive and nega-tive fields – as indicated by the arrows near ±65 mT – and shiftthe curve along the field axis so that Hupsweepc2 =−Hdownsweepc2 .The result is shown in Fig. S2b). We use the average of Hc2from up and down sweeps to determine the value of Hc2 usedfor data analysis. Hysteresis effects can still be observed, es-pecially near H∗c1, probably due to vortex pinning. For fieldssweeping from the Meissner state up to the vortex state, H∗c1 isclearly identified as the field where the susceptibility starts in-creasing from a constant value, which we determined by con-struction with two straight lines as shown in Fig. S2(c). Whenthe field is lowered in absolute value so that the Meissner stateis entered coming from the vortex state, vortices are slightlypinned and leave the sample slowly. However, at H = 0 theconstant value of the susceptibility is reached and we infer thatany pinning effect ends there.- 1 0 0 - 5 0 0 5 0 1 0 0- 1 0 0 1 0H pH v   Z F C  0  t o  1 2 0  m T   1 2 0  m T  t o  - 1 2 0  m T   - 1 2 0  m T  t o  1 2 0  m T�' (arb. u.)� 0 H  ( m T )T =  4 3  m KH c 2      P b  b a c k g r o u n dχ' (arb. u.)� 0 H  ( m T )H *c 1Figure S3. Ac magnetic susceptibility measurements at 43mK. Theblack line show the measurement measures with zero field cooling(ZFC) 0 mT to 120 mT, starting with the fully demagnetized super-conducting magnet. Blue and orange lines are the subsequent mea-surements from 120 mT to -120 mT (blue) and -120 mT to 120 mT(orange). The inset shows a zoom-in in the region where H∗c1 is de-fined. The shaded region is the background signal from the Pd of thesolder joint and the shield of the low temperature transformer.To exclude any effect of vortex pinning on the value ofH∗c1 in our measurement procedure, we also compared theresults to a zero-field-cooled measurement for 42 mK in fig.0 20 40 60 80 (mT) (arb. u.)* 0.06 K0.067 K0.075 K0.084 K0.103 K0.122 K0.152 K0.201 K0.251 K0.301 K0.352 K0.402 K0.453 K0.502 K0.552 K0.601 K0.652 K0.698 K0.747 K0.796 K0.846 K0.895 K0.946 K0.997 K1.064 K1.101 K1.154 K1.189 K1.225 K1.264 K1.303 K1.345 K1.433 KFigure S4. The complete data set of the magnetic field dependence ofχ ′(H) measurements at different temperatures. The arrows indicatethe feature where H∗c1,Hp,Hp and Hc2 are defined.S3. Here, the whole dilution refrigerator and superconductingmagnet were warm before starting the measurement so thatwe can exclude any effect of remanent fields. The upturn atthe beginning of the sweep is then likely related to the volt-age induced by the imperfect balance pickup coils when thebackground changes with small fields by a magnetisation ofsurrounding parts of the cryostat upon first application of themagnetic field. Apart from this difference below 4 mT, thesame signature near H∗c1 is observed as in the measurementscoming for high fields. The real part of the magnetic sus-ceptibility is flat below H∗c1 and then starts to increase quitesharply. Hence, we are confident that flux pinning does notaffect our value of Hc1 in the measurements where we start inthe field-induced normal state. Note that, since λ0 = 134 nmis of the size of the sample roughness given by the thicknessof the amorphous layer created by the ion beam of the or-der of 100 nm, the Bean-Levingston surface barrier effect [53]should not play a role here.Fig. S4 shows part of the field sweeps (only up sweeps andpositive fields) at all measured temperatures used to determinethe phase diagram shown in the main paper. These curves arealready shifted according to the procedure described above.80 5 1 0 1 5 2 0- 1 . 0 5- 1 . 0 0- 0 . 9 5- 0 . 9 0 S l a b S p h e r e�' (arb. u.)� 0 H  ( m T ) T  =  1 0 0  m K� 0 H  | |  cH *c 1H *c 1Figure S5. Comparison of χ ′(H) near Hc1 at 100 mK between aspherical sample and slab-geometry sample.The lower critical field and demagnetization factorsFigure S5 compares the measurement of χ ′(H) near Hc1 ofsamples with a spherical and slab geometry. The measure-ments were done under the same conditions of 5 Hz, 175 µTexcitation field, and 100 mK. The feature in χ ′(H) relatedto the H∗c1 depends strongly on the geometry of the sample.From this measurement we obtain that H∗c1,slab ≈ 7.5 mT andH∗c1,sphere ≈ 9.2 mT. If we take into account the demagnetiza-tion factor N:Hc1 =H∗c1N −1(S1)The demagnetization factor for a sphere is N = 1/3 and fora slab is described by [54]:N ≈ 4ab4ab+3c(a+b)(S2)The dimensions (a×b× c) of our slab sample are approx-imately (500 × 500 × 330)µm leading to N ≈ 0.5. Then,Hc1,sphere = 13.8 mT and Hc1,slab = 15 mT. Both values are inthe same order, within the experimental uncertainty on the es-timation of N.Thermodynamic critical field HcFor a consistency check, we compare the thermodynamiccritical field extracted from specific heat with the one ex-tracted from our values of Hc1 and Hc2. Hc is determined fromthe free-energy difference between the normal and the super-conducting states, ∆F = Fn −Fs, obtained from the C/T databy the integration of entropy difference [52]:0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4051 01 52 02 50 . 0 0 . 5 1 . 0 1 . 5 2 . 002 04 06 0 D e g u c h i  e t  a l M a e n o  e t  a lH c (mT)T  ( K )C p (mJ K-2  mol-1 )T  ( K )Figure S6. Thermodynamic critical field Hc estimated from specificheat measurement from Deguchi et. al. [19] and Maeno et. al. [55]∆F(T ) =µ0H2c (T )2=∫ TTc∫ T ′TcCs −CnT ′′ dT ′′dT ′. (S3)Here, Cn is the value of C/T at the normal state; this valuevaries from 38 to 40 mJK2mol−1, and Cs is the Cp measure-ment shown in the inset of Figure S6. If we extrapolated Hc to0 K, it is conservative to say that Hc(0) = (23±2) mT, whichis consistent with previous reports [3].We can also calculate Hc(0) using our measured values forHc1 and Hc2. Using Hc1Hc2= lnκGL+α(κGL)2κ2GL, where α(κ) is givenin the Ref. [40], we determine κGL = 1.92. This leads im-mediately to Hc(0) =Hc2√2κGL= 24.67 mT which is consistent,within experimental error, with the estimate from the analysisof the specific heat.Coherence length and penetration depth from the critical fieldsThe upper critical field is related to the superconducting co-herence length byHc2 =Φ02πξ 2On the other hand, the lower critical field is defined asHc1 =4πε1Φ0(S4)where ε1 is the free energy per unit length of a single vortexline. To determine ε1 requires the solution of a non-linearequation for the order parameter, while Hc2 is determinedby the linearized Ginzburg-Landau equations in the field. A9comprehensive analysis of the linear and nonlinear Ginzburg-Landau theory was performed by Brandt who findsHc1 =Φ04πλ 2 C(λξ)(S5)with C (κ) = logκ + α (κ). The function α (κ) was deter-mined numerically in Ref. [40] for all values κ > κc =1√2.An analytic expression, that reproduces the numerical findingwith an accuracy better than 10−3, was also given [40] :α (κ) = α∞ + exp(−c0 − c1 logκ − c2 (logκ)2), (S6)with α∞ = 0.49693, c0 = 0.41477, c1 = 0.775, and c2 =0.1303. This expression reproduces C (κc)≈ 1 at the onset oftype II superconductivity and C (κ ≫ 1)≈ logκ+0.49693, asdetermined in Ref. [41]. From the two critical fields, we firstconstructhc1 ≡Hc1Hc2=12κ2 (logκ +α (κ)) (S7)and determine κ and ξ using the expression for Hc2 and thendetermine λ = κξ .Formalism of the nonlocal electrodynamicsHorizontal and vertical line nodesWe consider gap functions on the Fermi surface that giverise to vertical or horizontal line nodes, respectively. In theformer case, we consider ∆(k) = ∆0 (coskx − cosky). In thelow-energy limit we can safely write ∆(ϕ) = ∆0 cos2ϕ in-stead, where ϕ is the polar angle of k in cylindrical coordi-nates. For the horizontal line nodes, we have in mind a pair-ing state like ∆(k) = ∆0 (sinkx ± isinky)sinkz. For our analy-sis only the magnitude of the gap |∆(k)|2 matters and we use|∆(k)|2 = ∆20 sin2 kz, i.e. we ignore the weak dependence ofthe gap amplitude on kx,y. The density of statesρ (ω) =∫BZd3k(2π)2 δ(ω −√ε2k +∆2k)(S8)of both pairing states is the same and given asρ (ω) =2ρFπRe(K(∆20ω2)), (S9)where K (x) is the elliptic integral of first kind. It holdsρ (ω ≪ ∆0) = ρF ω/∆0, ρ (ω ≫ ∆0) = ρF and ρ (ω ≈ ∆0) =ρFπlog 8∆0|ω−∆0|.In the local, London limit, the penetration depth is [56]λ−2L (T ) = λ−20 (1+δk (T )) (S10)withδk (T ) =2ρF∫∞0dωρ (ω)d f (ω)dω. (S11)It follows∆λ (T )λ0∣∣∣∣loc≡ λL (T )−λ0λ0=1√1+δk (T )−1. (S12)At small T holds ρ (ω) = ρFω∆0and one obtainsδk (T ≪ ∆0) = −2log2 T∆0which yields the celebratedresult of a nodal superconductor ∆λ (T≪∆0)λ0∣∣∣loc= log2 T∆0.In our numerical analysis, we will use the full expressionEq.S12.Non-local electrodynamicsThe nonlocal relation between a current j and a vector po-tential A (in London gauge: ∇ ·A= 0 together with A= 0 deepin the bulk and A ·n = 0 at a surface with normal vector n) isjα (q,ω) =−Kαβ (q,ω)Aβ (q,ω) . (S13)We consider a superconductor in the half space y > 0 andconsider the decay of the electromagnetic field along the y-direction. For a magnetic field B = ∇×A applied along thez-direction we consider currents along the x-direction. If wecombine this relation with Maxwell’s equation ∇×B = 4πc jwe get at ω = 0A′′x (y) =−4πc∫dy′Kxx(y− y′)Ax(y′). (S14)The penetration depth is commonly defined asλ (T ) =1Bz (y = 0)∫∞0B(y)dy =−Ax (y = 0)A′x (y = 0), (S15)since Bz (y) = A′x (y), where the primes are derivatives withrespect to y. For specular boundary conditions one finds afterFourier transformation [38]Ax (qy) =Bz (y = 0)q2 + 4πc Kxx (qy)(S16)Hence, it followsλ (T ) =2π∫∞0dqyq2y +4πc Kxx (qy,T ).. (S17)The London penetration depth is defined asλ−2L =4πcKxx (qy → 0) . (S18)10If the transverse current kernel Kxx (qy) depends weakly onmomentum it is safe to assume Kxx (qy) ≈ c4πλ−2L and per-forming the integral yields λ = λL as required. The other cru-cial length scale of the superconductor is the coherence lengthξ0 =vFπ∆0, (S19)with Fermi velocity vF and gap amplitude ∆0. Nonlocal ef-fects go back to Pippard [48] who’s kernel in momentumspace is discussed in Ref. [38]. The key observation is thatKxx(qy ≫ ξ−10)≈ 3c16 λ−2L (qtξ0)−1 decays for increasing mo-mentum, i.e. magnetic field configurations with wave lengthsmaller than the coherence length induce smaller screeningcurrents. Indeed, in the limit λL ≪ ξ0 the integral Eq.S17 isdominated by momenta qy > ξ−10 and yields the well knownresult of clean type I superconductors λ ∼ λL (ξ0/λL)1/3.Kosztin and Leggett [33] found that non-local effects arealso important in nodal superconductors, yet in a more subtleway, with different behavior of the kernel Kxx (qy,T ) for zeroand finite temperatures. At T = 0 the response is essentiallymomentum independent and one can writeKxx (qy,0) =c4πλ−2L (T = 0) . (S20)Hence, for T = 0 the actual penetration depth is the Londonlength and the electromagnetic response is local. For conve-nience we use λ0 ≡ λ (T = 0). The non-locality emerges atfinite temperatures where the electromagnetic kernel can bewritten as [33]4πcKxx (qy,T ) = λ−20 +(λ−2L (T )−λ−20)F(ξT qyπ).(S21)The characteristic length scale for the nonlocal response dueto nodes is the thermal de Broghlie wave length of masslessBogoliubov particles ξT = vF/(kBT ). For vertical line nodes,the function F (z) is given asFv (z) = 1− πz√8log2∫ 10dy f(π√8yz)√1− y2. (S22)As small z, i.e. in the local limit holds Fv (z ≪ 1) = 1 −π2z16√2log2while Fv (z ≫ 1) = 6ζ (3)π2 log2 z−2. The analogous resultfor horizontal line nodes isFh (z) = 1− πz2log2∫ 10dy f(π2yz)(1− y)2 (S23)Now holds Fh (z ≪ 1) = 1− πz12log2 and Fh (z ≫ 1) = π3log2 z−1.The key difference between horizontal and vertical line nodesstems from the distinct behavior at large z, i.e. for ξT qy ≫ 1is,where Fh (z ≫ 1) ∝ z−1 while Fv (z ≫ 1) ∝ z−2, respectively.Hence, the suppression of non-local currents by thermal exci-tations is stronger for vertical than for horizontal nodes.In order to gain qualitative insights, we first analyze theleading low-temperature regime, where nonlocal correctionsare small. For T ≪ ∆0 follows∆λ (T )λ0=∆λ (T )λ0∣∣∣∣loc× 4π∫∞0dqyλ0F( 1πξT qy)(1+λ 20 q2y)2 ,(S24)where the first term describes the T -dependence in the localapproximation and the second term is the important correc-tion. In the regime where ξT ≫ λ0 the penetration depth isdominated by F (z ≫ 1). In the case of vertical nodes holds∆λ (T )λ0∼ ∆λ (T )λ0∣∣∣∣loc∫∞ξ−1Tdqyλ01(ξT qy)2∼ ∆λ (T )λ0∣∣∣∣locλ0ξT, (S25)where we dropped factors of order unity. This is the resultgiven in the main text that yields ∆λ (T )/λ0 ∝ κT 2∆20. The sit-uation changes for horizontal nodes where∆λ (T )λ0∼ ∆λ (T )λ0∣∣∣∣loc∫λ−10ξ−1Tdqyλ01ξT qy∼ ∆λ (T )λ0∣∣∣∣locλ0ξTlogξTλ0, (S26)which yields ∆λ (T )/λ0 ∝ κT 2∆20log ∆0κT . These results allow fora qualitative understanding of the impact of nodal excitationson the electromagnetic response. In our numerical analysiswe avoided the expansion for small corrections to the zero-temperature limit and used instead∆λ (T )λ0= −2λ0π∫∞0dqyδk (T )F(1πvF qykBT)(1+λ 20 q2y)× 1(1+λ 20 q2y +δk (T )F(1πvF qykBT)) . (S27)which follows from Eq.(S21) together with Eq.(S10). Similarexpressions can also be obtained for diffuse boundaries.The dimensionless T -variation of the penetration depth∆λ (T )/λ0, plottet as function of the dimensionless temper-ature variable T/Tc, only depends on the T = 0 ratio κ0 =λ0/ξ0 and on 2∆0/(kBTc). The former is determined by oursimultaneous measurement of Hc1 and Hc2. For the latter weused the BCS result of 3.53 in the main text. In our anal-ysis, the gap amplitude ∆0 determines the slope of the gapnear the node which does not have to agree with the over-all gap maximum, as determined by spectroscopic means, inparticular for multiband superconductors. This slope alsodetermines the low-T dependence of the specific heat Cs =9ζ (3)ρF2T 2∆0, where we used the low-frequency behavior of thedensity of states, given earlier. A T -dependence Cs = αT 2of the heat capacity was indeed observed in Ref. [50] whereα ≈ 52.7mJ/(mol K3), while in the normal state Cn = γT withγ = 37.5 mJ/(molK2). Hence, the low-energy slope follows110 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 50 . 00 . 10 . 20 . 30 . 40 . 50 . 60 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 50 . 00 . 10 . 20 . 3( b )V e r t i c a l  l i n e  n o d e s  2 � 0  /  k B T c =  3 . 1 6  2 � 0  /  k B T c =  3 . 1 6�� /��0( T / T c ) 2H o r i z o n t a l  l i n e  n o d e s( a )V e r t i c a l  l i n e  n o d e s  2 � 0  /  k B T c =  3 . 1 6  2 � 0  /  k B T c =  4 . 5  2 � 0  /  k B T c =  3 . 1 6�� /��0( T / T c ) 2H o r i z o n t a l  l i n e  n o d e sFigure S7. (a) ∆λ (T )/λ0 vs (T/Tc)2, the red dashed and or-ange solid lines are the expected behavior of horizontal and verticalline nodes respectably in the non-local limit with 2∆0/kBTc = 3.16.(b) The black line is the best fit for horizontal line nodes with2∆0/kBTc = 4.5as ∆0 =27ζ (3)π2γαwhich yields 2∆0/(kBTc) = 3.16. In Fig. S7awe show the penetration depth due to nonlocal excitations forthis value of the gap for horizontal and vertical line nodes, re-spectively. The agreement between experiment and theory forvertical line nodes is even slightly better for this value of thegap amplitude. Finally, given the dependence of our results on2∆0/(kBTc) we also plot in Fig. S7b our best fit of the data tothe theory with horizontal line nodes (black curve). 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