Chiral singlet superconductivity in the weakly correlated metal LaPt3P
P. K. Biswas, S. K. Ghosh, J. Z. Zhao, D. A. Mayoh, N. D. Zhigadlo, Xiaofeng Xu, C. Baines, A. D. Hillier, G. Balakrishnan, M. R. Lees
CChiral singlet superconductivity in the weakly correlated metal LaPt P P. K. Biswas, ∗ S. K. Ghosh, † J. Z. Zhao, D. A. Mayoh, N. D. Zhigadlo,
5, 6
Xiaofeng Xu, C. Baines, A. D. Hillier, G. Balakrishnan, and M. R. Lees ISIS Pulsed Neutron and Muon Source, STFC Rutherford Appleton Laboratory,Harwell Campus, Didcot, Oxfordshire OX11 0QX, United Kingdom School of Physical Sciences, University of Kent, Canterbury CT2 7NH, United Kingdom Co-Innovation Center for New Energetic Materials,Southwest University of Science and Technology, Mianyang, 621010, China Physics Department, University of Warwick, Coventry, CV4 7AL, United Kingdom Laboratory for Solid State Physics, ETH Zurich, 8093 Zurich, Switzerland CrystMat Company, 8037 Zurich, Switzerland Department of Applied Physics, Zhejiang University of Technology, Hangzhou 310023,China Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland (Dated: January 26, 2021)
Topological superconductors (SCs) are novelphases of matter with nontrivial bulk topology.They host at their boundaries and vortex coreszero-energy Majorana bound states, potentiallyuseful in fault-tolerant quantum computation [1].Chiral SCs [2] are particular examples of topo-logical SCs with finite angular momentum Cooperpairs circulating around a unique chiral axis, thusspontaneously breaking time-reversal symmetry(TRS). They are rather scarce and usually featuretriplet pairing: best studied examples in bulk ma-terials are UPt and Sr RuO proposed to be f -wave and p -wave SCs respectively, although manyopen questions still remain [2]. Chiral triplet SCsare, however, topologically fragile with the gap-less Majorana modes weakly protected againstsymmetry preserving perturbations in contrast tochiral singlet SCs [3, 4]. Using muon spin relax-ation ( µ SR) measurements, here we report thatthe weakly correlated pnictide compound LaPt Phas the two key features of a chiral SC: sponta-neous magnetic fields inside the superconductingstate indicating broken TRS and low temperaturelinear behaviour in the superfluid density indi-cating line nodes in the order parameter. Usingsymmetry analysis, first principles band structurecalculation and mean-field theory, we unambigu-ously establish that the superconducting groundstate of LaPt P is chiral d -wave singlet. Cooper pairs in conventional SCs, such as the elemen-tal metals, form due to pairing of electrons by phonon-mediated attractive interaction into the most symmetric s -wave spin-singlet state [5]. In contrast, unconventionalSCs defined as having zero average onsite pairing am-plitude pose a pivotal challenge in resolving how super-conductivity emerges from a complex normal state. Theyusually require a long-range interaction [6] and have lowersymmetry Cooper pairs. A special class of unconven-tional SCs are the chiral SCs. A well established real-ization of a chiral p -wave triplet state is the A -phase of superfluid He [7]. In addition to UPt and Sr RuO ,the heavy fermion SC UTe is also proposed to be a chi-ral triplet SC [8]. The chiral singlet SCs are, however,extremely rare and are proposed to be realized withinthe hidden order phase of the strongly correlated heavyfermion SC URu Si [9] and in the locally noncentrosym-metric material SrPtAs [10] with many unresolved issues.LaPt P is a member of the platinum pnictide family ofSCs A Pt P ( A = Ca, Sr and La) with a centrosymmetricprimitive tetragonal structure [11]. Its T c = 1 . P ( T c = 8 . P ( T c = 6 . P come both fromtheory: first principles Migdal-Eliashberg-theory [12]and experiments: very low T c , unsaturated resistivityup to room temperature and a weak specific heat jumpat T c [11]. The chiral nature of superconductivity ofLaPt P with topologically protected Majorana Fermi-arc and Majorana flat-band, which we uncover here, fitsnicely with these characteristics.
Experimental results
We have performed a comprehensive analysis of thesuperconducting properties of LaPt P using the µ SRtechnique. Two sets of polycrystalline LaPt P speci-mens, referred to here as sample-A (from Warwick, UK)and sample-B (from ETH, Switzerland), were synthe-sized at two different laboratories by completely differentmethods. Zero-field (ZF), longitudinal-field (LF), andtransverse-field (TF) µ SR measurements were performedon these samples at two different muon facilities: sample-A in the MUSR spectrometer at the ISIS Pulsed Neutronand Muon Source, UK, and sample-B in the LTF spec-trometer at the Paul Scherrer Institut (PSI), Switzerland.ZF- µ SR measurements reveal spontaneous magneticfields arising just below T c ≈ . χ ) data for sample-B on the right axis of Fig. 1 b ) a r X i v : . [ c ond - m a t . s up r- c on ] J a n - 0 . 8- 0 . 6- 0 . 4- 0 . 20 . 00 3 6 9 1 2 1 50 . 0 50 . 1 00 . 1 50 . 2 00 . 2 5 c T e m p e r a t u r e ( K ) l ZF ( m s-1) T c = 1 . 1 K I S I SP S I I S I S0 . 0 7 5 K1 . 5 K0 . 1 5 K , L F 5 m T1 . 7 5 K , L F 5 m T
Asymmetry
T i m e ( m s ) Z F - m S RL F - m S R ab FIG. 1.
Evidence of TRS-breaking superconductivityin LaPt P by ZF- µ SR measurements. a ) ZF- µ SR timespectra collected at 75 mK and 1.5 K for sample-A of LaPt P.The solid lines are the fits to the data using Eq. 1. b ) Thetemperature dependence of the extracted λ ZF (left axis) forsample-A (ISIS) and sample-B (PSI) showing a clear increasein the muon spin relaxation rate below T c . The PSI datahas been shifted by 0 . µ s − to match the baseline valueof the ISIS data. Variation of the zero-field-cooled magneticsusceptibility ( χ ) on the right axis for sample-B. associated with a TRS breaking superconducting statein both samples of LaPt P, performed on different in-struments. Fig. 1 a shows representative ZF- µ SR timespectra of LaPt P collected at 75 mK (superconduct-ing state) and at 1 . T c show a clear increase in muon-spin relaxation rate compared to the data collected inthe normal state. To unravel the origin of the sponta-neous magnetism at low temperature, we collected ZF- µ SR time spectra over a range of temperatures across T c and extracted temperature dependence of the muon-spin relaxation rate by fitting the data with a GaussianKubo-Toyabe relaxation function G ( t ) [13] multiplied byan exponential decay: A ( t ) = A (0) G ( t )exp( − λ ZF t ) + A bg (1)where, A (0) and A bg are the initial and backgroundasymmetries of the ZF- µ SR time spectra, respectively. G ( t ) = + (cid:0) − σ t (cid:1) exp (cid:0) − σ t / (cid:1) . σ ZF and λ ZF represent the muon spin relaxation rates originating from the presence of nuclear and electronic moments in thesample, respectively. In the fitting, σ ZF is found tobe nearly temperature independent and hence fixed tothe average value of 0 . µ s − for sample-A and0 . µ s − for sample-B. The temperature dependenceof λ ZF is shown in Fig. 1 b . λ ZF has a distinct system-atic increase below T c for both the samples which im-plies that the effect is sample and spectrometer inde-pendent. Moreover, the effect can be suppressed veryeasily by a weak longitudinal field of 5 mT for boththe samples. It is shown in Fig. 1 a for sample-A. Thisstrongly suggests that the additional relaxation below T c is not due to rapidly fluctuating fields [14], but ratherassociated with very weak fields which are static or qua-sistatic on the time-scale of muon life-time. The spon-taneous static magnetic field arising just below T c isso intimately connected with superconductivity that wecan safely say its existence is direct evidence for TRS-breaking superconducting state in LaPt P. From thechange ∆ λ ZF = λ ZF ( T ≈ − λ ZF ( T > T c ) we can es-timate the corresponding spontaneous internal magneticfield at the muon site B int ≈ ∆ λ ZF /γ µ = 0 . . γ µ / (2 π ) = 13 .
55 kHz/G is the muon gyromagnetic ratio.
Asymmetry
T i m e ( m s ) a b m s )T e m p e r a t u r e ( K ) s ( m s-1) c Internal field (mT) T c FIG. 2.
Superconducting properties of LaPt P byTF- µ SR measurements.
TF- µ SR time spectra of LaPt Pcollected at a ) 1.3 K and b ) 70 mK for sample-A in a trans-verse field of 10 mT. The solid lines are the fits to the datausing Eq. 2. c ) The temperature dependence of the extracted σ (left panel) and internal field (right panel) of sample-A. s - w a v e p - w a v e c h i r a l d - w a v e r / r T e m p e r a t u r e ( K )
FIG. 3.
Evidence of chiral d -wave superconductivityin LaPt P. Superfluid density ( ρ ) of LaPt P as a functionof temperature normalized by its zero-temperature value ρ .The solid lines are fits to the data using different models ofgap symmetry. Inset shows the schematic representation ofthe nodes of the chiral d -wave state. We show the TF- µ SR time spectra for sample-A inFig. 2 a and Fig. 2 b at two different temperatures. Thespectrum in Fig. 2 a shows only weak relaxation mainlydue to the transverse (2/3) component of the weak nu-clear moments present in the material in the normalstate at 1 . b inthe superconducting state at 70 mK shows higher relax-ation due to the additional inhomogeneous field distribu-tion of the vortex lattice, formed in the superconductingmixed state of LaPt P. The spectra are analyzed usingthe Gaussian damped spin precession function [13]: A T F ( t ) = A (0) exp (cid:0) − σ t (cid:14)
2) cos ( γ µ (cid:104) B (cid:105) t + φ )+ A bg cos ( γ µ B bg t + φ ) . (2)Here A (0) and A bg are the initial asymmetries of themuons hitting and missing the sample respectively. (cid:104) B (cid:105) and B bg are the internal and background magnetic fields,respectively. φ is the initial phase and σ is the Gaussianmuon spin relaxation rate of the muon precession signal.The background signal is due to the muons implantedon the outer silver mask where the relaxation rate ofthe muon precession signal is negligible due to very weaknuclear moments in silver. Fig. 2 c shows the temper-ature dependence of σ and internal field of sample-A. σ ( T ) shows a change in slope at T = T c which keeps onincreasing with further lowering of temperature. Such anincrease in σ ( T ) just below T c indicates that the sampleis in the superconducting mixed state and the formationof vortex lattice has created an inhomogeneous field dis-tribution at the muon sites. The internal fields felt by themuons show a diamagnetic shift in the superconductingstate of LaPt P, a clear signature of bulk superconduc- tivity in this material.The true contribution of the vortex lattice field dis-tribution to the relaxation rate σ sc can be estimated as σ sc = ( σ − σ ) / , where σ nm = 0 . µ s − isthe nuclear magnetic dipolar contribution assumed to betemperature independent. Within the Ginzburg-Landautheory of the vortex state, σ sc is related to the Londonpenetration depth λ of a SC with high upper critical fieldby the Brandt equation [16]: σ sc ( T ) γ µ = 0 . λ ( T ) , (3)where Φ = 2 . × − Wb is the flux quantum. Thesuperfluid density ρ ∝ λ − . Fig. 3 shows the tempera-ture dependence of ρ for LaPt P. It clearly varies withtemperature down to the lowest temperature 70 mK andshows a linear increase below T c /
3. This nonconstantlow temperature behaviour is a signature of nodes in thesuperconducting gap.The pairing symmetry of LaPt P can be understoodby analysing the superfluid density data using differentmodels of the gap function ∆ k ( T ). For a given pairingmodel, we compute the superfluid density ( ρ ) as ρ = 1 + 2 (cid:28) (cid:90) ∞ ∆ k ( T ) E (cid:112) E − | ∆ k ( T ) | ∂f∂E dE (cid:29) FS . (4)Here, f = 1 / (cid:16) e EkBT (cid:17) is the Fermi function and (cid:104)(cid:105) FS represents an average over the Fermi surface (as-sumed to be spherical). We take ∆ k ( T ) = ∆ m ( T ) g ( k )where we assume a universal temperature dependence∆ m ( T ) = ∆ m (0) tanh (cid:104) . { .
018 ( T c /T − } . (cid:105) [17]and the function g ( k ) contains its angular dependence.We use three different pairing models: s -wave (singleuniform superconducting gap), p -wave (two point nodesat the two poles) and chiral d -wave (two point nodes atthe two poles and a line node at the equator as shownin the inset of Fig. 3). The fitting parameters are givenin the Supplemental Material. We note from Fig. 3that both the s -wave and the p -wave models lead tosaturation in ρ at low temperatures which is clearlynot the case for LaPt P and the chiral d -wave modelgives an excellent fit down to the lowest temperature.Nodal SCs are rare since the SC can gain condensationenergy by eliminating nodes in the gap. Thus thesimultaneous observation of nodal and TRS-breakingsuperconductivity makes LaPt P a unique material.
Discussion
We investigate the normal state properties of LaPt P bya detailed band structure calculation using density func-tional theory within the generalized gradient approxima-tion consistent with previous studies [12, 18]. LaPt Pis centrosymmetric with a paramagnetic normal staterespecting TRS. It has significant effects of spin-orbit k y k z -π-π/20π/2π -π -π/2 0 π/2 π a k x k y -π-π/20π/2π-π -π/2 0 π/2 π b (001) surface BZ ( ) s u rf a c e B Z Zero energyflat band MajoranaArc statek x k y k z Bulk BZ c d FIG. 4.
Properties of the normal and superconducting states of LaPt P. Projections of the four Fermi surfaces ofLaPt P with SOC on the y − z plane in a and x − y plane in b . The thickness of the lines are proportional to the contributionof the Fermi surfaces to the DOS at the Fermi level (green– 10 . . . chiral d -wave gap are shown by red dots in a and the line node reside on the x − y plane in b . c ) Schematicview of the Majorana Fermi-arc and the zero energy Majorana flat-band corresponding to the two Weyl point nodes and theline node respectively on the respective surface Brillouin zones (BZs) assuming a spherical Fermi surface. d ) Berry curvature F ( k ) corresponding to the two Weyl nodes on the x − z plane. Arrows show the direction of F ( k ) and the colour scale showsits magnitude = π arctan( | F ( k ) | ). ∆ = 0 . µ was chosen for clarity while a more realistic weak-coupling limit ∆ (cid:28) µ gives amore sharply peaked curvature at the Fermi surface. coupling (SOC) induced band splitting near the Fermilevel ( ∼
120 meV, most apparent along the
M X highsymmetry direction). Kramer’s degeneracy survives inthe presence of strong SOC due to centrosymmetry andSOC only produces small deformations in the Fermi sur-faces [19]. The shapes of the Fermi surfaces play an im-portant role in determining the thermodynamic proper-ties of the material. The projections of the four Fermisurfaces of LaPt P on the y − z and x − y plane are shownin Fig. 4 a and Fig. 4 b respectively with the Fermi surfacesheets having the most projected-DOS at the Fermi levelshown in blue and orange. It shows the multi-band na-ture of LaPt P with orbital contributions mostly comingfrom the 5 d orbitals of Pt and the 3 p orbitals of P.LaPt P has a nonsymmorphic space group P /mmm (No. 129) with point group D h . From the group the-oretical classification of the SC order parameters withinthe Ginzburg-Landau theory [15, 20], the only possiblesuperconducting instabilities with strong SOC which canbreak TRS spontaneously at T c correspond to the two2D irreducible representations, E g and E u , of D h [21].The superconducting ground state in the E g channel isa pseudospin chiral d-wave singlet state with gap func-tion ∆( k ) = ∆ k z ( k x + ik y ) where ∆ is an amplitudeindependent of k . While the E u order parameter is apseudospin nonunitary chiral p-wave triplet state with d -vector d ( k ) = [ c k z , ic k z , c ( k x + ik y )] where c and c are material dependent real constants independent of k .We compute the quasi-particle excitation spectrumfor the two TRS breaking states on a generic singleband spherical Fermi surface using the Bogoliubov-deGennes mean field theory [15, 20]. The chiral d -wave sin-glet state leads to an energy gap = | ∆ || k z | (cid:113) k x + k y .It has a line node at the “equator” for k z = 0 and two point nodes at the “north” and “south” poles(shown in Fig. 4 a ). The low temperature thermo-dynamic properties are, however, dominated by theline node because of its larger low energy DOS thanthe point nodes. The triplet state has an energygap = (cid:113) g ( k x , k y ) + 2 c k z − | c || k z | (cid:112) f ( k x , k y ) + c k z where f ( k x , k y ) = c ( k x + k y ). It has only two pointnodes at the two poles and no line nodes. Thus, the lowtemperature linear behaviour of the superfluid densityof LaPt P shown in Fig. 3 is only possible in the chiral d -wave state with a line node in contrast to the tripletstate with only point nodes which will give a quadraticbehaviour and saturation at low temperatures. ThusLaPt P is one of the rare unconventional SCs for whichwe can unambiguously identify the superconducting or-der parameter. The point nodes and the line node for thechiral d -wave state on the Fermi surface sheets of LaPt Pare shown in Fig. 4 a and Fig. 4 b .We now discuss the topological properties of the chiral d -wave state of LaPt P based on a generic single-bandspherical Fermi surface (chemical potential µ = k F / (2 m )where k F is the Fermi wave vector and m is the elec-tron mass) [7, 22]. However, topological protection ofthe nodes also ensures stability against multiband effects.The effective angular momentum of the Cooper pairs is L z = +1 (in units of (cid:126) ) with respect to the chiral c -axis.The equatorial line node acts as a vortex loop in mo-mentum space [23] and is topologically protected by a1D winding number w ( k x , k y ) = 1 for k x + k y < k F and= 0 otherwise. The nontrivial topology of the line nodeleads to two-fold degenerate zero-energy Majorana boundstates in a flat band on the (0 , ,
1) surface BZ as shownin Fig. 4 c . As a result, there is a diverging zero-energyDOS leading to a zero-bias conductance peak (which canbe really sharp [24]) measurable in STM. This inversionsymmetry protected line node is extra stable due to evenparity SC [3, 24]. The point nodes on the other hand areWeyl nodes and are impossible to gap out by symmetry-preserving perturbations. They act as a monopole andan anti-monopole of Berry flux as shown in Fig. 4 d andare characterized by a k z dependent topological invariant,the sliced Chern number C ( k z ) = L z for | k z | < k F with k z (cid:54) = 0 and = 0 otherwise (see the Supplemental Materialfor details). As a result, the (1 , ,
0) and (0 , ,
0) surfaceBZs each have a Majorana Fermi arc which can be probedby STM as shown in Fig. 4 c . There are two-fold degen-erate chiral surface states with linear dispersion carryingsurface currents leading to local magnetisation that maybe detectable using SQUID magnetometry. One of thekey signatures of chiral edge states is the anomalous ther-mal Hall effect (ATHE) which depends on the length ofthe Fermi arc in this case. Impurities in the bulk can,however, increase the ATHE signal by orders of magni-tude [25] over the edge contribution making it possibleto detect with current experimental technology [26]. Wealso note that a 90 ◦ rotation around the c -axis for thechiral d -wave state leads to a phase shift of π/ Acknowledgments : PKB gratefully acknowledges theISIS Pulsed Neutron and Muon Source of the UK Sci-ence & Technology Facilities Council (STFC) and PaulScherrer Institut (PSI) in Switzerland for access to themuon beam times. SKG thanks Jorge Quintanilla andAdhip Agarwala for stimulating discussions and acknowl-edges the Leverhulme Trust for support through the Lev-erhulme early career fellowship. The work at the Uni-versity of Warwick was funded by EPSRC,UK, GrantEP/T005963/1. XX is partially supported by the Na-tional Natural Science Foundation of China (Grants11974061, U1732162). NDZ thanks K. Povarov and ac-knowledges support from the Laboratory for Solid StatePhysics, ETH Zurich where synthesis studies were initi-ated.
SUPPLEMENTARY MATERIAL
In this Supplemental Material, we present details of thesynthesis, characterization measurements, experimentalmethods and data analysis of the LaPt P samples grownat Warwick, United Kingdom and at ETH, Switzerland.We also give additional band structure results, details ofthe symmetry analysis and topological properties of thechiral d -wave state of LaPt P. Synthesis and characterization of the sample grownat Warwick, United Kingdom
Polycrystalline LaPt P [11] samples (sample-A) weresynthesized by a solid state reaction method. Powders of elemental platinum, red phosphorus, and alkaline earth(lanthanum) were mixed in an argon-filled glove box, andsealed in a quartz tube filled with argon gas. The tubewas initially heated to 400 ◦ C and held at this temper-ature for 12 h in order to avoid rapid volatilization ofphosphorus, then reacted at 900 ◦ C for 72 h. The sin-tered pellet was reground and further annealed at 900 ◦ C within argon-filled quartz tubes for several days andfinally quenched into iced water.The room-temperature structure was determined viapowder x-ray diffraction (PXRD). PXRD was measuredusing a Bruker D5000 general purposed powder diffrac-tometer. The diffraction pattern is shown in Fig. 5. Re-itveld refinement was carried out using the TOPAS soft-ware package [28] which gave the parameters shown inTable I.
TABLE I. Crystallographic and Rietveld refinement parame-ters obtained on LaPt P.Space-group P nmm (No. 129)Formula units/unit cell (Z) 2Lattice parameter a (˚A) 5 . c (˚A) 5 . V cell (˚A ) 182 . The heat capacity in zero field was measured using aQuantum Design Physical Property Measurement Sys-tem (PPMS) with a He insert to get down to 0.5 K.The total specific heat C tot at low temperatures is madeup of several contributions, C tot = C el + C ph + C hyp (5)where C el is the electronic specific heat having the formin the normal state C el = γ n T (6)with γ n being the Sommerfeld coefficient, C ph is the spe-cific heat due to the phonons given by C ph = β T + β T (7)with β and β being temperature independent parame-ters, and C hyp is a contribution due to hyerfine splitting C hyp ∝ /T . (8)Fitting the normal state specific heat gives γ n =9 . β = 0 . and β = Intensity (a.u.) (cid:1) ( d e g r e e s )L a P t P FIG. 5.
Powder x-ray diffraction pattern of thesample-A of LaPt P at room temperature.
X-raydiffraction pattern of LaPt P at room temperature where thegreen, red and blue lines indicate the experimental data, thefit and the difference between the data and the fit, respec-tively. The orange dashes indicate the expected Bragg peaks.The inset shows the structure of a unit cell of LaPt P. S a m p l e A ( C tot -C ph) /T (mJ mol-1 K-2) T e m p e r a t u r e ( K )
FIG. 6.
Heat capacity of the sample-A of LaPt P inzero-field. ( C tot − C ph ) /T as a function of temperature.There is a small anomaly close to the expected superconduct-ing transition temperature that is masked by a large hyperfinecontribution. . µ J/mol-K . We then subtract the phonon contri-bution to the specific heat to plot the electronic specificheat including the hyperfine contribution. This is shownin the Fig. 6 for the sample-A of LaPt P, and is con-sistent with the previous measurement of Ref.[11]. Wenote that the specific heat has a small anomaly close tothe expected T c ≈ . m H = 0 . 5 m T c T ( K )L a P t P T c , o n s e t » FIG. 7.
Zero-field-cooled magnetic susceptibility of thesample-A of LaPt P as a function of temperature.
The magnetic susceptibility was measured using aQuantum Design Magnetic Property Measurement Sys-tem (MPMS) using an i-quantum He insert. As seenfrom Fig. 7 this sample has a relatively low Meissnerfraction ( ∼ Synthesis and characterization of the sample grownat ETH, Switzerland
A polycrystalline sample of LaPt P (sample-B) wassynthesized using the cubic anvil high-pressure and high-temperature technique. Starting powders of LaP and Ptof high purity (99.99%) were weighed according to thestoichiometric ratio, thoroughly ground, and enclosed ina boron nitride container, which was placed inside a py-rophyllite cube with a graphite heater. The details of ex-perimental setup can be found in Ref.[29]. All the workrelated to the sample preparation and the packing of thehigh pressure cell-assembly was performed in an argon-filled glove box. In a typical run, a pressure of 2 GPawas applied at room temperature. The temperature wasramped in 3 h to the maximum value of 1500 ◦ C, main-tained for 5 h, and then cooled to 1350 ◦ C over 5 h andfinally reduced to room temperature in 3 h. Afterward,the pressure was released, and the sample was removed.The sample exhibits a large diamagnetic response withthe superconducting transition temperature of 1.1 K.Susceptibility measurements were performed using aQuantum Design Magnetic Property Measurement Sys-tem (MPMS) by cooling the sample at base temperaturein zero field and then apply 7 mT magnetic field. Datawere collected while warming up the sample temperature.As shown in the main text, the temperature dependenceof the susceptibility data shows a bulk superconductingtransition with a T c at around 1.1 K. Moment (emu)
M a g n e t i c f i e l d ( m T ) T = 0 . 6 3 K m H c 1 = 1 m T FIG. 8.
Magnetic field dependence of the virgin mag-netisation curve for sample-B of LaPt P. We note thatthe lower critical field µ H c1 ≈ A virgin magnetisation curve was measured at 0.63 Kin a Quantum Design MPMS. A linear deviation of themagnetisation curve at low field region (see Fig. 8) showsthat the lower critical field H c1 of LaPt P is around 1mT. µ SR technique µ SR is a very sensitive local magnetic probe utilizingfully spin-polarized muons [13]. In a µ SR experimentpolarized muons are implanted into the host sample. Af-ter thermalization, each implanted muon decays (lifetime τ µ = 2 . µ s) into a positron (and two neutrinos) emittedpreferentially in the direction of the muon’s spin at thetime of decay. Using detectors appropriately positionedaround the sample, the decay positrons are detected andtime stamped. From the collected histograms, the asym-metry in the positron emission as a function of time, A ( t ),can be determined, which is directly proportional to thetime evolution of the muon spin polarization. µ SR measurements were performed on sample-A inthe MUSR spectrometer at the ISIS Pulsed Neutron andMuon Source, UK, and on sample-B in the LTF spec-trometer at the Paul Scherrer Institut (PSI), Switzer-land. The polycrystalline samples of LaPt P in the formof powder were mounted on high purity silver sampleholders. The samples were cooled from above T c to basetemperature in zero field for ZF- µ SR measurements, andin a field for the TF- µ SR measurements. The externalfield was 10 mT for the TF- µ SR measurements performedat ISIS and was 7 mT for the TF- µ SR measurements per-formed at PSI. ZF- µ SR measurements were performed intrue zero field, achieved by three sets of orthogonal coilsworking as an active compensation system which cancel any stray fields at the sample position down to 1.0 µ T.LF- µ SR measurements were also performed under simi-lar field-cooled conditions. The typical counting statisticswere ∼
40 and ∼
24 million muon decays per data pointat ISIS and PSI, respectively. The ZF-, LF- and TF- µ SRdata were analyzed using the equations given in the text.The zero temperature upper critical field for LaPt P, µ H c ≈ .
12 T which is much larger than the appliedtransverse fields in the TF- µ SR measurements. The de-tailed parameters for the analysis of superfluid densitydata from the TF- µ SR measurements for the two sam-ples using the different gap models mentioned in the maintext are given in the Table II.
Band structure
LaPt P crystallizes in a centrosymmetric primitivetetragonal crystal structure. The corresponding spacegroup is P4/nmm (No. 129) which is nonsymmorphic.The point group of the Bravais lattice is D h . The non-symmorphic symmetries within a unit cell include bothscrew axes and glide planes. We have performed detailedband structure calculations of LaPt P using density func-tional theory (DFT). The corresponding band structureresults with and without spin orbit coupling (SOC) areshown in Fig. 9(a) and Fig. 9(b) respectively. We notethat this material has significant splitting of bands dueto SOC [18]. The maximum band splitting caused bythe SOC near the Fermi level is estimated to be ∼ XCrySDen packages [30]. The Fermi surfaces with SOC are shown inFig. 10. We note that there are four Fermi surfaces withthe middle two shown in Fig. 10(b) and Fig. 10(c); andagain in Fig. 10(f) and Fig. 10(g) from a different view,contributing the most to the density of states (DOS) atthe Fermi level. This is seen from the projected DOSat the Fermi level shown in Fig. 11. Fig. 11(a) showsthe contributions of the different atomic orbitals to theDOS at the Fermi level. We note that Pt-5d orbitals con-tribute the most. Thus LaPt P is a multi-band system.Fig. 11(a) shows the contributions of the different Fermisurfaces to the DOS at the Fermi level.
Symmetry analysis
In this section we describe, the symmetry analysisof the possible superconducting order parameters forLaPt P. To proceed, we note the properties of the ma-terial: it is centrosymmetric, has nonsymmorphic sym-metries, has considerable effects of SOC, has multiple
TABLE II. Summary of the analysis of the superfluid density data for the two samples of LaPt P.Model g( θ, φ ) Gap type Reduced least-squared deviation ( χ r ) Fitted ∆ m (0) / ( k B T c ) s -wave 1 nodeless 13 .
025 1 . ± . p -wave sin( θ ) e iφ two point nodes 4 .
537 1 . ± . d -wave sin(2 θ ) e iφ two point nodes + a line node 2 .
238 1 . ± . Γ X M Γ Z A R Z Γ X M Γ Z A R Z E n e r gy ( e V ) -1.0-0.500.51.0 (a) (b) k y k x k z RAZ MXΓ
FIG. 9.
First principles band structure results of LaPt P. a) Band structure without SOC. b) Band structure withSOC. The primitive tetragonal Brillouin zone with the marked high symmetry points and directions used in the band structurecomputation is shown in the inset of (b). We note that SOC induces significant band splitting near the Fermi level especiallyfrom M to X. bands potentially participating in superconductivity, hasspontaneously broken TRS at T c and has line nodes dom-inating its thermodynamic behavior.The normal state symmetry group of the system isgiven by G = G ⊗ U (1) ⊗ T , where U (1) is the gaugesymmetry group, G is the group of symmetries contain-ing the point group symmetries of D h and spin rotationsymmetries in 3D of SO (3) and T is the group of time-reversal symmetry (TRS). The Ginzburg-Landau (GL)free energy of the system must be invariant under thissymmetry group.The D h point group has 8 one-dimensional irreduciblerepresentations (irreps) (4 of them have even parity andthe other 4 have odd parity) and 2 two dimensional ir-reps (one with even parity denoted by E g and the otherwith odd parity denoted by E u ). Centrosymmetry im-plies that this material has either purely triplet or purelysinglet superconducting instability in general. Further-more, a TRS breaking superconducting order parame-ter requires degenerate or multi-dimensional irreps. Thissystem can thus lead to such type of instability only inthe E g or the E u irrep. We will now focus only on these two irreps and construct possible superconducting orderparameters for the system. We consider strong SOC asuncovered by the band structure calculation of this ma-terial.The fourth order invariant corresponding to the 2 two-dimensional irreps E g and E u of D h gives the quarticorder term of the GL free energy [20, 31] to be f = β ( | η | + | η | ) + β | η + η | + β ( | η | + | η | ) (9)where ( η , η ) are the two complex components of thetwo-dimensional order parameters. This free energyneeds to be minimized with respect to both η and η . The nonequivalent solutions are: ( η , η ) = (1 , √ (1 ,
1) and √ (1 , i ). There is an extended region inthe parameter space where the states corresponding to( η , η ) = (1 , i ) is stabilized. The instabilities corre-sponding to this case spontaneously break TRS at T c due to a nontrivial phase difference between the two or-der parameter components.Then the even parity superconducting order parameter (a) (b) (c) (d)(e) (f) (g) (h) (i) FIG. 10.
Fermi surfaces of LaPt P with SOC.
Panels (a)–(d) are from a side view and the panels (e)–(h) are from thetop view for the four Fermi surface sheets and (i) shows a combined Fermi surface. belonging to E g is given by∆( k ) = ∆ k z ( k x + ik y ) (10)where ∆ is the real amplitude independent of k . This isa chiral d-wave singlet order parameter. The odd paritysuperconducting order parameter belonging to E u givesrise to the gap matrix ˆ∆( k ) = [ d ( k ) .(cid:126)σ ] iσ y where (cid:126)σ de-notes the three Pauli spin matrices and d ( k ) is the triplet d -vector given by d ( k ) = [ Ak z , iAk z , B ( k x + ik y )] . (11)Here, A and B are material dependent real constantsindependent of k and in general they are nonzero. Wenote that the values of A and B determine the orientationof the d -vector. For example, for A = 0 the d -vectorpoints along the c -axis and for B = 0 the d -vector points in the ab -plane. We also note that d ( k ) × d ∗ ( k ) = 2 iAk z ( Bk x ˆ x − Bk y ˆ y − Ak z ˆ z ) (12)which is nonzero in general. Hence, this superconductingstate is nonunitary chiral p-wave triplet state.The strong SOC case considered here implies that thesingle particle states are no longer the eigenstates of spinand we need to label them rather by pseudospins. Thepseudospin states are linear combinations of the spineigenstates. Since the pseudospin and the spin are closelyrelated, the even parity states correspond to pseudospinsinglet and the odd parity states correspond to pseu-dospin triplet states.We can now follow the standard Bogoliubov-de Gennesmean field theory [20] to compute the quasi-particle exci-tation energy spectrum for the two TRS breaking statesgiven in Eqn. (10) and Eqn. (11). The schematic view of0 -1 0 102468 Energy (eV) DO S [ / ( e V • u . c . )] (b)La 5 d P 3 p Pt 5 d Tot
Energy (eV) DO S [ / ( e V • u . c . )] FIG. 11.
Projected density of states (DOS) results.
Left panel shows the contributions of different orbitals to the DOS.We note that Pt 5d orbitals contribute the most to the DOS at the Fermi level. The right panel shows the DOS contributionsof the different Fermi surfaces. The blue is total and the other four correspond to the four Fermi surfaces. Their contributionsat the Fermi level are 10.3%, 43.4%, 39.5% and 6.3%.(a) (b)FIG. 12.
Polar plots of the excitation energy gaps. (a) The chiral d -wave singlet case and (b) the nonunitary chiral p -wave triplet case. In both the cases, point nodes appear at the two “poles”, while the singlet case has an additional line nodeat the “equator”. the excitation energy gaps for the two order parametersare shown in Fig. 12. Topological properties of the chiral singlet state
To discuss the topological properties of the nodal exci-tations for the chiral d -wave state with the gap function∆( k ) = ∆ k F k z ( k x + ik y ) (13) with ∆ being the pairing amplitude, we assume a sim-plified single band parabolic dispersion (in units of (cid:126) ) ξ ( k ) = k m − µ, (14)where m is the mass of an electron, µ = k F m is the chem-ical potential and k F is the Fermi wavevector. We notethat ∆( k ) ∼ Y ( θ, φ ) where Y lm ( θ, φ ) are the sphericalharmonics. Thus the Cooper pairs have an angular mo-mentum L z = +1 for this state.Then Bogoliubov-de Gennes Hamiltonian in the pseu-1dospin basis can be written as H = (cid:88) k Ψ † k H ( k )Ψ k (15)where Ψ k = ( c k ↑ , c †− k ↑ ) T with c k σ being the fermion an-nihilation operator with pseudospin flavor σ ∈ {↑ , ↓} . Wecan rewrite the BdG Hamiltonian as H ( k ) = N ( k ) · τττ (16)where τττ is the vector of the three Pauli matrices in theparticle-hole space and N ( k ) = (cid:110) ∆ k F k z k x , ∆ k F k z k y , ξ ( k ) (cid:111) is a pseudospin vector. The eigenvalues of the Hamilto-nian in Eqn. (16) are ± E ( k ) where E ( k ) = | N ( k ) | = (cid:112) ξ ( k ) + | ∆( k ) | . (17)Hence, the superconducting ground state has two pointnodes at the two poles of the Fermi surface k ± =(0 , , ± k F ) and a line node at the equator k z = 0 plane.The low energy Hamiltonian close to two point nodes canbe written as H ( k ) = ∆ k F ( p x τ x − p y τ y ) ± v F p z τ z (18)where we have defined p = ( k − k ± ). This is a WeylHamiltonian. Thus the two point nodes are also Weylnodes. As a result they are impossible to gap out sincethere is no fourth Pauli matrix which can come from amass term to gap out the nodes.The corresponding Bloch wave functions | u ± ( k ) (cid:105) arethe eigenfunctions of ˆn ( k ) .σσσ with eigenvalues ± ˆn ( k ) = N ( k ) / | N ( k ) | is the unit vector along the direc-tion of the pseudospin N ( k ). We note that this unit vec-tor ˆn ( k ) is well defined only when | N ( k ) | (cid:54) = 0 i.e. in thenodeless regions on the Fermi surface. In spherical co-ordinates, parametrizing ˆn ( k ) = [ n x ( k ) , n y ( k ) , n z ( k )] =[sin( θ ) cos( φ ) , sin( θ ) sin( φ ) , cos( θ )] we have | u − ( k ) (cid:105) = (cid:20) cos( θ ) e − iφ sin( θ ) (cid:21) and | u + ( k ) (cid:105) = (cid:20) sin( θ ) e − iφ − cos( θ ) (cid:21) . (19)Then from the negative energy occupied states | u − ( k ) (cid:105) the Berry connection is defined as A ( k ) = i (cid:104) u − ( k ) | ∇∇∇ k | u − ( k ) (cid:105) (20)and the corresponding Berry curvature is F ( k ) = ∇∇∇ k × A ( k ). In terms of the components of ˆn ( k ), it is given by F ( k ) = [ n y ( k ) {∇∇∇ k n z ( k ) ×∇∇∇ k n x ( k ) } − n x ( k ) {∇∇∇ k n z ( k ) ×∇∇∇ k n y ( k ) } ] / [2 { n x ( k ) + n y ( k ) } ].For the chiral d -wave case, F x ( k ) and F y ( k ) are oddfunctions of ( k y , k z ) and ( k x , k z ) respectively. Hence,there is no Berry flux along the x and y directions. Thenumber of field lines coming in and out of the ca and cb planes are the same. Whereas F z ( k ) is an even function of ( k x , k y ) and the flux through the ab plane as a functionof k z is Φ( k ) = (cid:90) dk x dk y F z ( k ) = 2 π C ( k z ) . (21) C ( k z ) is the ”sliced” Chern number (momentum depen-dent) of the effective 2D problem for a fixed k z . For agiven value of | k z | < k F , the Hamiltonian in Eqn. (16)describes an effective 2D problem with fully gapped weakcoupling BCS pairing and an effective chemical potential (cid:126) m ( k F − k z ) having the Chern number C ( k z ) = +1. For | k z | > k F , the effective chemical potential is negativeand describes a topologically trivial BEC state. Thus,the Weyl point nodes at (0 , , ± k F ) act as monopolesand anti-monopoles of the Berry curvature and the fluxthrough a sphere surrounding the monopole is 2 π andthat through the anti-monopole is − π . The topolog-ically protected Weyl nodes give rise to Majorana arcsurface states on the surface Brillouin zone correspondingto the (1 , ,
0) and (0 , ,
0) surfaces having chiral lineardispersions along y and x directions respectively. As a re-sult of the arc surface states the system shows anomalousthermal and spin Hall effects [4, 7, 22].The equatorial line node is characterized by a 1D wind-ing number. This can be defined in terms of the followingspectral symmetry [4, 22] of the Hamiltonian. We notethat the operatorΓ k = sin( φ k ) τ x + cos( φ k ) τ y (22)where tan( φ k ) = k y /k x anticommutes with the Hamilto-nian { H ( k ) , Γ k } = 0 . (23)As a result any eigenstate of the Hamiltonian H ( k ) withthe eigenvalue E k is also an eigenstate of the operatorΓ k with the eigenvalue − E k . Then with the help of thisspectral symmetry Γ k we define the winding number as w ( k ⊥ ) = − πi (cid:73) L d l T r (cid:2) Γ k H − ( k ) ∂ l H ( k ) (cid:3) , (24)where dl is the line element along a closed loop L encir-cling the line node and k ⊥ = ( k x , k y ). For this case thenwe have w ( k ⊥ ) = 1 ∀ k ⊥ < k F (25)= 0 otherwise . (26)We note that the winding number does not depend onthe angular momentum of the Cooper pairs. This non-trivial topology of the line node ensures the existence ofzero-energy surface Andreev bound states on the (0 , , ∗ [email protected] † [email protected][1] M. Sato and Y. Ando, “Topological superconductors:a review,” Reports on Progress in Physics , 076501(2017).[2] C. Kallin and J. Berlinsky, “Chiral superconductors,” Re-ports on Progress in Physics , 054502 (2016).[3] S. Kobayashi, K. Shiozaki, Y. Tanaka, and M. Sato,“Topological Blount’s theorem of odd-parity supercon-ductors,” Physical Review B , 024516 (2014).[4] P. Goswami and A. H. Nevidomskyy, “Topological Weylsuperconductor to diffusive thermal Hall metal crossoverin the B-phase of UPt ,” Physical Review B , 214504(2015).[5] M. Tinkham, Introduction to Superconductivity (McGraw-Hill Inc., 1996).[6] D. J. Scalapino, “A common thread: The pairing inter-action for unconventional superconductors,” Reviews ofModern Physics , 1383 (2012).[7] A. P. Schnyder and P. M. R. Brydon, “Topological sur-face states in nodal superconductors,” Journal of Physics:Condensed Matter , 243201 (2015).[8] L. Jiao, S. Howard, S. Ran, Z. Wang, J. O. Rodriguez,M. Sigrist, Z. Wang, N. P. Butch, and V. Madha-van, “Chiral superconductivity in heavy-fermion metalUTe ,” Nature , 523–527 (2020).[9] J. A. Mydosh and P. M. Oppeneer, “Colloquium: Hiddenorder, superconductivity, and magnetism: The unsolvedcase of URu Si ,” Reviews of Modern Physics , 1301–1322 (2011).[10] P. K. Biswas, H. Luetkens, T. Neupert, T. St¨urzer,C. Baines, G. Pascua, A. P. Schnyder, M. H. Fischer,J. Goryo, M. R. Lees, et al. , “Evidence for superconduc-tivity with broken time-reversal symmetry in locally non-centrosymmetric SrPtAs,” Physical Review B , 180503(2013).[11] T. Takayama, K. Kuwano, D. Hirai, Y. Katsura, A. Ya-mamoto, and H. Takagi, “Strong coupling superconduc-tivity at 8.4 K in an antiperovskite phosphide SrPt P,”Physical Review Letters , 237001 (2012).[12] A. Subedi, L. Ortenzi, and L. Boeri, “Electron-phononsuperconductivity in A Pt P ( A = Sr, Ca, La) com-pounds: From weak to strong coupling,” Physical ReviewB , 144504 (2013).[13] A. Yaouanc and P. D. De Reotier, Muon spin rotation, re-laxation, and resonance: applications to condensed mat-ter , Vol. 147 (Oxford University Press, 2011).[14] R. S. Hayano, Y. J. Uemura, J. Imazato, N. Nishida,K. Nagamine, T. Yamazaki, Y. Ishikawa, and H. Ya-suoka, “Spin fluctuations of itinerant electrons in MnSistudied by muon spin rotation and relaxation,” Journalof the Physical Society of Japan , 1773–1783 (1980).[15] S. K. Ghosh, M. Smidman, T. Shang, J. F. Annett, A. D.Hillier, J. Quintanilla, and H. Yuan, “Recent progress on superconductors with time-reversal symmetry breaking,”Journal of Physics: Condensed Matter , 033001 (2020).[16] E. H. Brandt, “Properties of the ideal Ginzburg-Landauvortex lattice,” Physical Review B , 054506 (2003).[17] A. Carrington and F. Manzano, “Magnetic penetrationdepth of MgB ,” Physica C: Superconductivity , 205–214 (2003).[18] H. Chen, X. Xu, C. Cao, and J. Dai, “First-principlescalculations of the electronic and phonon properties of A Pt P ( A = Ca, Sr, and La): Evidence for a charge-density-wave instability and a soft phonon,” Physical Re-view B , 125116 (2012).[19] S. Yip, “Noncentrosymmetric superconductors,” Annu.Rev. Condens. Matter Phys. , 15–33 (2014).[20] M. Sigrist and K. Ueda, “Phenomenological theory ofunconventional superconductivity,” Reviews of Modernphysics , 239 (1991).[21] Nonsymmorphic symmetries can give rise to additionalsymmetry-required nodes (other than the point groupsymmetry-required ones) on the Brillouin zone bound-aries along the high symmetry directions. The nonsym-morphic symmetries of LaPt P, however, can only gen-erate additional point nodes for the E g order parameterbut no additional nodes for the E u case [32].[22] P. Goswami and L. Balicas, “Topological properties ofpossible Weyl superconducting states of URu Si ,” arXivpreprint arXiv:1312.3632 (2013).[23] T. T. Heikkil¨a, N. B. Kopnin, and G. E. Volovik,“Flat bands in topological media,” JETP Letters , 233(2011).[24] S. Kobayashi, Y. Tanaka, and M. Sato, “Fragile surfacezero-energy flat bands in three-dimensional chiral super-conductors,” Physical Review B , 214514 (2015).[25] V. Ngampruetikorn and J. A. Sauls, “Impurity-inducedanomalous thermal Hall effect in chiral superconductors,”Physical Review Letters , 157002 (2020).[26] M. Hirschberger, R. Chisnell, Y. S. Lee, and N. P. Ong,“Thermal Hall effect of spin excitations in a kagome mag-net,” Physical Review Letters , 106603 (2015).[27] J. D. Strand, D. J. Van Harlingen, J. B. Kycia, andW. P. Halperin, “Evidence for complex superconductingorder parameter symmetry in the low-temperature phaseof UPt from Josephson interferometry,” Physical ReviewLetters , 197002 (2009).[28] A. A. Coelho, “ TOPAS and
TOPAS-Academic : an opti-mization program integrating computer algebra and crys-tallographic objects written in C++,” Journal of AppliedCrystallography , 210–218 (2018).[29] N. D. Zhigadlo, “High pressure crystal growth of the an-tiperovskite centrosymmetric superconductor SrPt P,”Journal of Crystal Growth , 94 – 98 (2016).[30] A. Kokalj, “Computer graphics and graphical user in-terfaces as tools in simulations of matter at the atomicscale,” Computational Materials Science , 155–168(2003).[31] J. F. Annett, “Symmetry of the order parameter for high-temperature superconductivity,” Adv. Phys. , 83–126(1990).[32] S. Sumita and Y. Yanase, “Unconventional supercon-ducting gap structure protected by space group symme-try,” Physical Review B97