Introduction

Cooper pairs in conventional superconductors (SCs), such as the elemental metals, form due to pairing of electrons by phonon-mediated attractive interaction into the most symmetric s-wave spin-singlet state¹. They have a nonzero onsite pairing amplitude in real-space. In contrast, unconventional SCs are defined as having zero onsite pairing amplitude in real-space². As a result, electrons in Cooper pairs of unconventional SCs avoid contact with each other to become energetically more favourable over conventional Cooper pairs, in strongly repulsive systems. Unconventional SCs pose a pivotal challenge in resolving how superconductivity emerges from a complex normal state. They usually require a long-range interaction³ and have lower symmetry Cooper pairs.

Chiral SCs belong to a special class of unconventional SCs having non-trivial topology and Cooper pairs with finite angular momentum. A well established realization of a chiral p-wave triplet superconducting state is in the A-phase of superfluid He^3 4. In bulk materials, perhaps the best studied examples are UPt₃⁵ and Sr₂RuO₄⁶. The long-held view of Sr₂RuO₄ being a chiral p-wave triplet SC⁷, however, has been called into question by recent NMR⁸ and neutron⁹ measurements, and a multicomponent chiral singlet order parameter has been suggested to be compatible with experiments¹⁰. UPt₃ is believed to realize a chiral f-wave triplet state, although many open questions still remain⁷. Recently, the heavy fermion SC UTe₂ has been proposed to be a chiral triplet SC¹¹. Chiral singlet SCs are also extremely rare, but may be realized within the hidden order phase of the strongly correlated heavy fermion SC URu₂Si₂¹² and in the locally noncentrosymmetric material SrPtAs¹³ although there are many unresolved issues for both these materials.

LaPt₃P is a member of the platinum pnictide family of SCs APt₃P (A = Ca, Sr and La) with a centrosymmetric primitive tetragonal structure¹⁴. Its T_c = 1.1 K is significantly lower than its other two isostructural counterparts SrPt₃P (T_c = 8.4 K) and CaPt₃P (T_c = 6.6 K)¹⁴, which are conventional Bardeen-Cooper-Schrieffer (BCS) SCs. Indications of the unconventional nature of the superconductivity in LaPt₃P come from a number of experimental observations: a very low T_c, unsaturated resistivity up to room temperature and a weak specific heat jump at T_c¹⁴. LaPt₃P also has a different electronic structure from the other two members in the family because La contributes one extra valence electron. Theoretical analysis based on first principles Migdal-Eliashberg-theory^15,16 found that the electron–phonon coupling in LaPt₃P is the weakest in the family, which can explain its low T_c. The weak jump in the specific heat which is masked by a possible hyperfine contribution at low temperatures¹⁴ (see also Supplementary Fig. 2), however, cannot be quantitatively captured.

Here, we show that the weakly correlated metal LaPt₃P spontaneously breaks time-reversal symmetry (TRS) in the superconducting state at T_c with line nodal behaviour at low temperatures based on extensive muon-spin relaxation (μSR) measurements. Using first principles theory, symmetry analysis and topological arguments, we establish that our experimental observations for LaPt₃P can be consistently explained by a chiral d-wave singlet superconducting ground state with topologically protected Majorana Fermi-arcs and a Majorana flat band.

Results

We have performed a comprehensive analysis of the superconducting properties of LaPt₃P using the μSR technique. Two sets of polycrystalline LaPt₃P specimens, referred to here as sample-A (from Warwick, UK) and sample-B (from ETH, Switzerland), were synthesized at two different laboratories by completely different methods (see Supplementary Note 1 and 2). Zero-field (ZF), longitudinal-field (LF) and transverse-field (TF) μSR measurements were performed on these samples at two different muon facilities: sample-A in the MUSR spectrometer at the ISIS Pulsed Neutron and Muon Source, UK, and sample-B in the LTF spectrometer at the Paul Scherrer Institut (PSI), Switzerland.

ZF-μSR results

ZF-μSR measurements reveal spontaneous magnetic fields arising just below T_c ≈ 1.1 K (example characterization is shown by the zero-field-cooled magnetic susceptibility (χ) data for sample-B on the right axis of Fig. 1b) associated with a TRS-breaking superconducting state in both samples of LaPt₃P, performed on different instruments. Figure 1a shows representative ZF-μSR time spectra of LaPt₃P collected at 75 mK (superconducting state) and at 1.5 K (normal state) on sample-A at ISIS. The data below T_c show a clear increase in muon-spin relaxation rate compared to the data collected in the normal state. To unravel the origin of the spontaneous magnetism at low temperature, we collected ZF-μSR 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 Gaussian Kubo-Toyabe relaxation function $$ \mathcal{G}\left(t\right)$$¹⁷ multiplied by an exponential decay:

where, A(0) and A_bg are the initial and background asymmetries of the ZF-μSR time spectra, respectively. $$ \mathcal{G}\left(t\right)=\frac{1}{3}+\frac{2}{3}\left(1-{\sigma }_{\mathrm{ZF}}^2{t}^2\right)\exp \left(-{\sigma }_{\mathrm{ZF}}^2{t}^2/2\right)$$. σ_ZF and λ_ZF represent the muon-spin relaxation rates originating from the presence of nuclear and electronic moments in the sample, respectively. The signal-to-background ratio A(0)/A_bg ≈ 0.40 (≈0.52) for sample-A (sample-B). In the fitting, σ_ZF is found to be nearly temperature independent and hence fixed to the average value of 0.071(4) μs⁻¹ for sample-A and 0.050(3) μs⁻¹ for sample-B. The temperature dependence of λ_ZF is shown in Fig. 1b. λ_ZF has a distinct systematic increase below T_c for both the samples which implies that the effect is sample and spectrometer independent. Moreover, the effect can be suppressed very easily by a weak longitudinal-field of 5 mT for both the samples as shown in Fig. 1a for sample-A. This strongly suggests that the additional relaxation below T_c is not due to rapidly fluctuating fields¹⁸, but rather associated with very weak fields which are static or quasistatic on the time-scale of muon life-time. The spontaneous static magnetic field arising just below T_c is so intimately connected with superconductivity that we can safely say its existence is direct evidence for TRS-breaking superconducting state in LaPt₃P. From the change Δλ_ZF = λ_ZF(T ≈ 0) − λ_ZF(T > T_c) we can estimate the corresponding spontaneous internal magnetic field at the muon site B_int ≈ Δλ_ZF/γ_μ = 0.22(4) G for sample-A and 0.18(2) G for sample-B, which are very similar to that of other TRS-breaking SCs¹⁹. Here, γ_μ/(2π) = 13.55 kHz/G is the muon gyromagnetic ratio.

Fig. 1

Evidence of TRS-breaking superconductivity in LaPt₃P by ZF-μSR measurements.

a ZF-μSR time spectra collected at 75 mK and 1.5 K for sample-A of LaPt₃P. The solid lines are the fits to the data using Eq. (1). b The temperature dependence of the extracted λ_ZF (left axis) for sample-A (ISIS) and sample-B (PSI) showing a clear increase in the muon-spin relaxation rate below T_c. The PSI data have been shifted by 0.004 μs⁻¹ to match the baseline value of the ISIS data. Variation of the zero-field-cooled magnetic susceptibility (χ) on the right axis for sample-B. The error bars in a and b show the standard deviations in the respective measurements.

TF-μSR results

We have shown the TF-μSR time spectra for sample-A in Fig. 2a and Fig. 2b at two different temperatures. The spectrum in Fig. 2a shows only weak relaxation mainly due to the transverse (2/3) component of the weak nuclear moments present in the material in the normal state at 1.3 K. In contrast, the spectrum in Fig. 2b in the superconducting state at 70 mK shows higher relaxation due to the additional inhomogeneous field distribution of the vortex lattice, formed in the superconducting mixed state of LaPt₃P. The spectra are analyzed using the Gaussian damped spin precession function¹⁷:

Here A(0) and A_bg are the initial asymmetries of the muons hitting and missing the sample, respectively. $$ ⟨B⟩$$ and B_bg are the internal and background magnetic fields, respectively. ϕ is the initial phase and σ is the Gaussian muon-spin relaxation rate of the muon precession signal. The background signal is due to the muons implanted on the outer silver mask where the relaxation rate of the muon precession signal is negligible due to very weak nuclear moments in silver. Figure 2c shows the temperature dependence of σ and internal field of sample-A. σ(T) shows a change in slope at T = T_c, which keeps on increasing with further lowering of temperature. Such an increase in σ(T) just below T_c indicates that the sample is in the superconducting mixed state and the formation of vortex lattice has created an inhomogeneous field distribution at the muon sites. The internal fields felt by the muons show a diamagnetic shift in the superconducting state of LaPt₃P, a clear signature of bulk superconductivity in this material. The decrease in the internal fields with decreasing temperature below T_c is an indication of a singlet superconducting ground state.

Fig. 2

Superconducting properties of LaPt₃P by TF-μSR measurements.

TF-μSR time spectra of LaPt₃P collected at a 1.3 K and b 70 mK for sample-A in a transverse field of 10 mT. The solid lines are the fits to the data using Eq. (2). c The temperature dependence of the extracted σ (left panel) and internal field (right panel) of sample-A. The error bars show the standard deviations in the TF-μSR measurements.

The true contribution of the vortex lattice field distribution to the relaxation rate σ_sc can be estimated as $$ {\sigma }_{\mathrm{sc}}={\left({\sigma }^2-{\sigma }_{\mathrm{nm}}^2\right)}^{1/2}$$, where σ_nm = 0.1459(4) μs⁻¹ is the nuclear magnetic dipolar contribution assumed to be temperature independent and was determined from the high-temperature fits. Within the Ginzburg-Landau theory of the vortex state, σ_sc is related to the London penetration depth λ of a SC with high upper critical field by the Brandt equation²⁰:

where Φ₀ = 2.068 × 10⁻¹⁵ Wb is the flux quantum. The superfluid density ρ ∝ λ⁻². Figure 3 shows the temperature dependence of ρ normalized by its zero-temperature value ρ₀ for LaPt₃P. It clearly varies with temperature down to the lowest temperature 70 mK and shows a linear increase below T_c/3. This non-constant low temperature behaviour is a signature of nodes in the superconducting gap.

Fig. 3

Evidence of chiral d-wave superconductivity in LaPt₃P.

Superfluid density (ρ) of LaPt₃P as a function of temperature normalized by its zero-temperature value ρ₀. The solid lines are fits to the data using different models of gap symmetry. Inset shows the schematic representation of the nodes of the chiral d-wave state. The error bars show the standard deviations in the TF-μSR measurements in the respective instruments.

The pairing symmetry of LaPt₃P can be understood by analysing the superfluid density data using different models of the gap function Δ_k(T). For a given pairing model, we compute the superfluid density (ρ) as

Here, $$ f=1/\left(1+e^{\frac{E}{k_{B}T}}\right)$$ is the Fermi function and 〈〉_FS represents an average over the Fermi surface (assumed to be spherical). We take Δ_k(T) = Δ_m(T)g(k) where we assume a universal temperature dependence $$ {\mathrm{\Delta }}_m\left(T\right)={\mathrm{\Delta }}_m\left(0\right)\tanh \left[1.82{\left\{1.018\left(T_{\mathrm{c}}/T-1\right)\right\}}^{0.51}\right]$$²¹ and the function g(k) contains its angular dependence. We use three different pairing models: s-wave (single uniform superconducting gap), p-wave (two point nodes at the two poles) and chiral d-wave (two point nodes at the two poles and a line node at the equator as shown in the inset of Fig. 3). The fitting parameters are given in the Supplementary Table 2. We note from Fig. 3 that both the s-wave and the p-wave models lead to saturation in ρ at low temperatures, which is clearly not the case for LaPt₃P and the chiral d-wave model gives an excellent fit down to the lowest temperature. Nodal SCs are rare since the SC can gain condensation energy by eliminating nodes in the gap. Thus the simultaneous observation of nodal and TRS-breaking superconductivity makes LaPt₃P a unique material.

Discussion

We investigate the normal state properties of LaPt₃P by a detailed band structure calculation using density functional theory within the generalized gradient approximation consistent with previous studies^15,22. LaPt₃P is centrosymmetric with a paramagnetic normal state respecting TRS. It has significant effects of spin-orbit coupling (SOC) induced band splitting near the Fermi level (~120 meV, most apparent along the MX high symmetry direction, see Supplementary Note 4). Kramer’s degeneracy survives in the presence of strong SOC due to centrosymmetry and SOC only produces small deformations in the Fermi surfaces²³. The shapes of the Fermi surfaces play an important role in determining the thermodynamic properties of the material. The projections of the four Fermi surfaces of LaPt₃P on the y−z and x−y plane are shown in Fig. 4a and Fig. 4b, respectively, with the Fermi surface sheets having the most projected-DOS at the Fermi level shown in blue and orange. It shows the multi-band nature of LaPt₃P with orbital contributions mostly coming from the 5d orbitals of Pt and the 3p orbitals of P.

Fig. 4

Properties of the normal and superconducting states of LaPt₃P.

Projections of the four Fermi surfaces of LaPt₃P with SOC on the y−z plane in a and x−y plane in b. The thickness of the lines are proportional to the contribution of the Fermi surfaces to the DOS at the Fermi level (green—10.3%, blue—43.4%, orange—40% and magenta—6.3%). The point nodes of the chiral d-wave gap are shown by red dots in a and the line node resides on the x−y plane in b. c Schematic view of the Majorana Fermi arc and the zero-energy Majorana flat band corresponding to the two Weyl point nodes and the line 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 shows its magnitude $$ =\frac{2}{\pi }\arctan \left(\mid \mathbf{F}\left(\mathbf{k}\right)\mid \right)$$. Δ₀ = 0.5 μ was chosen for clarity while a more realistic weak-coupling limit Δ₀ ≪ μ gives a more sharply peaked curvature at the Fermi surface.

LaPt₃P has a non-symmorphic space group P4/mmm (No. 129) with point group D_4h. From the group theoretical classification of the SC order parameters within the Ginzburg-Landau theory^19,24, the only possible superconducting instabilities with strong SOC, which can break TRS spontaneously at T_c correspond to the two 2D irreducible representations, E_g and E_u, of D_4h. Non-symmorphic symmetries can give rise to additional symmetry-required nodes on the Brillouin zone boundaries along the high symmetry directions. The non-symmorphic symmetries of LaPt₃P, however, can only generate additional point nodes for the E_g order parameter but no additional nodes for the E_u case²⁵. The superconducting ground state in the E_g channel is a pseudospin chiral d-wave singlet state with gap function Δ(k) = Δ₀ k_z(k_x + ik_y) where Δ₀ is a complex  amplitude independent of k. The E_u order parameter is a pseudospin nonunitary chiral p-wave triplet state with d-vector $$ \mathbf{d}\left(\mathbf{k}\right)=\left[c_1{k}_z,i{c}_1{k}_z,c_2\left(k_x+i{k}_y\right)\right]$$ where c₁ and c₂ are material dependent real constants independent of k.

We compute the quasi-particle excitation spectrum for the two TRS-breaking states on a generic single-band spherical Fermi surface using the Bogoliubov-de Gennes mean-field theory^19,24. The chiral d-wave singlet state leads to an energy gap given by $$ \mid {\mathrm{\Delta }}_0\mid \mid k_z\mid {\left(k_x^2+k_y^2\right)}^{1/2}$$. 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. 4a). The low temperature thermodynamic properties are, however, dominated by the line node because of its larger low energy DOS than the point nodes. The triplet state has an energy gap given by $$ {\left[g\left(k_x,k_y\right)+2c_1^2{k}_z^2-2\mid c_1\mid \mid k_z\mid {\left\{f\left(k_x,k_y\right)+c_1^2{k}_z^2\right\}}^{1/2}\right]}^{1/2}$$ where $$ f\left(k_x,k_y\right)=c_2^2\left(k_x^2+k_y^2\right)$$. It has only two point nodes at the two poles and no line nodes. Thus, the low temperature linear behaviour of the superfluid density of LaPt₃P shown in Fig. 3 is only possible in the chiral d-wave state with a line node in contrast to the triplet state with only point nodes, which will give a quadratic behaviour and saturation at low temperatures.

The preceding discussion assuming a generic Fermi surface can be adapted for the case of the inherently multi-band material LaPt₃P by considering the momentum dependence of the gap on the Fermi surfaces sheets neglecting interband pairing. We note from Fig. 4a and Fig. 4b that there are two important Fermi surface sheets in LaPt₃P, with the chiral d-wave state having the two point nodes on one of the Fermi surface sheets and a line node on the other. Thus LaPt₃P is one of the rare unconventional SCs for which we can unambiguously identify the superconducting order parameter.

The severe constraints on the possible pairing states as a result of the unique properties of LaPt₃P lead us to expect that our experimental observations will be consistent only with a chiral d-wave like order parameter belonging to the E_g channel even after considering pairing between bands in a multi-orbital picture¹⁰. It is also intriguing to think about the possible pairing mechanism giving rise to the chiral d-wave state in this material, which has a weakly correlated normal state, weak electron–phonon coupling and no spin fluctuations^15,16. These issues will be taken up in future investigations.

The topological properties of the chiral d-wave state of LaPt₃P are most naturally discussed considering a generic single-band spherical Fermi surface (chemical potential $$ \mu =k_F^2/\left(2m\right)$$ where k_F is the Fermi wave vector and m is the electron mass)^4,26. However, topological protection of the nodes²⁷ also ensures stability against multi-band effects assuming interband pairing strengths to be small. The effective angular momentum of the Cooper pairs is L_z = +1 (in units of ℏ) with respect to the chiral c-axis. The equatorial line node acts as a vortex loop in momentum space²⁸ and is topologically protected by a 1D winding number w(k_x, k_y) = 1 for $$ k_x^2+k_y^2<k_F^2$$ and = 0 otherwise. The non-trivial topology of the line node leads to two-fold degenerate zero-energy Majorana bound states in a flat band on the (0, 0, 1) surface BZ as shown in Fig. 4c. As a result, there is a diverging zero-energy DOS leading to a zero-bias conductance peak (which can be really sharp²⁹) measurable in STM. This inversion symmetry protected line node is extra stable due to even parity SC^29,30. The point nodes on the other hand are Weyl nodes and are impossible to gap out by symmetry-preserving perturbations. They act as a monopole and an anti-monopole of Berry flux as shown in Fig. 4d and are 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 ≠ 0 and = 0 otherwise (see Supplementary Note 6 for details). As a result, the (1, 0, 0) and (0, 1, 0) surface BZs each have a Majorana Fermi arc, which can be probed by STM as shown in Fig. 4c. There are two-fold degenerate chiral surface states with linear dispersion carrying surface currents leading to local magnetisation that can be detected using SQUID magnetometry. One of the key signatures of chiral edge states is the anomalous thermal Hall effect (ATHE), which depends on the length of the Fermi arc in this case. Impurities in the bulk can, however, increase the ATHE signal by orders of magnitude³¹ over the edge contribution making it possible to detect with current experimental technology³². We also note that a 90° rotation around the c-axis for the chiral d-wave state leads to a phase shift of π/2, which can be measured by corner Josephson junctions³³.

Methods