Discovery of nodal-line superconductivity in chiral crystals

TL;DR

This paper reports the discovery of topological nodal-line superconductivity in chiral LaIrSi crystals, revealing a new mechanism driven by isotropic spin-orbit coupling that broadens understanding of unconventional superconductivity in chiral materials.

Contribution

It introduces the first observation of nodal-line superconductivity in a chiral crystal and demonstrates a novel isotropic SOC mechanism distinct from previous models.

Findings

LaIrSi exhibits topological nodal-line superconductivity.

Substituting Rh with Ir enhances SOC and induces unconventional SC.

A new phase diagram for chiral superconductor engineering is proposed.

Abstract

Chiral crystals, whose key feature is the structural handedness, host exotic quantum phenomena driven by the interplay of band topology, spin-orbit coupling (SOC), and electronic correlations. Due to the limited availability of suitable chiral-crystal materials, their unconventional superconductivity (SC) remains largely unexplored. Here, we report the discovery of unconventional SC in the La(Rh,Ir)Si family of materials by combining muon-spin spectroscopy, band-structure calculations, and perturbation theory. This family, characterized by a double-helix chiral structure, hosts exotic multifold fermions that are absent in other topological chiral crystals. While LaRhSi behaves as a fully-gapped superconductor, the substitution of 4$d$-Rh by 5$d$-Ir significantly enhances the SOC and leads to the emergence of topological nodal-line SC in LaIrSi. The developed model shows that the nodal-line SC arises from an isotropic SOC with a specific strength. Such an exotic mechanism expands our conventional understanding of material candidates for unconventional SC, which typically rely on a significantly anisotropic SOC to promote the triplet pairing. Our work establishes a new type of phase diagram, which provides a comprehensive roadmap for identifying and engineering unconventional SC in chiral crystals. Furthermore, it calls for renewed investigations of unconventional SC in other widely studied superconductors with a chiral structure.

Equations

$$\begin{array}{rl}{\mathcal{H}}_{\text{perp}}(\mathbf{k})& =E_0(\mathbf{k})+(A_{1}k+A_3|k_x{k}_y{k}_z|/k)s_0\otimes {\tau }_z\\ & +{\lambda }_0\left({\lambda }_x{s}_x+{\lambda }_y{s}_y+{\lambda }_z{s}_z\right)\otimes {\tau }_0,\end{array}$$

$$E_{\text{BdG}}^{\alpha ,\beta }(\mathbf{k})=\pm \sqrt{(E_{\alpha ,\beta }(\mathbf{k})-E_f{)}^2+({\Delta }_s-\beta {\Delta }_{t}k{)}^2}.$$

$$\{\begin{array}{l}{E}_{\alpha ,+}(\mathbf{k})=E_f,\\ {\Delta }_s-{\Delta }_{t}k=0.\end{array}$$

$$\{\begin{array}{l}\alpha |k_x{k}_y{k}_z|=k(\widetilde{E_f}-\alpha {\tilde{A}}_{1}k+{\tilde{A}}_2{k}^2),\\ k=\frac{{\Delta }_s}{{\Delta }_t},\end{array}$$

$$x_{\min }^{+}\le \frac{{\Delta }_s}{{\Delta }_t}\le x_{\max }^{+},$$

$$x_{\min }^{+}$$

$$x_{\max }^{+}$$

$$x_{\min }^{-}\le \frac{{\Delta }_s}{{\Delta }_t}\le x_{\max }^{-},$$

$$x_{\min }^{-}$$

$$x_{\max }^{-}$$

$$E_0+\alpha (A_{1}k+A_3|k_x{k}_y{k}_z|/k)+|{\lambda }_0|=E_f,$$

$$k(\theta ,\varphi )=-\frac{2E_f^0}{-\alpha A_1+\sqrt{A_1^2+4A_2(\theta ,\varphi )E_f^0}},$$

$$\Delta (\theta ,\varphi )={\Delta }_s-{\tilde{\Delta }}_t\frac{k_{\text{min}}}{k(\theta ,\varphi )},$$

$$k_{\text{min}}=-\frac{2E_f^0}{-A_1+\sqrt{A_1^2+4A_2{E}_f^0}}.$$

$$\Delta (\theta ,\varphi )={\Delta }_s-{\tilde{\Delta }}_t\frac{A_1-\sqrt{A_1^2+4A_2(\theta ,\varphi )E_f^0}}{A_1-\sqrt{A_1^2+4A_2{E}_f^0}},$$

$$k_{\text{min}}=-\frac{2E_f^0}{A_1+\sqrt{A_1^2+4(A_2-\frac{A_3}{3\sqrt{3}})E_f^0}},$$

$$\Delta (\theta ,\varphi )={\Delta }_s-{\tilde{\Delta }}_t\frac{A_1+\sqrt{A_1^2+4A_2(\theta ,\varphi )E_f^0}}{A_1+\sqrt{A_1^2+4(A_2-\frac{A_3}{3\sqrt{3}})E_f^0}},$$

Full text

Tian Shang* Jianzhou Zhao* Lun-Hui Hu* Weikang Wu Keqi Xia Mukkattu O. Ajeesh Michael Nicklas Yang Xu Qingfeng Zhan Dariusz J. Gawryluk Ming Shi Toni Shiroka

{affiliations}

Prof. T. Shang

Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China

Chongqing Key Laboratory of Precision Optics, Chongqing Institute of East China Normal University, Chongqing 401120, China

Email Address: tshang@phy.ecnu.edu.cn

Prof. J. Zhao

Department of Physics, School of Science, Tianjin University, Tianjin 300354, China

Co-Innovation Center for New Energetic Materials, Southwest University of Science and Technology, Mianyang 621010, China

Email Address: jzzhao@swust.edu.cn

Prof. L. Hu, Prof. M. Shi

Center for Correlated Matter and School of Physics, Zhejiang University, Hangzhou 310058, China

Email Address: lunhui@zju.edu.cn

Prof. W. Wu

Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials (MOE), Shandong University, Jinan 250061, China

K. Xia, Prof. Y. Xu, Prof. Q. Zhan

Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China

Dr. M. O. Ajeesh, Dr. M. Nicklas

Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187 Dresden, Germany

Dr. D. J. Gawryluk

PSI Center for Neutron and Muon Sciences CNM, 5232 Villigen PSI, Switzerland

Dr. T. Shiroka

PSI Center for Neutron and Muon Sciences CNM, 5232 Villigen PSI, Switzerland

Laboratorium für Festkörperphysik, ETH Zürich, CH-8093 Zürich, Switzerland

1 Introduction

Chirality, a quality of asymmetry resulting from the absence of inversion, mirror or roto-inversion symmetries, is one of the pivotal concepts in various fields, including physics, chemistry, and biology [1, 2, 3, 4, 5, 6]. In solid-state physics, chirality usually refers to two main categories: static and dynamic. Static chirality pertains to the geometric arrangement of atoms in a crystal lattice, specifically structural chirality, while dynamic chirality is related to the spin or electronic properties [7, 8, 9, 10, 11, 12]. Electronic chirality is usually associated with the spin-momentum locking of particles or quasiparticles near degenerate points in momentum space (e.g., Weyl fermions) [13, 14, 15, 16, 17]. In contrast, spin (or magnetic) chirality characterizes the topological spin textures or domain walls in real space, such as magnetic skyrmions [18, 19, 20, 21, 22, 23, 24, 25]. The combination of structural and electronic chirality triggers the emergence of topological chiral crystals, which exhibit a plethora of exotic quantum phenomena. These include fermionic excitations characterized by large Chern numbers ($\lvert C\rvert>$ 1, with no analogues in high-energy physics), unique spin textures (e.g., parallel spin-momentum locking), giant helicoid Fermi arcs, a large topologically nontrivial energy window, chiral magnetic effects, and quantized circular photogalvanic effect [11, 8, 9, 2, 3, 26, 27, 28, 29, 30]. These chirality-induced effects promise new directions in fundamental science and new technological applications in spintronics, optoelectronics, and biochemistry.

Typical examples of topological chiral crystals are the cubic B20 materials with a noncentrosymmetric space group $P2_{1}3$ (No. 198). Among these materials, the most prominent ones are the transition-metal monosilicides or monoaluminides T$$X, where $T$ represents a transition metal and $X$ denotes Si, Ge, Sn, Al, and Ga. In particular, both CoSi and RhSi host a threefold spin-1 Weyl fermion with a topological charge $C$ = 2 at the center of the Brillouin zone (BZ) ($\Gamma$ point) and a fourfold charge-2 Weyl fermion with an opposite topological charge at the zone boundary ($R$ point) [31, 32, 33, 34]. Recent photoemission spectroscopy studies have confirmed the presence of unconventional multifold chiral fermions close to the Fermi energy, along with their associated extremely long helicoid Fermi arcs traversing the entire surface BZ [31, 32, 33, 35, 36, 37, 38, 34]. Moreover, spin-orbit coupling (SOC) has been shown to induce a spin splitting of the topological band degeneracies, effectively doubling the topological charge from $\lvert C\rvert=2$ to 4, as experimentally verified in PtAl, PtGa and PdGa [35, 36, 37, 38, 39]. Additionally, these chiral crystals serve as model systems for exploring topological bosonic excitations [40, 41].

Despite significant advancements in understanding their topological aspects, the role of many-body interactions in topological chiral crystals, and how their band topology affects the electronic correlations, remains largely unexplored and attracts increased attention. Notably, phenomena such as charge order and superconductivity (SC) have been reported in a few chiral crystals. The incommensurate charge-density-wave (CDW) order formed in the topological boundary states of the CoSi (001) surface is a striking example of chirality’s impact on the electronic states [42, 43], which, however, is missing on the other surfaces or in the bulk. The orientation of the CDW phase can be tuned by the chirality of the Fermi arcs, i.e., by the structural chirality [42, 43], and is closely related to the van Hove singularities [34], the latter possibly inducing SC as well [3, 8, 44]. So far, a limited number of chiral crystals has been found to exhibit SC [45, 46, 47, 48, 49], including some B20 chiral crystals, such as AuBe [50, 51], RhGe [52], and BiPdSe [53]. While most of them behave as conventional superconductors [54], only Li2Pt3B is known to exhibit spin-triplet pairing [47, 48], which also shows a gapless topological SC with Majorana surface modes [55]. It has recently been predicted that RhGe and PdBiSe host mixed-parity pairing and topological superconductivity, involving chiral fermions at time-reversal invariant momenta (TRIM) [44, 56]. In addition, topological SC arising from a $s_{+}\oplus s_{-}$ pairing has been theoretically investigated in RhSi [57]. Unfortunately, the superconducting nature of these chiral crystals has not yet been explored due to the absence of SC, the difficulty in synthesizing a material with bulk SC [52, 58, 53], and their extremely low $T_{c}$s ($<$ 0.2 K) [59, 60].

Despite previous studies of SC in chiral crystals, the chirality-driven mechanisms behind these novel superconducting states and the effect of SOC on the superconducting pairing have not been explored. Here, we report the discovery of unconventional SC in cubic La(Rh,Ir)Si chiral crystals and reveal the pivotal role of SOC in the superconducting pairing. Similar to the CoSi family of chiral crystals, there is also a large number of unconventional multifold fermions in La(Rh,Ir)Si, which significantly affect the band topology and superconducting states. Remarkably, while LaRhSi behaves as a fully-gapped superconductor, LaIrSi shows nodal-line SC. Contrary to the B20 transition-metal monosilicides or monoaluminides, the incorporation of La atoms introduces an additional chiral chain in La(Rh,Ir)Si, featuring a unique double-helix chiral structure. Density-functional-theory (DFT) calculations reveal that La(Rh,Ir)Si harbor more electronic bands near the Fermi energy than the CoSi family, pointing to a chirality-driven mechanism in the emergence of SC. Moreover, the substitution of 4$d$-Rh with 5$d$-Ir atoms enhances significantly the strength of SOC, culminating in a topological nodal-line superconducting state in LaIrSi. To understand how SC evolves from nodal to nodeless as the SOC is reduced, we develop a phenomenological theory which reveals that nodal-line SC is stabilized by an isotropic SOC with a well-defined strength, as well as a spin-independent $C_{3}$-wrapping term. Such an exotic mechanism for establishing triplet pairing goes beyond the conventional ones. The latter typically rely on either a significantly anisotropic SOC (in superconductors without an inversion center), or strong spin fluctuations (in magnetic superconductors). The nodal-line SC in LaIrSi is also different from that in cuprates, the latter hosting $d$-wave spin-singlet pairs. Our theoretical model suggests that a significant SOC is not required in order to induce a nodal SC. Consequently, the new mechanism can bypass the restrictions normally imposed on the materials expected to exhibit unconventional SC. The nodal-line phase diagram we propose here can serve as a preliminary guide in the search for unconventional SC in other chiral crystals.

2 Results and Discussion

Double-helix chiral structure. A detailed knowledge of the crystal structure is crucial because the symmetry principles inherent in it dictate the related phases within the framework of Landau theory [61, 62]. As depicted in Figure Discovery of nodal-line superconductivity in chiral crystalsa, a chiral crystal surface is characterized by an arrangement of atoms that does not coincide with its mirror image when reflected perpendicular to the surface plane. The binary AuBe alloy is a noncentrosymmetric superconductor with a B20 chiral structure [50], sharing the same space group $P2_{1}3$ (No. 198) with the CoSi family and La(Rh,Ir)Si, the latter being the focus of the current work. While AuBe is widely regarded as a conventional superconductor, with a dominant $s$-wave singlet pairing in the weak-coupling limit [51, 63], we uncover unconventional SC in La(Rh,Ir)Si. Such a striking difference between AuBe and La(Rh,Ir)Si hints at an intimate interplay between the crystal structure and the electronic correlations in materials with a chiral structure. Here, their cubic crystal structure contains a twofold screw axis and a threefold rotation axis. The CoSi family and AuBe exhibit a pair of sublattice distortions along the [111]-direction (i.e., the chiral axis). In the case of La(Rh,Ir)Si, the structure consists of two sublattices. The (Rh,Ir)Si sublattice resembles that of AuBe and CoSi family (see details in Table S1 in the Supporting Information), while the La sublattice forms a separate chiral chain. To elucidate this, both structures of the (Rh,Ir)Si sublattice and La(Rh,Ir)Si are depicted in Figure Discovery of nodal-line superconductivity in chiral crystalsa. The Rh/Ir and the Si atoms exhibit a clockwise arrangement when viewed along the [111]-direction, which presents a distinct structural chirality. Conversely, the La sublattice exhibits a chiral chain with an anticlockwise arrangement. Generally, the chirality can be distinguished by the handedness of the helix, either clockwise or anticlockwise, depending on the enantiomer. Considering that the (Rh,Ir)Si and La sublattices form chiral chains, but with opposite chirality, the La(Rh,Ir)Si family of materials can be classified as double-helix chiral crystals. Note that, to date, no SC has been found in the isostructural RhSi chiral crystal. However, due to the presence of two chiral sublattices, a double-helix chiral crystal may host a greater abundance of electronic states. This can account for the appearance of unconventional superconducting phenomena in La(Rh,Ir)Si.

Topological electronic band structure. We first performed band-structure calculations via density functional theory to investigate the impact of the aforementioned double-helix chiral structure on the electronic properties of La(Rh,Ir)Si. The density of states (DOS) close to the Fermi level ($E_{f}$) is dominated by the La-$5d$, Rh-$4d$ (or Ir-$5d$), and Si-$3p$ orbitals (see Figure S1, Supporting Information). In both LaRhSi and LaIrSi, the $d$-orbital of Rh/Ir and La atoms contributes over 22% and 15% of the total DOS, respectively, highlighting the important role of the La chiral chain. The electronic band structures of the RhSi sublattice, LaRhSi, and LaIrSi are shown in Figure Discovery of nodal-line superconductivity in chiral crystalsb-d. Similar to the CoSi family of chiral crystals, the electronic bands near the Fermi energy are dominated by the $d$ orbitals in the RhSi or IrSi sublattice (see band structures of B20 RhSi and IrSi chiral crystals in Figures S2 and S3 in the Supporting Information). Interestingly, despite sharing the same crystal structure, near the Fermi energy the band structures of LaRhSi and LaIrSi are distinct from that of the CoSi chiral family. This confirms the key role of the La chiral chain in the appearance of SC in these double-helix chiral materials. For example, without considering SOC, RhSi or IrSi host a threefold band crossing at the $\Gamma$ point, while a twofold and a threefold band crossing can be identified in LaRhSi or LaIrSi near the Fermi energy. By comparing the band structures of LaRhSi and LaIrSi — in the presence and absence of SOC — it turns out that SOC lifts the band degeneracies, except along the $X$–$M$ direction (Figure Discovery of nodal-line superconductivity in chiral crystalsc,d and Figures. S2 and S3, Supporting Information). The band splitting $E_{\mathrm{SOC}}$ caused by the SOC at the high-symmetry $R$ point is about 207 and 37 meV for LaIrSi and LaRhSi, respectively. Such a significant SOC effect in LaIrSi is mainly attributed to the 5$d$ orbitals of the La and Ir atoms near $E_{f}$. Compared to most other noncentrosymmetric superconductors, the $E_{\mathrm{SOC}}$ of LaIrSi is much higher [54]. Up to 10 bands are identified to cross the Fermi level of both materials, confirming their multiband character.

Due to their chiral structure, La(Rh,Ir)Si, similar to the CoSi family of materials [31, 32, 33, 35, 36, 37, 38, 34, 39], exhibit a rich set of topological features. These give rise to multifold Weyl fermions at high-symmetry momenta and spin-polarized Fermi arc states spanning the entire surface Brillouin zone (Figure Discovery of nodal-line superconductivity in chiral crystalse-g). For instance, at the zone boundary ($R$ point), a double Weyl fermion arises in the absence of SOC. When SOC is included, this excitation splits into a spin-1 state and a single Weyl point. Despite sharing similar symmetry constraints with the CoSi family, La(Rh,Ir)Si display distinct topological properties. In the absence of SOC, the band structure at the $\Gamma$ point hosts two kinds of topological excitations near the Fermi level, a twofold degeneracy with a topological charge of $C=+4$ and a threefold degeneracy with a topological charge of $C=-2$ (see Figures. S2 and S3, Supporting Information). Introducing SOC lifts the twofold degeneracy at the $\Gamma$ point and transforms it into a fourfold degeneracy with a topological charge $C=+2$, a hallmark topological feature absent in the CoSi family. Such distinct topological features arise from the additional electronic bands peculiar to the double-helix chiral structure of La(Rh,Ir)Si. The comparison of the band structures of La(Rh,Ir)Si with those of B20 (Rh,Ir)Si chiral crystals is presented in Figures. S2 and S3 in the Supporting Information.

Moreover, since the sum of topological charges in the Brillouin zone must equal zero, we identify a compensating topological charge around the $R$ point. This leads to a unique bulk-boundary correspondence, where the topological surface states form a continuous connection between the $\Gamma$ and $R$ points (or surface-projected $\bar{M}$ point). DFT calculations in La(Rh,Ir)Si reveal spin-polarized helicoid Fermi arcs spanning their entire surface Brillouin zone (Figure. Discovery of nodal-line superconductivity in chiral crystalsf,g). Since in this case the multifold fermions at the high-symmetry momentum points are very close to the Fermi level (i.e., $\sim 29$ meV below $E_{f}$ at the $\Gamma$ point), La(Rh,Ir)Si represent an ideal system for investigating the relationship among the chirality-induced band topology, SOC, and unconventional SC, as we discuss in detail below.

Characterization of superconductivity. Although both LaRhSi and LaIrSi compounds were known to exhibit SC a few decades ago [64, 65], and the superconducting transition temperature $T_{c}$ was reported, their superconducting properties remain largely unexplored. We synthesized LaRhSi and LaIrSi using the arc-melting method [64] and characterized their SC via magnetic susceptibility and electrical resistivity measurements. Both compounds crystallize in a cubic chiral structure with double helices (Figure. Discovery of nodal-line superconductivity in chiral crystalsa), confirmed by powder x-ray diffraction (XRD) refinements (Figure Discovery of nodal-line superconductivity in chiral crystalsa). Both LaRhSi and LaIrSi show metallic behavior in the studied temperature range (Figure S4, Supporting Information). The electrical resistivity in Figure Discovery of nodal-line superconductivity in chiral crystalsb drops to zero at $T_{c}=4.4$ K and 2.1 K for LaRhSi and LaIrSi, respectively, where the magnetic susceptibilities in Figure Discovery of nodal-line superconductivity in chiral crystalsc also show a clear diamagnetic response. The well-separated ZFC- and FC-curves indicate a type-II SC, consistent with the field-dependent magnetization shown in Figure Discovery of nodal-line superconductivity in chiral crystalsd. The upper critical fields determined from the electrical resistivity measured under various magnetic fields (Figure Discovery of nodal-line superconductivity in chiral crystalse,f) are \mu_{0}$$H_{\mathrm{c2}}(0) = 2.42(1) and 2.35(6) T for LaRhSi and LaIrSi, respectively (see details in Figure S6 and Note S1, Supporting Information). In addition, the positive curvature in the $H_{\mathrm{c2}}(T)$ data of both materials is most likely attributed to the multiband nature of their SC (Figure S6, Supporting Information). While this is consistent with the electronic band structures shown in Figure Discovery of nodal-line superconductivity in chiral crystalsc,d, it requires further investigation.

µSR and superconducting pairing. To investigate the superconducting nature of LaRhSi and LaIrSi, we first performed zero-field (ZF)-µSR to establish if time-reversal symmetry (TRS) is spontaneously broken in the superconducting state. ZF-µSR represents one of the most highly sensitive techniques for detecting weak magnetic fields (down to $\sim$ 0.01 mT) owing to the large muon gyromagnetic ratio (851.615 MHz/T) and the availability of nearly 100% spin-polarized muon beams [17, 66, 67, 68]. An increase in the zero-field muon-spin relaxation rate below the onset of SC provides direct evidence for a superconducting state with broken TRS [69, 70]. In both LaRhSi and LaIrSi, the nearly overlapping ZF-µSR spectra indicate the absence of a spontaneous field below $T_{c}$, confirming that TRS is preserved in the superconducting state of both compounds (Figure Discovery of nodal-line superconductivity in chiral crystalsa,b). Indeed, the derived muon-spin relaxation rates are almost identical in the superconducting and normal states, with their differences falling within the standard deviations (see Note S2, Supporting Information).

Next, we examine the SC pairing symmetries of LaRhSi and LaIrSi by detecting the temperature-dependent magnetic penetration depth via the transverse-field (TF) µSR technique. Since the superfluid density $\rho_{\mathrm{sc}}$ is proportional to the inverse square of the magnetic penetration depth (i.e., $\rho_{\mathrm{sc}}$ $\propto\lambda_{\mathrm{eff}}^{-2}$), the measured temperature-dependent $\lambda_{\mathrm{eff}}(T)$ reveals the nature of the superconducting pairing (see details in Notes S2 and S3, Supporting Information) [17, 66, 67, 68]. Due to the formation of the flux-line lattice (FLL), the TF-µSR spectra collected in the superconducting state of both LaRhSi and LaIrSi (Figure Discovery of nodal-line superconductivity in chiral crystalsc,d) show enhanced muon-spin relaxation rates, determined by the magnetic penetration depth. The inverse-square of the effective magnetic penetration depth $\lambda_{\mathrm{eff}}^{-2}(T)$ vs. the reduced temperature $T/T_{c}$ for LaRhSi and LaIrSi is shown in Figure Discovery of nodal-line superconductivity in chiral crystalse,f. The two $\lambda_{\mathrm{eff}}^{-2}(T)$ evolutions are clearly distinct. In LaRhSi, the temperature-invariant superfluid density at low temperatures suggests a fully-gapped superconducting state. By contrast, the strongly temperature-dependent superfluid density hints at the presence of low-energy excitations and thus, of gap nodes in LaIrSi. In this case, $\lambda_{\mathrm{eff}}(T)$ shows a subquadratic $T^{n}$ ($n\sim 1.0$–1.5) rather than a quadratic temperature dependence (see Figure S7 and Note S3, Supporting Information), which is consistent with the presence of line nodes. As shown by solid lines in Figure Discovery of nodal-line superconductivity in chiral crystalse,f, the $\rho_{\mathrm{sc}}(T)$ is well described by the proposed theoretical model involving mixed singlet-triplet pairing (see details below). While the singlet channel is dominant in LaRhSi, in LaIrSi, due to the significantly enhanced SOC (see Figure Discovery of nodal-line superconductivity in chiral crystals and Figures. S2 and S3, Supporting Information), the triplet component is larger than the singlet one, leading to the superconducting gap nodes.

Theory of nodal superconductivity. In chiral crystals, the coexistence of SC with chirality-induced multifold fermions can stabilize unconventional pairing symmetries, as exemplified by the nodal SC observed in LaIrSi. Our findings reveal that LaIrSi exhibits nodal SC with significantly stronger SOC than LaRhSi (see band-structure calculations in Figure Discovery of nodal-line superconductivity in chiral crystals and Figures. S2 and S3, Supporting Information), which instead remains fully gapped. This comparison highlights the crucial role of SOC in driving nodal SC in chiral crystals (see details also in Note S4, Supporting Information). In general, an enhanced SOC increases the degree of mixing between spin-singlet ($\Delta_{s}$) and spin-triplet ($\Delta_{t}$) components, a hallmark of noncentrosymmetric superconductors where broken inversion symmetry permits such a mixing [71, 54]. This mechanism has been invoked to explain nodal SC in systems like Li2Pt3B [47, 48], CePt3Si [72], and CaPtAs [73]. It is important to note that previous approaches are based on a topologically trivial Hamiltonian with isotropic Fermi surfaces, while the spin-triplet component exhibits significant anisotropy [74]. However, in a cubic lattice (e.g., LaIrSi), the leading-order spin-triplet pairing is constrained by the cubic symmetry to adopt an isotropic form, with a $d$-vector given by $d_{\boldsymbol{k}}\propto(k_{x},k_{y},k_{z})$.

Here, for the LaIrSi case, we consider both the topological Hamiltonian and the isotropic spin-triplet pairing. We focus on the four Fermi surfaces, which are relatively large and contribute most to the density of states, the latter being highly relevant to SC. An effective perturbation model around the $R$ point (Figure Discovery of nodal-line superconductivity in chiral crystals) was constructed, whose Hamiltonian is expressed as:

[TABLE]

where $E_{0}=A_{0}+A_{2}k^{2}$, $A_{3}$ represents the $C_{3}$-wrapping term, and $\lambda_{0}$ is the strength of SOC; $s$ and $\tau$ are the Pauli matrices of the spin and the projected orbital degrees of freedom, respectively; $(\lambda_{x},\lambda_{y},\lambda_{z})=\boldsymbol{k}/{k}$ with $\boldsymbol{k}=(k_{x},k_{y},k_{z})$. The energy dispersion are $E_{\alpha,\beta}=E_{0}+\alpha(A_{1}k+A_{3}|k_{x}k_{y}k_{z}|/k)+\beta|\lambda_{0}|$ with $\alpha,\beta=\pm$. Thus, the $A_{3}$ term implies that the Fermi surfaces are not isotropic but only symmetric under $C_{3}$. The band structure of the effective Hamiltonian, shown in blue lines, is presented in Figure Discovery of nodal-line superconductivity in chiral crystalsa. The three parameters $A_{0}$, $A_{1}$, and $A_{2}$ are extracted from the DFT calculations (red symbols in Figure Discovery of nodal-line superconductivity in chiral crystalsa). The obtained parameters are $A_{0}=0.536$ eV, $A_{1}=0.924~$ eV$\cdot$Å, $A_{2}$ = -$14.273~$ eV$\cdot$Å2, and $A_{3}=15~$ eV$\cdot$Å2

We then develop an effective Ginzburg-Landau theory for the stabilization of nodal-line SC in LaIrSi, which incorporates both the isotropic pairings ($\Delta_{s}$ and $\Delta_{t}$), SOC ($\lambda_{0}$), and the anisotropic wrapping term ($A_{3}$) (see details in Note S5 in the Supporting Information).With pairing potential $\Delta(\boldsymbol{k})=[\Delta_{s}+\Delta_{t}(\boldsymbol{k}\cdot\boldsymbol{s})]is_{y}\otimes\tau_{0}$, the superconducting quasi-particle spectrum reads

[TABLE]

Without loss of generality, here we consider the case with both $\Delta_{s}>0$ and $\Delta_{t}>0$. The superconducting gap nodes can only appear for the two $\beta=+$ bands, located at the intersection of the surfaces described by the equations $E_{\alpha,+}(\boldsymbol{k})=E_{f}$ and $\Delta_{s}-\Delta_{t}k=0$. After some algebra, the condition for nodal-line SC is obtained by imposing $x_{\text{min}}^{\pm}\leq\Delta_{s}/\Delta_{t}\leq x_{\text{max}}^{\pm}$ for the $E_{\pm,+}$ band, respectively (see details in Experimental Section and Note S6, Supporting Information). The evolution of $x_{\text{max}}^{-}$ and $x_{\text{min}}^{-}$ versus $\lambda_{0}$ and their difference versus $A_{3}$ for the $E_{-,+}$ band are presented in Figure Discovery of nodal-line superconductivity in chiral crystalsb and Figure Discovery of nodal-line superconductivity in chiral crystalsc, respectively. Both $x_{\text{max}}^{-}$ and $x_{\text{min}}^{-}$ increase with increasing SOC strength $\lambda_{0}$. Notably, the gray region that hosts the gap nodes in Figure Discovery of nodal-line superconductivity in chiral crystalsb, quantified by the difference between $x_{\text{max}}^{-}$ and $x_{\text{min}}^{-}$, appears to be nearly invariant with respect to $\lambda_{0}$. However, such a difference (i.e., $x_{\text{max}}^{-}$ – $x_{\text{min}}^{-}$) increases as the $C_{3}$-wrapping term $A_{3}$ increases (Figure Discovery of nodal-line superconductivity in chiral crystalsc). Such conclusions also apply to the $E_{+,+}$ band. Moreover, to better illustrate the position of the superconducting nodal lines on the Fermi surfaces, we project the superconducting gaps onto the $E_{\pm,+}$ bands, as depicted in Figure Discovery of nodal-line superconductivity in chiral crystalsd-f. For comparison, Figure Discovery of nodal-line superconductivity in chiral crystalsd shows the two Fermi surfaces corresponding to the $E_{\pm,+}$ bands. The gap on the $E_{+,+}$ band is nearly isotropic due to the dominant spin-singlet pairing (Figure Discovery of nodal-line superconductivity in chiral crystalse), while the gap on the $E_{-,+}$ band exhibits clear nodal lines (Figure Discovery of nodal-line superconductivity in chiral crystalsf). Note that, due to the chiral symmetry constraints, there is only one connected nodal line in each quadrant.

Next, we apply the above theoretical framework to the experimental results of the double-helix La(Rh,Ir)Si chiral materials. For convenience, we define $\tilde{\Delta}_{t}=\Delta_{t}k_{f}$. The temperature-dependent superfluid density $\rho_{\mathrm{sc}}(T)$ can be well fitted by our theoretical model (Figure Discovery of nodal-line superconductivity in chiral crystalse,f), where the dimensionless parameter $\Delta_{s}/\tilde{\Delta}_{t}\sim$ 1.5 and 0.9 were found to be in good agreement with the LaRhSi and LaIrSi data, respectively (see details in Experimental Section). The analysis of $\rho_{\mathrm{sc}}(T)$ using different $\Delta_{s}/\tilde{\Delta}_{t}$ ratios and other singlet- or triplet-pairing models are reported in Figures S8 and S9 (Supporting Information). Our theoretical model agrees well with the experimental observations. Thus, we conclude that the anisotropic Fermi surfaces with a nonzero $A_{3}$ term are crucial for achieving nodal-line SC in these cubic chiral crystals, in spite of the isotropic spin-singlet and spin-triplet pairings.

Discussion. We now extend our observations to other superconducting materials with a chiral space group $P2_{1}3$ (No. 198), thus providing insights for the discovery of new unconventional superconductors. The reported superconducting materials include binary compounds such as AuBe [50, 51], RhGe [44, 52], and ReSi [75], as well as ternary compounds such as PdBi(Se,Te) [56, 53, 76], BaPtP [59], and SbPtS [60]. Their electronic band structures and calculation details are presented in Figures S10–S18 and in Note S7 (Supporting Information). The superconducting transition temperatures $T_{c}$, together with the band-splitting $E_{\mathrm{SOC}}$ (due to the SOC at the high-symmetry points) are summarized in Table S3 (Supporting Information). According to our DFT calculations, multiple bands cross the Fermi level in all of these materials. Consequently, they can be classified as multiband superconductors. We have also demonstrated that, in the $\Delta_{s}/\Delta_{t}$ vs. $\lambda_{0}$ phase diagram, there are at least two regions where nodal-line SC can be stabilized (see colored regions in Figure Discovery of nodal-line superconductivity in chiral crystals). These findings may apply consistently also to other materials. First, LaIrSi and LaRhSi (indicated by star symbols) are two typical examples in the phase diagram, characterized by significantly different band splittings (207 vs. 37 meV) and located in a fully-gapped and a nodal-line region, respectively. This suggests the possibility of a crossover from nodal-line SC to fully-gapped SC in the LaIrxRh1-xSi family, if $\lambda_{0}$ can be systematically tuned by chemical substitution. Second, the binary compound AuBe, with $\lambda_{0}$ $\sim$ 145 meV, has been experimentally confirmed to be a fully-gapped superconductor [50, 51]. As such, it lies outside the nodal SC regions in the phase diagram. Third, ReSi, despite a large $\lambda_{0}$ $\sim$ 350 meV (comparable to LaIrSi), may also lie outside the nodal regions, presumably positioned in the lower right corner of the phase diagram. Finally, the binary RhGe has a $\lambda_{0}$ $\sim$ 84 meV, which is smaller than that of AuBe and ReSi. Surprisingly, RhGe appears to be located in the first nodal SC region and represents one of the most interesting cases to investigate.

Furthermore, the ternary compounds $TMX$ ($T$ = Co, Ni, Pd, Rh, Ir, and Pt, $M$ = P, As, Sb, and Bi, $X$ = S, Se, and Te) have the same crystal structure as La(Rh,Ir)Si [77]. Their $\lambda_{0}$s assume both small and large values, depending on the specific combination of elements. For instance, PdBiSe ($\lambda_{0}$ $\sim$ 193 meV) and PdBiTe ($\lambda_{0}$ $\sim$ 258 meV) become superconducting at $T_{c}$ = 1.5 and 2.1 K, respectively [53, 76]. In contrast, no physical properties have been reported for PtBiTe ($\lambda_{0}$ $\sim$ 318 meV), belonging to the same family and featuring a very large SOC (see Table S3, Supporting Information). Although the $T_{c}$s are relatively low [59, 60], SbPtS and BaPtP represent two extreme cases within this family, with $\lambda_{0}$ $\sim$ 72 and 321 meV, respectively. BaPtAs is another interesting case, with smaller $\lambda_{0}$ at the $R$ point compared to BaPtP, but similar $\lambda_{0}$ values at the $X$ and $\Gamma$ points. While the cubic LaIrSi-type phase of BaPtAs does not exhibit SC [78], its hexagonal form becomes superconducting below $\sim$3 K. Its sister compound, BaPtSb, shows the emergence of spontaneous fields, indicating a breaking of TRS in the SC state [79]. Except for AuBe and La(Rh,Ir)Si, the superconducting pairing symmetries of the aforementioned materials remain largely unexplored. It is noteworthy that, even in the fully-gapped regions of the phase diagram, unconventional SC can emerge when the SOC strength $\lambda_{0}$ is significant.

The phase diagram in Figure Discovery of nodal-line superconductivity in chiral crystals was constructed using the material-specific parameters of LaIrSi. Due to variations in the critical parameters, such as the $A_{3}$ term, which depends strongly on the symmetry of the crystal lattice and its electronic structure, an independent analysis is in high demand for other compounds. We emphasize that a combined theoretical and experimental effort, including spectroscopic and transport probes, is essential to fully resolve the unconventional superconducting phases across this chiral-crystal family. In conclusion, our results and the summary presented in Figure Discovery of nodal-line superconductivity in chiral crystals provide a comprehensive roadmap for identifying and exploring unconventional SC in chiral crystals.

Due to the structural chirality in real space, chiral crystals also exhibit chirality in momentum space. This leads to a series of exotic quantum phenomena [11, 8, 9, 2, 3, 26, 27, 28, 29, 30]. Yet, due to the limited availability of suitable chiral materials, their unconventional SC and its interplay with the topological electronic bands are barely explored. To date, there are a few theoretical works on the unconventional or topological SC in chiral crystals. The binary B20 CoSi family hosts multifold nodes split by the SOC at the same TRIM [31, 32, 33, 35, 36, 37, 38, 34, 39]. In RhGe, the superconducting pairing symmetries involving chiral Fermi pockets indicate their nontrivial characteristics. This makes RhGe a good candidate material for hosting unconventional and topological SC [44]. Unfortunately, no details of experimental studies have been reported in these B20 chiral crystals. In addition, the structural chirality may also help realize chiral SC too [80]. The La(Rh,Ir)Si family represents one of the valuable platforms for investigating the chirality-induced topological band topology and the interplay between weak correlations and geometric quantum effects. In this family, the correlated phases depend critically on both electron-electron interactions and the quantum geometry/band topology near the Fermi energy, where the chiral structure plays an important role. Different from the RhSi chiral crystal, the La(Rh,Ir)Si family has a double-helix chiral structure with the additional chiral sublattice of the La atoms. This sublattice contributes more electronic states around the Fermi surfaces with a large SOC and triggers the SC. Therefore, the La(Rh,Ir)Si family is one of the few material candidates that can be used to explore the interplay between structural chirality and unconventional SC — a field of study that combines topological band physics and correlated electron phenomena.

The topological properties of the CoSi family have been extensively studied. While sharing the same space group with the CoSi family of materials, La(Rh,Ir)Si exhibit a distinct topological band nature (Figure Discovery of nodal-line superconductivity in chiral crystals). Notably, the effect of SOC is significant in La(Rh,Ir)Si. In the CoSi family, the relatively weak SOC obscures the splitting between the spin-3/2 multifold and the Weyl node, making it challenging to resolve them via photoemission spectroscopy [31, 32, 33]. The pronounced SOC, on the other hand, causes a significant splitting in LaIrSi, which is likely to be detectable using advanced photoemission techniques. Further distinctions emerge in the electronic excitations. The spin-1 excitation with sixfold degeneracy is located at $\sim 138$ meV above the Fermi level in La(Rh,Ir)Si. Additionally, a fourfold degeneracy lying just $\sim$42 meV below the Fermi level and with a topological charge $C=+2$ resides at the zone center, but has never been observed in the CoSi family. These unconventional topological features, including the proximity of these states to the Fermi level, may serve as the driving forces behind the emergence of intrinsic topological superconductors and the long-sought Majorana quasiparticles. Finally, it would be interesting to investigate the chirality-related superconducting phenomena in LaIrSi-based devices. A prime example is the Josephson diode effect that requires simultaneous breaking of inversion and mirror symmetries. The La(Rh,Ir)Si family offers a natural platform to realize it. For example, TaS2 intercalated with chiral molecules shows a distinct superconducting diode effect, while it is absent in the pristine achiral TaS2 [81].

3 Conclusion

In summary, we observed unconventional superconductivity in the double-helix La(Rh,Ir)Si chiral crystals, which exhibit exotic multifold fermions near the Fermi level and giant helicoid Fermi arcs that span the entire surface Brillouin zone. Our work reveals two dominant factors for the emergence of novel superconducting phases in La(Rh,Ir)Si, distinct from the conventional superconducting behavior observed in AuBe, despite sharing the same chiral space group. The first factor is the double-helix chiral structure. This enhances the density of electronic states near the Fermi level due to the additional contributions of the La-$5d$ orbitals compared to the CoSi family of chiral crystals. The second factor involves tuning the ratio of spin-singlet to spin-triplet pairing via the spin-orbit coupling. This causes superconducitivity to evolve from a fully-gapped state (in LaRhSi) to a nodal-line state (in LaIrSi). Moreover, we develop a phenomenological theory that elucidates a new mechanism for nodal-line SC (here arising from the presence of anisotropic Fermi surfaces) and reproduces the experimental data remarkably well. Our work opens up a new avenue for exploring unconventional or topological superconductivity in chiral materials. At the same time, it calls for a re-examination of the interplay between SC and topologically non-trivial Fermi surfaces, an interplay that has hardly been considered in other noncentrosymmetric superconductors with a chiral crystal structure.

4 Experimental Section

\threesubsection

Materials synthesis and characterization

Polycrystalline LaRhSi and LaIrSi samples were prepared by stoichiometric arc melting of a La rod (99.9%, Alfa Aesar), Rh or Ir powders (99.9%, ChemPUR), and Si chunks (99.9999%, Alfa Aesar) in a high-purity argon atmosphere. To improve sample homogeneity, the ingots were flipped and re-melted more than six times. The crystal structure of the LaRhSi and LaIrSi samples was checked via powder x-ray diffraction at room temperature using a Bruker D8 diffractometer with Cu K$\alpha$ radiation. This confirmed the cubic noncentrosymmetric structure of LaRhSi and LaIrSi ($P$213, No. 198), which is also of chiral type. The electrical-resistivity and magnetic susceptibility measurements were performed on a Quantum Design physical property measurement system (PPMS) and a magnetic property measurement system (MPMS).

\threesubsection

µSR experiments

The µSR measurements were performed at the general-purpose surface-muon spectrometer (GPS) at the $\pi$M3 beam line and the multipurpose surface-muon spectrometer (Dolly) at the $\pi$E1 beamline of the Swiss muon source (SµS) at Paul Scherrer Institut (PSI) in Villigen, Switzerland. To exclude the possibility of stray magnetic fields during the ZF-µSR measurements, all the magnets were preliminarily degaussed, and we made use of an active field-compensating facility [82]. For the TF-µSR measurements, the applied magnetic field (i.e., 20 mT) was perpendicular to the muon-spin direction and the samples were cooled in an applied magnetic field down to the base temperature ($\sim$0.3 K). Both the ZF- and TF-µSR spectra were collected upon heating the samples.

\threesubsection

Electronic band-structure calculations

First-principles calculations were performed based on the density functional theory (DFT), as implemented in the Vienna ab-initio simulation package (VASP) package [83, 84]. Projector augmented wave (PAW) pseudo-potentials were adopted in the calculation [85, 86]. The generalized gradient approximation with the Perdew-Burke-Ernzerhof (PBE) realization [87] was used for the exchange-correlation functional. The valence electrons treated in the calculations include La ($5s^{2}6s^{2}5p^{6}5d^{1}$), Rh ($4d^{8}5s^{1}$), Ir ($5d^{8}6s^{1}$), and Si ($3s^{2}3p^{2}$). The kinetic energy cutoff was fixed to 400 eV. For the self-consistent calculations, the Brillouin zone (BZ) integration was performed on a $\Gamma$-centered mesh of $12\times 12\times 12$ $k$-points. The energy convergence criteria was set to 10*-7* eV. In order to study the topological electronic structure of LaIrSi, a Wannier tight binding Hamiltonian consisting of La-$5d$, La-$6s$, Ir-$5d$, Si-$3s$ and Si-$3p$ orbitals was constructed using the Wannier90 package [88]. The surface spectra (i.e., Fermi arcs) were calculated by using the iterative Green’s function method as implemented in the WannierTools package [89].

\threesubsection

Material phase diagram

In Figure Discovery of nodal-line superconductivity in chiral crystals, we qualitatively map out the phase diagram of unconventional SC in both simple- and in double-helix chiral crystals. Our analysis primarily focuses on identifying the regions where nodal-line SC emerges. To achieve this, we rely on the spectrum of the BdG Hamiltonian, as described by Equation 2. For the case with both $\Delta_{s}$ $>$ 0 and $\Delta_{t}$ $>$ 0, the superconducting nodes can only occur for the two $\beta=+$ bands, and the location of the nodes is determined by solving

[TABLE]

More explicitly, these conditions can be expressed as

[TABLE]

where we define $\tilde{E_{f}}=(E_{f}-A_{0}-|\lambda_{0}|)/A_{3}<0$, $\tilde{A}_{1}=A_{1}/A_{3}>0$, and $\tilde{A}_{2}=-A_{2}/A_{3}>0$. Because of the three glide symmetries about the $x$–$z$, $y$–$z$ and $x$–$y$ planes, it is sufficient to analyze the quadrant where $k_{x}>0,k_{y}>0,k_{z}>0$.

After performing straightforward calculations (whose details are provided in Note S6, Supporting Information), we derive the conditions for nodal-line SC:

• •

For the $E_{+,+}$ band (where $\alpha=+$). The condition for nodal-line SC is given by

[TABLE]

where

[TABLE]

Here, we define $\tilde{A}_{2}^{\prime}=\tilde{A}_{2}-\frac{\sqrt{3}}{9}$. In this work, we focus on the case with $\tilde{A}_{2}^{\prime}>0$.

• •

For the $E_{-,+}$ band ($\alpha=-$). Likewise, the condition for nodal-line SC is

[TABLE]

where

[TABLE]

Here, we define $\tilde{A}_{2}^{\prime\prime}=\tilde{A}_{2}+\frac{\sqrt{3}}{9}$. The dependence of $x_{\mathrm{min}}^{-}$ and $x_{\mathrm{max}}^{-}$ on $\lambda_{0}$ and $A_{3}$ is presented in Figure Discovery of nodal-line superconductivity in chiral crystals. Notably, as discussed in the main text, the nodal region expands as the $C_{3}$-wrapping term $A_{3}$ increases. The same features are found for the $E_{+,+}$ band ($\alpha=+$) (see Figure Discovery of nodal-line superconductivity in chiral crystals).

In the derivation of the nodal-line superconductivity scenario above, we treat $\Delta_{s}/\Delta_{t}$ as an independent tuning parameter, which can be determined either through self-consistent calculations or by fitting experimental data. Here, we present a general comparison between LaRhSi and LaIrSi. When $\lambda_{0}\to 0$, the ratio $\Delta_{s}/\Delta_{t}$ is relatively large (i.e., $\Delta_{s}/\Delta_{t}\gg x_{\text{max}}^{\pm}$), which explains the fully-gapped superconducting state observed in LaRhSi (a system with weak SOC). As $\lambda_{0}$ increases, we expect that the ratio $\Delta_{s}/\Delta_{t}$ decreases, leading to the emergence of nodal SC in the $E_{+,-}$ band. Furthermore, our calculations reveal that the $E_{-,-}$ band can also exhibit nodal SC when $\lambda_{0}$ increases further. This suggests that the relatively strong SOC in LaIrSi reduces $\Delta_{s}/\Delta_{t}$, thereby enabling the possibility of nodal SC.

\threesubsection

Approximation to fit the superfluid density

Experimentally, the ratio $\Delta_{s}/\Delta_{t}$ can be estimated directly from the measured superfluid density data. In the previous section, the ratio $\Delta_{s}/\Delta_{t}$ was expressed in units of [Å*-1*], since the dimensions of both $[\Delta_{s}]$ and $[\Delta_{t}\times k_{f}]$ are [eV], where $k_{f}$ is the Fermi momentum. This formulation is more general for analyzing the phase diagram (Figure Discovery of nodal-line superconductivity in chiral crystals), as it accounts for both the inner and outer Fermi surfaces. However, to fit the experimental data, a dimensionless parameter $\Delta_{s}/\tilde{\Delta}_{t}$ with $\tilde{\Delta}_{t}=\Delta_{t}k_{f}$ is more convenient, as will be illustrated below.

In addition, the presence of an anisotropic band structure or Fermi surfaces complicates the analysis, making a simple fitting approach impractical. To address this (i.e., $k_{f}$ is not isotropic), we devise an approximation by mapping the non-spherical Fermi surface onto a spherical Fermi surface. While this approximation simplifies the geometry, it introduces an anisotropic gap function. First, we solve the equation

[TABLE]

for the Fermi momentum $k$, which leads to the expression

[TABLE]

where we define $E_{f}^{0}=E_{f}-A_{0}-|\lambda_{0}|$ and $A_{2}(\theta,\phi)=A_{2}+\alpha\frac{A_{3}}{4}|\sin 2\theta\sin 2\phi\sin\theta|$. Note that, $A_{0}<0$, $A_{1}>0$, $A_{2}<0$ and $A_{3}>0$, so that $E_{f}^{0}<0$ and $A_{2}(\theta,\phi)\leq A_{2}+\frac{\sqrt{3}}{9}A_{3}<0$.

Then, we figure out the pairing gap projected onto the Fermi surfaces. For the intra-band pairing gap, we have

[TABLE]

where we define $k_{\text{min}}$ as the smallest Fermi momentum of $k(\theta,\phi)$ for $\theta\in[0,\pi)$ and $\phi\in[0,2\pi)$. The value of $k_{\text{min}}$ varies across different bands. For the $E_{+,-}$ band (where $\alpha=+$), it is given by

[TABLE]

Finally, by substituting Equation 12 and Equation 14 into Equation 13, we obtain the fitting gap function for the $E_{+,-}$ band as

[TABLE]

where $A_{2}(\theta,\phi)=A_{2}+\frac{A_{3}}{4}|\sin 2\theta\sin 2\phi\sin\theta|$.

Likewise, for the $E_{-,-}$ band ($\alpha=-$), $k_{\text{min}}$ is given by

[TABLE]

because $|\sin 2\theta\sin 2\phi\sin\theta|_{\mathrm{max}}=4/(3\sqrt{3})$. Then, the fitting gap function for the $E_{-,-}$ band is

[TABLE]

where $A_{2}(\theta,\phi)=A_{2}-\frac{A_{3}}{4}|\sin 2\theta\sin 2\phi\sin\theta|$.

The key parameters obtained from the fit are $E_{f}=0$, $A_{0}=0.536\;\text{eV}$, $A_{1}=0.924\;\text{eV}$, $A_{2}=-14.273\;\text{eV}\cdot$Å2, $\lambda_{0}|_{\text{LaRhSi}}=0.1\;\text{eV}$, $\lambda_{0}|_{\text{LaIrSi}}=0.4\;\text{eV}$, and $A_{3}=15\;\text{eV}\cdot$Å2. These best-fit results for $\Delta_{s}/\tilde{\Delta}_{t}$ are presented in the main text (see Figure Discovery of nodal-line superconductivity in chiral crystals). Our model nicely captures the evolution of fully-gapped to nodal-line SC in the La(Rh,Ir)Si family.

**Conflict of Interest

**The authors declare no conflict of interest.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

**Supporting Information

**Supporting Information is available from the Wiley Online Library or from the author.

**Acknowledgements

**We acknowledge the allocation of beam time at S$\mu$S (Dolly and GPS spectrometers) and thank Sudeep K. Ghosh for useful discussions. T.S. acknowledges support by the National Natural Science Foundation of China (Grant Nos. 12374105 and 12350710785), the Natural Science Foundation of Shanghai (Grant Nos. 21ZR1420500 and 21JC1402300), the Natural Science Foundation of Chongqing (Grant No. 2022NSCQ-MSX1468), and the Fundamental Research Funds for the Central Universities.