The exceptionally low-symmetry crystal structures of the time-reversal symmetry-breaking superconductors ${\mathrm{LaNiC}}_2$ and ${\mathrm{LaNiGa}}_2$ lead to an internally antisymmetric nonunitary triplet state as the only possibility compatible with experiments. We argue that this state has a distinct signature: a double-peak structure in the density of states (DOS) which resolves in the spin channel in a particular way. We construct a detailed model of ${\mathrm{LaNiGa}}_2$ capturing its electronic band structure and magnetic properties ab initio. The pairing mechanism is described via a single adjustable parameter. The latter is fixed by the critical temperature $T_c$ allowing parameter-free predictions. We compute the electronic specific heat and find excellent agreement with experiment. The size of the ordered moment in the superconducting state is compatible with zero-field muon spin relaxation experiments and the predicted spin-resolved DOS suggests the spin splitting is within the reach of present experimental technology.

The ${\mathbb{Z}}_2$ topological metals $R{\mathrm{V}}_3{\mathrm{Sb}}_5$ ($R$ = K, Rb, Cs) with a layered kagome structure provide a unique opportunity to investigate the interplay between charge order, superconductivity, and topology. Here, we report muon-spin relaxation/rotation ($\mu \mathrm{SR}$) measurements performed on $\mathrm{Cs}{\mathrm{V}}_3{\mathrm{Sb}}_5$ across a broad temperature range, in order to uncover the nature of the charge density wave order and superconductivity in this material. From zero-field $\mu \mathrm{SR}$, we find that spontaneous magnetic fields appear below 50 K, which is well below the charge density wave transition ($T^{*}\sim 93$ K). We show that these spontaneous fields are dynamic in nature making it difficult to associate them with a hidden static order. The superconducting state of $\mathrm{Cs}{\mathrm{V}}_3{\mathrm{Sb}}_5$ is found to preserve time-reversal symmetry, and the transverse-field $\mu \mathrm{SR}$ results are consistent with a superconducting state that has two fully open gaps.

Chiral superconductors are novel topological materials with finite angular momentum Cooper pairs circulating around a unique chiral axis, thereby spontaneously breaking time-reversal symmetry. They are rather scarce and usually feature triplet pairing: a canonical example is the chiral p-wave state realized in the A-phase of superfluid He³. Chiral triplet superconductors are, however, topologically fragile with the corresponding gapless boundary modes only weakly protected against symmetry-preserving perturbations in contrast to their singlet counterparts. Using muon spin relaxation measurements, here we report that the weakly correlated pnictide compound LaPt₃P has the two key features of a chiral superconductor: spontaneous magnetic fields inside the superconducting state indicating broken time-reversal symmetry and low temperature linear behaviour in the superfluid density indicating line nodes in the order parameter. Using symmetry analysis, first principles band structure calculation and mean-field theory, we unambiguously establish that the superconducting ground state of LaPt₃P is a chiral d-wave singlet.

We report the discovery of ${\mathrm{Zr}}_3\mathrm{Ir}$ as a structurally different type of unconventional noncentrosymmetric superconductor (with $T_c=2.3$ K), here investigated mostly via muon-spin rotation/relaxation ($\mu \mathrm{SR}$) techniques. Its superconductivity was characterized using magnetic susceptibility, electrical resistivity, and heat capacity measurements. The low-temperature superfluid density, determined via transverse-field $\mu \mathrm{SR}$ and electronic specific heat, suggests a fully gapped superconducting state. The spontaneous magnetic fields, revealed by zero-field $\mu \mathrm{SR}$ below $T_c$, indicate the breaking of time-reversal symmetry in ${\mathrm{Zr}}_3\mathrm{Ir}$ and hence the unconventional nature of its superconductivity. By using symmetry arguments and electronic-structure calculations we obtain a superconducting order parameter that is fully compatible with the experimental observations. Hence, our results clearly suggest that ${\mathrm{Zr}}_3\mathrm{Ir}$ represents a structurally different member of noncentrosymmetric superconductors with broken time-reversal symmetry.

Topological semimetals are three dimensional materials with symmetry-protected massless bulk excitations. As a special case, Weyl nodal-line semimetals are realized in materials having either no inversion or broken time-reversal symmetry and feature bulk nodal lines. The 111-family, including LaNiSi, LaPtSi and LaPtGe materials (all lacking inversion symmetry), belongs to this class. Here, by combining muon-spin rotation and relaxation with thermodynamic measurements, we find that these materials exhibit a fully-gapped superconducting ground state, while spontaneously breaking time-reversal symmetry at the superconducting transition. Since time-reversal symmetry is essential for protecting the normal-state topology, its breaking upon entering the superconducting state should remarkably result in a topological phase transition. By developing a minimal model for the normal-state band structure and assuming a purely spin-triplet pairing, we show that the superconducting properties across this family can be described accurately. Our results demonstrate that the 111 materials reported here provide an ideal test-bed for investigating the rich interplay between the exotic properties of Weyl nodal-line fermions and unconventional superconductivity.

Superconductors with broken time-reversal symmetry represent arguably one of the most promising venues for realizing highly sought-after topological superconductivity that is vital to fault-tolerant quantum computation. Here, by using extensive muon-spin relaxation and rotation measurements, we report that the isostructural silicide superconductors (Ta, Nb)OsSi spontaneously break time-reversal symmetry at the superconducting transition while surprisingly showing a fully gapped superconductivity characteristic of conventional superconductors. The first-principles calculations show that (Ta, Nb)OsSi are three-dimensional Dirac semimetals protected by nonsymmorphic symmetries. Taking advantage of the exceptional low symmetry crystal structure of these materials, we have performed detailed theoretical calculations to establish that the superconducting ground state for both (Ta, Nb)OsSi is most likely a nonunitary triplet state.