Time Reversal Symmetry Breaking and {\it Fragile Magnetic Superconductors}

TL;DR

This work examines time-reversal symmetry breaking in a class of low-$T_c$ conventional metals identified as fragile magnetic superconductors, where $cmu$SR detects tiny spontaneous fields near $T_c$ and prompts questions about whether TRSB is intrinsic or muon-induced. It develops a multiscale framework for the muon’s near-field environment, incorporating spin and orbital polarization, induced supercurrents, quantum positional uncertainty, and Kondo/YSR-type impurity physics, to assess how the implanted muon couples to the superconducting order parameter. A detailed case study of LaNiGa$_2$—a material with a non-symmorphic Cmcm symmetry and proposed nonunitary triplet, as well as exotic singlet scenarios tied to Dirac-point degeneracies—highlights how symmetry, SOC, and degeneracies shape TRSB interpretations and potential pairing states. The analysis suggests that conventional electron-phonon–driven singlet pairing can accommodate many observed SC properties and that the tiny TRSB fields may arise from near-field muon effects or subtle valley/degeneracy physics, underscoring the need for careful disentanglement of intrinsic order from muon-induced perturbations. Overall, the paper provides a comprehensive framework to interpret μSR TRSB signals, clarifies the role of muon perturbations in fragile magnetic superconductors, and points to Dirac-point–based singlet scenarios as a viable alternative to a purely triplet explanation for LaNiGa$_2$.

Abstract

Roughly twenty reports (as of 2025) of time-reversal-symmetry breaking (TRSB) states in low critical temperature (T$_c$) superconducting (SC), otherwise conventional Fermi liquid, metals have emerged primarily from muon spin relaxation ($μ$SR) data. The detected fields, inferred from the current interpretation of depolarization data, are similar in magnitude and not far above the lower limit of detection, corresponding to magnetizations of no more than 10$^{-3}$ $μ_B$/atom. These materials comprise a new class of {\it fragile magnetic superconductors} modeled as triplet pairing. The measured SC state properties, excepting only the fields detected below T$_c$, are representative of low T$_c$ singlet BCS SCs, not showing unusual coherence lengths or critical fields. While it is recognized that the muon does affect the sample by displacing nearby atoms and impacting magnetic interaction parameters, the measurement process, changing the system from sample $\rightarrow$ sample+$μ^+$ thereby breaking TRS, may deserve further scrutiny. This overview provides a survey of the environment of the muon, from the normal state to the superfluid state, where the induced supercurrent and Yu-Shiba-Rusinov gap states provide coupling of the muon moment to the superfluid. The unusual topological superconductor LaNiGa$_2$, currently modeled as non-unitary triplet, is used as a case study. Supposing that the prevailing $μ$SR inference of a small spontaneous field within the bulk of theSC obtains, the current picture of (possibly non-unitary) triplet pairing is discussed and an attractive alternative for LaNiGa$_2$ is noted.

Figures (5)

• Figure 1: Plot of constant magnetic field lines of a point dipole oriented along the $\hat{z}$ direction of this plot, plotted in the $x$-$z$ plane; distances along the axes are in arbitrary units relative to the magnitude of the point dipole at the origin. The large arrow indicates the direction of the dipole. The lines with arrows indicate the direction of the field at that point. The darker blue shading indicates larger magnitudes of the $\vec{B}$ field. The radial modulation as $1/r^3$ is evident.
• Figure 2: (a) The structure of LaNiGa$_2$ from Badger et al., determined from single crystal XRD. (b) The structure obtained in 1982 by Yarmolyuk and Grin Grin1982 from powder XRD. The difference lies in (i) the Ni-Ga layer in the center (and top and bottom) layer of this plot, with Ni repositioning, and (ii) repositioning of Ga between the La layers, together resulting in a non-symmorphic operation and the $Cmcm$ space group.
• Figure 3: Top panel: the five Fermi surfaces of $Cmcm$ LaNiGa$_2$, illustrating the degeneracies on the node surface (the pink plane in the lower panel). The green and red Fermi surfaces (FS2 and FS3) merge together on the light green loop, while the blue and brown Fermi surfaces (FS4 and FS5) merge on the vertical blue line. The red dots, denoted by "protected from SOC," pinpoint where the loop of degeneracies cross the $Z$-$T$ lines, leading to 3D Dirac point character at those dots. Bottom panel: symmetry labels of $Cmcm$ Brillouin zone, with the node surface shown in pink.
• Figure 4: Angular distribution of the direction of emission of the positron upon muon decay into a positron and two neutrinos, given by $W(\theta)\propto 1+A\cos\theta$. The factor $A$ depends on the energy of the emitted positron. Two positron energies are pictured: dashed line, 25 MeV near minimum, $A\approx 0$; full red line, 50 MeV near maximum, $A\approx 1$. The arrow denotes the muon polarization direction upon decay. For reference: the rest mass of the muon is $207\times 0.511$ MeV $\sim$ 100 MeV, which is what is available for the decay products. See text for further discussion.
• Figure 5: Sketch of the quasiparticle (Bogoliubov-de Gennes) band structure along one dispersive band direction of the triplet, topological superconductivity model for LaNiGa$_2$.Quan2022 Graphs for three values of the "triplet strength" $\omega$ are displayed. For $\omega$=1, the non-symmorphic band sticking at the zone boundary, hence two nodes, persists into the then gapless superconducting state. In the nonunitary regime $\omega<1$ the degeneracy is broken, giving a gap to quasiparticle excitation.