Symmetry-Based Approach to Superconducting Nodes: Unification of Compatibility Conditions and Gapless Point Classifications

Determination of the symmetry property of superconducting gaps has been a central issue in studies to understand the mechanisms of unconventional superconductivity. Although it is often difficult to completely achieve the aforementioned goal, the existence of superconducting nodes, one of the few important experimental signatures of unconventional superconductivity, plays a vital role in exploring the possibility of unconventional superconductivity. The interplay between superconducting nodes and topology has been actively investigated, and intensive research in the past decade has revealed various intriguing nodes out of the scope of the pioneering work to classify superconducting order parameters based on the point groups. However, a systematic and unified description of superconducting nodes for arbitrary symmetry settings is still elusive. In this paper, we develop a systematic framework to comprehensively classify superconducting nodes pinned to any line in momentum space. While most previous studies are based on the homotopy theory, our theory is on the basis of the symmetry-based analysis of band topology, which enables systematic diagnoses of nodes in all nonmagnetic and magnetic space groups. Furthermore, our framework can readily provide a highly effective scheme to detect nodes in a given superconductor by using density-functional theory and assuming symmetry properties of Cooper pairs (called pairing symmetries). There are two main advantages of our method. One is that, while the symmetry-indicator theory cannot be directly applied to systems violating compatibility conditions, our diagnostic scheme focuses on such systems, which results in a complement of the symmetry-indicator theory. The other is that, although our method does not predict any pairing symmetry itself, our framework can diagnose not only the positions but also the dimensions of nodes such as line or surface nodes, which can reduce candidates of pairing symmetries. We substantiate the power of our method through the time-reversal broken and noncentrosymmetric superconductor CaPtAs. Our work establishes a unified theory for understanding superconducting nodes and facilitates determining superconducting gaps in materials combined with experimental observations.