[Paper Review] Symmetry-based approach to nodal structures: Unification of compatibility relations and point-node classifications

Seishiro Ono, Ken Shiozaki|arXiv (Cornell University)|Feb 15, 2021

Iron-based superconductors research110 references6 citations

TL;DR

This paper presents a symmetry-based framework to systematically classify superconducting nodes pinned to lines in momentum space across all magnetic space groups, unifying prior compatibility relations and point-node classifications. By leveraging group-theoretical analysis of band topology instead of homotopy theory, the method enables efficient node detection in materials using density functional theory, validated in CaPtAs, and establishes a unified theory for identifying nodal structures in unconventional superconductors.

ABSTRACT

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 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 have been 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 magnetic space groups. Furthermore, our framework can readily provide a highly effective scheme to detect nodes in a given material by using density functional theory, which elucidates the symmetry property of superconductivity. 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.

Motivation & Objective

• To develop a systematic and unified theory for classifying superconducting nodes pinned to lines in momentum space, beyond existing point-group-based classifications.
• To overcome the limitations of homotopy-theory-based approaches by introducing a symmetry-based analysis of band topology applicable to all magnetic space groups.
• To provide a practical, computationally efficient scheme for detecting superconducting nodes in real materials using density functional theory.
• To unify compatibility relations and point-node classifications into a single theoretical framework applicable to both centrosymmetric and noncentrosymmetric superconductors.
• To validate the method in a time-reversal-breaking, noncentrosymmetric superconductor (CaPtAs) to demonstrate its predictive power and experimental relevance.

Proposed method

• The framework is built on group-theoretical analysis of band topology, focusing on the symmetry properties of the superconducting gap function in momentum space.
• It classifies nodes based on the irreducible representations of the little co-group at high-symmetry lines in the Brillouin zone, enabling systematic diagnosis of nodal structures.
• The method avoids reliance on homotopy theory by directly analyzing the symmetry constraints on the gap function, allowing application to all magnetic space groups.
• It integrates with first-principles calculations via density functional theory to predict node locations and symmetry properties in real materials.
• The approach identifies the topological invariants associated with nodal lines by analyzing the compatibility of irreducible representations along symmetry lines.
• The framework enables automated diagnosis of nodal structures by mapping the symmetry of the gap function to the topological classification of the band structure.

Experimental results

Research questions

• RQ1 How can superconducting nodes pinned to lines in momentum space be systematically classified across all magnetic space groups?
• RQ2 What is the relationship between symmetry-based band topology and the topological classification of nodal superconductors?
• RQ3 Can a unified framework be developed that subsumes existing compatibility relations and point-node classifications?
• RQ4 How can such a framework be practically implemented in first-principles calculations to predict nodes in real materials?
• RQ5 To what extent can this method detect and characterize nodes in time-reversal-breaking, noncentrosymmetric superconductors like CaPtAs?

Key findings

• The proposed symmetry-based framework successfully classifies superconducting nodes pinned to lines in momentum space for all magnetic space groups, unifying prior classification schemes.
• The method enables efficient and systematic diagnosis of nodal structures in materials using density functional theory, providing a practical tool for node detection.
• The framework reveals that symmetry constraints on the gap function, derived from irreducible representations of the little co-group, determine the existence and topology of nodal lines.
• In CaPtAs, the method correctly predicts the presence of line nodes consistent with experimental observations, validating its predictive power.
• The approach demonstrates that symmetry-based analysis of band topology is more effective than homotopy theory for classifying nodes in complex, noncentrosymmetric systems.
• The theory provides a clear, systematic pathway to link the symmetry of the superconducting gap to the topological nature of the nodal structure in unconventional superconductors.