Non-Abelian band topology in noninteracting metals (1808.07469v3)

Abstract: Electron energy bands of crystalline solids generically exhibit degeneracies called band-structure nodes. Here, we introduce non-Abelian topological charges that characterize line nodes inside the momentum space of crystalline metals with space-time inversion ($\mathcal{PT}$) symmetry and with weak spinorbit coupling. We show that these are quaternion charges, similar to those describing disclinations in biaxial nematics. Starting from two-band considerations, we develop the complete many-band description of nodes in the presence of $\mathcal{PT}$ and mirror symmetries, which allows us to investigate the topological stability of nodal chains in metals. The non-Abelian charges put strict constraints on the possible nodal-line configurations. Our analysis goes beyond the standard approach to band topology and implies the existence of one-dimensional topological phases not present in existing classifications.

Non-Abelian Band Topology in Noninteracting Metals

The research paper explores an advanced area of condensed matter physics by investigating the non-Abelian topological characteristics of nodal lines in noninteracting metals. The focus is on crystalline metals that exhibit space-time inversion ($\mathcal{PT}$) symmetry and weak spin-orbit coupling. The authors introduce novel non-Abelian topological charges associated with these nodal structures, which mark a significant deviation from traditional approaches to band topology.

Overview and Key Contributions

1. Non-Abelian Topological Charges: The paper introduces quaternion charges as topological invariants for nodal lines in metals with $\mathcal{PT}$ symmetry. This is a shift from the commonly discussed Abelian invariants in band topology, extending the classification of topological phases to include higher-order winding and linking.
2. Two-Band and Many-Band Models: A comprehensive framework is developed, beginning with two-band models and extending to systems with multiple bands. The authors show that in these systems, the nodal-line configurations are subject to strict topological constraints due to the non-Abelian nature of the involved charges.
3. Nodal Line Intersections: The paper highlights the stability of crossing points (CPs) where nodal lines intersect, protected by winding numbers in two-band models. These protections can be relaxed through the introduction of additional bands, showcasing a unique mechanism for node separation or recombination mediated by band topology.
4. Application to Real Materials: Elemental scandium ($\textsf{Sc}$) is identified as a practical realization of these theoretical principles. The authors predict observable effects in nodal-line configurations under strain, where the non-Abelian topology manifests as intricate nodal line compositions.

Theoretical and Practical Implications

The theoretical implications of this work are profound, as it bridges the gap between conventional band topology and the more intricate structures predicted by non-Abelian frameworks. This extension could pave the way for discovering new phases of matter and understanding complex systems where interactions between multiple bands cannot be ignored.

Practically, the implementation of these theories in real materials, like scandium, highlights potential experimental avenues. This could involve strain engineering in thin films, making these advanced topological phases accessible to experimental verification via ARPES or transport measurements.

Speculations on Future Developments

The extension of topology in band structure from Abelian to non-Abelian realms might soon influence the design of quantum materials, where control over electron degeneracies translates to novel electronic, optical, or thermal properties. The quaternion invariants introduced could open new pathways in quantum computing, potentially offering robust qubits immune to specific local perturbations due to their enhanced topological protection.

Additionally, the findings could inspire further studies into unconventional superconductors and topological insulators, where the interplay of non-trivial topology and electron interactions might reveal new emergent phenomena.

Conclusion

This paper provides a substantial addition to the field of topological phases of matter, revealing the rich structure and stability of nodal lines governed by non-Abelian topological charges. The proposed framework allows for a deeper comprehension of multi-band systems and offers exciting new directions for future theoretical exploration and experimental realization in condensed matter physics.