Higher-Order Topology in Bismuth (1802.02585v2)

Abstract: The mathematical field of topology has become a framework to describe the low-energy electronic structure of crystalline solids. A typical feature of a bulk insulating three-dimensional topological crystal are conducting two-dimensional surface states. This constitutes the topological bulk-boundary correspondence. Here, we establish that the electronic structure of bismuth, an element consistently described as bulk topologically trivial, is in fact topological and follows a generalized bulk-boundary correspondence of higher-order: not the surfaces of the crystal, but its hinges host topologically protected conducting modes. These hinge modes are protected against localization by time-reversal symmetry locally, and globally by the three-fold rotational symmetry and inversion symmetry of the bismuth crystal. We support our claim theoretically and experimentally. Our theoretical analysis is based on symmetry arguments, topological indices, first-principle calculations, and the recently introduced framework of topological quantum chemistry. We provide supporting evidence from two complementary experimental techniques. With scanning-tunneling spectroscopy, we probe the unique signatures of the rotational symmetry of the one-dimensional states located at step edges of the crystal surface. With Josephson interferometry, we demonstrate their universal topological contribution to the electronic transport. Our work establishes bismuth as a higher-order topological insulator.

• The paper demonstrates that bismuth is a higher-order topological insulator characterized by conducting hinge states instead of traditional surface states.
• It employs symmetry-based indices and first-principle calculations alongside STM/STS techniques to validate the presence and robustness of hinge modes.
• Experimental validations via Josephson interferometry reveal ballistic hinge conduction, underscoring its potential for low-dissipation electronic applications.

Investigation of Higher-Order Topology in Bismuth

The exploration of topological phases has significantly expanded beyond conventional topological insulators, with recent advancements in the domain of higher-order topological insulators (HOTIs). The research paper on higher-order topology in bismuth presents an intricate analysis and experimental confirmation of bismuth as a HOTI, challenging the previous classification of the element as topologically trivial. This work elucidates the presence of topologically protected conducting states localized not on the surfaces, but on the hinges of the bismuth crystal structure.

Key Insights

1. Higher-Order Topological Classification: The work centers around classifying bismuth as a higher-order topological insulator. Traditionally, topological insulators are characterized by conducting surface states; however, in HOTIs, these conducting states are restricted to lower-dimensional boundaries (e.g., edges or hinges) due to the preservation of spatial symmetries. The paper uses symmetry arguments alongside topological quantum chemistry to establish this classification distinctly, where hinge modes become the highlight.
2. Symmetry Protection: The hinge states in bismuth are protected by a combination of time-reversal symmetry (TRS) and spatial symmetries—specifically, the threefold rotational symmetry and inversion symmetry. Such protection mechanisms ensure the robustness of these modes against localization, a haLLMark of topological protection.
3. Theoretical and Experimental Frameworks: The authors substantiate their claims through comprehensive theoretical frameworks and experimental modalities. Theoretically, symmetry-based topological indices and first-principle calculations provide a substantial underpinning for their assertions. Experimentally, techniques such as scanning tunneling microscopy/spectroscopy (STM/STS) and Josephson interferometry elucidate the localized topological states along step edges and hinge channels in bismuth nanowires.
4. Bulk-Boundary Correspondence: The paper reinforces the concept of bulk-boundary correspondence in the context of higher-order topology, where the existence of one-dimensional hinge modes is derived from the bulk's topological characteristics. This is a significant extension of the established topological principles generalizing the correspondence to hinge-localized modes rather than surface-localized modes.
5. Experimental Validation and Observations: Through STM experiments, the paper identifies localized states along certain step edges of the bismuth (111) surface, while Josephson interferometry confirms the ballistic nature of supercurrents along the hinge states in bismuth nanowires—providing tangible evidence for the theoretical predictions made.

Numerical Results and Observations

The empirical evidence presented shows pronounced conductance at specific edges observed through differentials in STM spectroscopy. The Josephson interferometry reveals periodic oscillations of critical current, matching theoretical predictions for interference between paths along nanowire edges, thus aligning well with the expected hinge modes' contribution.

Implications and Future Directions

The implications of categorizing bismuth as a HOTI are profound. For practical applications, this finding opens up avenues in the development of robust electronic components with reduced dissipation due to hinge-protected conduction. In theoretical paradigms, it prompts a reevaluation of other materials that may have been misclassified under older topological frameworks. Furthermore, the methodology enhances the potential for discovering and designing new materials manifesting higher-order topological properties.

As the field progresses, one of the intriguing prospects involves employing proximitized superconductivity to facilitate topological quantum computations using these hinge states as qubits. The hinge states' spin-momentum locking could also pave the way for advancements in spintronics. Thus, further exploration in the manipulation of these systems could lead to breakthrough technologies in both quantum computing and condensed matter physics. The research sets a benchmark in understanding complex topological phases, providing keen insights for future exploration in the topology of crystalline materials.