Revealing Higher-Order Topological Bulk-boundary Correspondence in Bismuth Crystal with Spin-helical Hinge State Loop and Proximity Superconductivity

By combining scanning tunneling microscopy, first-principles calculations, and global symmetry analysis on bismuth crystals grown on superconducting V3Si, this study provides direct evidence of higher-order topological bulk-boundary correspondence through the observation of spin-helical hinge state loops and proximity-induced superconductivity, establishing bismuth as a promising platform for realizing topological superconductivity and Majorana quasiparticles.

The Gist

The Big Idea: Finding the "Hidden Roads" on a Crystal Mountain

Imagine you have a crystal of Bismuth (a shiny, silvery metal). In the world of physics, this crystal isn't just a solid block; it's a complex landscape with different "facets" (flat sides) and "hinges" (the sharp edges where two sides meet).

For a long time, scientists knew that some materials have special "highways" for electrons that run along their surfaces. This is like a road that exists only on the skin of an apple. However, a newer theory called Higher-Order Topology predicted something stranger: that in certain crystals, the highways don't run on the flat skin, but instead run along the sharp edges (hinges) where the sides meet.

Think of a cube.

• Normal Topology: The "highways" (electrons) run all over the flat faces of the cube.
• Higher-Order Topology: The flat faces are dead ends (no traffic). The only place electrons can flow freely is along the 12 edges of the cube, forming a continuous loop around the entire object.

What the Scientists Did

The team at Fudan University and other institutions wanted to prove this "edge highway" theory exists in real life, specifically in Bismuth crystals. They faced a huge challenge: to prove the highway forms a complete loop, they had to check every single edge of the crystal, not just one or two.

Here is how they did it, step-by-step:

1. Building the Crystal Mountain
They grew tiny, mesoscopic crystals of Bismuth on top of a superconducting material (V3Si). Think of this as planting a tiny mountain on a special, super-conductive floor.

2. Mapping the Terrain with a "Super-Microscope"
They used a tool called a Scanning Tunneling Microscope (STM). Imagine a needle so sharp it can feel individual atoms. They used this needle to "listen" to the electrons.

• The Discovery: When they scanned the flat sides of the crystal, the electrons were quiet (gapped). But when they scanned the sharp hinges, they found a loud, distinct signal. It was like finding a busy highway running along the edge of a quiet cliff.
• The Loop: They checked all five different types of edges on their crystal. They found that these "highways" existed on specific edges and, crucially, they connected to form a closed loop circling the entire crystal. This confirmed the "Higher-Order" theory: the traffic flows around the perimeter, not across the face.

3. Proving the "Spin-Helical" Nature (The One-Way Street)
The theory predicted these edge highways are "spin-helical." This is a fancy way of saying the electrons have a built-in traffic rule: Spin determines direction.

• Analogy: Imagine a two-lane road where cars with red hats must drive clockwise, and cars with blue hats must drive counter-clockwise. They can never crash into each other because they are forced to stay in their lanes.
• The Test: To prove this, the scientists dropped tiny magnetic "boulders" (Iron clusters) onto the edges.
  • On a normal road, a boulder would cause traffic to stop or bounce back randomly.
  • On this special "spin-helical" road, the magnetic boulder forced the electrons to do something unexpected: they had to flip their "hat" (spin) to get around the obstacle. The scientists saw this "spin-flip" scattering in their data. This proved the electrons were indeed following the strict, one-way traffic rules of a topological highway.

4. The Superconducting Connection
Because the Bismuth crystal was sitting on a superconductor, the "super-power" of the floor leaked up into the edge highways.

• Analogy: Imagine the highway is made of a special material that suddenly becomes frictionless (superconducting) because it's touching a super-conductive floor.
• The Result: The electrons flowing along the edge not only moved without resistance but also gained a new, exotic property. The paper suggests this setup creates the perfect conditions for Majorana quasiparticles.
• What is a Majorana? Think of it as a particle that is its own mirror image. In the world of quantum computing, these are the "holy grail" because they are incredibly stable and could be used to build computers that don't crash easily.

The Conclusion

The paper claims to have provided the first complete proof that this "Higher-Order Topology" exists in Bismuth.

• They didn't just find a highway on one edge; they mapped the entire loop.
• They proved the highway has spin-helical traffic rules (using magnetic scatterers).
• They showed that this highway can carry superconducting currents, making it a potential playground for creating Majorana particles.

In short, they took a theoretical map of a "loop highway" on a crystal and successfully drove a car around the entire track, proving the map was real.

Technical Summary

Technical Summary: Revealing Higher-Order Topological Bulk-boundary Correspondence in Bismuth Crystal

Problem Statement
Topological materials are traditionally defined by the bulk-boundary correspondence, where nontrivial bulk band topology guarantees gapless metallic states on all surfaces. While this has been verified for first-order three-dimensional topological insulators (3D TIs), the concept has recently been generalized to higher-order topological insulators (HOTIs). In second-order 3D TIs, the bulk and surfaces are gapped, but one-dimensional (1D) gapless hinge states emerge at the intersections of specific facets. A complete experimental verification of this higher-order topology has been lacking because it requires probing all crystal boundaries to confirm that the hinge states form a closed loop encircling the crystal. Previous studies on candidates like bismuth (Bi) often observed density-of-states (DOS) peaks at edges but failed to resolve the characteristic dispersive 1D bands with spin-helical structures, nor did they demonstrate the global connectivity of these states.

Methodology
The authors investigated mesoscopic bismuth (Bi) crystals grown on a superconducting V$_3$Si(111) substrate. The study employed a multi-faceted approach:

1. Sample Fabrication: Bi crystals were deposited via thermal evaporation on V$_3$Si and annealed to form well-defined mesoscopic structures with lateral dimensions of ~150 nm and heights of ~60 nm.
2. Scanning Tunneling Microscopy (STM): Low-temperature STM was used to characterize the crystal morphology, identifying three low-index facets ({111}, {110}, {100}) and five distinct types of hinges formed by their intersections.
3. Quasiparticle Interference (QPI): To probe the electronic structure, the authors acquired $dI/dV$ line spectra along the hinges. Fast-Fourier transforms (FFT) of real-space $dI/dV$ distributions were used to visualize scattering vectors ($q$) and determine band dispersion.
4. Magnetic Scattering Test: To verify the topological nature (specifically spin-helical structure), Fe clusters were deposited as magnetic scatterers. The emergence of new scattering channels ($q^*$) indicative of spin-flip scattering was used to distinguish topological states from trivial ones.
5. Theoretical Modeling:
   • First-Principles Calculation: Due to the size of the mesoscopic crystals, a Hamiltonian Graph Neural Network (HamGNN) model was trained on DFT data to calculate the band structures of large prism models containing the specific hinge geometries.
   • Symmetry Analysis: A global symmetry analysis based on the $D_{3d}$ point group and topological classification theory was performed to predict the distribution of gapless states and mass terms on the crystal surface.
6. Proximity Effect: Superconducting gap measurements were conducted to observe the proximity-induced superconductivity in the hinge states.

Key Results

• Identification of Hinge States: The authors identified five types of hinges on Bi crystals. $dI/dV$ spectra on these hinges showed distinct DOS peaks (90–160 meV) absent on the facets, indicating localized 1D states.
• Dispersive 1D Bands: QPI measurements revealed hole-like dispersions for all hinge types. The band tops ($q=0$) aligned with the DOS peaks, confirming the 1D nature of the states.
• Spin-Helical Nature: Upon depositing Fe clusters, new scattering vectors ($q^*$) appeared selectively on type-1 and type-4 hinges, corresponding to spin-flip scattering. Type-2 and type-3 hinges showed no such changes, suggesting they are topologically trivial. Type-5 hinges were analyzed via symmetry.
• Global Loop Formation: Combining experimental QPI data, HamGNN calculations, and symmetry analysis, the authors demonstrated that the topological hinge states (types 1, 4, and 5) form a continuous closed loop encircling the crystal. This loop separates the crystal surface into two regions with opposite mass terms (gap signs), consistent with the theoretical prediction for a second-order TI.
• Proximity Superconductivity: A superconducting gap (~0.4–0.5 meV) was observed in the topological hinge states due to proximity with the V$_3$Si substrate. Notably, the coherence peaks were enhanced at the hinge regions, indicating a distinct superconducting gap opening in the hinge states compared to the bulk or surface states.

Significance and Claims
The paper claims to provide the first complete verification of higher-order bulk-boundary correspondence in a practical material. By examining a complete set of hinges sufficient to construct a 3D crystal, the authors confirm that the topological hinge states in bismuth form a closed loop, thereby experimentally validating the second-order topology of Bi.

Furthermore, the observation of proximity-induced superconductivity in these spin-helical hinge states establishes the Bi/V$_3$Si system as a hybrid platform. According to the Fu-Kane model, such a system is a promising candidate for hosting Majorana quasiparticles at the endpoints of the hinge states (potentially induced by magnetic clusters), which are essential for topological quantum computation. The study bridges the gap between theoretical predictions of higher-order topology and experimental reality by resolving the specific dispersive and spin-helical characteristics of the hinge states and demonstrating their global connectivity.