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A First-principles Study of Weyl Nodal Loop and Multiple Sets of Weyl Points in Trigonal PtBi $_2$

This first-principles study investigates the robust Weyl nodal loop and multiple sets of Weyl points in trigonal $\gamma$ -PtBi $_2$ , revealing that while key topological features near the Fermi energy remain stable, the specific number and location of additional Weyl points depend sensitively on the magnitude of Bi-layer buckling, alongside predictions of new Fermi arc features at higher energies.

Original authors: Lin-Lin Wang

Published 2026-10-05

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CC BY 4.0

Original authors: Lin-Lin Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the hidden world of solid materials, electrons do not always behave like the tiny, predictable particles we imagine. Under the right conditions, they can organize themselves into exotic states that defy ordinary rules, creating materials with properties that seem almost impossible. One such state is found in a class of crystals known as topological semimetals. In these materials, the energy levels of electrons cross each other in specific ways, forming points in space where the usual separation between conducting and non-conducting states vanishes. These crossing points act like portals, allowing electrons to move with unusual freedom and creating unique patterns on the surface of the crystal. When these patterns appear, they are called Fermi arcs, and they are the visible signature of the material's hidden, twisted geometry. Scientists are intensely interested in these materials because they might host a rare form of superconductivity, a state where electricity flows without any resistance, potentially leading to new technologies for computing and energy. However, understanding exactly how these materials work requires knowing their precise atomic structure, as even the tiniest shift in the position of atoms can change the entire electronic landscape.

A researcher has taken a deep dive into a specific crystal called trigonal PtBi2 to understand how its internal structure shapes these exotic electronic states. This material has recently captured attention because it appears to host both surface superconductivity and the telltale Fermi arcs on its surface. The scientist used powerful computer simulations to map out the paths electrons take through the crystal, looking for the specific points where energy bands cross. They focused on a subtle but critical detail: the way the layers of bismuth atoms within the crystal are stacked. In some reports, these layers are slightly buckled, or bent, while in others, the bend is less pronounced. The researcher wanted to know if this small difference in the physical shape of the crystal would change the fundamental electronic features that make the material so interesting.

By running detailed calculations based on the laws of quantum mechanics, they explored the entire interior of the crystal's repeating unit. They found that the material contains a large, continuous loop of crossing points, known as a Weyl nodal loop, which remains stable regardless of how much the bismuth layers are bent. This loop is protected by the crystal's symmetry, acting as a robust backbone for the material's electronic structure. They also identified a specific set of crossing points located just above the energy level where electrons usually sit, which corresponds to the Fermi arcs seen in experiments. These features were consistent across all the different structural models they tested, confirming that the basic topological nature of the material is solid and reliable.

However, the story changed when they looked at the other sets of crossing points found deeper within the crystal's energy landscape. The number and location of these additional points proved to be extremely sensitive to the exact amount of buckling in the bismuth layers. When the researcher used the structural parameters from a 2020 study, which showed a larger bend in the layers, their simulations revealed only two sets of these extra points. In contrast, when they used parameters from a 2014 study with a smaller bend, the number of sets increased to five. This finding suggests that the specific details of how the atoms are arranged can create or destroy these electronic features, explaining why different experiments might see different things. The researcher also predicted that at slightly higher energy levels, new patterns of surface electrons would appear, including additional arcs that point outward from the main features. These new patterns, along with the variations in how electrons scatter off impurities on the surface, offer a more complete picture of what scientists should expect to see when they look at this material under a microscope.

The work provides a clear explanation for why experimental observations of this material have varied. It shows that while the core topological features are strong and unchanging, the finer details of the electronic structure depend heavily on the precise atomic arrangement. By mapping out these differences, the study helps researchers understand which structural variations lead to which electronic behaviors. The researcher also predicted new features at higher energies that have not yet been seen in the lab, offering a roadmap for future experiments to verify these theoretical discoveries. Ultimately, this research clarifies that the behavior of electrons in PtBi2 is not just a fixed property of the chemical formula, but a delicate balance determined by the exact geometry of the atoms inside.

Technical Summary: First-principles Study of Weyl Nodal Loop and Multiple Sets of Weyl Points in Trigonal PtBi2

Problem Statement
The trigonal phase of Platinum Bismuthide ( $\gamma$ -PtBi $_2$ ) has recently attracted significant attention due to the coexistence of surface superconductivity and Fermi arcs, suggesting a potential platform for topological superconductivity. While previous studies have identified Weyl points (WPs) just above the Fermi energy ( $E_F$ ) and associated them with observed Fermi arcs, the full topological landscape of the material remains incompletely mapped. A critical issue is the discrepancy between two experimentally reported structural parameters for $\gamma$ -PtBi $_2$ , specifically regarding the magnitude of the buckling in the Bismuth (Bi) layers. It is unclear how sensitive the topological features—such as the Weyl nodal loop (WNL) and the number and location of Weyl point sets—are to these structural variations. Furthermore, a comprehensive mapping of band crossings across the full Brillouin zone (BZ) and the resulting surface states (Fermi arcs and quasi-particle interference) under these different structural conditions has not been fully established.

Methodology
The author employed first-principles calculations based on Density Functional Theory (DFT) using the PBE and modified Becke-Johnson (mBJ) exchange-correlation functionals, implemented in VASP with spin-orbit coupling (SOC). To ensure accuracy, the study utilized:

Structural Models: Three distinct structural parameter sets were analyzed: the 2014 experimental report (Ref. 1, less Bi-layer buckling), the 2020 experimental report (Ref. 12, more Bi-layer buckling), and an averaged set of these parameters. Additionally, internal atomic positions were relaxed using various functionals (PBE, D3, optB86b, r2SCAN+rVV10) with fixed experimental lattice constants to validate the structural differences.
Topological Analysis: Maximally localized Wannier functions were constructed to generate tight-binding models reproducing the bulk band structure within $E_F \pm 2$ eV. These models were used to efficiently map the full BZ for band crossings, identifying Weyl nodal loops and Weyl points.
Surface and Scattering Calculations: The surface spectral function, 2D Fermi surfaces (FS), and quasi-particle interference (QPI) patterns were calculated using the surface Green's function method (WannierTools) for semi-infinite surfaces. Calculations were performed for both A- and B-terminations of the (0001) surface.

Key Results

Structural Sensitivity of Weyl Points: The study confirms that the Weyl nodal loop (WNL) on the $\Gamma$ $Γ$ -K-A plane and the primary set of Weyl points (Set 1) located just above $E_F$ $E_{F}$ along the $\Gamma$ $Γ$ - $M$ $M$ direction are robust features, protected by mirror symmetry and the main band inversion, regardless of the Bi-layer buckling magnitude. However, the number and energy positions of additional sets of Weyl points are highly sensitive to the structural parameters.
- Using the averaged structure, six sets of WPs were identified.
- Using the "more buckling" structure (Ref. 12), only two sets of WPs were found.
- Using the "less buckling" structure (Ref. 1), five sets of WPs were found, including a new set (Set 2) appearing just below $E_F$ .
Fermi Arcs and QPI Evolution: The 2D Fermi surfaces and QPI patterns were calculated at energies near $E_F$ $E_{F}$ ( $\pm 0.012$ $\pm 0.012$ eV).
- The primary Fermi arcs (FAs) associated with Set 1 WPs and the bulk gap region (located at $\sim 2/3$ along the $\Gamma$ - $M$ direction) show good agreement with existing Angle-Resolved Photoemission Spectroscopy (ARPES) and experimental QPI data.
- Variations in Bi-layer buckling lead to distinct secondary features. For the "less buckling" structure, the FAs connect differently, and a secondary set of FAs appears along the $\Gamma$ - $K$ direction, connected by a distinct scattering vector ( $q_5$ ).
- The QPI patterns reflect these changes, with the "less buckling" case showing more separated scattering vectors ( $q_3, q_4$ ) and a weak secondary vector, aligning with recent experimental QPI measurements taken at the quasiparticle edge of the superconducting gap.
New High-Energy Features: At higher energies ( $E_F + 0.100$ eV), the calculations predict new Fermi arc features not previously reported. Specifically, additional outward-pointing Fermi arcs aligned along the inner hexagon of the bulk band projection around the $\bar{\Gamma}$ point are observed for Set 1 WPs. These high-energy FAs and their corresponding QPI patterns differ significantly from those near $E_F$ .

Significance and Claims
The paper claims that its detailed mapping provides a better understanding of the experimental observations in $\gamma$ -PtBi $_2$ , particularly explaining why different samples might exhibit varying topological signatures. The author emphasizes that while the WNL and the primary Weyl points near $E_F$ are robust against structural variations, the existence of multiple sets of Weyl points at higher energies is a sensitive function of the Bi-layer buckling. This sensitivity suggests that small off-stoichiometry or strain, which alter the buckling magnitude, can significantly change the topological landscape.

The study concludes by predicting new Fermi arc features and related QPI patterns at $E_F + 100$ meV. The author states that these predictions require future experimental verification. The work does not propose new applications but rather clarifies the structural dependence of the topological properties in a material already of interest for topological superconductivity research.

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