🔬 materials science

Single Pair of Charge-two Weyl Fermions in Chiral Boron Allotropes

This study identifies two stable nonmagnetic chiral boron allotropes, HDSBC-B $_{20}$ and CR-B $_{12}$ , as the first electronic materials to host a single pair of charge-two Weyl fermions by leveraging the interplay between time-reversal and crystallographic rotation symmetries to circumvent conventional node-quartet constraints.

Original authors: Hui-Jing Zheng, Yan Gao, Yanfeng Ge, Yong Liu, Zhong-Yi Lu

Published 2026-02-27

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CC BY 4.0

Original authors: Hui-Jing Zheng, Yan Gao, Yanfeng Ge, Yong Liu, Zhong-Yi Lu

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

Imagine the world of solid materials as a vast, bustling city. In this city, electrons are the citizens, and they usually follow strict traffic laws, moving in predictable lanes. But sometimes, in special materials called Weyl Semimetals, the traffic laws break down. The electrons behave like massless, ghostly particles called Weyl fermions, and the city map (momentum space) develops "hubs" or "portals" where these particles can teleport. These portals are called Weyl Points.

For a long time, physicists had a major headache trying to build the simplest version of this city.

The Problem: The "Four-Door" Rule

In most non-magnetic materials, nature has a strict rule (the Nielsen-Ninomiya theorem): you can't have just one pair of these portals. If you open one door, you are forced to open three others to keep the balance. It's like trying to build a house with only two doors; the universe insists you must have at least four. This makes studying the "purest" form of these particles very difficult because the extra doors create too much noise and confusion.

Until now, the only way to get a single pair of doors was to use magnetic materials. But magnets are picky; they often only work at freezing temperatures or require very specific, delicate setups.

The Solution: A New Kind of Boron City

This paper introduces two new, stable forms of Boron (a light element found in pencils and glass) that act as the "perfect city" for these particles. The researchers found two specific crystal structures:

HDSBC-B20: A chiral (handed) structure that looks like a twisted spiral staircase made of boron chains.
CR-B12: A cage-like structure made of boron icosahedrons (like tiny, hollow soccer balls).

These materials are special because they are non-magnetic (they work at room temperature) and light (they don't have heavy atoms that mess up the physics).

The Magic Trick: The "Charge-Two" Portal

In these boron crystals, the researchers found a way to break the "Four-Door" rule. Instead of having four weak portals, they created one single pair of "Super Portals."

Think of a normal Weyl point as a gentle hill where a ball rolls down in a straight line. These new "Charge-Two" Weyl points are like a double-helix slide.

If you go straight up or down the slide, it's a straight line (linear).
If you try to go sideways, the slide curves like a parabola (quadratic).

Because these "Super Portals" carry double the charge, they are much stronger and more stable. The crystal's rotation symmetry acts like a guardian, locking exactly one pair of these portals in place and forbidding any others from appearing.

The "Handedness" Connection

One of the coolest features of the spiral boron (HDSBC-B20) is its connection to chirality (handedness).

Imagine a left-handed screw and a right-handed screw.
In this material, if you build the crystal as a left-handed screw, the "Super Portal" will have a negative charge.
If you build it as a right-handed screw, the charge flips to positive.

It's as if the physical shape of the building dictates the direction of the traffic flow inside. This gives scientists a new way to control these particles just by twisting the material's structure.

The "Fingerprints": Fermi Arcs

How do we know these portals exist? The paper predicts that on the surface of these crystals, the electrons will form Fermi Arcs.

Imagine the bulk of the crystal is a 3D sphere. The Weyl points are inside.
On the surface, the electrons don't form a closed loop like a normal ring; instead, they form a long, open bridge (an arc) connecting the two portals.
Because these are "Charge-Two" portals, the bridges are extra long, stretching all the way across the surface. They are like massive, glowing highways that are impossible to miss if you look at the material with the right microscope.

Why This Matters

This discovery is a game-changer for three reasons:

Simplicity: It's the first time we've found a non-magnetic material with the absolute minimum number of Weyl points (just one pair). It's the "cleanest" lab for studying these exotic particles.
Accessibility: Since it's made of light boron and isn't magnetic, it works at room temperature. We don't need giant fridges to study it.
Control: The link between the crystal's twist (chirality) and the particle's charge opens the door to new types of electronics and optical devices that could be controlled by the material's shape.

In short, the researchers have built a new, stable, and simple "playground" in the world of boron where the most fundamental rules of quantum physics can be observed clearly, without the noise of magnets or heavy atoms.

1. Problem Statement

The realization of a minimal Weyl semimetal (WSM)—specifically one hosting only a single pair of Weyl points (WPs)—has been a major challenge in condensed matter physics.

The Constraint: In nonmagnetic systems, Time-Reversal Symmetry ( $\mathcal{T}$ ) enforces that Weyl points appear in pairs at $\mathbf{k}$ and $-\mathbf{k}$ with identical chirality. To satisfy the Nielsen-Ninomiya no-go theorem (net topological charge neutrality), a minimum of four WPs (two pairs) is required in nonmagnetic materials.
Current Limitations:
- Magnetic Systems: Can host a single pair by breaking $\mathcal{T}$ , but this often requires low temperatures or specific magnetic ordering.
- Bosonic Systems: Single-pair WPs have been realized in photonic/phononic crystals, but these lack electronic degrees of freedom for charge transport.
- Electronic Systems: Previous attempts to find minimal WSMs in nonmagnetic electronic materials have failed, often due to strong Spin-Orbit Coupling (SOC) causing Kramers degeneracy or the presence of multiple WP pairs and trivial Fermi pockets.
The Goal: Identify a nonmagnetic, light-element electronic material that hosts a single pair of double-Weyl points (topological charge $|C|=2$ ) in the spinless regime, circumventing the "four-node" barrier.

2. Methodology

The authors employed a combination of first-principles calculations and symmetry analysis:

Materials Screening: Focused on boron allotropes due to boron's light atomic mass (intrinsically weak SOC), which allows the system to be treated in the spinless regime ( $\mathcal{T}^2 = +1$ ).
Computational Tools:
- DFT: Used VASP with the Projector Augmented Wave (PAW) method and PBE-GGA exchange-correlation functional.
- Stability Checks: Verified dynamical stability via phonon dispersions (PHONOPY), thermal stability via ab initio molecular dynamics (AIMD), and mechanical stability via elastic stiffness constants.
- Topological Analysis: Calculated band structures, projected density of states (PDOS), Berry curvature, and topological charges (Chern numbers) using WannierTools and maximally localized Wannier functions.
- Effective Modeling: Constructed low-energy $\mathbf{k} \cdot \mathbf{p}$ effective Hamiltonians to confirm the symmetry protection and dispersion relations of the Weyl points.

3. Key Contributions

The study identifies two stable, nonmagnetic boron allotropes as the first electronic materials to realize a minimal single-pair double-Weyl configuration:

HDSBC-B20: A chiral tetragonal structure ( $P4_3$ or $P4_1$ ) featuring interconnected single-atom-width spiraling chains and double-atom-width helical chains.
CR-B12: A rhombohedral cage-like structure ( $R32$ ) composed of a new type of $B_{12}$ icosahedral cages, distinct from the conventional $\alpha$ - $B_{12}$ .

Novel Mechanism: The authors demonstrate that the interplay between Time-Reversal Symmetry ( $\mathcal{T}$ ) and specific crystallographic rotation symmetries ( $C_4$ for HDSBC-B20 and $C_3$ for CR-B12) stabilizes exactly one pair of charge-2 WPs pinned at Time-Reversal-Invariant Momenta (TRIMs). This mechanism circumvents the conventional requirement for four nodes.

4. Key Results

A. Structural and Thermodynamic Stability

HDSBC-B20: Dynamically stable (no imaginary phonon modes) and thermally stable up to 300 K.
CR-B12: Dynamically stable and thermally stable up to 900 K. Its formation energy (-6.52 eV/atom) is lower than many proposed topological boron phases and even lower than experimentally synthesized $\alpha$ -Ga boron, suggesting high synthetic feasibility.

B. Electronic Structure and Topology

Spinless Regime: Due to weak SOC in boron, the system is accurately described without SOC. Explicit SOC calculations show only negligible gaps (0.08–1.19 meV) at the WPs, preserving the topological nature at experimental temperatures.
Minimal Node Count: Both materials exhibit band crossings exclusively at two distinct TRIMs (one pair of WPs), with a finite gap everywhere else.
- HDSBC-B20: WPs located at $\Gamma$ ( $W_1$ ) and $M$ ( $W_2$ ).
- CR-B12: WPs located at $\Gamma$ ( $W_1$ ) and $T$ ( $W_2$ ).
Double-Weyl Nature ( $|C|=2$ ):
- Dispersion: The bands exhibit linear dispersion along the rotation axis ( $k_z$ ) and quadratic dispersion in the perpendicular plane ( $k_x-k_y$ ).
- Topological Charge: Integration of Berry curvature confirms charges of $C = \pm 2$ .
- Symmetry Protection: The degeneracy is protected by $C_4$ (for HDSBC-B20) or $C_3$ (for CR-B12) rotation combined with $\mathcal{T}$ .

C. Surface States and Chirality

Double Fermi Arcs: The materials host exceptionally long double Fermi arcs spanning the surface Brillouin zone, connecting the projections of the WPs with opposite charges. These are direct experimental signatures.
Chirality Control (HDSBC-B20): The sign of the topological charge is strictly dictated by the structural handedness (left- vs. right-handed enantiomers). Inverting the crystal structure reverses the chirality of the WPs ( $C = -2 \leftrightarrow +2$ ), establishing a direct link between real-space geometry and momentum-space topology.

5. Significance

Paradigm Shift: This work moves the search for minimal Weyl semimetals from magnetic or bosonic systems to nonmagnetic, light-element electronic platforms.
Experimental Accessibility: The predicted materials are stable at room temperature, have clean electronic structures (no trivial Fermi pockets), and possess long Fermi arcs, making them ideal candidates for experimental verification via ARPES and STM.
Fundamental Physics: It validates the theoretical possibility of a single-pair $|C|=2$ Weyl configuration in electronic systems, providing a "conceptually clean" setting to study bulk-boundary correspondence and unconventional topological phenomena (e.g., enhanced circular photogalvanic effects).
Application Potential: The direct correlation between structural chirality and topological charge in HDSBC-B20 opens new avenues for chirality-dependent optoelectronic devices and transport phenomena.