Symmetry-protected four double-Weyl fermions and their topological phase transitions in nonmagnetic crystals

This paper establishes a rigorous symmetry framework identifying 28 space groups that host exactly four double-Weyl fermions, proposes the chiral carbon allotrope THRLN-C$_{32}$ as a material realization, and elucidates its strain-driven topological phase transitions into various exotic or trivial states.

The Gist

Imagine the universe of materials as a giant, bustling city. In this city, electrons are the commuters, and the "roads" they travel on are defined by the material's atomic structure. Usually, these roads are smooth and predictable. But sometimes, in special materials called Weyl semimetals, the roads twist into impossible knots called Weyl points.

Think of a Weyl point as a magical traffic circle where the rules of physics get a little weird. Electrons passing through these circles act like massless particles, moving incredibly fast and behaving in ways that create cool, exotic effects (like one-way streets for electricity).

Here is the story of this paper, broken down into simple concepts:

1. The Problem: Too Much Traffic

For a long time, scientists knew how to build these magical cities, but they always had a problem: too many traffic circles.

• In most materials, you get dozens or even hundreds of these Weyl points scattered everywhere.
• It's like trying to study a single specific car in a massive, chaotic traffic jam. It's hard to see the unique behavior of the electrons because there are too many of them getting in the way.
• Scientists wanted a "minimalist" city with the absolute fewest number of traffic circles possible to study them clearly.

2. The Goal: The "Four-Point" Dream

In a normal city (without magnetic fields), physics rules say you can't have just one or two of these circles; they must come in pairs. The absolute minimum for a standard city is four circles.

• Most materials found so far have exactly four "standard" circles.
• But this paper is about something rarer: Four "Super-Circles."
• These aren't just normal circles; they are Double-Weyl points. Imagine a standard circle is a simple roundabout. A "Double-Weyl" point is a double-decker, spiraling roundabout that is twice as powerful and twists the electrons in a much more complex way.
• Until now, no one knew exactly how to build a city with exactly four of these super-circles, or if it was even possible to keep them from multiplying into a chaotic mess.

3. The Blueprint: The Symmetry Rulebook

The authors acted like master architects. They wrote a strict rulebook (based on mathematical symmetry) to figure out exactly which building codes (crystal structures) allow for exactly four of these super-circles and no more.

• They scanned through 230 different types of building codes (space groups).
• They found that only 28 specific codes allow for this perfect "four super-circle" setup. It's like finding that only 28 specific Lego instructions can build a tower that doesn't fall over.

4. The Discovery: The "THRLN-C32" Crystal

Using this rulebook, they designed a brand-new material made entirely of carbon (the same element as pencil lead and diamonds).

• They named it THRLN-C32.
• Imagine taking a standard carbon nanotube (a tiny straw) and weaving it together with helical (spiral) rings of carbon atoms.
• The result is a 3D, chiral (handed) structure. It comes in two versions: Left-Handed and Right-Handed, like your hands. They are mirror images of each other but cannot be superimposed.
• This material is stable, strong, and perfectly hosts exactly four of those "Double-Weyl" super-circles right at the energy level where electrons flow.

5. The Magic: Closed-Loop Roads

In normal Weyl materials, the surface of the material has "Fermi arcs"—think of them as open roads that start at one point and end at another, like a dead-end street.

• In this new THRLN-C32 material, because the super-circles are so powerful, the surface roads form closed loops.
• Imagine a race track that circles back on itself perfectly. This is a unique signature that scientists can look for to prove they found this special material.

6. The Remote Control: Strain as a Switch

The most exciting part is that this material is like a shape-shifter. The authors showed that if you squeeze or stretch the material (apply "strain"), you can change its entire personality:

• Squeeze it hard (Hydrostatic Pressure): The four super-circles crash into each other and disappear. The material becomes a boring, normal insulator (like plastic).
• Stretch it gently (Uniaxial Strain): The super-circles split apart! They break down into "Three-Terminal Complexes." It's like a double-decker bus splitting into a regular bus and two motorcycles. The material becomes a mix of different exotic particles.
• Stretch it the wrong way (Breaking Symmetry): If you stretch it unevenly, the special "Double" nature is lost. The four super-circles turn into eight regular, standard Weyl points. The magic "double" power is gone, and it becomes a standard Weyl semimetal.

Why Does This Matter?

This paper is a huge step forward because:

1. It provides a map: It tells scientists exactly where to look for these rare materials.
2. It offers a clean lab: By having only four points, scientists can finally study the "pure" physics of these particles without noise.
3. It's tunable: We can use physical pressure to switch the material between different quantum states, which is a dream for future quantum computers and ultra-fast electronics.

In a nutshell: The authors wrote the instruction manual for building a perfect, minimalist quantum city with exactly four super-powered traffic circles, found a carbon-based material that fits the bill, and showed that we can use a "remote control" (stretching the material) to change the traffic patterns at will.

Technical Summary

1. Problem Statement

Weyl semimetals (WSMs) are topological states characterized by Weyl points (WPs) acting as monopoles of Berry curvature. While conventional WPs carry a topological charge (Chern number) of $|C|=1$, higher-order "unconventional" WPs with $|C| \ge 2$ (e.g., double-Weyl points, DWPs) offer amplified topological responses.

• The Challenge: Realizing WSMs with the minimal number of WPs is crucial for isolating intrinsic topological signatures, as most known materials possess a large, complex number of WPs that obscure fundamental physics.
• The Gap: In nonmagnetic (time-reversal invariant) systems, the theoretical minimum for conventional WPs is four. However, the specific symmetry requirements and material realizations for a configuration of exactly four unconventional double-Weyl points ($|C|=2$) remain unresolved. A rigorous symmetry framework to restrict the existence of exactly four DWPs in nonmagnetic crystals has been lacking.

2. Methodology

The authors employed a multi-step approach combining group theory, first-principles calculations, and topological analysis:

• Symmetry Classification: They established rigorous algebraic constraints to stabilize exactly four symmetry-protected DWPs. By analyzing the "encyclopedia of emergent particles" across all 230 space groups (SGs), they identified the specific conditions under which a space group can host exactly four DWPs without generating additional symmetry-related nodes.
• Material Design: Guided by the symmetry constraints, they proposed a novel chiral carbon allotrope, THRLN-C32 (tetragon–hexagon ring-linked nanotube), utilizing $sp^2$-$sp^3$ hybridization.
• First-Principles Calculations: Using Density Functional Theory (DFT) and Density Functional Perturbation Theory (DFPT) (Quantum ESPRESSO), they calculated electronic band structures, phonon dispersions, and elastic constants to verify structural and dynamical stability.
• Topological Characterization: They constructed $k \cdot p$ effective Hamiltonians, calculated Wannier charge centers (WCCs) to determine Chern numbers, and simulated surface states (Fermi arcs) using WannierTools.
• Strain Engineering: They simulated the effects of hydrostatic and uniaxial strains to map the topological phase transitions and evolution of the Weyl nodes.

3. Key Contributions

• Definitive Symmetry Framework: The study establishes that exactly four symmetry-protected $|C|=2$ DWPs in nonmagnetic systems (both spinless and spinful) are mathematically restricted to only 28 specific space groups. This provides a complete search space for ideal double-Weyl semimetals.
• Discovery of THRLN-C32: The authors identified a specific chiral carbon allotrope, THRLN-C32, as an ideal candidate. It crystallizes in chiral tetragonal space groups ($P4_322$ and $P4_122$) and hosts exactly four DWPs located almost precisely at the Fermi level ($E_F$).
• Unified Phase Transition Landscape: They mapped a complete evolution landscape where the pristine four-DWP state can transform into:
  1. Two exotic Three-Terminal Weyl Complexes (TTWCs) under symmetry-preserving strain.
  2. Eight conventional $|C|=1$ WPs under symmetry-breaking strain.
  3. A trivial insulator under extreme compression.

4. Key Results

A. Structural and Electronic Properties of THRLN-C32

• Structure: The material consists of interlinked (2,2) carbon nanotubes, helical tetragonal ring chains, and helical hexagonal ring chains. It exists as left-handed ($l$-THRLN-C32) and right-handed ($r$-THRLN-C32) enantiomers.
• Stability: Phonon spectra show no imaginary frequencies, and elastic constants satisfy Born stability criteria, confirming dynamical and mechanical stability.
• Band Structure: The conduction and valence bands cross exclusively along the high-symmetry $\Gamma$-$Z$ line.
• Double-Weyl Nature: The crossings exhibit linear dispersion along the $k_z$ axis but quadratic dispersion in the $k_x$-$k_y$ plane.
• Chern Numbers: WCC calculations confirm the nodes carry charges of $C = +2$ and $C = -2$. The macroscopic structural chirality dictates the sign of the topological charges (inversion reverses the signs).

B. Topological Surface States

• Closed-Loop Fermi Arcs: Unlike the open arcs of conventional WSMs, the projection of the four DWPs onto the (100) and (110) surfaces creates extended or closed-loop Fermi arcs.
  • On the (100) surface, arcs form longitudinally connected closed loops.
  • On the (110) surface, nearest-neighbor pairing creates compact, independent closed Fermi rings.
• Experimental Accessibility: The DWPs are located within $\pm 12$ meV of the Fermi level, making these exotic surface features directly observable via Angle-Resolved Photoemission Spectroscopy (ARPES) without doping.

C. Strain-Driven Phase Transitions

The system exhibits a hierarchical stability dictated by $C_4$ rotational symmetry:

1. Symmetry-Preserving Strain (Hydrostatic -7 GPa or Uniaxial +6% along c-axis):
   • The $C_4$ symmetry is maintained.
   • The four DWPs dissociate into two Three-Terminal Weyl Complexes (TTWCs).
   • Each complex consists of one $|C|=2$ node flanked by two $|C|=1$ nodes.
   • Surface states evolve into superimposed double arcs and quadruple spiral arcs.
2. Symmetry-Breaking Strain (Uniaxial +2% along a/b-axis):
   • The $C_4$ symmetry is broken.
   • The four DWPs degenerate into eight conventional $|C|=1$ WPs located at generic $k$ points.
   • Surface states revert to standard, unclosed open Fermi arcs.
3. Extreme Compression (Hydrostatic +25 GPa):
   • The nodes annihilate, opening a global bandgap and driving the system into a trivial insulator.

5. Significance

• Theoretical Framework: This work provides the first rigorous classification for minimal double-Weyl semimetals in nonmagnetic crystals, restricting the search to 28 space groups. This framework is applicable not only to electronic systems but also to bosonic quasiparticles (phonons and photons).
• Material Platform: THRLN-C32 serves as a pristine, experimentally accessible platform for studying chiral topological physics, offering a clear signature of macroscopic chirality linked to momentum-space topology.
• Dynamic Tunability: The demonstration of "topological fractionalization" (splitting higher-order charges into lower-order complexes) via strain offers a new mechanism for dynamically reconfiguring quantum states.
• Experimental Impact: The unique closed-loop Fermi arcs and the proximity of the WPs to the Fermi level provide unambiguous spectroscopic signatures, facilitating the experimental verification of unconventional Weyl fermions.