🔬 condensed matter

Second Chern Number Protected Gapless Vortex Modes in Superconducting Weyl Semimetals

This paper demonstrates that superconducting Weyl semimetals host $\vert n\chi \vert$ robust gapless vortex modes protected by a second Chern number in a synthetic four-dimensional space, arising from an imbalanced pumping of flux bound states between Chern insulators at opposite $k_z$ momenta.

Original authors: Zhicheng Zhang, Jiangping Hu

Published 2026-10-20

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CC BY 4.0

Original authors: Zhicheng Zhang, Jiangping Hu

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 architecture of certain solid materials, electrons do not simply flow like water in a pipe; they move in ways that are dictated by the deep geometry of the material itself. Some of these materials, known as topological insulators and semimetals, possess a unique property where their interior acts as an insulator, blocking the flow of electricity, while their surface or edges conduct it perfectly. This behavior is protected by mathematical rules that are so robust they cannot be easily broken by impurities or defects. When scientists combine these exotic materials with superconductors—materials that conduct electricity with zero resistance—they create a new hybrid state. In this state, the electrons pair up and move in a coordinated fashion that can trap special particles called Majorana zero modes. These particles are of intense interest because they could serve as the building blocks for quantum computers that are immune to the errors that plague current technology. The challenge has been to find a reliable way to create and control these modes, particularly within the vortex lines that form when a magnetic field penetrates a superconductor.

A team of researchers has now explored how these vortex lines behave inside a specific class of materials called Weyl semimetals, which are known for their unique, cone-shaped energy structures. By studying both linear and quadratic versions of these materials, the scientists discovered a precise rule that determines how many special, gapless energy channels appear inside the vortex. They found that the number of these channels is not random but is directly tied to two specific properties of the material: the strength of the magnetic twist, known as vorticity, and the "handedness" or chirality of the material's internal structure. In their simulations, they observed that if the magnetic twist is one unit and the material's handedness is one, a single channel appears. If the handedness is doubled, two channels appear. If the handedness is reversed, the direction of the flow in these channels flips. This relationship holds true even when the material is not perfectly symmetrical, suggesting that these channels are a fundamental feature of the physics involved rather than a fragile artifact of a perfect crystal.

To understand why this happens, the researchers looked at the material as a stack of two-dimensional layers. In a normal state, these layers act like insulators with a specific topological number. When the material becomes a superconductor and a magnetic vortex is introduced, the researchers found that the system effectively pumps energy states from one layer to another. Because the layers above and below the center of the material have opposite topological properties, this pumping process is unbalanced. This imbalance forces a specific number of energy channels to cross the energy gap, creating the gapless modes. The researchers confirmed this picture by calculating a complex mathematical quantity known as a second Chern number, which acts as a topological fingerprint for the entire system in a four-dimensional space. Their calculations showed that this number perfectly predicts the number of channels observed in the vortex, matching the product of the magnetic twist and the material's handedness.

The team also tested whether these channels would survive if the material lost some of its perfect rotational symmetry, a common occurrence in real-world crystals. They introduced disturbances that broke the symmetry and found that while some of the channels might merge or shift, the core gapless modes remained intact and robust. This resilience is crucial, as it implies that these special states are not just theoretical curiosities but could be stable enough to be observed in actual experiments. The findings provide a clear physical picture of how these exotic states emerge, moving beyond previous studies that focused only on simpler, linear versions of these materials. By establishing that the number and direction of these channels are governed by a simple, robust rule, the work offers a solid foundation for future efforts to engineer and detect Majorana modes in superconducting Weyl semimetals, bringing the dream of fault-tolerant quantum computing one step closer to reality.

Technical Summary: Second Chern Number Protected Gapless Vortex Modes in Superconducting Weyl Semimetals

Problem Statement
The study of vortex bound states (VBSs) in superconducting (SC) topological materials has been a focal point for identifying Majorana zero modes (MZMs) and non-Abelian statistics. While VBSs in SC topological insulators, iron-based superconductors, and SC Dirac semimetals have been extensively investigated, the VBSs in SC Weyl semimetals (WSMs) with time-reversal symmetry (TRS) remain less explored, particularly regarding quadratic Weyl semimetals. Previous theoretical work established that in TRS-symmetric WSMs with BCS pairing, gapless vortex modes appear if specific constant- $k_z$ planes contain a net chirality of Weyl points. However, the precise physical origin of these gapless modes, their characterization in terms of topological invariants, and their robustness against symmetry-breaking perturbations were not fully understood. Specifically, the relationship between the number of gapless modes, the Weyl point chirality ( $\chi$ ), and the vortex vorticity ( $n$ ) lacked a unified topological framework.

Methodology
The authors employed a combination of analytical field theory and numerical lattice simulations to investigate SC linear and quadratic Weyl semimetals.

Model Construction: The authors formulated $k \cdot p$ effective Hamiltonians for linear ( $\chi = 1$ ) and quadratic ( $\chi = -2$ ) Weyl semimetals possessing TRS and specific rotational symmetries ( $C_4$ and $C_6$ , respectively). These models were extended to include superconductivity via the Bogoliubov-de Gennes (BdG) formalism, introducing an $s$ -wave singlet pairing potential with a vortex line along the $z$ -direction carrying vorticity $n$ .
Numerical Simulation: To calculate the VBSs, the authors utilized tight-binding lattice models corresponding to the $k \cdot p$ Hamiltonians. They applied open boundary conditions in the $x$ and $y$ directions while maintaining translational symmetry along $z$ (making $k_z$ a good quantum number). The vortex was modeled by adding a phase factor $e^{in\theta}$ to the pairing potential.
Topological Analysis: The authors derived an effective low-energy BdG Hamiltonian near the Weyl points. By treating the vortex phase $\phi$ as a synthetic dimension, they constructed a four-dimensional parameter space $(q_x, q_y, q_z, \phi)$ . Within this space, they calculated the second Chern number ( $C_2$ ) to characterize the topological nature of the gapless modes.
Physical Mechanism Investigation: To explain the emergence of multiple gapless modes, the authors analyzed the $\pi$ -flux pumping of Chern insulators. They examined the spectral flow of flux-bound states in two-dimensional slices of the WSM at $k_z$ and $-k_z$ as the magnetic flux increases.
Robustness Test: The stability of the gapless modes was tested by introducing rotational symmetry-breaking terms into the Hamiltonian, observing whether the gapless nature of the modes near the Fermi energy persisted.

Key Contributions and Results

Quantification of Gapless Modes: The study demonstrates that for a vortex with vorticity $n$ and a Weyl point with chirality $\chi$ , there are exactly $|n\chi|$ gapless vortex modes localized around the $k_z$ component of each Weyl point. The propagation direction of these modes is determined by the sign of the product $-n\chi$ .
- For linear WSMs ( $\chi=1$ ) with $n=1$ , there is 1 gapless mode per Weyl point.
- For quadratic WSMs ( $\chi=-2$ ) with $n=1$ , there are 2 gapless modes per Weyl point.
- Numerical results confirm that doubling the vorticity ( $n=2$ ) or the chirality magnitude doubles the number of modes.
Topological Characterization via Second Chern Number: The authors identify the product $-n\chi$ as the second Chern number ( $C_2$ ) defined in a synthetic four-dimensional space spanned by the momentum components and the vortex phase. By deriving the effective BdG Hamiltonian and performing the integration over this 4D space, they analytically prove that $C_2 = -n\chi$ . This provides a rigorous topological invariant characterizing the gapless vortex modes.
Physical Origin via Flux Pumping: The paper proposes a physical picture where the gapless modes arise from the "imbalanced pumping" of flux-bound states between Chern insulators at $k_z$ and $-k_z$ . Due to TRS, the Chern numbers at $k_z$ and $-k_z$ are opposite. When flux is inserted, flux-bound states are pumped across the Fermi level. The imbalance in the number of states pumped from occupied to unoccupied (or vice versa) between the $k_z$ and $-k_z$ planes results in a net number of zero-energy modes, quantified by $|n\chi|$ .
Robustness Against Symmetry Breaking: The authors show that the multiple gapless modes are robust against rotational symmetry-breaking perturbations. While such perturbations can gap out the intersections of counter-propagating modes, the primary gapless modes near the Fermi energy remain intact. This indicates that the existence of these modes relies on the underlying topological structure rather than specific rotational symmetries.

Significance
The paper establishes a direct link between the topological properties of Weyl semimetals (chirality) and the vortex physics in their superconducting counterparts (vorticity). By characterizing the gapless vortex modes with the second Chern number, the work provides a unified topological framework for understanding VBSs in TRS-symmetric WSMs. The proposed "flux pumping" picture offers an intuitive mechanism for the emergence of multiple gapless modes, suggesting that this mechanism may be a generic feature applicable to other superconducting topological materials, such as Dirac semimetals and topological insulators. The findings bridge the gap between theoretical predictions of topological invariants and the observable spectral properties of vortex cores in these complex materials.

Technical Summary: Second Chern Number Protected Gapless Vortex Modes in Superconducting Weyl Semimetals

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