🔬 mesoscale physics

Multiple Majorana zero modes realization based on superconducting topological crystalline metal ZrRuAs

This paper proposes a feasible platform for realizing multiple Majorana zero modes by integrating the superconducting topological crystalline metal ZrRuAs with a ferromagnetic insulator, which leverages mirror symmetry and spin splitting to generate a topological superconducting phase with a Chern number of -2.

Original authors: Xiaoxu Wang, Jinyu Zou, Gang Xu

Published 2026-09-14

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CC BY 4.0

Original authors: Xiaoxu Wang, Jinyu Zou, Gang Xu

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 quest to build a computer that can solve problems far beyond the reach of today's machines, scientists are looking for a very specific kind of building block. They need a particle that is its own antiparticle, a rare entity known as a Majorana fermion. In the world of solid materials, these particles appear not as fundamental atoms, but as ghostly excitations called Majorana zero modes. These modes are special because they are incredibly stable; they do not easily break apart or lose their information to the noisy environment that plagues current quantum computers. If scientists could gather four of these modes together, they could form a "topological qubit," the basic unit of a fault-tolerant quantum computer that would not require constant error correction. The challenge has always been finding a material that naturally hosts these modes, or creating a setup where they can be reliably generated and controlled. For years, researchers have tried to build artificial structures by stacking different materials, but these interfaces are often messy and difficult to perfect.

A team of researchers has now proposed a new, simpler path forward by identifying a specific material that might hold the key. They focused on a class of substances called superconducting topological crystalline metals. These are unique because they possess two powerful traits at once: they conduct electricity with zero resistance, and they have a special internal structure that protects certain electronic states from being destroyed. The researchers began by creating a theoretical model to show how such a material, when combined with a magnetic influence, could generate not just one, but multiple Majorana zero modes simultaneously. Their calculations suggested that the number of these modes depends on a specific property of the material's symmetry, a kind of geometric fingerprint. If the material has a high enough symmetry value, it should host multiple modes, which is a significant advantage for building complex quantum circuits.

To move from theory to reality, the team turned their attention to a real-world material that had already been synthesized in a laboratory: a compound made of zirconium, ruthenium, and arsenic, known as ZrRuAs. Using powerful computer simulations based on the laws of quantum mechanics, they analyzed the atomic structure and electronic behavior of this crystal. They found that ZrRuAs fits the description of their ideal candidate perfectly. It is a metal that becomes superconducting at very low temperatures, and its internal symmetry protects two distinct "cones" of electrons on its surface. These surface cones are the playground where the Majorana modes would appear. The simulations showed that the material has a specific symmetry value of two, which theoretically predicts the existence of two pairs of these special modes.

The final piece of the puzzle involved figuring out how to activate these modes. In its natural state, the material is not yet in the right phase to host the modes. The researchers proposed placing the ZrRuAs crystal next to a magnetic insulator, a material that is magnetic but does not conduct electricity. They selected a specific magnetic insulator made of gadolinium and iodine for their model. When these two materials are placed together, the difference in their electrical properties shifts the energy levels of the electrons on the surface of the ZrRuAs. This shift, combined with the magnetic influence from the insulator, breaks a specific symmetry in the system and opens a gap in the electron energy spectrum. The team calculated that under these conditions, the material enters a new state of matter: a topological superconductor. In this state, the two surface cones transform, giving rise to two distinct branches of Majorana zero modes.

The researchers mapped out the exact conditions required to reach this state, showing that it is achievable with realistic magnetic fields and electrical controls. They found that the energy levels needed to trigger this transformation are very close to the natural energy of the electrons in the material, meaning the setup does not require extreme or impossible adjustments. Because the magnetic direction of the insulator can be easily tuned, scientists could potentially control the behavior of these modes with precision. This work suggests that the combination of ZrRuAs and a magnetic insulator offers a clean, controllable platform for studying these elusive particles. By providing a material that naturally hosts multiple modes without the need for complex, multi-layered artificial structures, this discovery could accelerate the development of stable quantum computers, turning a long-standing theoretical possibility into a tangible experimental reality.

Technical Summary: Multiple Majorana Zero Modes Realization Based on Superconducting Topological Crystalline Metal ZrRuAs

Problem Statement
The realization of fault-tolerant quantum computing relies heavily on Majorana zero modes (MZMs), which possess non-Abelian braiding statistics. While topological superconductors are the primary platform for MZMs, natural examples are scarce. Existing artificial strategies face significant hurdles: heterostructures (e.g., superconductor/topological insulator) suffer from interfacial complexity and pairing mechanism uncertainties, while intrinsic superconducting topological metals (like Fe(Se,Te)) typically host only single MZMs in magnetic vortices. Although proximity-coupled topological crystalline insulators have been proposed to host multiple MZMs, experimental realization remains hindered by interface complexity. Consequently, there is a critical need for a material platform that intrinsically combines crystalline symmetry-protected topological bands with superconductivity to facilitate the realization of multiple MZMs with enhanced controllability.

Methodology
The authors employ a dual approach combining minimal model analysis and first-principles calculations:

Minimal Model Construction: A 3D tight-binding model is constructed on a trigonal lattice with mirror symmetry ( $M_{1\bar{1}0}$ ). The model incorporates $s$ -wave superconductivity, spin-orbit coupling (SOC), and a time-reversal symmetry ( $T$ ) breaking perturbation (spin splitting $B_z$ ). The authors analyze the Bogoliubov–de Gennes (BdG) Hamiltonian to determine the topological phase transitions, specifically focusing on the relationship between the mirror Chern number ( $C_M$ ) of the normal state and the resulting BdG Chern number ( $N$ ) in the superconducting state.
First-Principles Calculations: Density Functional Theory (DFT) calculations are performed using the Vienna ab initio Simulation Package (VASP) with the PBE-GGA functional. The study identifies the experimentally synthesized superconductor ZrRuAs as a candidate. The electronic structure is analyzed to confirm the presence of topological bands and surface Dirac cones.
Heterostructure Simulation: To induce the necessary symmetry breaking, a heterostructure of Ferromagnetic Insulator (FMI) and ZrRuAs is modeled. The work function difference between the materials (specifically GdI $_2$ and ZrRuAs) is used to calculate the induced onsite energy shift. A slab BdG Hamiltonian is constructed to map the topological phase diagram in the space of chemical potential ( $\tilde{\mu}$ ) and spin splitting ( $B_z$ ).

Key Contributions and Results

Theoretical Mechanism for Multiple MZMs: The model analysis demonstrates that the interplay between $s$ -wave superconductivity, mirror symmetry-protected multiple surface Dirac cones, and spin splitting leads to a high BdG Chern number. Specifically, the topological superconducting phase is characterized by $N = \pm C_M$ , where $C_M$ is the mirror Chern number of the underlying topological crystalline metal. For a system with $C_M = 2$ , the resulting phase supports two branches of MZMs.
Identification of ZrRuAs: First-principles calculations identify ZrRuAs as a superconducting topological crystalline metal (STCM). It possesses a mirror Chern number $C_M = 2$ protected by $M_{1\bar{1}0}$ symmetry, hosting two symmetry-protected surface Dirac cones on the (001) surface. The material exhibits intrinsic $s$ -wave superconductivity with a transition temperature of $\approx 7.9$ K and a gap of $\approx 1.2$ meV.
Realization in FMI/ZrRuAs Heterostructure: The study proposes a GdI $_2$ /ZrRuAs heterostructure. The work function difference (GdI $_2$ at 3.71 eV vs. ZrRuAs at 4.81 eV) induces an onsite energy shift of approximately 0.55 eV, moving the surface Dirac cones to 77 meV relative to the Fermi level.
Topological Phase Diagram: Calculations of the slab BdG Hamiltonian reveal a topological superconducting phase with $N = -2$ when the spin splitting $B_z$ exceeds the critical boundary defined by $B_z = \sqrt{\tilde{\mu}^2 + \Delta^2}$ . For the specific parameters of ZrRuAs ( $\Delta = 1.2$ meV), this phase is accessible with moderate magnetic fields. The phase boundary is confirmed via Wilson loop calculations.

Significance and Claims
The paper claims that the proposed FMI/STCM heterostructure offers a feasible platform for investigating multiple MZMs and realizing novel topological qubits. The significance lies in:

Structural Simplicity: Unlike complex multi-layer heterostructures, this approach utilizes an intrinsic superconductor (ZrRuAs) coupled with a simple ferromagnetic insulator.
Tunability: The topological phase can be controlled by adjusting the chemical potential (via gate voltage) or the magnetization direction in the FMI, which can break mirror symmetry to provide precise control over MZM evolution.
Generalizability: The authors note that the mechanism is generalizable to any mirror Chern number $C_M$ . As long as the combined mirror and time-reversal symmetry ( $M_T$ ) is preserved on the surface, the system can host $N = \pm C_M$ chiral topological superconducting phases, corresponding to $Z$ multiple Majorana modes at the boundaries.

The work establishes ZrRuAs as a promising candidate material for experimental realization, leveraging its intrinsic properties to overcome the scarcity of natural topological superconductors capable of hosting multiple MZMs.

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