🔬 mesoscale physics

Altermagnetism Induced Bogoliubov Fermi Surfaces Form Topological Superconductivity

This paper proposes a novel route to topological superconductivity in altermagnetic topological insulators proximitized by s-wave superconductors, where altermagnetism-induced anisotropic Bogoliubov Fermi surfaces enable the realization and tunable transition of Majorana zero modes via quantum confinement and vortex engineering.

Original authors: Bo Fu, Chang-An Li, Björn Trauzettel

Published 2026-09-03

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CC BY 4.0

Original authors: Bo Fu, Chang-An Li, Björn Trauzettel

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 the next generation of quantum computers, scientists are hunting for a very specific kind of particle: the Majorana zero mode. These elusive entities are not ordinary particles but rather "quasiparticles" that emerge in certain materials, behaving as if they are their own antiparticles. This unique property makes them incredibly stable and resistant to the environmental noise that usually destroys delicate quantum information. For years, researchers have tried to trap these modes by combining superconductors, which conduct electricity without resistance, with materials that have strong magnetic properties. However, this approach has faced a significant hurdle. Traditional magnets, like the ones found in a refrigerator, create stray magnetic fields that tend to crush the superconducting state, making it difficult to keep the two necessary ingredients working together. Furthermore, the complex magnetic arrangements required to create these modes often demand precise external magnetic fields, adding layers of difficulty and instability to the experiment.

A new study proposes a way to bypass these obstacles by utilizing a recently discovered class of materials called altermagnets. Unlike standard magnets, altermagnets possess a peculiar internal structure where the magnetic spins are arranged in a way that cancels out the overall magnetic field, leaving no stray fields to disrupt superconductivity. Yet, despite having zero net magnetism, these materials still split the energy levels of electrons based on their momentum, a feature that is crucial for manipulating quantum states. The researchers, working with theoretical models, suggest that by placing an altermagnet in contact with a conventional superconductor, they can create a hybrid system that naturally hosts these stable Majorana modes without needing any external magnetic fields. This work opens a new path toward engineering topological superconductivity, a state of matter that could serve as the foundation for fault-tolerant quantum computing.

The team, led by physicists at Great Bay University and the University of Würzburg, focused on a specific type of material known as a topological insulator, which conducts electricity only on its surface while acting as an insulator in its interior. They proposed adding a layer of altermagnetic order to this insulator and then bringing it close to a superconductor. In this setup, the altermagnetic order acts like a subtle, internal switch that shifts the energy of electrons depending on the direction they are moving. When the superconductor pairs up electrons to form a superconducting state, this directional shift prevents the electrons from pairing up in the usual way on certain surfaces. Instead of opening a full energy gap that blocks all movement, the system forms what the researchers call "Bogoliubov Fermi surfaces." These are essentially lines or surfaces within the material where electrons can still move freely, even though the rest of the material is in a superconducting state.

The researchers found that the behavior of these surfaces is highly dependent on the specific crystal face of the material. On some sides of the crystal, the altermagnetic order shifts the electron energy significantly, creating these gapless surfaces. On other sides, the effect vanishes, and the material remains fully gapped. This difference is the key to the discovery. By shaping the material into a thin, triangular prism wire, the researchers showed that the electrons confined within this narrow geometry would be forced to interact with these gapless surfaces in a specific way. As the strength of the internal altermagnetic order is adjusted, the system undergoes a topological phase transition. In this new phase, the free-moving electrons are forced to localize at the very ends of the wire, forming the sought-after Majorana zero modes. This process is driven entirely by the internal properties of the material and the geometry of the wire, requiring no external magnetic fields to trigger the transition.

Beyond simple wires, the study also explored what happens when a magnetic vortex—a swirling line of magnetic flux—is introduced into the material. In conventional systems, Majorana modes are expected to sit right at the center of such a vortex. However, the researchers discovered that in their altermagnetic system, the internal magnetic order acts as a powerful control knob. By tuning the strength of the altermagnetic order, they could force the Majorana modes to migrate. Instead of staying at the center of the vortex, the modes would shift and relocate to the physical outer boundaries of the material. This ability to move these quantum states from the core of a vortex to the surface of the material offers a new method for manipulating them. The researchers suggest that this spatial shift could be observed using scanning tunneling microscopy, a technique that allows scientists to image individual atoms and electrons on a surface.

The study relies on theoretical models and computer simulations to demonstrate that these effects are possible. The researchers identified specific materials, such as a compound containing europium, indium, and arsenic, which are known to exhibit the necessary magnetic structures. They calculated that the energy scales required to observe these phenomena are well within the range of current experimental capabilities. The proposed mechanism does not require fine-tuning of parameters to extreme precision; rather, it relies on the natural hierarchy of energy scales found in these materials. The work suggests that by combining altermagnets with superconductors, scientists can create a robust platform for generating and controlling Majorana zero modes. This approach eliminates the need for external magnetic fields, which are often a source of instability, and provides a clean, controllable environment for these exotic quantum states. The findings offer a clear roadmap for experimentalists to test these ideas, potentially bringing the dream of stable quantum computing a step closer to reality.

Technical Summary: Altermagnetism Induced Bogoliubov Fermi Surfaces Form Topological Superconductivity

Problem Statement
The discovery of altermagnets (AMs)—magnetic phases with zero net magnetization but momentum-dependent spin splitting—has opened new avenues for manipulating electronic states without the detrimental stray fields associated with ferromagnets. While significant progress has been made in understanding AM-superconductor (SC) hybrids regarding bulk phenomena (e.g., Andreev reflection, Josephson effects), the influence of altermagnetism on topological surface states remains largely unexplored. A critical open question is whether the unique anisotropic features of AMs can be harnessed to engineer topological surface states that yield topological superconductivity and Majorana zero modes (MZMs) without external magnetic fields.

Methodology
The authors propose a theoretical framework based on a 3D lattice model of an altermagnetic topological insulator (AMTI) proximitized by an $s$ -wave superconductor.

Hamiltonian Construction: The system is described using a Bogoliubov–de Gennes (BdG) Hamiltonian in the Nambu basis. The normal-state Hamiltonian incorporates a $d$ -wave altermagnetic order term, $t_{AM}(\cos k_x - \cos k_y)\rho_z \sigma_z$ , which breaks time-reversal symmetry ( $T$ ) and four-fold rotation symmetry ( $C_{4z}$ ) individually but preserves their combined operation ( $C_{4z}T$ ).
Phase Diagram Analysis: The authors analyze the topological phase diagram of the AMTI in the parameter space of mass ( $m$ ) and altermagnetic strength ( $t_{AM}$ ), identifying phases including strong/weak topological insulators (STI/WTI), normal insulators (NI), and topological nodal-ring semimetals (TNRS).
Surface State Modeling: Using a $k \cdot p$ expansion near high-symmetry points, the authors derive effective Hamiltonians for specific crystal facets (e.g., (100), (010), (110)). They demonstrate that the altermagnetic order acts as a momentum-dependent Zeeman field, inducing anisotropic shifts in Dirac cones on specific facets while leaving others intact.
Superconducting Proximity: The introduction of $s$ -wave pairing is analyzed to determine the formation of Bogoliubov Fermi surfaces (BFSs). The authors calculate local density of states (LDOS) and energy spectra for quasi-1D geometries (triangular prism nanowires) and vortex line configurations using tight-binding numerical diagonalization and analytical low-energy effective theories.
Topological Invariants: The topological nature of the phases is characterized using $Z_2$ invariants for quasi-1D systems and winding numbers ( $w_{3D}$ , $w_{1D}$ ) for bulk and vortex systems, considering symmetries such as particle-hole symmetry ( $\Xi$ ) and chiral symmetry ( $C$ ).

Key Contributions and Results

Formation of Crystal-Facet-Dependent Bogoliubov Fermi Surfaces (BFSs):
The study reveals that the interplay between altermagnetic spin splitting and $s$ -wave superconductivity suppresses conventional singlet pairing between time-reversed states. This leads to the formation of gapless BFSs. Crucially, these BFSs are anisotropic and facet-dependent:
- On facets where the altermagnetic term acts as a Zeeman field (e.g., (010) and (100) surfaces in the STI phase), the Dirac cones are shifted, leading to gapless BFSs when the altermagnetic strength exceeds a critical value ( $t_{AM} > t_c^s$ ).
- On facets where the projection of the altermagnetic term vanishes (e.g., (110) surfaces), a full superconducting gap remains.
Topological Superconductivity in Quasi-1D Nanowires:
The authors propose a quasi-1D triangular prism nanowire geometry bounded by (110), ( $\bar{1}$ 10), and (010) facets.
- Quantum confinement discretizes the surface BFSs on the (010) facet into energy levels within the superconducting gap.
- The altermagnetic order strength ( $t_{AM}$ ) and the wire width ( $L_x$ ) act as control knobs to drive topological phase transitions.
- In non-trivial phases, the system hosts pairs of MZMs localized at the ends of the nanowire, protected by a gap comparable to the bulk superconducting gap $\Delta$ .
Altermagnetic Control of Vortex-Induced MZMs:
The paper investigates a vortex line along the $z$ -axis in the AMTI/SC hybrid.
- The system exhibits distinct topological phases characterized by winding numbers $(w_{+i}, w_{-i})$ in the presence of the vortex.
- A unique "vortex phase transition" is identified where increasing $t_{AM}$ drives the system from a phase where MZMs are localized at the vortex core to a phase where MZMs are localized at the physical side surfaces.
- This transition is driven by the altermagnetic order shifting the topological character from the vortex line to the surface BFSs, a phenomenon not observed in conventional superconducting TIs.
Material Realization and Feasibility:
The authors identify candidate materials such as $KV_2Se_2O$ , $Rb_{1-\delta}V_2Te_2O$ , $Mn_5Si_3$ , and $EuIn_2As_2$ .
- For $EuIn_2As_2$ , the authors extend the analysis to a potential $g$ -wave altermagnetic topological insulator phase in the $afmc$ magnetic configuration, demonstrating that similar mechanisms (momentum-dependent Dirac cone shifts and BFS formation) apply, further broadening the scope of the proposal.
- Energy scale analysis suggests that the required hierarchy ( $t_c^s < t_{AM} < t_c^b$ ) is naturally satisfied in these materials, where $t_c^s$ is pairing-limited and $t_c^b$ is bandwidth-limited.

Significance and Claims
The paper claims to provide a fundamentally new avenue for realizing topological superconductivity by exploiting the unique interplay between $d$ -wave (and potentially $g$ -wave) altermagnetism, topology, and $s$ -wave superconductivity. The significance of this work lies in three main aspects:

Stray-Field-Free Operation: The proposal utilizes the zero net magnetization of AMs, eliminating the need for external magnetic fields and avoiding the stray fields that typically suppress superconductivity and hinder scalability.
Anisotropy-Driven Engineering: It demonstrates that crystal-facet-dependent anisotropy can be used to engineer topological surface states, a mechanism distinct from previous proposals relying on uniform magnetic fields or ferromagnetic order.
Spatial Control of MZMs: The work introduces a novel mechanism for manipulating MZMs, showing that altermagnetic order can drive a topological phase transition that relocates MZMs from vortex cores to physical boundaries. This offers a new degree of freedom for controlling Majorana modes in topological quantum devices.

The authors conclude that their predictions, including facet-dependent superconducting gaps and the spatial shift of MZMs, are experimentally accessible via scanning tunneling microscopy/spectroscopy (STM/STS) or angle-resolved photoemission spectroscopy (ARPES) in heterostructures combining AMTIs and conventional superconductors.

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