Topological superconductors and Majorana fermions based on $X$-wave magnets with $X=p,d,f,g,i$

In the quiet world of solid materials, electrons usually behave like a chaotic crowd, bumping into one another and scattering in every direction. But under the right conditions, they can synchronize into a single, fluid state known as a superconductor, where electricity flows without any resistance at all. For decades, scientists have known that this state is fragile; it usually collapses the moment a magnetic field is introduced, which tends to disrupt the delicate pairing of electrons. However, a newer class of magnetic materials has emerged that defies this expectation. These materials, called altermagnets and odd-parity magnets, possess a unique internal structure where the magnetic forces cancel out to zero overall, yet they still split the energy levels of electrons based on their spin. This creates a hidden order that does not destroy superconductivity but instead reshapes it, potentially unlocking a new frontier in physics where electricity could carry information in ways that are immune to the noise and errors that plague current technology.

A researcher at the University of Tokyo has now mapped out exactly how these exotic magnets interact with superconductors, revealing a landscape of possibilities that depends entirely on the specific shape of the magnetic order. By building detailed computer models of these materials on both square and triangular grids, the researcher explored what happens when different types of magnets are paired with superconducting electrons. They focused on a family of magnets labeled by the letters p, d, f, g, and i, which describe the complex geometric patterns of their internal magnetic fields. The study found that these magnets do not all behave the same way; instead, they split into two distinct camps with fundamentally different rules. One group, the altermagnets, which includes the d, g, and i types, breaks a fundamental symmetry of time, while the other group, the odd-parity magnets, which includes the p and f types, preserves it. This single difference dictates whether the resulting superconductor will host one kind of exotic particle or another.

When the researcher coupled the superconductor with the altermagnets, they discovered that the magnetic order actively stabilized a swirling, chiral superconducting state. In this state, the electrons pair up in a way that creates a full energy gap, meaning there are no loose electrons to scatter and cause resistance. More importantly, this state is topological, a term that describes a property of the material that is robust against small changes, much like a knot that cannot be untied without cutting the rope. In these systems, the edges of the material become highways for special particles called Majorana fermions. These particles are their own antiparticles and can travel along the edge without scattering. The researcher found that for the d-wave and g-wave altermagnets, this edge state is characterized by a single channel of flow, but for the i-wave altermagnet, the situation is even more remarkable. In the i-wave case, two distinct channels of these edge states emerge simultaneously, a feature that could be crucial for building more complex quantum devices.

In contrast, when the superconductor was paired with the odd-parity magnets, the outcome was different. Because these magnets preserve time-reversal symmetry, they do not force the superconductor into a swirling chiral state. Instead, they stabilize a different kind of topological order where the Majorana particles do not just flow along the edge in a single line but form flat bands of zero-energy states. These flat bands are essentially a wide, stable platform of these exotic particles that exists between the gaps in the material's energy spectrum. The researcher noted that even though the magnetic order in these systems inevitably causes a tiny mixing between the superconducting states, the flat bands remain remarkably robust. This resilience suggests that these states could be reliably created and maintained, even in imperfect real-world conditions.

The study also clarified why some superconducting states disappear in the presence of strong magnetic order while others thrive. For the altermagnets, the traditional s-wave superconductivity, which is the most common and isotropic form, is quickly destroyed as the magnetic strength increases. This happens because the magnetic splitting forces electrons with the same spin to occupy the same momentum states, which contradicts the requirement for the standard electron pairs to have opposite spins. However, the chiral p-wave state, which pairs electrons with the same spin, is not only compatible with this magnetic environment but is actually stabilized by it. The researcher observed that as the magnetic strength grew, the material would transition from a standard superconductor to this exotic chiral state, a shift that was clearly visible in their phase diagrams.

By systematically varying the chemical potential, which controls the number of electrons in the system, and the strength of the interactions between them, the researcher constructed a comprehensive map of these phases. They found that the specific lattice structure of the material, whether it was a square grid or a triangular one, played a decisive role in which superconducting state would emerge. For instance, the f-wave superconductivity, which involves a more complex geometric pattern, was found to be stable only on the triangular lattice. The results confirm that the interplay between these new magnetic orders and superconductivity is not a simple matter of one destroying the other, but rather a rich dialogue that can produce entirely new states of matter. The work provides a theoretical blueprint for identifying which materials might host these elusive Majorana particles, guiding future experiments toward the specific combinations of magnetic order and lattice geometry that are most likely to succeed.