On March 27, 1938, Italian physicist Ettore Majorana vanished without a trace at the age of 31. Intellectually brilliant yet emotionally conflicted, Majorana was considered a genius. His professor, Enrico Fermi, winner of the 1938 Nobel Prize in Physics, placed him on the same level of greatness as Isaac Newton (1642–1727). The previous year, in 1937, Majorana published the article "Teoria simmetrica dell'elettrone e del positrone." It was one of the few texts by Majorana that survived his tendency to tear up and throw away everything he wrote. In the article, which offered a solution to the Dirac equation (which unites quantum mechanics and special relativity and predicts the existence of antimatter), the young Italian physicist presented the idea of a particle identical to its antiparticle. This particle-antiparticle pair is currently known as Majorana fermions.
The 1937 paper was read in small circles of experts and considered mathematically elegant but disconnected from the real world. Its importance only became clear in the 1950s and 1960s in the context of neutrino physics and, especially, in condensed matter physics from the 2000s onward. Researchers have found that, while Majorana fermions have not been confirmed by experiment, an analog could arise in solid materials, particularly in certain types of superconductors. This analogue is not composed of real particles but of quasiparticles, or collective excitations of the system, which behave as if the Majorana fermion were present.
Majorana states typically appear at the ends of superconducting wires or chains, which are materials that conduct electric current without resistance below a critical temperature. The electronic excitations are mathematically decomposed into two "half-fermions," each located at one end. The non-local quantum conjugate that they form corresponds to a zero-energy state in that it does not alter the total energy of the system.
Today, Majorana bound states are one of the central pillars of research in topological quantum computing. The search for quantum computers that can operate reliably, even in the presence of noise, defects, and environmental fluctuations, has led physicists from different fields to study these unusual quantum excitations. These excitations are of interest because they could form the basis of topological qubits, an alternative quantum computing architecture that promises greater robustness against external disturbances.
"In conventional quantum platforms, information is encoded in local degrees of freedom and, as a result, becomes extremely sensitive to microscopic imperfections, leading to rapid loss of coherence. In systems that host Majorana states, however, quantum information is stored non-locally, distributed across spatially separated regions of the device and protected by the system's global topological properties, reducing dependence on local details and making these states particularly promising candidates for implementing more stable qubits," says Poliana Heiffig Penteado , a researcher at the São Carlos Institute of Physics at the University of São Paulo (IFSC-USP) in Brazil.
Penteado, together with José Carlos Egues de Menezes , a full professor at the IFSC-USP, and former doctoral student Rodrigo Abreu Dourado , coordinated a study published as an "Editors' Suggestion" in the journal Physical Review B.
The study investigated methods for creating and stabilizing Majorana states for potential applications in quantum computing. "We studied a theoretical model known as a Kitaev chain, which can be implemented in practice using arrays of quantum dots coupled to superconductors. In very short chains, especially those with only two quantum dots, it's possible to obtain states similar to Majorana states, but only under specific conditions involving the fine-tuning of system parameters. The main objective of the study was to understand what happens when we increase the number of quantum dots in the chain. The study demonstrated that what previously appeared as an isolated point of stability gradually evolves into a 'topological island,' an extensive region of parameters where Majorana states remain protected," Egues reports.
Currently, there is an intense international debate on how to unequivocally distinguish genuine Majoranas from trivial quantum excitations that produce similar signals in experiments. This study contributes to that debate. It starts with a theoretical model known as the Kitaev chain. Russian physicist Alexei Kitaev proposed this model in 2001. Kitaev was born in Moscow in 1962 and is currently based at the California Institute of Technology (Caltech). Kitaev showed that a one-dimensional system of electrons coupled by superconducting pairing can enter a topological phase in which an electron fermion decomposes mathematically into two spatially separated Majorana modes located at the ends of the system. The non-local quantum state formed by this pair has zero energy relative to the ground state. The Kitaev chain provides the conceptual foundation for producing topological qubits that are protected against local perturbations.
"The problem is that, in very short chains, such as those formed by just two quantum dots, these states only appear under extremely specific conditions, known as sweet spots. Any small fluctuation in the system's parameters causes the energy of these states to no longer be exactly zero, destroying the desired protection and making experimental observation impossible," says Egues.
The researchers' idea was to increase the number of quantum dots in the chain and observe the outcome. "We showed that, as the chain grows longer, the sweet spots cease to be isolated points and begin to cluster together, forming a continuous region in space. For sufficiently long chains with about 20 quantum dots or more, this region becomes a true topological island. Within it, Majorana states remain strictly at zero energy, well-localized at the ends of the chain and resistant even to the presence of random fluctuations in the system's parameters. This behavior marks the transition from a fragile regime to a genuinely topological regime, in which protection no longer depends on fine-tuning," Penteado explains.
In addition to theoretically characterizing this transition, the study proposed a concrete method for detecting topological islands in experiments. Inspired by a previous study by the IFSC-USP group that received support from FAPESP and has become a classic reference in the field, the researchers coupled a quantum dot connected to metal contacts laterally to the chain (see Figure 1) and measured the electrical conductance of the system (a measure of how easily electric current flows through a material and the inverse of electrical resistance). When a Majorana state is present and protected, the conductance takes on a quantized value, forming a plateau around zero voltage. This plateau serves as a robust electrical signature of the topological state.
Furthermore, the study showed that conductance is directly related to the exchange statistics of Majorana fermions ("braiding"), demonstrating that its square operator, γ₂, is ½ and not zero as with ordinary fermions, which obey the Pauli principle (see Figure 2a). Majorana edge states in condensed matter systems do not follow a Fermi-Dirac distribution.
"The beauty of the result was linking a simple electrical measurement – conductance – to a fundamental property of the particle [its statistics]. If the conductance takes on this quantized value, it indicates that the current is being carried by a Majorana mode, and not by a trivial excitation," argues Egues. Penteado adds that this point is crucial in a research field marked by controversy: "From the very first experiments, it was clear that other phenomena, such as the Kondo effect, can generate similar signals. The challenge has always been to show that the conductance peak actually came from a Majorana. Our work contributes precisely to that distinction."
Although the study is theoretical, it used realistic parameters drawn from recent experiments and aligns directly with international efforts to build topological qubits. Companies like Microsoft have invested heavily in this line of research, hoping that Majoranas could enable more stable quantum computers. The Brazilian researchers' work shows that perfect control of the parameters is not necessary to observe robust Majorana states. Increasing the size of the system is sufficient, and this is already within the reach of current experimental platforms.
