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Majorana Edge-to-bound State Transition in Interferometric Superconductor Rings

This paper demonstrates a strategy to drive topological edge states into vortex bound states in 2M-WS $_2$ superconducting rings by tuning the coherence length relative to the annulus width, a transition confirmed by Little-Parks oscillations and simulations that offers new insights for flux manipulation in topological quantum computing.

Original authors: Xiangyu Bi, Aoshen Yang, Hongzi Cao, Ganyu Chen, Yuqiang Fang, Zeya Li, Qizheng Sun, Junwei Huang, Da Wang, Fuqiang Huang, Toshiya Ideue, Qianghua Wang, Yoshihiro Iwasa, Hongtao Yuan

Published 2026-10-26

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CC BY 4.0

Original authors: Xiangyu Bi, Aoshen Yang, Hongzi Cao, Ganyu Chen, Yuqiang Fang, Zeya Li, Qizheng Sun, Junwei Huang, Da Wang, Fuqiang Huang, Toshiya Ideue, Qianghua Wang, Yoshihiro Iwasa, Hongtao Yuan

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 world of quantum physics, scientists are constantly searching for ways to build computers that are immune to the errors that plague today's machines. One promising path involves a special kind of material called a topological superconductor. Inside these materials, electricity flows without resistance, and if you introduce a tiny magnetic field, it creates a swirling vortex. Theoretically, the center of these vortices can trap a unique particle known as a Majorana zero mode. These particles are special because they are their own antiparticles and can be used to store information in a way that is naturally protected from outside noise. To use them for computing, researchers need to be able to move these particles around and braid them together, much like weaving strands of rope, to perform calculations. However, controlling exactly where these vortices sit inside a tiny ring of material has been a difficult challenge, as they tend to behave in ways that are hard to predict or pin down.

A team of researchers has now demonstrated a way to control this behavior by simply changing the temperature. Working with a material called 2M-WS2, which acts as a topological superconductor, they created tiny rings, or nano-rings, that are only a few hundred nanometers wide. They observed how magnetic fields interact with these rings as they cooled them down. At higher temperatures, the magnetic field lines, or flux quanta, behaved as expected: they gathered together in the empty space in the very center of the ring, like a bundle of sticks. This is a known state where the magnetic field avoids the solid part of the ring entirely. However, as the researchers lowered the temperature, something unexpected happened. The magnetic field lines stopped staying in the center. Instead, they moved into the solid wall of the ring itself, spreading out along the circular path like beads on a necklace.

This shift from a central bundle to a necklace-like arrangement was detected by measuring the electrical resistance of the ring while applying a magnetic field. The researchers looked for a specific pattern in the resistance, known as the Little-Parks effect, which oscillates as the magnetic field changes. When the magnetic flux was in the center, the pattern repeated at a predictable, short interval. But when the temperature dropped and the flux moved into the ring wall, the pattern changed dramatically. The interval between the peaks became much larger, reaching approximately 70 mT. This enlargement was the key signature that the magnetic flux had rearranged itself from a central cluster into a series of individual vortices distributed along the ring's wall. The team confirmed this by comparing their results with computer simulations, which showed that this transition only happens when the material has specific properties that allow the magnetic vortices to get "stuck" or pinned in place along the wall, rather than being pushed back to the center.

The significance of this finding goes beyond just moving magnetic fields. The researchers used advanced calculations to show that when these vortices sit in the ring wall, their cores host the Majorana zero modes they were looking for. In the central bundle configuration, these special states exist as extended waves along the edges of the ring. But in the necklace configuration, they become tightly confined, localized points right in the middle of each vortex. This transition from an extended state to a localized one is crucial because it creates a stable environment where these quantum particles can be manipulated. The team also tested other materials, such as niobium and niobium diselenide, which are standard superconductors. They found that these materials did not show this necklace behavior, even under similar conditions. This proved that the effect relies on a specific combination of properties found in the 2M-WS2 material, particularly its ability to pin magnetic vortices strongly.

This work provides a new strategy for managing the building blocks of future quantum computers. By simply adjusting the temperature, scientists can now switch a device between two different states: one where the magnetic flux is in the center, and another where it is distributed along the ring wall, carrying the desired quantum states with it. The researchers suggest that this ability to control the position of these vortices could allow for the braiding of Majorana particles, a necessary step for creating fault-tolerant quantum gates. While the study focuses on the physics of the transition itself, it lays the groundwork for designing devices that can read and write quantum information by manipulating these flux configurations. The findings offer a clear, experimental path toward understanding how to harness these elusive quantum states for practical use in topological quantum computing.

Technical Summary: Majorana Edge-to-Bound State Transition in Interferometric Superconductor Rings

Problem Statement
In topological superconductors, manipulating magnetic flux quanta is essential for vortex braiding and topological quantum computing. Theoretical models suggest that flux quanta in interferometric superconducting rings can exist in two distinct configurations depending on the ratio between the annulus width ( $w$ ) and the superconducting coherence length ( $\xi_{ab}$ ). When $w < \xi_{ab}$ , fluxes form a "flux bundle" confined to the central hollow area, supporting topologically protected edge states at the inner and outer rims. Conversely, when $w > \xi_{ab}$ , fluxes are predicted to penetrate the annulus wall, forming a "flux necklace" configuration. In this necklace state, vortex cores are expected to host localized Majorana zero modes. However, experimentally realizing and verifying this flux-configuration-driven transition from extended edge states to localized vortex bound states has remained elusive.

Methodology
The authors utilized the topological superconductor candidate 2M-WS $_2$ to fabricate interferometric nano-ring devices. The material was chosen for its intrinsic topological superconductivity, high transition temperature ( $T_c \approx 8.8$ K), and the presence of charge stripe phases that act as natural pinning centers for flux quanta.

Device Design: Nano-rings with an annulus width ( $w$ ) of approximately 200 nm were fabricated. This dimension was specifically designed to straddle the critical condition ( $\xi_{ab} \approx w/5$ ) required for the formation of a flux necklace, allowing the transition to be driven simply by tuning temperature (which modulates $\xi_{ab}$ ).
Experimental Probes: The team performed temperature-dependent magnetoresistance ( $R_{xx}$ ) measurements under out-of-plane magnetic fields to observe Little–Parks oscillations. The oscillation period serves as a sensitive probe for the spatial distribution of flux quanta.
Theoretical Modeling:
- Time-Dependent Ginzburg–Landau (TDGL) simulations were used to model the evolution of the superconducting order parameter and flux distribution under varying geometric ratios ( $w/\xi_{ab}$ ) and magnetic fields.
- Bogoliubov–de Gennes (BdG) calculations were employed to compute the local density of states (LDOS) and verify the presence of Majorana bound states in the vortex cores of the flux necklace configuration.
Control Experiments: Comparative studies were conducted on conventional s-wave superconductors (Nb and NbSe $_2$ ) to isolate the roles of the Ginzburg–Landau parameter ( $\kappa$ ) and flux-pinning energy ( $U_p$ ).

Key Results

Coherence-Length-Driven Transition: The experiments demonstrated a clear transition in flux configuration driven by temperature.
- High Temperature ( $T > 6.8$ K): Here, $\xi_{ab} > w$ . The observed Little–Parks oscillation period (~4.5 mT) matched the theoretical expectation for a flux bundle confined to the ring center, corresponding to edge states at the annulus rims.
- Low Temperature ( $T < 4.4$ K): Here, $\xi_{ab} < w$ . The oscillation period dramatically increased to ~70 mT. This "unconventional" period indicates that flux quanta are no longer in the center but are distributed along the annulus wall in a flux necklace configuration.
Theoretical Validation: TDGL simulations confirmed that the transition from a flux bundle to a flux necklace occurs when the ratio $w/\xi_{ab}$ exceeds a critical threshold (approx. 5). The simulations also showed that large values for both the Ginzburg–Landau parameter ( $\kappa$ ) and flux-pinning energy ( $U_p$ ) are vital for stabilizing the necklace configuration.
Majorana Bound States: BdG calculations on the flux necklace configuration revealed cusp-like zero-energy peaks localized at the vortex cores within the annulus wall. This supports the theoretical prediction that the flux necklace hosts localized Majorana bound states, distinct from the extended edge states found in the flux bundle regime.
Material Specificity: Control experiments on Nb and NbSe $_2$ nano-rings showed only conventional Little–Parks oscillations (flux bundle) across all temperatures. This confirms that the transition requires specific material properties: 2M-WS $_2$ possesses both a large $\kappa$ and sufficient $U_p$ (aided by charge stripes), whereas Nb lacks a large enough $\kappa$ and NbSe $_2$ lacks sufficient $U_p$ to stabilize the necklace state.

Significance and Claims
The paper claims to provide a novel strategy for manipulating flux configurations in topological superconductors. By demonstrating that the transition from edge states to vortex bound states can be driven by tuning the relative scale of coherence length and annulus width, the authors establish a method to access localized Majorana zero modes.

Flux Braiding: The discrete arrangement of Majorana bound states along the annulus wall in the flux necklace configuration provides a natural platform for the adiabatic braiding of non-Abelian anyons, a key operation for topological quantum gates.
Readout Mechanism: The switchable flux configuration transition serves as a sensitive flux interferometer, potentially enabling the readout of Majorana parity for topological qubit measurement.
Universality: The study suggests that this flux necklace phenomenon could be generalized to other type-II topological superconductors (e.g., Fe(Te,Se), UTe $_2$ , $\beta$ -Bi $_2$ Pd) provided they satisfy the conditions of large $\kappa$ and large $U_p$ .

The work bridges the gap between theoretical predictions of flux configurations and experimental observation, offering a pathway toward realizing fault-tolerant topological quantum computing through flux control.

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