Manipulating the topological spin of Majoranas

The non-Abelian exchange statistics of Majorana zero modes make them interesting for both technological applications and fundamental research. Unlike their non-Abelian counterpart, the Abelian contribution, $e^{iθ}$, where $θ$ is directly related to the Majorana’s topological spin, is often neglected. However, the Abelian exchange phase and hence the topological spin can differ from system to system. For vortices in topological superconductors, the Abelian exchange phase is interpreted as an Aharonov-Casher phase arising from a vortex encircling a $e/4$ charge. In this work, we show how this fractional charge, and hence the topological spin, can be manipulated through the control of device geometry, introducing an additional control knob for topological quantum computing. To probe this effect, we propose a vortex interference experiment that reveals the presence of this fractional charge through shifts in the critical current.

💡 Research Summary

The paper investigates a largely overlooked component of Majorana zero‑mode (MZM) exchange statistics: the Abelian phase factor (e^{i\theta}), which is directly tied to the anyon’s topological spin (s=\theta/2\pi). While the non‑Abelian part (U_{0}) is universal across platforms, the Abelian contribution can vary because it originates from an Aharonov‑Casher (AC) phase acquired when a vortex encircles a fractional electric charge bound to another vortex. The authors demonstrate that this fractional charge, and therefore the topological spin, can be engineered by tailoring device geometry and material parameters, providing a new “knob” for topological quantum computation (TQC).

Theoretical framework
Starting from a generic mean‑field Bogoliubov‑de Gennes Hamiltonian for a proximitized topological material, the authors introduce a vortex through a phase winding of the superconducting order parameter. By performing a unitary transformation that changes the global superconducting phase (\varphi_{0}) by (2\pi), they relate the accumulated Berry phase to the total fermion number operator, showing that the Abelian exchange phase equals the AC phase (\phi_{\text{AC}}=2\pi q/e) where (q) is the charge bound to the vortex core. Consequently, a charge (q=e/4) yields (\theta=\pi/2) (topological spin (s=1/4)).

2‑D topological‑superconductor model
In a Rashba‑spin‑orbit, Zeeman‑split, (s)-wave proximitized 2‑D electron gas, numerical BdG calculations reveal that the MZM and the bound charge co‑localize at the vortex core. The charge magnitude depends continuously on microscopic parameters (spin‑orbit strength, chemical potential, pairing amplitude) and is not quantized to (e/4). Thus, in purely 2‑D platforms the Abelian phase can be tuned only indirectly via fine‑grained material engineering.

3‑D TI/FMI heterostructure
The central advance is the analysis of a three‑dimensional topological insulator (TI) sandwiched between a ferromagnetic insulator (FMI) on the top surface and an (s)-wave superconductor on the bottom. Two mass terms appear: a magnetic gap (\beta) (top) and a superconducting gap (\Delta_{0}) (bottom). Solving the BdG equations shows a striking spatial separation: when (\beta\gg\Delta_{0}) a fractional charge (-e/4) localizes on the top surface while the MZM resides on the bottom surface; when (\Delta_{0}\gg\beta) the charge disappears and only the MZM remains. By varying the ratio (\Delta_{0}/\beta) through geometric design (film thickness, interface area) or external gating, the bound charge can be continuously switched between (\pm e/4) and zero, thereby continuously tuning the topological spin from (\pm 1/4) to 0.

Control mechanisms
The authors propose several practical knobs: (i) adjusting the thickness of the TI slab to modify the overlap between the magnetic and superconducting proximity effects; (ii) patterning the vortex lattice to change inter‑vortex distance, which influences charge screening; (iii) applying gate voltages to shift the chemical potential, thereby altering (\Delta_{0}) relative to (\beta). These controls enable deterministic manipulation of the Abelian phase without affecting the universal non‑Abelian part.

Experimental proposal
To detect the fractional charge, a vortex‑interference interferometer is suggested. Two vortices are placed in a superconducting loop containing a Josephson junction. The critical current (I_{c}) of the loop depends on the total AC phase accumulated around the loop. When the bound charge is (+e/4) the AC phase contributes (\pi/2), leading to a minimum of (I_{c}); when the charge is (-e/4) the phase flips sign, giving a maximum. By sweeping magnetic field or gate voltage and measuring the resulting shifts in (I_{c}), one can directly read out the topological spin. The authors provide analytical expressions for the current‑phase relation and numerical simulations showing the expected modulation amplitude.

Critical assessment
The work is conceptually strong: it identifies a concrete, experimentally accessible signature of the Abelian exchange phase and offers a realistic route to control it. However, several challenges remain. The assumption of a perfectly localized (e/4) charge may be compromised by screening from bulk carriers or disorder, potentially reducing the AC phase. Fabricating high‑quality TI/FMI/SC trilayers with independently tunable (\beta) and (\Delta_{0}) is non‑trivial; interface roughness could blur the spatial separation of charge and MZM. Detecting the predicted (\sim 1%) changes in (I_{c}) demands low‑noise cryogenic measurement setups. Complementary probes such as scanning single‑electron transistors or charge‑sensitive STM could strengthen the evidence.

Outlook
If realized, the ability to imprint a controllable Abelian phase on top of the universal non‑Abelian braid would expand the computational space of TQC, allowing for phase‑gate operations that are otherwise difficult to implement. Moreover, the geometric control paradigm could be extended to other platforms (e.g., proximitized transition‑metal dichalcogenides or quantum‑Hall–superconductor hybrids). The paper thus opens a promising avenue for both fundamental anyon physics and practical quantum‑information processing.