Manipulating the topological spin of Majoranas

TL;DR

The paper investigates how the Abelian part of Majorana exchange, i.e., the topological spin, can be manipulated by geometry in vortex-bound Majorana systems. It establishes a direct link between the topological spin and the fractional Fermi-sea charge bound to a vortex through the Berry connection, with $2\mathcal{A}=\langle \hat{N}\rangle$ and $4s\equiv\langle \hat{N}\rangle\pmod{2}$, and demonstrates this across 2D and 3D topological insulator platforms. A key result is that in 2D-like TI systems the bound charge is model-dependent, whereas in genuine 3D TI heterostructures the charge can be quantized to $-e/4$ and spatially separated from the Majorana zero mode, enabling geometry-driven control over the topological spin. The authors propose a vortex-interference experiment based on the Aharonov-Casher effect to read out the fractional charge and hence the topological spin, offering a practical route to enhanced braiding operations for topological quantum computation.

Abstract

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.

Figures (8)

• Figure 1: S/3D TI/FMI proximity system. By proximity, a magnetic energy gap (an s-wave superconductive pair potential) is introduced in a 3D topological insulator from the top (bottom) by a feromagnetic insulator (s-wave superconductor). A localized $-e/4$ Fermi-sea charge accumulates around the vortex core (carrying $\Phi_0^{\rm SC}=h/2e$ flux) at the top interface and Majorana zero modes appear around the vortex core at the bottom interface ($\gamma_{1}$) and the outer-surface ($\gamma_{2}$). The magnetic proximity system can, without loss of generality, be replaced by an intrinsic magnetic 3D TI.
• Figure 2: Charge and MZM densities in 2D (top row) and 3D (bottom row).a) Charge and MZM densities as a result of a $\Phi_0^{\text{sc}}=h/2e$ vortex in a 2D TI system. Colorbars for the charge $q(\mathbf{r})$ and zero mode density $|\psi(\mathbf{r})|^2$ are shared across the figure. b) Integrated charge density as a function of system size $r$ in units of the lattice space, $a$, for the two band QAH model (green squares) and four band model being the 2D limit of the 3D TI (orange circles). Inset: relative deviation of integrated charge from $-e/4$. c-d) Similar to a-b) but for a 3D S/3D TI/FMI heterostructure and with additional markers for the charge integrated on the bottom surface(red triangle) and the top surface(blue triangle).
• Figure 3: Robustness in a FMI/3D TI/S heterostructure.a) Center charge localized within a radius of $R+2a$ around the vortex core, and b) total charge as a function of the ratio of $\beta$ over $\Delta_0$ on the top and bottom surface of the 3D TI. c) The ensemble ($N_{\rm ensemble}=1000$) averaged accumulated bottom (red down triangle), top (blue up triangle), and total (orange circle) charge as a function of chemical potential disorder strength normalized by the hopping strength in panel. Shaded colors correspond to one standard deviation from their mean values. d) Charge (same markers as in panel c) as a function of the chemical potential.
• Figure 4: Experimental signatures.a) A double Josephson junction set-up of two superconductors $\mathrm{S_{1,2}}$ coupled via a superconducting charge island $\mathrm{S}^\prime$' consisting of the S/3D TI/FMI heterostructure in Fig. \ref{['fig:3D_TI_with_vortex']}. The island gets charged through a flux modulation line and optionally an electrostatic gate. b) Mean subtracted critical current response as a function of the number of admitted vortices on the island for $q_{\mathrm{gate}}$ = 0 (solid blue line), 0.5e (dashed red line), and 1e (dash-dot green line), here $E_{\mathrm{J}}/E_{\mathrm{C}}=1.25$. c) Strength of oscillating critical current signal Eq. \ref{['eq:total charge interference']}, relative to its mean as a function of the $E_\mathrm{J}/E_\mathrm{C}$. d) Relative critical current signal as a function of the function of the number of admitted vortices and $V_\mathrm{gate}$ (which changes $q_\mathrm{gate}$), here $E_{\mathrm{J}}/E_{\mathrm{C}}=10^{-2}$.
• Figure S1: Fractional charges and Berry connection. Twice the Berry connection $2\mathcal{A}$ (orange circles) and the number operator expectation value $\langle \hat{N}\rangle$ (dashed gray line) as a function of superconducting phase $\phi_0$, for the four band 2D model. Inset: deviation of ${2\mathcal{A}}$ from $\langle \hat{N}\rangle$.