Scheme for braiding Majorana zero modes in vortices using an STT-matrix

TL;DR

This work addresses the challenge of braiding Majorana zero modes in vortices of two-dimensional topological superconductors by introducing a programmable STT-matrix that generates localized stray fields to deterministically move and braid vortices. The authors develop a coupled TDGL and TDBdG simulation framework, embedding the STT texture into the TDGL equations and evolving MZMs through the BdG spectrum to quantify braiding fidelity and fusion energy splitting. They show that single-vortex control, four-MZM braiding, and MZM fusion are achievable, while fidelity is sensitive to finite MZM coupling $E_M$ and nonadiabatic vortex dynamics; these effects can be mitigated by optimizing STT spacing $L_s$ and vortex separation $L_v$. The platform promises a versatile, scalable path toward fault-tolerant topological quantum computation using vortex-hosted MZMs, with broad compatibility across different TSC realizations and material platforms.

Abstract

Majorana zero modes (MZMs), promising for topological quantum computation, are naturally hosted in vortices of two-dimensional topological superconductors (TSCs). However, precise control and braiding of these vortex-bound MZMs remains a significant challenge. This work proposes and numerically demonstrates a novel braiding scheme utilizing a programmable matrix of spin transfer torque (STT) devices (STT-matrix) integrated with a TSC layer. By selectively activating individual STT elements, their localized stray fields enable deterministic manipulation, including driving, braiding, and fusion, of superconducting vortices and their associated MZMs. We establish a comprehensive simulation framework combining finite element analysis for STT-induced vortex formation, time-dependent Ginzburg-Landau equations for vortex dynamics and time-dependent Bogoliubov-de Gennes equations for MZM evolution. Simulations confirm the STT-matrix's capability for high-fidelity vortex manipulation and demonstrate MZM braiding dynamics. We quantify the impact of vortex acceleration and finite MZM coupling on braiding fidelity, showing it can be optimized by adjusting STT spacing and vortex separation. Furthermore, we demonstrate controlled MZM fusion and measure the resultant energy splitting. This STT-matrix-based approach offers a highly versatile, scalable, and potentially practical platform for operating MZMs within TSC vortices, advancing towards fault-tolerant topological quantum computation.

Figures (6)

• Figure 1: (Color online) Device structures and MZMs in TSC. (a) Schematic diagram illustrating the device with two layers: the control layer, consisting of an STT-matrix, and the material layer, comprising the TSC. The STTs can be individually switched ON or OFF, as indicated by the red dashed rectangle. In the ON state, a vortex in the TSC layer can be driven or dragged via the local magnetic stray field generated by each active STT element. The inset shows the simulated stray field of an isolated STT. (b)-(c) Probability density distribution of MZMs in a TSC with one and two vortices, respectively. To highlight the MZM edge mode, logarithmic scale is used to display probability. Here, the TSC is modeled as a square geometry with open boundary conditions. (d)-(e) Line-cuts along the x-axis taken at $y=0$ corresponding to (b) and (c), respectively. Parameters used for calculations include $\tilde{W}=0.3$, $\tilde{v_F} = 0.5$, $\Delta_0 = 1$, $\xi = 0.25$, $d=1$, lattice size $L_x =L_y =30$.
• Figure 2: (Color online) Vortex dynamics induced by STTs. (a) Sketch of a single vortex's motion from the position of STT-A to STT-B shown in Fig. \ref{['figDevice']}(a). The vortex is represented by the order parameter. (b) The operation sequences of the two STTs. (c) The velocity of the vortex $v$ versus the time $\tau$ at different STTs' distance $L_s$. (d) The maximum velocity $v_{max}$ versus $L_s$. The value of $v_{max}$ saturates when $L_s \gtrsim R_v$. The parameters used in the calculations are $\kappa=4$, $b_0=1.2$, $R_v=1.7$, $\sigma=1$, and $\xi=0.25$. Note that spatial parameters and temporal parameters are rescaled by $\lambda$ and $\xi^2/D$, respectively. Other parameters are the same values as in Fig. \ref{['figDevice']}.
• Figure 3: (Color online) Braiding dynamics of MZMs facilitated by the STT-matrix. (a) Worldlines illustrating the braiding operator $\mathbb{B}_X=b_2b_2$ for four MZMs $\gamma_{i=1,2,3,4}$. (b) Cooper pair density distribution $\rho(x,y)$ in TSC layer with depicted vortices, where the two middle vortices are driven along a programmable path indicated by arrows using STTs. Vortices associated with $\gamma_{2,3}$ are overlaid on the background following the braiding trajectory. (c) Evolution of $P_{|\pm E_i\rangle}, i=1,2$ (defined in the text) at $L_v=9$ and $L_s=3$. (d) Evolution of $P_{|\pm E_i\rangle}, i=1,2$ at $L_v=18$ and $L_s=4.5$. (e) Variation of both $P_{\rm MZM}$ and $P_{|+E_1\rangle}$ as functions of $L_v$, while keeping $L_s$ fixed at 3. (f) Dependence of both $P_{\rm MZM}$ and $P_{|+E_1\rangle}$ on varying values of $L_s$, with fixed value of $L_v = 18$. The state $|-E_1\rangle$ is chosen as the initial state in all the calculations. Other parameters are the same values as in Fig. \ref{['figDevice']}.
• Figure 4: (Color online) Fusion of two MZMs within vortices induced STTs. (a) Worldlines depicting the fusion process of two MZMs are presented. (b) Simulated Cooper pair density $\rho(x,y)$ illustrates the dynamics of vortex fusion. (c) Energy splitting $E_M$ between two MZMs against their separation distance $L_M$. The inset displays the probability density distribution of MZMs at $L_M \approx \lambda$.
• Figure A1: (Color online) FEA simulation of an STT device. (a) Simulated stray field of an STT device in the ON and OFF states. The linecuts are the $B_z$-component of the stray field along a horizontal axis located 40 nm below the STT: the $B_z$ in the ON state (blue line), the Gaussian fitted $B_z$ in the ON state (dotted black line), and the $B_z$ in the OFF state (red line). The inset depicts a 3D simulation of the STT in the ON state. The magnetization intensities of the reference layer and free layer are set to $1.7\times10^6$ A/m Zhou2019. (b) Simulated Cooper pair density $\rho(x,y)$ by the FEA data [the blue line in (a)]. The current density vectors surrounding the vortex are indicated by white arrows.