Towards Scalable Braiding: Topological Superconductivity Unlocked under Arbitrary Magnetic Field Directions in Curved Planar Josephson Junctions

Abstract

The non-Abelian statistics of Majorana zero modes (MZMs) are central to fault-tolerant topological quantum computing. Planar Josephson junctions provide a particularly versatile platform for realizing robust topological superconductivity hosting MZMs over a broad parameter space. However, it is generally believed that such topological superconductivity is restricted to a narrow range of in-plane magnetic field orientations, posing a major obstacle to scalable and noncollinear junction-network architectures. Here, we uncover that the apparent suppression of MZMs under misaligned fields does not arise from the destruction of topological superconductivity itself, but instead originates from emergent shifted bulk states at other momenta that obscure the global excitation gap and MZMs. By introducing spatial modulations along the junction to scatter and gap out these bulk states, we restore a global topological gap and recover MZMs for arbitrary in-plane magnetic field orientations. Remarkably, such modulations can be naturally realized by transforming a straight junction into a curved geometry, rendering the topological gap robust against field misalignment and enabling MZMs survival in complex junction networks. Building on this robustness, we propose a scalable protocol for MZMs braiding and fusion using gate or superconducting-phase control, opening new routes toward scalable topological quantum computing.

Figures (4)

• Figure 1: (a) Schematic of the PJJ with $\vec{B}$ misaligned by $\beta$ relative to the interface between S (blue) and N (yellow) regions with phase difference, $\phi=\phi_1-\phi_2$, where MZMs (stars) can emerge at the ends of the N region. (b)–(d) Spin texture on the Fermi surface for a 1D system at $\vec{B}=0$, $\vec{B}\|x$, and $\vec{B}\|y$. (e)–(g) Same as (b)–(d), but for a PJJ (2D system). The canted spin texture (red) enabling effective $p-$wave pairing are indicated by the green dashed lines. (h)–(j) Corresponding PJJ spectra for (e)–(g), where $\Delta_{\Gamma}$ denotes the gap at $k=0$ and $\Delta_{G}$ the global gap. (k) $\beta$ dependence of $\Delta_{\Gamma}$ and $\Delta_{G}$, where the parameters are $L=6500$ nm, $W_{N}=100$ nm, $W_{S1}=W_{S2}=300$ nm, $\mu=2$ meV, $B=0.3$ T and $\phi=\pi$. Other parameters are specified in the main text.
• Figure 2: (a) Same as Fig. \ref{['fig1']}, but with a gate array (orange and green) producing a periodic step modulation of the chemical potential with amplitude $V$ and period $l_0$. (b) Energy spectrum of (a) at $\mu=2.2$ meV, $V=0$ and $\vec{B}=0$, exhibiting a gap closing at momentum $k_0=0.36\pi/a$. (c) Folded spectrum of (b) (orange) in the mini Brillouin zone with superlattice period $l_0=\pi/k_0$; a finite $V$ opens a gap at $k_0$ (blue) (d) Same geometry as (a), but with a spatially noncollinear magnetic texture generated by a sinusoidal modulation of $\beta$ along the $x$ direction. (e) Energy spectrum of (d) for $\mu=7$ meV, $\vec{B}\|y$ ($B=0.4\,$T, $\beta=\pi/2$). (f) Same as (e), but with $\beta=\sin(2\pi x/l_\beta)$ ($l_\beta = 1300$ nm), which opens a gap (blue) compared to the uniform $\beta=\pi/2$ case (orange). (g) Same as (a), but for a CJJ with a sinusoidal N region of period $l=1300$ nm and amplitude $w=200$ nm. (h) Same as (e), but at $\mu=8$ meV. (i) Same as (h), but for the CJJ, where a global gap opens (blue) compared to the SJJ spectrum (orange). Other parameters are taken from Fig. \ref{['fig1']}.
• Figure 3: (a)-(b) Phase diagram of a SJJ, plotted as the product $Q\Delta_G$, for $\beta=0$ and $\beta=\pi/2$. (c)-(d) Same as (a)-(b) but for a CJJ, where the gapped topological region suppressed at $\beta=\pi/2$ in (b) is recovered in (d). (e)-(f) Energy spectra of the CJJ at $\phi=\pi$ for $\beta=0$ and $\beta=\pi/2$; insets show the probability density of the zero-energy modes at $B=0.8$ T (black dots). (g)-(h) Critical current $I_c$ (brown) and ground-state phase $\phi_{GS}$ (green) versus $B$ for the CJJ at $\beta=0$ and $\beta=\pi/2$. (i) $\beta$ dependence of $\Delta_G$ in the CJJ. (j) Polar map of $\Delta_G(\beta)$ at $B=0.5$ T. Other parameters are taken from Fig. \ref{['fig2']}.
• Figure 4: (a) Schematic of gate-controlled MZMs braiding in a CJJ T junction, where the superconducting phase difference between adjacent S regions is fixed at $2\pi/3$. The chemical potential in each branch is tuned by mini-gate voltages ($V_l$, $V_r$, and $V_b$) between topological ($V_+$, orange) and trivial ($V_-$, green) regimes, enabling MZMs exchange through the three steps I–III indicated by red arrows. (b)–(f) Evolution of the calculated MZMs probability density $\rho$ (color scale) and the corresponding energy spectrum during steps I–III, with $V_+=4$ meV and $V_-=10$ meV. (g) Schematic of phase-controlled MZMs exchange in a CJJ cross junction, implemented by tuning the superconducting phases $\phi_{1-4}$ according to steps ➀-➃. (h)–(l) Evolution of the calculated $\rho$ during steps ➀–➃. Other parameters are taken from Fig. \ref{['fig3']}