Emergence of multiple zero modes bound to vortices in extended topological Josephson junctions

TL;DR

This work shows that the Fu–Kane effective theory for extended Josephson junctions on a TI surface breaks down when the Majorana edge-mode velocity vanishes, revealing the emergence of additional zero-energy Dirac cones at finite momentum. By analyzing both the traditional low-energy projection and a Fourier-mode spectral-mmatrix approach in a Corbino geometry, the authors demonstrate that multiple zero-energy bound states can appear at a Josephson vortex, with the number of zero modes growing as parameters tune $v$ to zero. These extra modes are symmetry-protected only in the presence of certain symmetries and can be lifted to near-zero CdGM states by symmetry-breaking perturbations, impacting experimental observables such as the Josephson current and microwave absorption. The findings underscore the need to go beyond the Fu–Kane model for accurate interpretation of experiments and motivate further studies on vortex lattices, disorder, and the relation to 1D topological invariants.

Abstract

We study planar Josephson junctions formed on the surface of a three-dimensional topological insulator (Fu-Kane proposal) and examine the experimentally relevant parameter regimes in which the effective velocity of the emergent one-dimensional Majorana modes approaches zero. We show that the frequently employed Fu-Kane effective theory breaks down in this case. As parameters like the chemical potential or the width of the junction are tuned, instances of vanishing effective velocity mark the emergence of additional 'Dirac cones' at zero energy and finite momentum. If the junction is subjected to an external magnetic field, Josephson vortices may then bind a number of zero modes in addition to the topological Majorana mode. The additional zero modes are 'symmetry-protected' and can be lifted by a broken mirror symmetry (which is to be expected in realistic scenarios) as well as by an in-plane magnetization (or Zeeman field). We note that the ensuing presence of additional low-energy Andreev states can significantly contribute to measured quantities like the Josephson current or microwave absorption spectra.

Figures (6)

• Figure 1: Sketch of the superconductor-ferromagnet-superconductor (SMS) junction on the surface a three-dimensional topological insulator (TI) with width $W$ and length $L$. The external magnetic field $\vec{B} = B\hat{e}_z$ leads to a running phase difference $\varphi(y) = 2\pi y/l_B+\varphi_0$ (sketched below) between the superconductors with magnetic length $l_B \propto 1/B$. The dotted green lines indicate the positions of Josephson vortices, separated by a distance of $l_B$.
• Figure 2: Dispersion of the one-dimensional bound states following from the Hamiltonian \ref{['eq:fukanehamiltonianwithk']} for $\varphi=\pi$ (blue) and $\varphi=1.05\pi$ (orange in the insets) with $W=0.1\,v_F/\Delta_0$, $\mu_S = 100\Delta_0$ and (a) $\mu_N = 0$, (b) $\mu_N = 31\Delta_0 \approx \mu_{N,1}^{(0)}$, (c)-(f) $\mu_N = 50\Delta_0$. Additionally in panel (d) an asymmetry has been introduced to the junction $\mu_S(x>W/2) = 1.2\mu_S(x<-W/2)$. In the panels (a)-(d) it holds $\vec{M}=0$, whereas for (e) ($M_y=\Delta_0$) and (f) ($M_x = 0.8\Delta_0$), an in-plane magnetization has been added. In (b) at $\mu_N\approx \mu_{N,1}^{(0)}$, where the effective velocity (i.e. the slope at $k=0$) becomes zero, the non-linear corrections to the previously linear dispersion (shown in (a), where the naive effective theory is still an adequate description) become apparent. Increasing $\mu_N$ further to what is shown in (c), two additional Dirac cones, i.e. linearly dispersing low-energy modes, at finite momenta appear. For $\varphi\neq \pi$ a gap opens in each case. The asymmetry in panel (d) results in gaps at the two additional Dirac points (but not at $k=0$) even for $\varphi=\pi$. A finite value of $M_y$ 'tilts' the picture as shown in (e), such that the Dirac cones at finite $k$ are no longer centered at $E=0$. It is here interesting to note that, due to this tilting, sufficiently large values of $M_y$ may change the nature of the low-$k$ modes from counter- to co-propagating. A finite value of $M_x$ on the other hand, shown in (f), only acts like an additional phase difference and results in the gap closings being shifted to a value $\varphi\neq\pi$.
• Figure 3: Centers of one-dimensional Dirac cones, determined by the values of $k$ for which $E(k)=0$, as a function of $\mu_N$ following from \ref{['eq:fukanehamiltonianwithk']} for $W=0.1\,v_F/\Delta_0$, $\vec{M}=0$, $\mu_S = 100\Delta_0$. As $\mu_N$ grows, new Dirac cones emerge periodically, corresponding to instances where the effective velocity at $k=0$ vanishes.
• Figure 4: SMS junction on the surface of a 3D TI in an external magnetic field, as depicted in Fig. \ref{['fig:sketchSMSinmagneticfield']}, closed into a Corbino ring geometry such that $y$ and $y+L$ correspond to the same point along the junction. The gauge is chosen for the phase of the inner superconductor to be 0 while the phase of the outer superconducting ring is given by $\varphi(y) = \varphi(y+L)(\text{mod}\,2\pi)$. The magnetic flux $\Phi = n\Phi_0$ is quantized such that $L = n l_B$, $n\in\mathbb{N}$.
• Figure 5: Bound state spectrum as a function of $\mu_N$ for $W=0.1\,v_F/\Delta_0$, $\vec{M}=0$, $\mu_S = 100\Delta_0$ and $L=l_B = 8\,v_F/\Delta_0$ ($\Phi = \Phi_0$). Shown in color is the absolute value of the lowest-lying eigenvalue $\lambda_M(E)$ of the spectral matrix $M(E)$ for the respective energy. Bright spots indicate the presence of an eigenmode. At $\mu_{N,1}^{(c)}\approx 55\Delta_0$, we observe that the first excited state merges with the zero-energy state, in accordance with the arguments presented in Sec. \ref{['sec:translationallyinvariant']}, resulting in three-fold degenerate zero modes. The inset shows the prediction as derived in Ref. piasotski_topological_2024 according to the naive effective theory.