Generic Theory for Majorana Zero Modes in 2D Superconductors

It is well known that non-Abelian Majorana zero modes (MZM) are located at vortex cores in a $p_x+𝒾p_y$ topological superconductor, which can be realized in a 2D spin-orbit coupled system with a single Fermi surface and by proximity coupling to an $s$-wave superconductor. Here we show that the existence of non-Abelian MZMs is unrelated to the bulk topology of a 2D superconductor, and propose that such exotic modes can result in a much broader range of superconductors, being topological or trivial. For a generic 2D system with multiple Fermi surfaces that is gapped out by superconducting pairings, we show that at least a single MZM survives if there are only an odd number of Fermi surfaces of which the corresponding superconducting orders have vortices; such a MZM is protected by an emergent Chern-Simons invariant, irrespective of the bulk topology of the superconductor. This result enriches new experimental schemes for realizing non-Abelian MZMs. In particular, we propose a minimal scheme to realize the MZMs in a 2D superconducting Dirac semimetal with trivial bulk topology, which can be well achieved based on recent cold-atom experiments.

Figure 1

(a) The band structure of 2D topological Dirac metal with two Dirac points located at ${\mathbf{Q}}_{\pm }$; the gray thick loops around two Dirac points represent the Fermi surfaces, and the color represents the average value of the spin component $⟨s_z⟩$. (b) Schematic of the spin orientations, shown by blue arrows, at the Fermi surfaces around the Dirac points. Parameters are $t_{x,y}=t_{\mathrm{so}}=m_z=1$, $\mu =0.8$, and the corresponding Dirac node momenta ${\mathbf{Q}}_{+}=-{\mathbf{Q}}_{-}=(0,2\pi /3)$.

Figure 2

(a) Mean field phase diagram of the Dirac-Hubbard Hamiltonian versus attractive Hubbard interaction $U$ and chemical potential $\mu$. In the “Dirac metal” phase, all ${\mathrm{\Delta }}_{\mathbf{q}}=0$; in the narrow “gapless” region, ${\mathrm{\Delta }}_{2{\mathbf{Q}}_{\pm }}$ are finite but not strong enough to fully gap the system. In the “PDW” phase, the system is fully gapped by finite PDW. In the “${\mathrm{PDW}}^{\prime }$” phase at large $U$, BCS and CDW orders of small magnitude also appear and accompany the PDW. (b) Magnitude of PDW order ${\mathrm{\Delta }}_{2{\mathbf{Q}}_{\pm }}$. (See [37] for the magnitude of BCS ${\mathrm{\Delta }}_0$ and CDW ${\rho }_{2\mathbf{Q},\sigma }$.) The parameters for numerics are the same as those in Fig. 1.

Figure 3

(a) Wave function density $\sum _{s=\uparrow ,\downarrow }|{\psi }_s(\mathbf{r}){|}^2$ for MZMs computed in the Dirac-Hubbard model, with $\mu =0.8$, $U=5.5$, which gives self-consistent pairing ${\mathrm{\Delta }}_{2{\mathbf{Q}}_{\pm }}=0.23$ (corresponding to a SC gap of ${\mathrm{\Delta }}_{\mathrm{gap}}=0.28$), and a trivial 2D bulk. System size $N_x=N_y/2=60$. The vortices with opposite unit vorticities are located at (30,30) and (30,90) and the vortex field $e^{𝒾\theta (\mathbf{r})}$ is attached only to ${\mathrm{\Delta }}_{2{\mathbf{Q}}_{+}}$. (b) Schematic of the physical origin of MZMs at vortex cores. The vortex core can be viewed approximately as the open boundary of $r$ dimension in the 3D space spanned by $(r,k_{\parallel },\varphi )$. (c),(d) Energy spectrum for the half-vortex (c) and full-vortex (d) regime. The MZMs are obtained in the former case. Other parameter conditions are the same as in Fig. 1.