Angular momentum of vortex-core Majorana zero modes

TL;DR

The paper analyzes vortex-core Majorana zero modes in a $d+id$ superconductor on a three-dimensional topological-insulator surface, uncovering that these MZMs can carry a nontrivial angular-momentum flavor $\ell$ determined by the windings of the Dirac cone $n_X$, the order parameter $n_\Delta$, and the vortex $n_V$, independent of the Chern number. It presents a combined continuum ($d+id$ Dirac) and lattice BdG treatment, validated by exact diagonalization and kernel polynomial method, to classify and characterize the MZMs and their angular momentum. The study reveals that the MZM flavor is constrained by a rotational symmetry $R_{\mathrm{tot}}$ and that the angular momentum is not fixed by the Chern number, with a rich set of scenarios across different windings and lattice transitions. However, topological protection is not as robust as previously claimed: the small topological gap $\Delta_F$ and the presence of Caroli–de Gennes–Matricon states (poisoning) yield limited stability and localization of the vortex-core MZMs. These findings emphasize both the fundamental interest of MZM flavors and the practical challenges for detecting and manipulating such modes in realistic systems.

Abstract

Majorana zero modes (MZMs) are highly sought-after states with a possible application in quantum computation. Here, we show that vortex-core MZMs can carry a nontrivial angular momentum. This establishes new `flavors' of Majorana modes, independent of the Chern classification of topological superconductors. The MZM angular momentum is explicitly calculated for a microscopic model of a $d+id$ superconductor placed on a three-dimensional topological insulator ($d+id+\phantom{}$Dirac model) using both exact diagonalization and the Chebyshev expansion. We classify all possible quantum numbers of MZMs depending on the windings of the order parameter and underlying normal state. The topological protection of the MZM is set by the bulk gap, quasiparticle poisoning by trivial in-gap states, and its localization length. All these severely limit the stability of MZMs in the $d+id+\phantom{}$Dirac model, in contrast to earlier claims. Nevertheless, the possibility of having different flavors of MZM - in the form of angular momentum or something else - can provide a unique path forward for the study of MZMs.

Figures (9)

• Figure 1: Origin of vortex Majorana zero modes (MZMs) in a chiral superconductor. (a) A superconductor with a nonzero Chern number has chiral edge states. (b) The Chern number derives from a winding of the complex order-parameter phase around the Fermi surface in momentum space. (c) In real space, a vortex adds an additional winding to the order parameter. Behaving as a void in the superconductor, the vortex core must have edge modes around the void (blue arrows).
• Figure 2: (a) Energy of the edge modes around a puncture in a topological $p+ip$ superconductor on a square lattice, vs angular momentum $L_z$. For odd Chern number $C$, $L_z$ is half-integer in the absence of vortex ($n_V = 0$) and integer in the presence of a vortex of vorticity $n_V = \pm 1$, $L_z=0$ corresponding to a MZM. (Inset) Example of edge mode. (b) Vortex MZM carrying an angular winding and without amplitude at the center of the vortex when $n_V$ has the same sign as $C$. (c) MZM carrying no winding and with amplitude at the center of the vortex when $n_V$ and $C$ have opposite signs.
• Figure 3: Spin-resolved LDOS of the MZM near the vortex center for all values of $n_X$, $n_\Delta$, and $n_V$, calculated with $L=201$ at negative $\mu$. Similar results are obtained at positive $\mu$. The columns labeled $n_\uparrow$ ($n_\downarrow$) show $|u_\uparrow(\mathbf{r})|^2$ [$|u_\downarrow(\mathbf{r})|^2$] and the numbers on the images are the numerically determined values of $n_\uparrow$ and $n_\downarrow$. The color scale covers the whole data range in all images.
• Figure 4: Total (left panels) and spin-resolved (center and right panels) LDOS at $\mathbf{r}=(0,0)$ and $\mathbf{r}=(1,0)$ for Dirac-cone winding $n_X=-1$, pairing winding $n_\Delta=-2$, and vorticity (a) $n_V=+1$ and (b) $n_V=-1$. At $\mathbf{r}=(0,0)$, there is no spin-$\uparrow$ intensity in the vortex and neither spin-$\uparrow$, nor spin-$\downarrow$ intensity in the antivortex. The insets display the total LDOS in the vortex- and antivortex-core region. Calculations are made with $L=1001$ using the KPM at order $N=20000$. The horizontal bar in (a) indicates the energy resolution of the calculation.
• Figure 5: Low-energy spectrum of the lattice Hamiltonian, Eq. \ref{['Eq:LatticeHBdGMomentumSpace']}, for (a) $s$-wave pairing ($\Delta_\mathbf{k}\equiv0.3v$) and (b) $d\pm id$ pairing ($\Delta_d=0.3v$, $\Delta'_d=0.2v$). The tiny spectral gap in (b) is given approximately by Eq. \ref{['Eq:SmallGap']}, as better seen by zooming in (c). The other model parameters are $m=0.5$ and $\mu=-0.3$. The thin lines show the spectrum for $m=0$.