Antichiral edge states and Bogoliubov Fermi surfaces in a two-dimensional proximity-induced superconductor

TL;DR

The paper identifies a new 2D topological superconducting phase in a Rashba-coupled electron gas proximity-coupled to an $s$-wave superconductor when an in-plane magnetic field is applied. It derives an effective low-energy Bogoliubov–de Gennes Hamiltonian that yields a pair of Dirac-like cones and antichiral Majorana edge states propagating along edges perpendicular to the field, coexisting with a single pair of Bogoliubov Fermi surfaces. The topological character is captured via a Wilson-loop–like invariant, and the edge modes are described by a Jackiw–Rebbi-type continuum model with a characteristic velocity $v$ tied to the SOC. The predicted signatures appear in the current-phase relation of a wide Josephson junction, including a phase-locked jump associated with the zero mode and a diode/anomalous Josephson effect, offering experimental routes in Al/InAs and related platforms. The work highlights a weak-topology-like phase with robust antichiral Majorana channels and Bogoliubov Fermi surfaces, expanding the landscape of 2D topological superconductivity in proximitized systems.

Abstract

We show that a magnetic field parallel to the plane of a two-dimensional electron gas with Rashba spin orbit coupling in proximity to a superconductor leads to a topological phase in coexistence with a single pair of Bogoliubov Fermi surfaces. This phase hosts antichiral edge states of co-propagating Majorana fermions and are spatially localized at the opposite edges of the sample, perpendicular to the magnetic field. We discuss the characteristic signatures in the current-phase relation of a Josephson junction formed by two reservoirs in the topological phase.

Figures (5)

• Figure 1: Antichiral edge states: A 2D electron system in proximity with an $s$-wave superconductor is placed in the $x,y$ plane. The Rashba SOC is in the plane as well as the external magnetic field ${\bf B}=(B_x,B_y,0)$. A topological phase with antichiral edge states along ${\bf n}_{||}$ exists for $\mu^2 \leq V^2-\Delta_0^2$ within a range of angles satisfying $|{\bf n}_{||} \cdot {\bf n}_V|< \Delta_0/V<1$ (see text). (a) and (b) illustrate two different configurations of the magnetic field and the antichiral edge states. The existence of the topological phase is accompanied by the emergence of a single pair of Bogoliubov Fermi Surfaces in the spectrum. (c) BdG spectrum obtained within the lattice model for the magnetic field oriented along ${\bf n}_x$. The projected plane-cuts correspond to $(k_x=0,k_y,E)$ and $(k_x,k_y=0,E)$ surfaces.
• Figure 2: Edge states and topological invariant:(a) Bogoliubov-de Gennes spectrum of a ribbon of $N_x=200$ lattice sites with open boundary conditions (OBC) in $x$ and periodic boundary conditions (PBC) along $y$, as a function of $k_y$ in the topological phase for the configuration of Fig. \ref{['fig:spec']}(c), magnetic field along ${\bf n}_x$. The spectrum shows the dispersion relation of the edge states connecting the cones above and below the Fermi level. (b) Topological invariant $\theta (k_y)$ obtained from the numerical evaluation of the eigenvalues of the Wilson loop defined in Eq. (\ref{['eq:wilson_loop']})(see text). Parameters are: $t=50\Delta_0$, $\lambda=2.8\Delta_0$, $V=(2\Delta_0,0,0)$, $\mu=0$.
• Figure 3: Mode localization and robustness: Spatial probability distribution of the lowest positive energy eigenstate $E\approx 0$ within the topological phase, for a lattice with $N_x\times N_y$ sites (labeled with $l_x,l_y$). (a) OBC in the $x$-direction and PBC in the $y$-direction. (b) OBC are considered in both directions. In both cases $N_x=200$ sites. In (a), $N_y^{(PBC)}=300$, $E/t=9.1\times10^{-5}$. In (b), $N_y^{(OBC)}=1000$ and $E/t=5.6\times10^{-5}$. Other parameters are the same as in Fig. \ref{['fig:Wilson-loop']}.
• Figure 4: Josephson junction: Current-phase relation (CPR) $J(\phi)$ relative to the critical current $J_c$ at zero temperature for different magnetic field orientations on the plane with polar angle $\varphi$ (see inset). The two superconductors are modeled by the lattice model with the same parameters as in Fig. \ref{['fig:Wilson-loop']}. The junction has 200 $k_y$ channels connected through a row of sites with a hopping $t_{\rm J}=t/2$ (see Ref. sm). The contribution to the CPR arising from the transverse mode with $k_y=0$ is shown with dashed lines.
• Figure S1: Disk symmetry: Spatial probability distribution of the lowest positive energy eigenstate $E\approx0$ for an square lattice with $N=100000$ sites embeded in a circular shape with radius $R=90$ sites, $V=(1.1\Delta_0, 0, 0)$, $E/t=1.7\times10^{-5}$. The rest of the parameters are the same as in the main text.