Tunable Majorana corner states in a two-dimensional second-order topological superconductor induced by magnetic fields

A two-dimensional second-order topological superconductor exhibits a finite gap in both bulk and edges, with the nontrivial topology manifesting itself through Majorana zero modes localized at the corners, i.e., Majorana corner states. We investigate a time-reversal-invariant topological superconductor in two dimensions and demonstrate that an in-plane magnetic field could transform it into a second-order topological superconductor. A detailed analysis reveals that the magnetic field gives rise to mass terms which take distinct values among the edges, and Majorana corner states naturally emerge at the intersection of two adjacent edges with opposite masses. With the rotation of the magnetic field, Majorana corner states localized around the boundary may hop from one corner to a neighboring one and eventually make a full circle around the system when the field rotates by $2\pi$. In the end, we briefly discuss physical realizations of this system.

Figure 1

(Left) Schematic plot of a 2D $p\pm ip$ superconductor, which features counterpropagating edge modes, with spin-up modes denoted in magenta and spin-down modes in blue. Right panel: an effective description of the four edges with a 1D theory, where the letters $D^{\prime },A^{\prime }$, and $B^{\prime }$ correspond to their counterparts $D,A$, and $B$ in the left panel. In presence of an in-plane Zeeman field, the mass gap $m$ induced varies across edges and MZMs emerge at corners where $m$ reverses sign and forms a kink.

Figure 2

Variations of the edge gap with the orientation $\theta$ of the Zeeman field for a $80\times 80$ lattice. Blue (dot dashed) line represents one of the two zero modes, magenta (solid) line reflects the edge gap, and black (dashed) line are edge gaps determined by the minimum of mass terms in Eq. (7) for the four edges. $t=1,\mu =3,\mathrm{\Delta }=1,\varphi =0$, and $V=0.5$. (b) An inversion-breaking term ${\mathrm{\Delta }}_s{\tau }_y$ is added, with ${\mathrm{\Delta }}_s=-0.15$.

Figure 3

Probability density distribution of MZMs for a $20\times 20$ lattice. Insets are the mass profile on 1D system shown in the right panel of Fig. 1, with dotted line representing zero masses and red dots denoting the corners. $t=1,\mu =3,\mathrm{\Delta }=1,\varphi =0$, and $V=0.5$. For (e)–(h), the term $\mathrm{\Delta }{\tau }_y$ is added, with ${\mathrm{\Delta }}_s=-0.15$.

Figure 4

Phase for edge modes with the lowest positive energy along the boundary sites of a $80\times 80$ lattice. The phase for each component is given by its argument relative to that of site $B$. The values shown on vertical axis are expressed in unit of $\pi$. $t=1,\mu =3,\mathrm{\Delta }=1,\varphi =0$, and $V=0.5$.

Figure 5

Probability density plot of MZMs for a $30\times 30$ lattice introduced in Ref. [39] with an additional in-plane magnetic field applied. Parameters in the lattice Hamiltonian therein are chosen to be $t=1,{\lambda }_R=1,\mu =-1,\text{and}{\mathrm{\Delta }}_0=-{\mathrm{\Delta }}_1=-2$. Strength of the Zeeman field $V=1$.

Figure 6

An arbitrary edge of the system considered in the main text. Two sets of Cartesian coordinate systems, $O-k_x{k}_y$ and $O-k_{x^{\prime }}{k}_{y^{\prime }}$ are shown, related by an in-plane rotation of $\phi$. Axis $k_{y^{\prime }}$ is along the direction of the edge.