Perfectly localized Majorana corner modes in fermionic lattices

Focusing on examples of Majorana zero modes on the corners of a two-dimensional lattice, we introduce a method to find parameter regions where the Majorana modes are perfectly localized on a single site. Such a limit allows us to study the dimerization structure of the sparse bulk Hamiltonian that results in the higher-order topology of the system. Furthermore, such limits typically provide an analytical understanding of the system energy scales. Based on the dimerization structure we extract from the two-dimensional model, we identify a more general stacking procedure to construct Majorana zero modes in arbitrary corners of a $d$-dimensional hypercube, which we demonstrate explicitly in $d\le 3$.

Figure 1

Construction of the dimerization structure in the parameter regimes of Eq. (4), starting from (a) a pair of topological and trivial Kitaev chains, which are (b) stacked on top of each other and then coupled as in (c) to gap out the intermediary Majorana modes to obtain a 2D bulk model with corner modes in adjacent corners. Panel (d) shows the gapped bulk bands for $t_0=0.5d_y, t_y=0.3d_y, s_x=2d_y$, and panel (e) shows the eigenvalues for a square lattice with $7\times 7$ unit cells with open boundary conditions for the same parameters as in (d). The corner modes (in green) are at zero energy (by construction), and the spectrum shows the gapped bulk (blue) and edge (yellow) bands.

Figure 2

Each of the two sheets in panel (a) features two isolated corner modes. By introducing an additional interlayer coupling [see panel (b)], a pair of corner modes becomes gapped without closing a bulk or edge gap. This results in a 2D bilayer model with isolated Majorana corner modes on opposite corners. In panels (c) and (d), we show the bulk bands and spectrum of a $7\times 7$ lattice for ${\mu }_1={\mu }_2=t_{\perp }, t_{y,1}=t_{x,2}=2.5t_{\perp }, t_{x,1}=t_{y,2}=1.5t_{\perp }, t_{x,1}^{\prime }=t_{y,2}^{\prime }=\lambda =0$.

Figure 3

Adiabatic deformation to connect different geometric configurations: moving isolated corner modes from adjacent to opposite corners. (a) The couplings that will be varied (with $t_0$ acting as reference energy scale). (b) The spectrum of the finite system as a function of time, which suggests a gap closing at time $T=0.5$. (c,d) The spectrum of ribbon geometries along the $x$ and $y$ directions, respectively, at time $T=0.5$. This shows that the gap of an edge along the $y$ axis closes at $T=0.5$.

Figure 4

From (a) to (c), we present the strategy to couple a pair of 2D Majorana corner mode models to obtain a 3D bulk model with corner modes in adjacent corners. Panel (d) shows the gapped bulk bands, and in panel (e) we show the eigenvalues for a square with $7\times 7\times 7$ unit cells with open boundary conditions. The corner modes (in red) are at zero energy, and the spectrum shows the eigenvalues of clearly gapped bulk (blue) bands. The parameters used are ${\alpha }_1=-0.9t_x, {\mu }_1={\mu }_2=-0.25t_x, t_y=t_y^{\prime }=0.45t_x, t_z=-1.2t_x$.