Edge Currents as a Signature of Flatbands in Topological Superconductors

We study nondegenerate flatbands at the surfaces of noncentrosymmetric topological superconductors by exact diagonalization of Bogoliubov–de Gennes Hamiltonians. We show that these states are strongly spin polarized and acquire a chiral dispersion when placed in contact with a ferromagnetic insulator. This chiral mode carries a large edge current which displays a singular dependence on the exchange-field strength. The contribution of other edge states to the current is comparably weak. We hence propose that the observation of the edge current can serve as a test of the presence of nondegenerate flatbands.

Figure 1

Left column: Momentum-resolved LDOS on a log scale for the ten outermost layers at the (10) edge of the ($d_{xy}+p$)-wave NCS with (a) $q=0.25$ and (c) $q=0.75$, and for the ($s+p$)-wave NCS with (e) $q=0.25$ and (g) $q=0.75$. In the right column, we present the corresponding $x$ component of the momentum-resolved spin LDOS for the outermost layer in units of $\hslash /20$ on a linear scale. The $z$ component of the momentum-resolved spin LDOS is provided in Fig. S1 of the SM [29].

Figure 2

Momentum-resolved LDOS at the (10) edge of (a),(b) the ($d_{xy}+p$)-wave and (c),(d) the ($s+p$)-wave NCSs in the presence of an exchange field along the $x$ axis with $H_{\mathrm{ex}}^x=0.4$, applied to the edge layer $n=1$. In the left column, we have $q=0.25$ (majority triplet), while in the right column, we plot results for $q=0.75$ (majority singlet). As in Fig. 1, we use a log color scale.

Figure 3

Top row: Zero-temperature edge current $I_y$ as a function of the exchange field $H_{\mathrm{ex}}^x$ for various values of the singlet-triplet parameter $q$ for (a) the ($d_{xy}+p$)-wave and (b) the ($s+p$)-wave NCSs. Bottom row: $I_y$ as a function of the singlet-triplet parameter $q$ for various exchange fields along the $x$ axis, applied to the leading edge of (c) the ($d_{xy}+p$)-wave and (d) the ($s+p$)-wave NCSs. For comparison, the edge current of a chiral $p$-wave superconductor without an exchange field or SOC is about 0.64 in our units.

Figure 4

Momentum-resolved edge current $I_y(k_y)$ for various values of singlet-triplet parameter $q$ in (a) the ($d_{xy}+p$)-wave and (b) the ($s+p$)-wave NCSs. The exchange field is $H_{\mathrm{ex}}^x=0.2$. The vertical gray lines indicate the projections of the edges of the two spin-orbit-split Fermi surfaces $|k_y|=k_{F,\pm }$. Note that the ($s+p$)-wave results are multiplied by 10 for clarity.