Role of topology and symmetry for the edge currents of a two-dimensional superconductor

The bulk-boundary correspondence guarantees topologically protected edge states in a two-dimensional topological superconductor. Unlike in topological insulators, these edge states are, however, not connected to a quantized (spin) current as the electron number is not conserved in a Bogolyubov–de Gennes Hamiltonian. Still, edge currents are in general present. Here we use the two-dimensional Rashba system as an example to systematically analyze the effect symmetry reductions have on the order-parameter mixing and the edge properties in a superconductor of Altland-Zirnbauer class DIII (time-reversal-symmetry preserving) and D (time-reversal-symmetry breaking). In particular, we employ both Ginzburg-Landau and microscopic modeling to analyze the bulk superconducting properties and edge currents appearing in a strip geometry. We find edge (spin) currents independent of bulk topology and associated topological edge states which evolve continuously even when going through a phase transition into a topological state. Our findings emphasize the importance of symmetry over topology for the understanding of the nonquantized edge currents.

Figure 1

Coupling scheme for the different order-parameter components. In parentheses are the respective irreps of $D_{4h}$.

Figure 2

(Left) Band structure of the Rasbha model studied with the dispersion along a high-symmetry axis. We have used $t=1$, $t^{\prime }=0.25$, $\alpha ={\alpha }^{\prime }=0.5$, and $\mu =-3$. The inset shows the corresponding Fermi surfaces. (Right) Zoom of the band-crossing region with the same parameters apart from $\alpha ={\alpha }^{\prime }=h=0$ (solid line), only $h=0$ (dotted line), and $h=0.125$ (dashed line).

Figure 3

(Top) Order parameter coefficients for different ratios of the interaction strengths at constant $T=0.5T_c$. $T_c$ is defined by $U_s=-3.5$, $U_p=0$. Single-particle parameters as in Fig. 2. (Bottom) Interaction strengths for different ratios at constant $T_c$.

Figure 4

Phase diagram for the inversion-symmetry-broken case with a topologically trivial (green) and a nontrivial, helical phase (blue). Note that there is a finite region (gray) in between, where the gap has point nodes. Inset: Point nodes at the critical ratios corresponding to the trivial and the topological phase.

Figure 5

Order parameter coefficients for different ratios of the interaction strengths at $T=0.5T_c$. $T_c$ is defined by $U_s=-3.5$, $U_p=0$.

Figure 6

(Top) Integrated current over half of the strip as a function of $U_p/U_s$. The insets show examples of the spectrum along $k_x$ in the trivial and the topological phase; the latter featuring counterpropagating edge states. (Bottom) Charge and spin currents for the trivial phase (left) and the helical phase (right) with parameters as for the corresponding insets.

Figure 7

(Top) Integrated current over half of the strip as a function of the chemical potential $\mu$ with $h=0.125$. The interaction strengths are kept constant at $U_s=-2.5$ and $U_p=-5.0$. The insets show the spectrum for the trivial ($\mu =-4.8$, left) and the topological ($\mu =-4.95$, right) phase along $k_x$; the latter featuring chiral edge states. (Bottom) Charge and spin currents for the trivial (left) and the chiral (right) phases with parameters as for the corresponding insets.

Figure 8

Strip geometry.

Figure 9

Comparison of wave-function profiles (lower panel) for the topologically trivial (upper left spectrum) and the nontrivial helical (upper right spectrum) superconducting state in the strip geometry. The two crosses denote the energy and momentum of the states shown in the lower panel. The dashed line, which decays exponentially from both edges, is for comparison.

Figure 10

Contributions to the spin current in the topologically trivial (top) and nontrivial helical (bottom) case. The left side visualizes the contribution of eigenstates (averaged over four consecutive energies for each $k_x$ as a smoothing procedure) to the spin current ${\mathcal{J}}^{x,z}(E,k_x)$ for one of the two edges. The right side shows the momentum-integrated contribution corresponding to the energy-resolved spin current ${\mathcal{J}}^{x,z}\left(E\right)$.

Figure 11

Contributions to the charge current in the topologically trivial (top) and nontrivial chiral (bottom) case. The left side visualizes the contribution of individual eigenstates to the charge current ${\mathcal{J}}^{x,0}(E,k_x)$ for one of the two edges. The right side shows the momentum-integrated contribution, the energy-resolved charge current ${\mathcal{J}}^{x,0}\left(E\right)$.