Properties of edge states in a spin-triplet two-band superconductor

Motivated by Sr${}_2$RuO${}_4$, the magnetic properties of edge states in a two-band spin-triplet superconductor with electronlike and holelike Fermi surfaces are investigated assuming chiral $p$-wave pairing symmetry. The two bands correspond to the $\alpha$-$\beta$ bands of Sr${}_2$RuO${}_4$ and are modeled within a tight-binding model including interorbital hybridization and spin-orbit coupling effects. Including superconductivity, the quasiparticle spectrum is determined by means of a self-consistent Bogoliubov–de Gennes calculation. While a full quasiparticle excitation gap appears in the bulk, gapless states form at the edges that produce spontaneous spin and/or charge currents. The spin current is the result of the specific band structure, while the charge current originates from the superconducting condensate. Together they induce a small spin polarization at the edge. Furthermore, onsite Coulomb repulsion is included to show that the edge states are unstable against the formation of a Stoner-type spin polarization of the edge states. Through spin-orbit coupling, the current- and the correlation-induced magnetism are coupled to the orientation of the chirality of the superconducting condensate. We speculate that this type of phenomenon could yield a compensation of the magnetic fields induced by currents and also explain the negative result in the recent experimental search for chiral edge currents.

Figure 1

Lattice structure of $L$-leg ribbon with two $p_x$ and $p_y$ orbitals. $t$ ($t^{\prime }$) stands for the hopping integral between the same (different) orbitals.

Figure 2

Fermi surfaces of two-dimensional bulk system for $t^{\prime }=0.1t$ and $\lambda =0.1t$ (solid line) and $t^{\prime }=\lambda =0$ (dashed line), in which particle number $n=8/3$ and $U_a=U_r=J_r=0$.

Figure 3

Spin-dependent current in normal phase for $L=100$, $t^{\prime }=\lambda =0.1t$, $U_r=J_r=0$, and $n=8/3$. The solid (dashed) line stands for spin-up (-down) current. The inset shows magnification near the edge.

Figure 4

Spin-dependent chirality.

Figure 5

Order parameters of the superconducting state as a function of leg index $l$ for $\lambda =0.1t$ and $U_r=J_r=0$. The black (red) lines stand for $U_r=0.0$ ($U_r=0.5t$). The inset shows magnification near the edge.

Figure 6

Energy dispersions near low-energy region; (a) $U_a=-1.25t$ and (b) $U_a=-1.5t$ for $U_r=J_r=0$. Red (blue) line stands for edge state from $l=1$ ($l=L$).

Figure 7

Spin-dependent currents as a function of $l$ for $\lambda =0.1t$ and $U_r=J_r=0$. The black (red) lines stand for $\mathit{d}=\widehat{z}(k_x+i{k}_y)$ [$\mathit{d}=\widehat{z}(k_x-i{k}_y)$] state and the solid (dashed) lines stand for spin-up (-down) contributions. The inset shows each component of the spin-up current for the $\mathit{d}=\widehat{z}(k_x+i{k}_y)$ state.

Figure 8

Spin and charge currents as a function of $l$ for $\lambda =0.1t$ and $U_r=J_r=0$. The inset shows the flow direction of the chiral charge current near the edges.

Figure 9

Sum of spin current for various $\lambda$. The inset shows the sum of charge current for $U_r=J_r=0$.

Figure 10

Spin and charge currents for with/without repulsive interactions near an edge for $\lambda =0.1t$.

Figure 11

Spin polarization for various choices of $\lambda$ as a function of $l$. (a) and (b) stand for $m_{l{p}_x}$ and $m_{l{p}_y}$, respectively. $U_r=0.5t$ and $J_r=0$. The insets show the magnifications near edge.

Figure 12

Spin polarization for several choices of $U_r$. (a) and (b) stand for $m_{l{p}_x}$ and $m_{l{p}_y}$, respectively. $\lambda =0.1t$ and $J_r=0$. Both insets show the magnifications near the edge.

Figure 13

Sum of spontaneous magnetic field for several choices of repulsive interaction. Solid (dashed) lines stand for the magnetic field from the charge current (spin polarization) ($\lambda =0.1t$ and $J_r=0$). Note no screening currents have been included in this calculation.