Topological and edge state properties of a three-band model for Sr${}_2$RuO${}_4$

Modeling the spin-triplet superconductor Sr${}_2$RuO${}_4$ through a three-orbital tight-binding model, we investigate topological properties and edge states assuming chiral $p$-wave pairing. In concordance with experiments, the three Fermi surfaces consist of two electronlike and one holelike surface, corresponding to the $\alpha$, $\beta$, and $\gamma$ band on the level of a two-dimensional system. The quasiparticle spectra and other physical quantities of the superconducting phase are calculated by means of a self-consistent Bogoliubov-de Gennes approach for a ribbon-shaped system. While a full quasiparticle excitation gap is realized in the bulk system, at the edges gapless states appear, some of which have linear and others of which have a nearly flat dispersion around zero energy. This study shows the interplay between spin-orbit-coupling-induced spin currents, chiral edge currents, and correlation-driven surface magnetism. The topological nature of the chiral $p$-wave state manifests itself in the $\gamma$ band characterized by an integer Chern number. As the $\gamma$ band is close to a Lifshitz transition in Sr${}_2$RuO${}_4$, changing the sign of the Chern number, the topological nature may be rather fragile.

Figure 1

Lattice structure of $L$-leg ribbon with three orbitals: $d_{yx}$, $d_{zx}$, and $d_{xy}$. $t$ ($t^{\prime }$) stands for the hopping integral between same (different) orbitals. $t_z$ ($t_z^{\prime }$) stands for the hopping integral between $\gamma$ orbitals. $U_a$ represents the attractive interaction between the nearest-neighbor sites.

Figure 2

Fermi surfaces of two-dimensional bulk system for $t^{\prime }=0.1t$, $t_z=0.7t$, $t_z^{\prime }=0.3t$, $\Delta \varepsilon =0.065t$, and $\lambda =0.1t$ with $U_a=U_r=J_r=0$. The black, blue, and red lines stand for the $\alpha$, $\beta$, and $\gamma$ bands, respectively. The thin dashed lines stand for the Fermi surface of $\alpha$-$\beta$ bands without $t^{\prime }$ and $\lambda$.

Figure 3

Gap functions as a function of leg index $l$ in $\mathit{d}=\widehat{z}(k_x+i{k}_y)$ superconducting state for $U_r=J_r=0$.

Figure 4

Spectral functions at $l=1$ for $U_r=J_r=0$. (a) Total spectral function. (b) Each component of spectral function.

Figure 5

Spectral functions of the $y$ orbitals for various $l$ near an edge for $U_r=J_r=0$.

Figure 6

Energy dispersion in the low-energy sector of the superconducting state; (a) bulk and (b) ribbon for $U_r=J_r=0$.

Figure 7

Spin polarization of the $y$ orbitals for several choices of $U_r$ near an edge. The inset shows spin polarizations of the $x$ orbitals (solid line) and $z$ orbitals (dashed line).

Figure 8

(a) Energy dispersions and (b) spectral functions at $l=1$ in the low-energy region for several choices of $U_r$ for $J_r=0$. The solid (dashed) line stands for the spectrum of the electron up (down) spin in the lower panel.

Figure 9

Spin polarization of the $y$ orbitals for several choices of $J_r$ near an edge for $U_r=t$. The inset shows spin polarizations of the $x$ orbital (solid line) and $z$ orbitals (dashed line).

Figure 10

Spin polarization of the $y$ orbital for several choices of $\lambda$ near an edge for $U_r=0.5t$ and $J_r=0$. The inset shows spin polarizations of the $x$ orbital (solid line) and $z$ orbital (dashed line).

Figure 11

(a) Spin $\uparrow$ currents for $\mathit{d}=\widehat{z}(k_x+i{k}_y)$ state as a function of leg index $l$ for $\lambda =0.1t$ and $U_r=J_r=0$. (b) Magnification of (a). Solid lines stand for spin $\uparrow$ and $\downarrow$ electron for $\lambda =0.1t$, and open (closed) circle represents the current for $\lambda =0$.

Figure 12

Charge and spin currents as a function of leg index $l$ for $U_r=J_r=0$. The insets show the magnifications near an edge.

Figure 13

Charge current as a function of leg index $l$ near an edge for several choices of $U_r$ and $J_r$. The inset shows spin current.

Figure 14

Spontaneous magnetic field for several choices of $U_r$ for $\lambda =0.1$ and $J_r=0$. The solid [dashed] line stands for $B_z^r\left(l\right)$ [$B_z^c\left(l\right)$].

Figure 15

Topological number as a function of chemical potential $\mu$ for several choices of $\lambda$ with $\mathit{d}=\widehat{z}(k_x+i{k}_y)$: (a) two-band and (b) three-band models.

Figure 16

Energy dispersion in the low-energy sector of the superconducting state at $\mu =1.2t$.