Majorana zero modes in dislocations of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$

We study the topologically protected Majorana zero modes induced by lattice dislocations in chiral topological superconductors. Dislocations provide a new way to realize Majorana modes at zero magnetic field. In particular, we study several different types of dislocations in the candidate material ${\mathrm{Sr}}_2{\mathrm{RuO}}_4.$ We also discuss the properties of linked dislocation lines and linked dislocation and flux lines. Various experimental consequences are predicted which provide an approach to determine the nature of the superconducting phase of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4.$

Figure 1

(a) The Brillouin zone of a 3D SC. The primary weak topological invariants $\mathbf{n}=(n_x,n_y,n_z)$ are defined as the Chern numbers in the three independent planes (colored green, blue, and red), and the secondary weak topological invariants $\nu =({\nu }_x,{\nu }_y,{\nu }_z)$ are defined as the 1D $Z_2$ indices along the three perpendicular lines colored in red. (b) Illustration that a topological superconductor with topological invariants $\mathbf{n}=(0,0,1),\nu =(1,1,0)$, such as ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$, is equivalent to decoupled layers of chiral superconductors (with the red line and arrow labeling the chiral edge states) and 1D wires along $x$ and $y$ directions with Majorana zero modes at end points.

Figure 2

(Left) Fermi-surface structure of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ showing the Fermi surfaces coming from the $d_{xz}$, $d_{yz}$, and $d_{xy}$ bands with orbital mixing. (Right) Low-energy quasiparticle spectrum of superconducting ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ for a geometry with open boundary conditions in the $y$ direction and periodic boundary conditions on the $x$ direction. The states in red-dashed lines are edge states arising from the quasi-1D bands; notice they exist at $k_x=0$ and $k_x=\pi .$ The green dash-dotted lines are the edge states arising from the quasi-2D band; notice they exist only around $k_x=0.$

Figure 3

(a) Illustration of an edge dislocation line with Burgers' vector $\mathbf{b}$ in a TSC consisting of decoupled layers stacked along the direction of topological index vector $\mathbf{n}$. (b) Illustration of an edge dislocation line with Burgers' vector $\mathbf{b}$ and direction $\mathbf{l}$ in a TSC consisting of decoupled 1D wires along the direction of secondary weak topological index vector $\nu$. The red dots at the end of wires represent Majorana zero modes. (c) Illustration of the Majorana zero modes induced by linking of two edge dislocations with ${\mathbf{b}}_1=\widehat{z}$ and ${\mathbf{b}}_2=-\widehat{x}$ in the decoupled plane limit. There is a Majorana zero mode at each dislocation line, indicated by the red dot and red line.

Figure 4

Spatial profile of the sum of the probability densities for the two lowest energy (closest to zero energy) bound states at the ends of a dislocation that stretches from $x=40$ to $x=120$ on the line $y=50$. The total lattice size is $160\times 100$. The asymmetry in the density profile is due to finite-size error introduced by discretization.