Square-root topological semimetals

We propose topological semimetals generated by the square-root operation for tight-binding models in two and three dimensions, which we call square-root topological semimetals. The square-root topological semimetals host topological band touching at finite energies, whose topological protection is inherited from the squared Hamiltonian. Such a topological character is also reflected in emergence of boundary modes with finite energies. Specifically, focusing on topological properties of squared Hamiltonian in class AIII, we reveal that a decorated honeycomb (decorated diamond) model hosts finite-energy Dirac cones (nodal lines). We also propose a realization of a square-root topological semimetal in a spring-mass model, where robustness of finite-energy Dirac points against the change of tension is elucidated.

Figure 1

(a) A decorated honeycomb lattice and (b) the band structure with $t=1$. The finite-energy Dirac points are denoted by green circles.

Figure 2

(a) The dispersions for the system with the zigzag edge. The cyan lines represent the flat edge modes. (b) The winding number defined for $q_{\mathit{k}}^{\left(\mathrm{H}\right)}$ in Eq. (11). We set $t=1$.

Figure 3

(a) A decorated diamond lattice and (b) the band structure with $t=1$. The finite-energy nodal lines are denoted by green ellipses.

Figure 4

(a) The dispersions of the decorated diamond model with a slab geometry. The cyan lines represent the flat surface modes. (b) The map of the winding number defined for $q_{\mathit{k}}^{\left(\mathrm{D}\right)}$ in Eq. (16). We set $t=1$.

Figure 5

(a) A schematic figure of the spring-mass decorated honeycomb model. We have introduced potential force arising from the dents of the floor so that the diagonal elements of $\mathrm{\Gamma }\left(\mathit{k}\right)$ become ${\left[\mathrm{\Gamma }\left(\mathit{k}\right)\right]}_{ii}=3\kappa (1-\eta /2)$ for $i=1,\cdots ,10$. Dispersion relations of the spring-mass decorated honeycomb model for $\kappa =m=1$ with (b) $\eta =0$ and (c) $\eta =0.25$. Green circles denote the finite-energy Dirac points.

Figure 6

(a) The band structure for the decorated honeycomb model with $t=1$ and $V=0.5$. The finite-energy Dirac points are denoted by green circles. (b) The dispersions for the cylinder with the same parameters. The cyan lines represent the flat edge modes.