Time-reversal symmetric topological metal

Topological metals possess gapless band structures accompanied by nontrivial edge states. Topological metals are created from the topological insulators by adjusting the magnetic flux. Here we propose the time-reversal symmetry protected topological metal without employing the magnetic flux. Topological metallic phase presents although the Chern number vanishes. In the topologically nontrivial phase, three different cases in the metallic phase are distinguished from the band-touching and in-gap features of the topological edge states. These findings shed light on the time-reversal symmetric topological metals and greatly simplify the realization of topological metals.

Figure 1

Schematic of the 2D square lattice. The unit cell consists of two sublattices $A$ and $B$ in cyan and magenta as indicated inside the dashed blue rectangle. $\lambda , \kappa$, and $m$ are the couplings between the sublattices.

Figure 2

Phase diagram of $h\left(\mathbf{k}\right)$ in the $\kappa -\lambda$ parameter space. Regions with different colors indicate different phases: topological metal (TM) in purple, ordinary metal (OM) in green, topological insulator (TI) in cyan, and ordinary insulator (OI) in yellow. The black solid lines indicate the Wilson transitions. The red dashed lines indicate the topological phase transition with two DPs, while the blue dashed line indicates the topological phase transition with four DPs. The brown solid lines are the boundary that characterizes the positional relationship between the edge states and bulk bands, indicating three different cases in the topological metallic or insulator phase.

Figure 3

Band structures of three cases in the insulator phase at (a) $\kappa =1,\lambda =-0.5$, (b) $\kappa =2,\lambda =-1$, (c) $\kappa =5,\lambda =-1$ and three cases in the metallic phase at (d) $\kappa =1,\lambda =0.5$, (e) $\kappa =2,\lambda =1$, (f) $\kappa =5,\lambda =1$ as indicated by the black solid dots in the phase diagram of Fig. 2. The energy band is depicted for the OBC in the $x$ direction and the PBC in the $y$ direction. The other system parameter is $m=1$. The red line indicates the edge state.

Figure 4

Plot of the total inverse participation ratio along the momentum $k_y$ as the counterpart of Fig. 3. The red line represents the TIPR including the localized edge state, and the black line represents the TIPR of extended states in bulk. (a) $\kappa =1,\lambda =-0.5$, (b) $\kappa =2,\lambda =-1$, (c) $\kappa =5,\lambda =-1$, (d) $\kappa =1,\lambda =0.5$, (e) $\kappa =2,\lambda =1$, (f) $\kappa =5,\lambda =1$.

Figure 5

Plot of the local density of state near the zero energy as the counterpart of Fig. 3. (a) $\kappa =1,\lambda =-0.5$, (b) $\kappa =2,\lambda =-1$, (c) $\kappa =5,\lambda =-1$, (d) $\kappa =1,\lambda =0.5$, (e) $\kappa =2,\lambda =1$, (f) $\kappa =5,\lambda =1$. Here we take $\alpha =2$ and lattice size is $30\times 30$. The color bar indicates the probability of wave function and the maximum is renormalized to 1.

Figure 6

Topological phase diagram of the time-reversal symmetric 2D square lattice. The $\left|\lambda \right|<\left|\kappa \right|$ region in purple is the topologically nontrivial phase; in contrast, the $\left|\lambda \right|>\left|\kappa \right|$ region in green is the trivial phase. The Zak phase ${\mathrm{\Theta }}_x$ distinguishes different topological phases.