Hidden-Symmetry-Protected Topological Semimetals on a Square Lattice

We study a two-dimensional fermionic square lattice, which supports the existence of a two-dimensional Weyl semimetal, quantum anomalous Hall effect, and $2\pi$-flux topological semimetal in different parameter ranges. We show that the band degenerate points of the two-dimensional Weyl semimetal and $2\pi$-flux topological semimetal are protected by two distinct novel hidden symmetries, which both correspond to antiunitary composite operations. When these hidden symmetries are broken, a gap opens between the conduction and valence bands, turning the system into a insulator. With appropriate parameters, a quantum anomalous Hall effect emerges. The degenerate point at the boundary between the quantum anomalous Hall insulator and trivial band insulator is also protected by the hidden symmetry.

Figure 1

(a) Schematic of the lattices. The arrows represent the accompanying phases of hopping $\gamma$ and the dashed lines indicate a $\pi$ accompanying phase; the blue (black) and green (gray) filled circles represent the lattice sites of sublattices $A$ and $B$, respectively. (b) The first Brillouin zone, which is surrounded by red solid lines. Here, the green (gray) filled circles represent symmetry points denoted by $\Gamma$, $M$, and $X_{1,2}$, respectively.

Figure 2

The energy dispersions for (a) a two-dimensional Weyl semimetal with $\gamma =\pi /4$, $t_1=0$, $v=0$; (b) quantum anomalous Hall insulator with $\gamma =\pi /4$, $t_1=0.8t$, $v=0.2t$; (c) the boundary between the quantum anomalous Hall insulator and trivial band insulator with $\gamma =\pi /4$, $t_1=0.5t$, $v=2t$; (d) $2\pi$-flux topological semimetal with $\gamma =0$, $t_1=t$ and $v=0$.

Figure 3

(a) Phase diagram for insulators. Here, QAH and BI represent quantum anomalous Hall insulator and trivial band insulator, respectively. (b) The edge states of the quantum anomalous Hall phase for $\gamma =\pi /4$, $t_1=0.8t$ and $v=0.2t$. Red solid line and green dashed line represent edge states at two opposite edges, respectively.