Novel Topological Phase with a Zero Berry Curvature

We present a two-dimensional (2D) lattice model that exhibits a nontrivial topological phase in the absence of the Berry curvature. Instead, the Berry connection provides the topological nontrivial phase in the model, whose integration over the momentum space, the so-called 2D Zak phase, yields a fractional wave polarization in each direction. These fractional wave polarizations manifest themselves as degenerated edge states with opposite parities in the model.

Figure 1

(a) Schematic of 2D SSH model on a square lattice with symmetric hopping in each direction. Red and black bonds represent the intracelluar and intercelluar hoppings, respectively. The amplitudes of intracelluar and intercelluar hoppings are $w$ and $v$, respectively. There are four atoms in one unit cell. (b) First BZ of the square lattice.

Figure 2

Bulk spectra of 2D SSH model. From the lowest to the highest the energy bands are $s$, $p_x$, $p_y$, and $d_{xy}$, accordingly. The two $p$ bands are degenerate at $C_{4v}$ invariant points. Topological phase transition (band inversion) happens when an intracellular hopping equals an intercellular one. Bulk spectra before and after the band inversion is shown in (a), where the parities at high symmetric points before and after the band inversion are marked by “$\pm$” with and without parenthesis, respectively. The four bands touch at $X$ ($Y$) and $\mathrm{\Gamma }$ points when $w=v=1.0$, which is shown in (b).

Figure 3

(a) Edge states naturally appear when the intracellular hopping is 0. There are four types of atoms (marked as different colors in a unit cell) in the edge; thus, there should be four edge states in total. Ribbon spectra for (b) $w=2.0$, $v=1.0$, and (c) $w=1.0$, $v=2.0$, respectively. After the band inversion, topological edge states marked as blue lines appear in the two topological gaps, which are characterized by the two dimensional Zak phase $(\pi ,\pi )$ accompanying the fractional wave polarization $(1/2,1/2)$ for each topological gap as marked by the right side of the panel. The topologically trivial gaps have 2D Zak phase (0,0). Wave function of edge states at $k=0.2\pi$ is shown in (d). The edge states are doubly degenerate due to the inversion symmetry along the $x$ direction, which are symmetric and antisymmetric along the $x$ direction. The spatial distribution of the degenerate edge states are shown in the inset of (d).

Figure 4

Topological phase diagram of the 2D SSH model. The system enters topological nontrivial phase when $|w|<|v|$. The nontrivial topological phases are characterized by the nonzero 2D Zak phase $(\pi ,\pi )$.