Scattering matrix invariants of Floquet topological insulators

Similar to static systems, periodically driven systems can host a variety of topologically nontrivial phases. Unlike the case of static Hamiltonians, the topological indices of bulk Floquet bands may fail to describe the presence and robustness of edge states, prompting the search for new invariants. We develop a unified description of topological phases and their invariants in driven systems by using scattering theory. We show that scattering matrix invariants correctly describe the topological phase, even when all bulk Floquet bands are trivial. Additionally, we use scattering theory to introduce and analyze new periodically driven phases, such as weak topological Floquet insulators, for which invariants were previously unknown. We highlight some of their similarities with static systems, including robustness to disorder, as well as some of the features unique to driven systems, showing that the weak phase may be destroyed by breaking translational symmetry not in space, but in time.

Figure 1

Band structure of the Floquet operator (6) in an infinite strip geometry (infinite in $y,L=21$ sites in the $x$ direction). Parameters are $\alpha =2\pi /3,J_x=2\pi /3$, and $J_y=\pi ,2\pi ,3\pi$ in the left, middle, and right panels, respectively. Red (dark gray) indicates states localized on the left boundary and blue (light gray) indicates states on the right. From left to right, the bulk gap at $\varepsilon =\pi$ shows zero, one, or two pairs of counterpropagating edge modes, separated by a momentum difference $\mathrm{\Delta }{k}_y=\pi$ (horizontal arrows).

Figure 2

Side view of the Hamiltonian (10), consisting of two copies of (1) labeled by $j=1,2$ (top and bottom layers). One of the three unit cells shown is marked by a box. The coupling within the top and bottom layers is marked with orange and red lines, respectively, while that between the layers is shown in blue. Notice that chiral symmetry is preserved, as there is no coupling between layers in the same unit cell.

Figure 3

Band structures of the model (10) in the same geometry as Fig. 1. We set $\alpha =J_x=2\pi /3,J_y=2\pi$, and $J_c=0.6$. In the left, middle, and right panels, $\delta =0.7\pi ,0.85\pi$, and $0.98\pi$, respectively. Sliding one pair of edge modes on top of the other (horizontal arrows) causes them to annihilate and opens gaps in the spectrum. This indicates they are protected by a ${\mathbb{Z}}_2$ invariant.

Figure 4

Two possible ways in which the phase of $det\tilde{r}$ can evolve as the twist angle is advanced from $\varphi =0$ to $\varphi =\pi$. The solid and dashed lines indicate topologically different scenarios, since they cannot be deformed into each other as long as the bulk mobility gap remains open $\left(det\tilde{r}\ne 0\right)$. The presence of protected $\pi$-crossings signals a topologically nontrivial phase in our choice of gauge.

Figure 5

Log-log plot of the edge transmission of model (6) at $\varepsilon =\pi$ as a function of system length $L$ for bond (red) and on-site (blue) disorder, with $W=40,\alpha =J_x=2\pi /3$, and $J_y=2\pi$. For on-site disorder we have used a disorder strength $U=1.8$, while for bond disorder it was set to $U=0.45$. In each case, 600–800 independent disorder realizations were averaged over. The solid and dashed lines show the expected algebraic and exponential decay of transmission, respectively.

Figure 6

Setup and driving protocol of Hamiltonian (18). We consider an $L\times W$ hexagonal lattice (sublattices $A$ and $B$), with Bravais vectors $a_x$ and $a_y$. When $J_u=0$, only one of the hopping amplitudes $J_{x,y,z}$ (blue, green, and red, respectively) is turned on at any given time. For $J_s=\pi /2$, particles at any site in the bulk return to their original position after two driving periods, while those on the edges form chiral propagating modes, as shown by the arrows.

Figure 7

Band structures of the model (18) at $J_s=\pi /2$ in a strip geometry (infinite along $a_x,W=20$). In panel (a), $J_u=0$ and there are two flat bulk bands at quasienergies $\varepsilon =\pm \pi /2$ as well as one chiral edge mode present at all quasienergies. In panel (b) we set $J_u=0.25$, such that there is one chiral mode at $\varepsilon =\pi$ and a pair of counterpropagating modes at $\varepsilon =0$. The strong and weak invariants at $\varepsilon =\pi$ read $(C_{\mathrm{SM}},{\nu }_{x,\pi },{\nu }_{y,\pi })=(-1,1,1)$ in both panels. At $\varepsilon =0$, the system in panel (a) is in a $(-1,-1,-1)$ phase, while panel (b) has $(0,-1,-1)$. The color scale denotes the eigenstate intensity on the first and last $10\%$ of lattice sites.

Figure 8

Band structure of the model (18) in an infinite strip geometry with boundaries along the $a_x$ (top, $W=30$) and $a_y$ (bottom, $L=30$) directions. We use $J_s=\pi /2$ and inequivalent uniform components of the hopping amplitudes, $J_{u,x}=-0.05,J_{u,y}=J_{u,z}=0.5$. In the right panels, the BZ is folded in quasienergy by changing the stroboscopic part of $J_x$ to $J_{s,x}=\pi /2+0.6$ on every second period. The invariants of the prefolding phases (left panels), $(C_{\mathrm{SM}},{\nu }_{x,\pi },{\nu }_{y,\pi })=(0,-1,-1)$ at $\varepsilon =0$ and $(0,-1,1)$ at $\varepsilon =\pi$, combine to form a new gapped phase with $(0,1,-1)$ at $\varepsilon =0$. The color scale denotes the eigenstate intensity on the first and last $10\%$ of the lattice sites.