Universal driving protocol for symmetry-protected Floquet topological phases

We propose a universal driving protocol for the realization of symmetry-protected topological phases in $2+1$-dimensional Floquet systems. Our proposal is based on the theoretical analysis of the possible symmetries of a square lattice model with pairwise nearest-neighbor coupling terms. Among the eight possible symmetry operators we identify the two relevant choices for topological phases with either time-reversal, chiral, or particle-hole symmetry. From the corresponding symmetry conditions on the protocol parameters, we obtain the universal driving protocol where each of the symmetries can be realized or broken individually. We provide specific parameter values for the different cases, and demonstrate the existence of symmetry-protected copropagating and counterpropagating topological boundary states. The driving protocol especially allows us to switch between bosonic and fermionic time-reversal symmetry, and thus between a trivial and nontrivial symmetry-protected topological phase, through continuous variation of a parameter.

Figure 1

Eight examples of patterns of compatible pairwise couplings in the square lattice model. The couplings are indicated by lines between adjacent sites. The labels below each pattern follow the notation in the text, with the four sites $A$ (filled red), $B$ (filled blue), $C$ (open red), and $D$ (open blue) as shown in the leftmost pattern.

Figure 2

Graphical representation of the eight symmetry operators $S_1,\cdots ,S_8$ for the Hamiltonian (1). The arrows in the upper row indicate how each symmetry operator maps a site onto itself, or onto one of its eight neighbors. The lines in the lower row indicate the compatible pairwise coupling terms.

Figure 3

Two variants of the driving protocol A, which consist of a cyclic six-step sequence of the first five coupling patterns (a)—(e) from Fig. 1. (Left) In this variant, the protocol consists of the sequence $\left(\mathrm{b}\right)\to \left(\mathrm{a}\right)\to \left(\mathrm{c}\right)\to \left(\mathrm{d}\right)\to \left(\mathrm{a}\right)\to \left(\mathrm{e}\right)$. (Right) In this variant, the protocol consists of the sequence $\left(\mathrm{b}\right)\to \left(\mathrm{a}\right)\to \left(\mathrm{d}\right)\to \left(\mathrm{c}\right)\to \left(\mathrm{a}\right)\to \left(\mathrm{e}\right)$. As shown in the text, both variants are equivalent.

Figure 4

Patterns of motion during one cycle of driving protocol A at perfect coupling, on a finite lattice of $6\times 4$ unit cells (one unit cell is shown as a gray rhomboid). The lattice comprises only entire unit cells, such that the boundaries are compatible with the symmetry operators $S_1$ and $S_8$. The left and right panel correspond to the two variants of the protocol in Fig. 3. For the “left” variant, the coupling in the horizontal steps 2, 5 is equal to $\pm J_p$, for the “right” variant it is equal to zero.

Figure 5

Dispersion of bulk (solid) and boundary (dashed) states for perfect coupling with fermionic time-reversal (left) or chiral (right) symmetry. Parameter values can be deduced from the corresponding columns in Table 3, setting $J=J_p$ and $\mathrm{\Delta }=0$. Here and in Figs. 6–8. we show the states on one boundary of a semi-infinite ribbon, and do not include the states on the opposite boundary.

Figure 6

Floquet bands and boundary states for fermionic (top row) and bosonic (bottom row) time-reversal symmetry, using the parameters from Table 3. (Left column) Blue arcs indicate the (fourfold degenerate) Floquet bands, red arcs the gaps. Quasienergies $\varepsilon$ are plotted on the circle $\varepsilon \mapsto e^{-\mathrm{i}\varepsilon }$. Included are the respective (Kane-Mele $\mathrm{KM}$ or Chern number $C$) invariants of the bands, and the $W_{\mathrm{tr}}$ invariant or the $W_3$ invariant associated with the gap. (Central and right columns) Floquet bands (solid) and boundary state dispersion (dashed), as a function of momentum $k_x$ or $k_y$ for a boundary along the $x$ or $y$ direction.

Figure 7

Switching between bosonic (left and central) and fermionic (central and right) time-reversal symmetry through continuous variation of the parameter $\delta$ (see text). The panels show the boundary state dispersion. The protocol parameters for the negative and positive $\delta =\pm \pi /T$ used here agree with Fig. 6.

Figure 8

Same as Fig. 6, now for chiral (top) and particle-hole symmetry (bottom) and parameters from Table 3. The relevant bulk invariants are here the Chern numbers $C$ of the Floquet bands, and the symmetry-adapted $W_{\mathrm{ch}}$ and $W_{\mathrm{ph}}^{\alpha }$ invariants from Ref. [32].

Figure 9

Propagation of boundary states in the vicinity of a corner, starting from a “red” $A$ site or a “blue” $B$ site. Open black circles indicate the lattice sites. Shown is the (squared) wave function amplitude, with colors according to the two color bars, after zero ($t=0$), three ($t=3T$), or eight cycles ($t=8T$) of driving protocol A.

Figure 10

(Left) Six-step driving protocol B for particle-hole symmetry ${\mathrm{\Pi }}^2=-1$. (Central) Patterns of motion during one cycle at perfect coupling. (Right) Floquet bands and boundary states for the set of parameters from Table 4.

Figure 11

The square lattice can be viewed as a centered square lattice, or the union of a “red” and “blue” square lattice.

Figure 12

The eight coupling patterns with parallel diagonal pairwise couplings. Patterns (f1) and (g1) correspond to patterns (f) and (g) in Fig. 1.

Figure 13

A universal driving protocol that contains protocol A (right variant in Fig. 3) and protocol B, but violates the constraints from Sec. 2.

Figure 14

Pairwise couplings (solid lines) on a square lattice with a two-element (a filled and an open circle) unit cell, and two choices for translation symmetry.

Figure 15

Options for a symmetry operator for fermionic time-reversal symmetry on the square lattice from Fig. 14 (top row) and the compatible pairwise couplings (bottom row).