Floquet Weyl phases in a three-dimensional network model

We study the topological properties of three-dimensional (3D) Floquet band structures, which are defined using unitary evolution matrices rather than Hamiltonians. Previously, two-dimensional band structures of this sort have been shown to exhibit anomalous topological behaviors, such as topologically nontrivial zero-Chern-number phases. We show that the band structure of a 3D network model can exhibit Weyl phases, which feature “Fermi arc” surface states like those found in Weyl semimetals. Tuning the network's coupling parameters can induce transitions between Weyl phases and various topologically distinct gapped phases. We identify a connection between the topology of the gapped phases and the topology of Weyl point trajectories in $k$ space. The model is feasible to realize in custom electromagnetic networks, where the Weyl point trajectories can be probed by scattering parameter measurements.

Figure 1

(a) One unit cell of the 3D network model. The thick black lines are directed links, with the arrows indicating the directions of wave propagation. The colored circles represent coupling nodes, lying at the center of each face of the cube. The incident wave amplitudes are labeled ${\psi }_{\mu }^{\pm }$. (b) Weyl point positions in $k$ space, for ${\theta }_x={\theta }_y=\pi /8, {\theta }_z=3\pi /8$. The Weyl points consist of two pairs, at $\varphi =0$ and $\varphi =\pi$, all lying on the plane $k_x=k_y$ (shaded region). Each Weyl point is labeled by its topological charge. (c) The quasienergy band structure plotted along the $k$-space cross section $k_x=k_y$.

Figure 2

(a) Quasienergy band structure for the 3D network in a slab geometry with thickness $N=16$. The coupling parameters are ${\theta }_x={\theta }_y=\pi /8, {\theta }_z=3\pi /8$. (b) Closeup view of the band structure near $\varphi =\pi$, showing bulk Weyl cones and Fermi arc surface states. (c), (d) Cross sections of the band structure at (c) $\varphi =0$ and (d) $\varphi =\pi$; circles indicate Weyl points with positive ($+$) and negative ($-$) topological charges, calculated using the Berry flux on the band below each Weyl point.

Figure 3

(a)–(c) Heat maps of the reflection phase ${\omega }_{-}$ in a gapped phase. We take ${\theta }_x={\theta }_y=3\pi /8$ and ${\theta }_z=\pi /8$ [green star in Fig. 5], and quasienergy $\varphi =\pi$. The slab is taken in the (a) $yz$, (b) $xz$, and (c) $xy$ planes. Dashed lines are contours of ${\omega }_{-}=0$. (d) Trajectory of the Weyl points when traversing a Weyl phase from a $(0,0,0)$ conventional insulator phase to a $(0,0,1)$ weak topological insulator phase, along the path indicated in Fig. 5. The white (pink) dots show the $k$-space positions where the Weyl points are created (annihilated), at the phase boundary with the conventional insulator (weak topological insulator).

Figure 4

Heat maps of the reflection phases ${\omega }_{\pm }$ in a Weyl phase. We take ${\theta }_x={\theta }_y=\pi /8$ and ${\theta }_z=3\pi /8$ [blue star in Fig. 5], and an $xy$ slab. In (a) and (b), we fix the quasienergy at $\varphi =0$ and plot (a) the reflection phase ${\omega }_{-}$ off the lower surface, and (b) the reflection phase ${\omega }_{+}$ off the upper surface. In (c) and (d), we fix the quasienergy at $\varphi =\pi$ and plot (c) ${\omega }_{-}$ and (d) ${\omega }_{+}$. The projection of each Weyl point onto the plane coincides with a phase singularity; the circles labeled $\pm$ indicate the Weyl point's topological charge. The dashed lines are contours of ${\omega }_{\pm }=0$.

Figure 5

Phase diagram of the 3D network for two different sections of the parameter space $\{{\theta }_x,{\theta }_y,{\theta }_z\}$: (a) ${\theta }_x={\theta }_y$ and (b) ${\theta }_z=\pi /4$. The white regions are conventional insulator phases; the pink regions are $\mathcal{T}$-broken weak topological insulator phases characterized by nonzero winding number triplets $(n_x,n_y,n_z)$; and the cyan regions are gapless Weyl phases. Along the red lines and dots, the band-touching points form line nodes. The dotted line is the path producing the Weyl point trajectories in Fig. 3. Green and blue stars indicate the parameters for Figs. 3 and 4, respectively.

Figure 6

Quasienergy band structure for the 3D network in various slab geometries: (a) $yz$, (b) $xz$, and (c) $xy$. In all cases, the slab thickness is $N=16$, and the couplings are ${\theta }_x={\theta }_y=3\pi /8$ and ${\theta }_z=\pi /8$, indicated by the green star in the phase diagram, i.e., a $(0,0,1)$ gapped phase.

Figure 7

Schematic of a 1D network model.