Floquet chiral hinge modes and their interplay with Weyl physics in a three-dimensional lattice

We demonstrate that a three-dimensional time-periodically driven (Floquet) lattice can exhibit chiral hinge states and describe their interplay with Weyl physics. A peculiar type of the hinge states is enforced by the repeated boundary reflections with lateral Goos-Hänchen–type shifts occurring at the second-order boundaries of our system. Such chiral hinge modes coexist in a wide range of parameter regimes with Fermi-arc surface states connecting a pair of Weyl points in a two-band model. We find numerically that these modes still preserve their locality along the hinge and their chiral nature in the presence of local defects and other parameter changes. We trace the robustness of such chiral hinge modes to special band structure unique in a Floquet system allowing all the eigenstates to be localized in quasi-one-dimensional regions parallel to each other when open hinge boundaries are introduced. The implementation of a model featuring both the second-order Floquet skin effect and the Weyl physics is straightforward with ultracold atoms in optical superlattices.

Figure 1

(a) Bonds connected during the driving steps 1 to 6. (b) Bulk trajectories within a Floquet cycle at the fine-tuned point $\varphi =\pi /2$. Depending on the starting sublattice, particles will travel in opposite directions along the cubic diagonal $\mathit{d}=(1,-1,1)$ within a period. (c) Trajectories at the same $\varphi =\pi /2$ but with a surface termination (open boundary). Particles starting from sublattice $A$ (or $B$) near the $y=1$ (or $x=1$) surface would have their dynamics impeded by the open boundary at a certain driving step. That results in a switch of sublattice after a Floquet cycle, and therefore the direction of the trajectory is reversed after the reflection from the surface. (d) The hinge formed by two intersecting terminating surfaces renders unidirectional modes at $\varphi =\pi /2$. The figure shows two of the modes closest to the hinge starting at a site of $B$ (lower plot) or $A$ (upper plot) sublattices directly at the hinge. Each color for the arrow denotes one Floquet cycle. The lower panel of (d) shows the mode situated directly at the hinge which undergoes a unidirectional motion along $z$ within a period.

Figure 2

Table showing energy spectra and Floquet modes. The phases of the system are indicated in the vertical axis on the left. The table's rows refer to the driving parameters $\varphi =\pi /8,\pi /3,\pi /2$, corresponding to the metallic phase, the Weyl semimetal/hinge phase, and the fine-tuned point, respectively. The columns represent periodic and open boundary conditions (PBC and OBC) along the specified directions. The system sizes are $L_x=L_y=40$ in column (1), 20 in (2), and 16 in (3). Columns (1) and (2) contain energy spectra projected onto $k_z$. Although the spectra for fully periodic and open boundary conditions in $x$ and $z$ directions are almost identical in the metallic phase ($\varphi =\pi /8$), they are very different in the WSM and hinge phase ($\varphi =\pi /3$) and at fine tuning ($\varphi =\pi /2$). The difference is a result of the formation of chiral hinge modes for open boundary conditions in $x$ and $z$ directions (column 2). This can be inferred also from the dot size reflecting the inverse participation ratio with respect to the site basis as a measure for localization, as well as from the color code indicating the mean distance to two opposite hinges and interpolating from blue for one hinge over black near the center of the system to red for the other hinge. The green dots mark the Weyl points. We also plot the real-space densities of various Floquet modes for open boundary conditions along $x$ and $y$ (column 2) or all (column 3) directions.

Figure 3

(a) The different phases can be distinguished by the presence or absence of Weyl points at quasienergy $\pi$. For $\varphi =\pi /8$, in the metallic phase ($\varphi <\pi /6$), no Weyl points are present. At the transition $\varphi =\pi /6$, the band-touching point appears at $\mathit{k}=\mathit{0}$ (green circle). For $\pi /6<\varphi <\pi /2$ the band touching splits into two Weyl points of opposite charge (green circles with $\pm$ sign) that separate along the diagonal $k_x=-k_y=k_z$, as shown for $\varphi =\pi /3$. At fine tuning $\varphi =\pi /2$ the dispersion becomes flat in two directions and the Weyl points disappear to reappear for $\varphi >\pi /2$ with reversed charges (signs in green dots correspond to the limit $\varphi =\pi /2-0$). (b)–(d) Quasienergy spectra versus $k_x,k_y$ for periodic boundary conditions in $x$ and $y$ and either periodic (b), (d) or open (c) boundary conditions in the $z$ direction. A surface Fermi arc can be observed for $\varphi =\pi /3$ by comparing the spectra with periodic (b) and open (c) boundary conditions the $z$ direction. The orange surface denotes the contour formed by quasienergies closest to $E=\pi$ at each $(k_x,k_y)$. For $\varphi =\pi /6$ (d) a pair of Weyl points reduces to a single band-touching point at the quasienergy $E=\pi$.

Figure 4

(a) Anisotropic 1D-like dispersion along the coordinate $k_x-k_y+k_z$ at the fine-tuned point $\varphi =\pi /2$. (b) For a small detuning, $\varphi =\pi /2-0.1$, the dispersion is no longer completely flat in other two directions, and a pair of nonequivalent Weyl points is formed along the diagonal at $\mathit{k}=\pm k_0(1,-1,1)$. Yet, the dispersion remains highly anisotropic.

Figure 5

An example of a fine-tuned stroboscopic motion of a particle at the lower-left hinge. The picture shows the projection of the particle's trajectory in the $xy$ plane. The sites of the $B$ and $A$ sublattices are marked in blue and red, respectively. The particle is initially at the site of the sublattice $B$ characterized by the coordinates $x=M+1$ and $y=1$, with even $M=4$. This corresponds to the lower row ($y=1$) and the fifth column ($x=5$). Subsequently, the particle undergoes the stroboscopic evolution described by Eqs. (B3, B4, B5, B6, B7, B8, B9) in Appendix pp2. Dashed lines with arrows show stroboscopic reflections from the planes $x=1$ or $y=1$. Bulk ballistic trajectories over one or two driving periods are indicated by thin or thick solid arrows. The particle returns to its initial site after $2M+1=9$ periods.

Figure 6

Like in Fig. 5 the particle is initially at the site of the sublattice $B$ with $x=M+1$ and $y=1$, but now with odd $M=5$. The particle returns to the initial site after $2M+1=11$ periods.

Figure 7

The dynamics of a particle initially localized at the site $(x,y,z)=(1,1,16)$ for a system of $16\times 16\times 16$ sites with full open boundary conditions and fine tuned $\varphi =\pi /2$. The squared wave function at different times is reflected in the opacity of the plotted dots. Different colors indicate time. (a) Without defect. (b) In the presence of a potential defect of energy $\mathrm{\Delta }=3\pi$ at the two sites marked by the green tube. Additionally, we plot the squared wave function of one hinge mode. (c) Snapshots of the time evolution in the presence of the defect at different times.

Figure 8

As Fig. 7, but for $\varphi =0.9\times \pi /2$, away from fine tuning.

Figure 9

Density distribution of a particle initially localized at the corner site $(x,y,z)=(1,1,16)$ after an evolution over 100 driving cycles for the non-fine-tuned parameter $\varphi =0.9\times \pi /2$, without defect (a) and with a defect (b) [corresponding to the parameters of Fig. 8]. The densities are representative for late-time states in the limit $t\to \infty$ as similar distribution is found also after 1000 driving cycles.

Figure 10

The stepwise modulation of the dimensionless superlattice amplitudes ${\alpha }_{ab}$ according to the protocol given in the table gives rise to different dimerizations of the cubic lattice in each driving step, that enables tunneling along the desired bonds.

Figure 11

The dynamics of particles in a box potential with different softness of the boundary. The initial density distribution takes a Gaussian profile spreading over several lattice sites. The parameters are $\varphi =\pi /2,V_{b}T/6\hslash =7.5\pi$. The initial Gaussian profile has the center $(x_0,y_0,z_0)=(4,4,12)$ and width $s_0=0.75$.

Figure 12

Simulation of gap measurements using Stückelberg interferometry. (a1), (a2) Contours for quasienergy gap at $E=0$, for $\varphi =\pi /8$ and $\pi /3$, respectively. (b) The gaps at $E\sim 0$ and $E\sim \pi$, and the measured gap which takes the smaller one of the two.

Figure 13

The Berry curvatures for the surfaces wrapping a Weyl point. Adding the net Berry curvatures up we have $2\pi$.