Floquet Hopf Insulators

We predict the existence of a Floquet topological insulator in three-dimensional two-band systems, the Floquet Hopf insulator, which possesses two distinct topological invariants. One is the Hopf $\mathbb{Z}$ invariant, a linking number characterizing the (nondriven) Hopf topological insulator. The second invariant is an intrinsically Floquet ${\mathbb{Z}}_2$ invariant, and represents a condensed matter realization of the topology underlying the Witten anomaly in particle physics. Both invariants arise from topological defects in the system’s time evolution, subject to a process in which defects at different quasienergies exchange even amounts of topological charge. Their contrasting classifications lead to a measurable physical consequence, namely, an unusual bulk-boundary correspondence where gapless edge modes are topologically protected, but may exist at either 0 or $\pi$ quasienergy. Our results represent a phase of matter beyond the conventional classification of Floquet topological insulators.

Figure 1

Depiction of the Floquet Hopf insulator’s two topological invariants. (a) The “static” $\mathbb{Z}$ invariant is the Hopf invariant of the Floquet Hamiltonian $H_F(\mathbf{k})$, corresponding to the linking number of the preimages of two points (blue, red) on the Bloch sphere. For nearby points, this equals the twisting of the Jacobian (colored arrows) along a single preimage. (b) The “Floquet” ${\mathbb{Z}}_2$ invariant classifies the micromotion operator $U_m(\mathbf{k},t)\in \mathrm{SU}(2)$ and is similarly interpreted as the Jacobian twisting (dashed black arrows) along a preimage, with a reduced classification due to the larger dimensionality.

Figure 2

(a) A point Hopf defect (black point) has quadratic dispersion, functioning as a strand crossing that changes the linking number of any two eigenvectors’ preimages (red and blue). A loop Hopf defect (black loop) has linear dispersion, and can occur along a former preimage. The defect charge is defined on a surface (gray, shaded) enclosing the defect. (b) Two Floquet evolutions with different defect charges but the same topological invariants, which are connected by a smooth deformation $\lambda \in [0,1]$ preserving the Floquet unitary’s band gaps. (c) The deformation is a $\pi$ rotation of the three-sphere parametrized by $(\mathbf{n},\xi )$. Images of time slices representing the initial 0 defect (yellow), $\pi$ defect (blue), trivial Hopf invariant (gray), and Hopf invariant 1 (red) are displayed before and after the rotation. (d) During the deformation, the 0 defects (black outline) and $\pi$ defects (solid black) become loops that link in the Brillouin zone, at which point their individual charges are undefined. The total charge $h_0+h_{\pi }$ is conserved, corresponding to the static $\mathbb{Z}$ invariant. Arrows indicate increasing $\lambda$.

Figure 3

Numerical calculation of the Floquet invariant, static invariant, and 0 and $\pi$ defect charges (a) across a phase transition $(h_S,h_F)=(1,0)\to (0,1)$ ($\kappa =0\to 1$) and (b) along the smooth deformation exchanging defect charge.

Figure 4

Quasienergy spectra of the Floquet unitary at various smooth boundaries between Floquet Hopf insulator phases, solved via exact diagonalization [38]. Quasienergies are colored according to their eigenstates’ average distance from the edge region, from localized at the edge (red) to far from the edge (black). (a) The boundary between $(h_S,h_F)=(0,1)$ and the trivial phase $(h_S,h_F)=(0,0)$ features gapless edge modes across both band gaps despite the Floquet Hamiltonian being trivial. (b) In contrast, we find no gapless edge modes between phases with different topological defect charges $(h_0,h_{\pi })=(1,-1)$ and $(-1,1)$, but the same topological invariants $(h_S,h_F)=(0,1\text{mod}2)$, demonstrating the ${\mathbb{Z}}_2$ classification of the Floquet invariant. (c),(d) Two different boundaries between the same two phases, $(h_S,h_F)=(2,0)$ and the trivial phase $(h_S,h_F)=(0,0)$, featuring gapless edge modes across either the 0 or $\pi$ gap.