Floquet Insulators and Lattice Fermions

Floquet insulators are periodically driven quantum systems that can host novel topological phases as a function of the drive parameters. These new phases exhibit features reminiscent of fermion doubling in discrete-time lattice fermion theories. We make this suggestion concrete by mapping the spectrum of a noninteracting $(1+1)\mathrm{D}$ Floquet insulator for certain drive parameters onto that of a discrete-time lattice fermion theory with a time-independent Hamiltonian. The resulting Hamiltonian is distinct from the Floquet Hamiltonian that generates stroboscopic dynamics. It can take the form of a discrete-time Su-Schrieffer-Heeger model with half the number of spatial sites of the original model, or of a $(1+1)\mathrm{D}$ Wilson-Dirac theory with one quarter of the spatial sites.

Figure 1

Schematic of the mapping between stroboscopic Floquet dynamics and time-independent lattice Hamiltonians. The spectrum of the stroboscopic dynamics corresponding to Eq. (6), shown in (a), can be mapped onto that of a discrete-time Su-Schrieffer-Heeger (b) or Wilson-Dirac (c) theory. The number of spatial sites in (b) is halved with respect to (a) to accommodate the additional degrees of freedom due to fermion doubling. The number of sites are further halved in (c) to accommodate the spinful nature of the Wilson-Dirac fermions (depicted via arrows inside the lattice sites).

Figure 2

Phase diagram of the Floquet model (6). The phases are labeled by the presence or absence of zero and $\pi$ modes localized to boundaries. This Letter focuses on the vertical dashed line at $(t_0/T)=(\pi /4)$, which passes through the trivial and $0\pi$ phases.

Figure 3

Spectrum of the Floquet model and its mapping onto the lattice spectrum for $\eta =\pi /20$. (a) Quasienergy spectrum of Eq. (6) along the line $(t_0/T)=(\pi /4)$ [see Eq. (7)]. The quasienergy values highlighted in blue are those corresponding to crystal momenta $k$ in the interval $[(\pi /4),(3\pi /4))$ (gray vertical lines), denoted $\tilde{\epsilon }$ in the text. (b) Sine-transformed quasienergy spectrum entering Eq. (8). (c) Pole positions $p_0$ with momenta assigned according to the SSH mapping. The orange bands denote doublers, which correspond to the quasienergies $(\pi /T)-\tilde{\epsilon }$.

Figure 4

Finite-size scaling of the maximum difference between discrete-time spectra $p_0$ and quasienergy spectra $\epsilon$ (a) for OBC and (b) for OBC with a domain wall where $\eta$ switches sign. In both panels we compare to the SSH spectrum and fix $\eta =\pi /8$, considering system sizes $N=100,200,\dots ,900$. Dashed lines indicate best fits to power laws in $1/N$ that are consistent with the expected scaling.