Complex classes of periodically driven topological lattice systems

Periodically driven (Floquet) crystals are described by their quasienergy spectrum. Their topological properties are characterized by invariants attached to the gaps of this spectrum. In this article, we define such invariants in all space dimensions, both in the case where no symmetry is present and in the case where the unitary chiral symmetry is present. When no symmetry is present, a $\mathbb{Z}$-valued invariant can be defined in each gap in all even space dimensions. This invariant does not capture all the properties of a system where chiral symmetry is present. In even space dimension, chiral symmetry puts constraints on its values in different gaps. In odd space dimension, chiral symmetry also enables to define a $\mathbb{Z}$-valued invariant, only in the chiral gaps 0 and $\pi$. We relate both gap invariants to the standard invariants characterizing the quasienergy bands of the system. Examples in one and three dimensions are discussed.

Figure 1

Lattice of the (driven) SSH model.

Figure 2

Quasienergy spectrum of a finite system (with edges) of the driven SSH model for $J_1=3/2,J_2=1,A=6$ and $T=2.35$. The undriven system with $A=0$ is trivial, but the driven phase at this point of the parameter space the driven system is topologically nontrivial, as $G_{\pi }\left[U\right]=1$ (and $G_0\left[U\right]=0$ as this gap is trivial). As a consequence, edge states appear in the quasienergy spectrum of the finite system (in blue, as opposed to the bulk bands which are in red), only in gap $\pi$, and can be related to the static chiral invariant of the effective Hamiltonian. The quasienergy spectrum is obtained by diagonalization in the Sambe space truncated to 19 sidebands for a system of length 80. The invariants were computed from the bulk Hamiltonian by direct integration. The code used for all computations is available in Ref. [26].

Figure 3

Quasienergy spectrum of a finite system (with edges) of the driven SSH model for $J_1=1,J_2=3/2,A=6$, and $T=2.35$. In this point of the parameter space, the driven system is topological and anomalous, as $G_0\left[U\right]=G_{\pi }\left[U\right]=1$. As a consequence, edge states appear in the quasienergy spectrum of the finite system (in blue, as opposed to the bulk bands, which are in red), both in gap 0 and in gap $\pi$. See Fig. 2 for details on the computation.

Figure 4

Projection along $k_x$ of the quasienergy spectrum of a finite system (with edges) of the 3D driven chiral model for $\delta =1/2,m_0=1.75,m_1=1$, and $\omega =5$. Bands are in gray. The driven system is topological, with $G_0\left[U\right]=1$ and $G_{\pi }\left[U\right]=-2$. As a consequence, surface states with linear dispersion (Dirac cones) appear in the quasienergy spectrum of the finite system. In accordance with the bulk topological invariants, for each (bottom and top) surface boundary of the system, there are one Dirac cone in bulk gap 0 and two Dirac cones in bulk gap $\pi$. The quasienergy spectrum is obtained by diagonalization in the Sambe space truncated to 7 sidebands for a system of length 15. The invariants were computed from the bulk Hamiltonian by direct integration (e.g., we obtain here $G_0=0.99$ and $G_{\pi }=-1.97$, which are rounded to the values given in the main text). The code used for all computations is available in Ref. [26].

Figure 5

Three-dimensional view of the Dirac cone in bulk gap 0 of the quasienergy spectrum of Fig. 4. The dispersion is clearly linear in the neighborhood of the Dirac cone (located approximately at $k_x=-2.62$ and $k_y=-\pi$). The cone then merges in the bulk bands (shown in Fig. 4).