Topological gaps without masses in driven graphene-like systems

We illustrate the possibility of realizing band gaps in graphene-like systems that fall outside the existing classification of gapped Dirac Hamiltonians in terms of masses. As our primary example we consider a band gap arising due to time-dependent distortions of the honeycomb lattice. By means of an exact, invertible, and transport-preserving mapping to a time-independent Hamiltonian, we show that the system exhibits Chern-insulating phases with quantized Hall conductivities $\pm e^2/h$. The chirality of the corresponding gapless edge modes is controllable by both the frequency of the driving and the manner in which sublattice symmetry is broken by the dynamical lattice modulations. Finally, we discuss a promising possible realization of this physics in photonic lattices.

Figure 1

Schematic of the low-energy band structure of the Hamiltonian (5) with $M\left(\mathit{p}\right)$ given by Eq. (18). When only one sublattice is excited (i.e., ${\Delta }_B=0$), the gap opens along a circle of radius $p_0$ in momentum space.

Figure 2

(a) Phase diagram of graphene in the presence of the LO/LA phonon modes. Chern-insulating regions with $C=\pm 1$ are separated by a gap-closing transition along the line $|c_{+}^B|=|c_{+}^A|$, i.e., when the phonon modes of Eqs. (6) are excited with equal amplitudes. (b) Schematic phase diagram for the Hamiltonian ${\mathcal{H}}_{\eta \Omega ,\mathit{p}}$ of Eq. (24). When $\alpha =0$, the $d$-wave and Haldane phases are separated by a gap-closing transition at $2\eta /\Omega =\pm 1$. However, when $\alpha \ne 0$, this critical point can be avoided and the two phases can be connected without closing the gap. To the right of the dashed line, the location $p_{\min }$ of the minimum gap size is identically zero, while to the left $p_{\min }>0$.

Figure 3

Motion of the honeycomb lattice in the asymmetric case where the $A$ and $B$ sublattices are excited with different amplitudes. The faint circles indicate the equilibrium positions of the ions, around which the driven ions rotate along the dashed circles with chirality indicated by the blue arrows. The boxed region indicates the tripled unit cell of the driven system, and the black arrows indicate the initial phases $-{\varphi }_i$ of the inequivalent lattice sites labeled by $i=1,2,3$.