Modulating a photonic graphene lattice by the Lorentz-like transformation

Photonic lattice modulations, such as lattice distortions achieved by stress-strain engineering, have been crucial for the development of various fields of photonics, in particular, topological photonics. Here we modulate the Hermitian and non-Hermitian photonic graphene lattices by utilizing a Lorentz-like transformation applied to the lattice sites. The modulated lattice may fall into two categories, according to the symmetry axis of the lattice. If the axis is aligned in the diagonal direction through the unit cells, the lattice belongs to case II; otherwise, it belongs to case I. For the case-I transformed lattice, the Dirac cones always disappear from the band structure, whereas for the case-II transformed lattice, the Dirac cones only disappear if the transformation angle $\ensuremath{\theta}$ is larger than that of the regular lattice, $\ensuremath{\theta}=\ensuremath{\pi}/2$. The Lorentz-like transformation always helps to restore the parity-time ($\mathcal{PT}$) symmetry in the non-Hermitian case-I photonic graphene lattices. In the case II of transformed non-Hermitian lattices, the $\mathcal{PT}$ symmetry can be restored as well, but only for $\ensuremath{\theta}>\ensuremath{\pi}/2$. When $\ensuremath{\theta}<\ensuremath{\pi}/2$, one obtains exceptional lines in the band structure of the lattice. These results demonstrate that the Lorentz-like transformation can serve as an effective tool to modulate photonic lattices and thus may inspire novel ideas in the manipulation of photonic topological insulators.