Line nodes, Dirac points, and Lifshitz transition in two-dimensional nonsymmorphic photonic crystals

Topological phase transitions, which have fascinated generations of physicists, are always demarcated by gap closures. In this work, we propose very simple two-dimensional photonic crystal lattices with gap closures, i.e., band degeneracies protected by nonsymmorphic symmetry. Our photonic structures are relatively easy to fabricate, consisting of two inequivalent dielectric cylinders per unit cell. Along high-symmetry directions, they exhibit line degeneracies protected by glide-reflection symmetry and time-reversal symmetry, which we explicitly demonstrate for $pg,pmg,pgg$, and $p4g$ nonsymmorphic groups. They also exhibit point degeneracies (Dirac points) protected by a $Z_2$ topological number associated only with crystalline symmetry. Strikingly, the robust protection of $pg$ symmetry allows a Lifshitz transition to a type-II Dirac cone across a wide range of experimentally accessible parameters, thus providing a convenient route for realizing anomalous refraction. Further potential applications include a stoplight device based on electrically induced strain that dynamically switches the lattice symmetry from $pgg$ to the higher $p4g$ symmetry. This controls the coalescence of Dirac points and hence the group velocity within the crystal.

Figure 1

The $s$, $p_x$, $p_y$, and $2s$ orbitals above each elliptical cylinder in the photonic lattice unit cell. Note that they are neither isotropic nor aligned with the $x$ and $y$ axes, like the ellipses themselves.

Figure 2

(a)–(d) The lattice structure, nodal positions, and band dispersion for the four lattices we considered, with $pg$, $pmg$, $p4g$, and $pgg$ symmetry, respectively. Results from our minimal effective TB model (colored lines, see Appendix pp1) agree closely with comsol multiphysics simulation results (black dotted lines). (a) The $pg$ bands from the $p_x$ and $p_y$ orbitals. There exists a Dirac point (DP) $P_1$ along $Y\text{−}M$ and a nodal line along $M\text{−}X$ protected by $\left\{m_y\right|{\tau }_x\}$. (b) By rotating the ellipses in the $pg$ lattice with to respect an additional mirror symmetry $m_x$, we obtain the $pmg$ lattice. The nodal line remains unchanged, but the DP $P_2$ is now along $\mathrm{\Gamma }\text{−}Y$. (c) The lattice with $p4g$ symmetry, which has mirror symmetry along two diagonals and an additional glide-reflection symmetry $\left\{m_x\right|{\tau }_y\}$. The nodal lines persist, but there is now a doubly degenerate DP $P_3$ at the $\mathrm{\Gamma }$ point. (d) The $pgg$ lattice obtained by breaking the $C_4$ symmetry of the $p4g$ lattice through arbitrary rotation of the ellipses. The previously doubly degenerate DP decomposes into two singly degenerate DPs between $\mathrm{\Gamma }$ and $\pm X$, one of which is visible here.

Figure 3

$\gamma \left(k_p\right)$ from the tight-binding models for the (a) $pg$, (b) $pmg$, (c) $p4g$, and (d) $pgg$ lattices, which are quantized to zero or $\pi$. Protected Dirac points exist at its discontinuities, where the gap has to close. In the $p4g$ case, the Dirac points coalesce, making the nontrivial $Z_2$ region into a sharp spike. Edge states from simulation results of the (e) $pmg$ lattice and (f) $pgg$ lattice are shown, which agree well with those from the tight-binding models with open boundary conditions (see details in Appendix pp3). Indeed, the edge states exists across $Z_2$ nontrivial regions in (b) and (d), which are terminated by point degeneracies.

Figure 4

The $pg$ (a) lattice and (b) band structure hosting type-II (tilted over) Dirac points. (c) Phase diagram for Lifshitz transition dependence on dielectric constant ${\epsilon }_r$ and aspect ratio $r_2/r_1$ of the ellipses, with $r_1=0.16a$ and orientation angles $\pm {80}^{\circ }$. Type-I (type-II) regions are marked in red (blue), while no DP exists in the white regions.

Figure 5

(a) Three-dimensional frequency plot near a type-II Dirac point, with characteristic cross-shaped isofrequency lines due to its tilt. (b) Right: Anomalous refraction with angle ${\varphi }_1\approx {\varphi }_2$ from a type-II DP. Left: The refraction angles are aligned with the refracted group velocities, which are determined by the normals of the isofrequency lines. The gray ray at angle ${\varphi }_3$ results from another band unrelated to the type-II Dirac point. These rays intersect with the incident wave vector at the original tangential wave vector component $k_y$.

Figure 6

(a) Stoplight device with group velocity controlled by external applied voltage. An electric field deforms the silicone elastomer elliptic cylinders and breaks the $p4g$ symmetry to $pgg$, thereby modifying the dispersion. (b) The good agreement between comsol multiphysics simulation results and the effective Hamiltonian in Eq. (9).

Figure 7

Edge states from simulation results of the (a) $pmg$ lattice and (c) $pgg$ lattice, which agree well with those from the (b) $pmg$ and (d) $pgg$ tight-binding models with open boundary conditions. Indeed, the edge states exists across $Z_2$ nontrivial regions in Figs. 3 and 3 of the main text, which are terminated by point degeneracies.

Figure 8

The configurations of $\mathbf{h}$ near the DPs of lattices with $pmg$, $p4g$, and $pgg$ symmetries. (a) and (b) depict the two DPs ($\pm P_2$ between $\mathrm{\Gamma }$ to $\pm Y$) of the $pmg$ lattice, with windings $w=-1$ and $w=1$. (c) The winding $w=-2\mathbf{h}$ around the quadratic degeneracy ($P_3$) at $\mathrm{\Gamma }$ for the $p4g$ symmetric lattice. (d) With $p4g$ broken to $pgg$, the above $w=-2$ degeneracy splits into to two DPs ($\pm P_4$) along $-X$ to $X$, each with winding $w=-1$.

Figure 9

(a) Model structure for the simulation. Scattering boundary conditions (SBC) are applied to the left and right sides, while periodic boundary conditions (PBC) are applied to the top and bottom sides. (b) Lights with angles ${\varphi }_1$ and ${\varphi }_2$ are from a type-II DP. The light with angle ${\varphi }_3$ is from another band. (c) The eigenstates of three refraction lights. (d) Comparison between the simulation result and the superposition result; note their very good numerical agreement.