Effective theory of quadratic degeneracies

We present an effective theory for the Bloch functions of a two-dimensional square lattice near a quadratic degeneracy point. The degeneracy is protected by the symmetries of the crystal, and breaking these symmetries can either open a band gap or split the degeneracy into a pair of linear degeneracies that are continuable to Dirac points. A degeneracy of this type occurs between the second and third transverse magnetic bands of a photonic crystal formed by a square lattice of dielectric rods. We show that the theory agrees with numerically computed photonic band structures and yields the correct Chern numbers induced by parity breaking.

Figure 1

(Color online) TM band structure of a two-dimensional photonic crystal formed by a square lattice of dielectric rods. (a) Fully symmetric crystal with rod radius of $0.15a$ (where $a$ is the lattice constant) and $ϵ=20$. The rods are embedded in air and $\mu =1$ everywhere. The quadratic degeneracy between the second and third TM bands is indicated with an arrow. Inset: the crystal structure in real space. (b) Parity-broken crystal, with off-diagonal permeability component ${\mu }_{xy}=0.1i$. Inset: band structure near the lifted degeneracy, with dots showing numerical solutions of the exact Maxwell equations (Ref. 9). The solid lines show the analytic approximation of Eq. (5), where ${\lambda }_0$, $\beta$, and $\gamma$ are calculated from the symmetric system and ${\alpha }_2$ is calculated separately based on the proportionality constant with $\left|{\mu }_{xy}\right|$ (see Fig. 4).

Figure 2

(Color online) Photonic band structures for time-reversible lattices of dielectric rods with radius $r_0$ and permittivity $ϵ$, plotted against the $k$-space displacement from the corner of the Brillouin zone, $\vec{\kappa }\equiv \vec{k}-\left(\pi /a,\pi /a\right)$, along the line $\vec{\kappa }=\left({\kappa }_x,-{\kappa }_x\right)$. The dots show numerical data (Ref. 8); the solid lines show the analytic results after fitting the free parameters in Eq. (4) to the numerical data. Note that the band structure is independent of the parameter $\beta$ along this given line. In (a) and (b), we take $r_0=0.25a$ and $ϵ=16.26$, with fitted value $\gamma \simeq 0$. For (a), $\theta =0$ (square lattice) and ${\alpha }_3=0$; for (b), $\theta =1.2\times {10}^{-4}$ rad with fitted value ${\alpha }_3=3.7\times {10}^{-5}{\left(\pi /a\right)}^2$. In (c)–(e), we take $r_0=0.2a$ and $ϵ=9.92$, with fitted value $\gamma \simeq -0.5$. In (c), $\theta =0$ and ${\alpha }_3=0$; for (d), $\theta ={10}^{-4}$ rad with fitted value ${\alpha }_3=1.8\times {10}^{-5}{\left(\pi /a\right)}^2$. In (e), a three dimensional plot of the band structure of system (d) is shown.

Figure 3

(Color online) Values of $\gamma$ and $\beta$ obtained from least-squares fits of Eq. (5) to band structures computed numerically along the lines $k_y=\pi /a$ and $k_x=-k_y$. The photonic crystal consists of a square lattice of rods with radius $r_0$ and permittivity $ϵ$, embedded in air, with $\mu =1$ everywhere.

Figure 4

(Color online) Plots of ${\alpha }_1$, ${\alpha }_2$, and ${\alpha }_3$ for two different photonic crystals. In the left column, the dielectric rods have radius $r_0=0.1a$ (where $a$ is the lattice constant) and permittivity $ϵ=15$. In the right column, $r_0=0.3a$ and $ϵ=20$. We obtain ${\alpha }_1$ from the location of the degeneracies $\vec{\kappa }=\pm \left(0,\sqrt{{\alpha }_1/\beta }\right)$ for lattice stretching parameter $\delta a_x/a$. We obtain ${\alpha }_2$ by performing a nonlinear least-squares fit of Eq. (5) to the computed $\mathcal{T}$-broken band structures along the line ${\kappa }_x=-{\kappa }_y$ using the values of $\gamma$ found in Fig. 3 (which were obtained using the symmetric lattice); it is plotted against the imaginary off-diagonal permeability component, ${\mu }_{xy}=i\eta$ $$. We obtain ${\alpha }_3$ from the location of the degeneracies $\vec{\kappa }=\pm \left(\sqrt{{\alpha }_3/2},-\sqrt{{\alpha }_3/2}\right)$ for lattice distortion angle $\theta$.