${\mathbb{Z}}_2$ topological index for continuous photonic materials

Electronic topological insulators are one of the breakthroughs of 21st century condensed matter physics. So far, the search for a light counterpart of an electronic time-reversal invariant topological insulator has remained elusive. This is due to the fundamentally different natures of light and matter and the different spins of photons and electrons. Here, it is shown that the theory of electronic topological insulators has a genuine analog in the context of light wave propagation in time-reversal invariant continuous materials. We introduce a gauge invariant ${\mathbb{Z}}_2$ index that depends on the global properties of the photonic band structure and is robust to any continuous weak variation of the material parameters that preserves the time-reversal invariance. A nontrivial ${\mathbb{Z}}_2$ index has two possible causes: (i) the lack of smoothness of the pseudo-Hamiltonian in the $\mathbf{k}\to \infty$ limit and (ii) the entanglement between positive and negative frequency eigenmode branches. In particular, it is proven that electric-type plasmas and magnetic-type plasmas are topologically inequivalent for a fixed wave polarization. We propose a bulk-edge correspondence that links the number of edge modes with the topological invariants of two continuous bulk materials and present detailed numerical examples that illustrate the application of the theory.

Figure 1

(a) The wave vector space can be regarded as the Riemann sphere. Each point of the sphere is projected into a single point of the plane through the stereographic projection. Half of the wave vector space (e.g., the semi-space $k_y>0$) is mapped into half of the Riemann sphere surface: the effective Brillouin zone (EBZ). The boundary of the EBZ is represented in the Riemann sphere by a circle ($\partial \mathrm{EBZ}$). (b) Integration contours used in Eq. (11). (c) Generic band diagram of a lossless material with the time-reversal and reality symmetries, showing both the positive and negative parts of the frequency spectrum.

Figure 2

Topological transition from an ENG material ($\tau =0$) to an MNG material ($\tau =1$) (${\omega }_{0m}=0.5{\omega }_{pe};{\omega }_{1m}=2.0{\omega }_{pe}$). The purple dashed (black dotted-dashed) lines represent the longitudinal modes associated with $\mu =0$ ($\varepsilon =0$). The green solid lines are associated with the transverse waves. The insets show the topological invariants for the TM-polarized waves. For $\tau =0.15$, the bandgap closes, and the ${\mathbb{Z}}_2$ topological index of the upper (lower) bands changes value.

Figure 3

Dispersion of the edge modes at an interface between an ENG material and an MNG material. The dashed horizontal gray lines delimit the common bandgap of the two bulk materials. The solid green lines represent the edge modes dispersion. The TE and TM branches within the bandgap are identified by the arrows and meet at the $k_x=0$ point. The dashed blue (dotted-dashed black) lines represent the dispersion of the transverse modes of the bulk MNG (ENG) materials with material dispersion as in Eq. (17) and (a) ${\omega }_{0m}=0.5{\omega }_{pe}$; ${\omega }_{1m}=2.0{\omega }_{pe}$ and (b) ${\omega }_{0m}=0.8{\omega }_{pe}$; ${\omega }_{1m}=2.0{\omega }_{pe}$.

Figure 4

Spin assisted unidirectional excitation of edge modes at an MNG-ENG interface. The background material is an MNG material with ${\omega }_{0m}=0.5{\omega }_{pe};{\omega }_{1m}=1.2{\omega }_{pe}$, and the kite-shaped object is made of a ENG material with plasma frequency ${\omega }_{pe}$. The frequency of operation is $0.95{\omega }_{pe}$, and ${\lambda }_0$ is the corresponding vacuum wavelength. The light source is positioned at $\left(-2.5{\lambda }_0,0\right)$ and excites a TE-polarized edge mode that propagates in the counterclockwise direction. (a) Time snapshot of $E_z$ for a lossless structure. The bluish (reddish) colors represent positive (negative) values of the field. The green color represents a vanishing field. (b) Time snapshot of $E_z$ when a lossy circular object with $R=0.5{\lambda }_0$ is centered at $\left(x_{obj},y_{obj}\right)=\left(2.51{\lambda }_0,0\right)$. The parameters of the lossy object are the same as those of the MNG background except that the imaginary parts of the material parameters are ${\varepsilon }^{\prime \prime }={\mu }^{\prime \prime }=0.3$.

Figure 5

Topological transition from an Ω medium ($\tau =0$) to an ENG material ($\tau =1$) (${\omega }_0=0.5{\omega }_{pe}, {\omega }_e={\omega }_0, {\omega }_m=1.55{\omega }_0$, and ${\omega }_{\mathrm{\Omega }}=0.9{\omega }_e{\omega }_m/{\omega }_0$). The purple dashed (black dotted-dashed) lines represent the dispersion of the longitudinal-type modes associated with $\mu =0$ ($\varepsilon =0$). The green solid lines represent the doubly degenerate transverse-type modes.

Figure 6

Dispersion of the edge modes at an interface between an ENG material and an Ω medium. The shaded yellow regions delimit the common bandgaps wherein the topological edge modes are predicted to occur. The solid green lines represent the edge modes dispersion. The dashed blue (dotted-dashed black) lines represent the dispersion of the transverse-type modes of the bulk Ω material (ENG material). The two dashed horizontal lines represent the dispersion of longitudinal-type modes of the Ω medium. (a) ${\omega }_0=0.5{\omega }_{pe}, {\omega }_e={\omega }_0, {\omega }_m=1.55{\omega }_0$, and ${\omega }_{\mathrm{\Omega }}=0.9{\omega }_e{\omega }_m/{\omega }_0$. (b) ${\omega }_0=0.5{\omega }_{pe}, {\omega }_e={\omega }_0, {\omega }_m=0.9{\omega }_0$, and ${\omega }_{\mathrm{\Omega }}=0.7{\omega }_e{\omega }_m/{\omega }_0$.