# Topological Photonic Media and the Possibility of Toroidal Electromagnetic Wavepackets
## 4 Topological Photonic Media  
Based on the pioneering studies on quantum Hall effect in 2D electron systems [26; 27; 28], a topological invariant, known as Chern number or index, is assigned to an isolated band via integration of its Berry curvature over the entire first Brillouin zone. Depending on total Chern numbers of bulk bands below a bulk gap, exotic states localize at the edge of a finite system and form edge bands traversing over the bulk gap [29; 30]. Furthermore, each state works a one-way waveguide; therefore, is called a chiral edge state. For a Chern number of an isolated band to be nonzero, time-reversal symmetry breaking is necessary. This symmetry may be broken not only by an externally applied magnetic field but also by a spontaneously induced magnetic order [28]. When the quantum Hall effect is induced by the latter mechanism, it is sometimes called the spontaneous quantum Hall effect, as distinguished from the original one. To realize a similar situation in photon systems, the above mechanism requires isolated bands to form with band gaps stemming from time-reversal symmetry breaking. For instance, a magnetic body of a complex permittivity tensor with imaginary off-diagonal components breaks time-reversal symmetry for the photon system. At least the necessary conditions are satisfied by designing a 2D periodic structure made of such a material to form isolated photonic bands. This analogy was the basis for a photonic version of the quantum Hall system theoretically proposed [31; 32; 33] and experimentally confirmed [34; 35]. Presently, a clear-cut demonstration has also been made on nonreciprocal lasing from chiral edge modes surrounding a network of topological cavities in arbitrary geometry [36]. In reverse, a one-way propagation of a chiral edge mode inevitably breaks the time-reversal symmetry. (Time-reversal symmetry breaking is a necessary condition for the presence of a single chiral edge state; conversely, the presence of a single chiral edge state is a sufficient condition for the symmetry breaking.) Figure 3: (**a**) Y-shaped waveguide composed of the photonic crystal in Figure 2. The relative permittivity of the gray region of block layer around the waveguide is set to be 9. (**b**) \(z\)-component of magnetic field \(H_{z}\). (**c**) Spectra of the transmissions to the upper right (blue line) and to the lower right (red line) branches. The broken lines represent the transmission spectra for a vacuum Y-shaped region. Fortunately, however, chiral edge states and their time-reversal partners can simultaneously exist in a single system preserving time-reversal symmetry in its entirety [37]. As an extension of quantum Hall system to the case with time-reversal symmetry, an insulator with a topologically-protected pair of edge states has been proposed [38]. Such insulator and edge states are respectively called topological insulator and helical edge states [39]. The propagation direction of a helical edge state is selectively governed by its spin polarization. In a naive picture, topological insulator is understood as a superposition of a spontaneous quantum Hall system spin-polarized in a specific direction and its time-reversal partner spin-polarized in the opposite direction which is necessary to maintain time-reversal symmetry. Based on this situation, it was initially called quantum spin Hall system. More precisely, the parity of the numbers of Kramers pairs of helical edge states is critical [38] and corresponds to the topological invariant called \(\mathbb{Z}_{2}\) index. As the index can be calculated by means of the bulk states of a system with periodic boundary conditions, topological insulators can be distinguished from non-topological insulators even with bulk information alone. Furthermore, topological crystalline insulators were proposed by introducing combinations of crystalline symmetries and time-reversal symmetry [40], extending to their photonic version [41; 42; 43].