# Topological Photonic Media and the Possibility of Toroidal Electromagnetic Wavepackets
## 5 Electronic State with Twisted Spin-polarization
The Hamiltonian of the topological insulator part is represented by  
\[H_{\rm 2 DTI} = \sum_{\mathbf{r}}(-1)^{\mathbf{r}}t_{\rm n}\left[\left(c_{\mathbf{r}+a_{1}}^{ \dagger}e^{i\frac{\pi}{4}(-1)^{\mathbf{r}}\sigma_{3}}c_{\mathbf{r}}-c_{\mathbf{r}+a_{2}}^{ \dagger}e^{-i\frac{\pi}{4}(-1)^{\mathbf{r}}\sigma_{3}}c_{\mathbf{r}}\right)+{\rm H.c.}\right] \tag{12}\] \[+\sum_{\mathbf{r}}(-1)^{\mathbf{r}}t_{\rm nn}\left[\left(c_{\mathbf{r}+a_{1}+ a_{2}}^{\dagger}c_{\mathbf{r}}+c_{\mathbf{r}-a_{1}+a_{2}}^{\dagger}c_{\mathbf{r}}\right)+{ \rm H.c.}\right]+\sum_{\mathbf{r}}(-1)^{\mathbf{r}}v_{\rm s}c_{\mathbf{r}}^{\dagger}c_{\mathbf{r}},\]  
where \((-1)^{\mathbf{r}}=(-1)^{n_{1}+n_{2}}=\pm 1\) for A and B sub-lattices, respectively. The Hamiltonian \(H_{\rm 2 DTI}\) is time-reversal invariant as a whole as well as \(H_{\rm 2 DC}\), because each of the spin sectors is a time-reversal partner of the other. In the parameter range \(4|t_{\rm nn}|>|v_{\rm s}|\), each of the spin sectors comes in a quantum Hall phase. The spin-resolved quantized Hall conductances have a common absolute value and different signs. They cancel each other so as to preserve time-reversal symmetry. A homogeneously spin-polarized incident wavepacket is illustrated in Figure 7a. The packet is incident from the conductor (left) perpendicularly to the boundary (red vertical line); its spin polarization is uniformly pointing in the incident direction. On the other hand, the spin-polarization state of the reflected wavepacket is depicted in Figure 7b. The spin density of the top white area faces in this side of the page, whereas that of the bottom black area faces the back. Polarization in the vicinity of the wavepacket center has the same state as the pre-incidence. (See Figure 7c for details about the correspondence between spin density and color space.) These results suggest that an electronic state with twisted spin-polarization can be generated from a homogeneously spin-polarized state via topological interface between a conductor and a topological insulator, where helical edge states run along the boundary. The concept of topological insulator extends to the three-dimensional (3D) system, where helical surface/interface states traversing bulk band gaps emerge [49; 50]. Figure 8a shows a conceptual diagram of idealistic helical surface states. The green balls depict the electrons, whose propagation direction and spin angular momentum are indicated by each set of green and black arrows, respectively. For example,  
Figure 7: (**a**) An incident wavepacket with homogeneous polarization along the \(x\)-direction and (**b**) a reflected wavepacket with twisted spin texture. (**c**) Relative correspondence between spin density \(S\) and hue, lightness, and saturation (HLS) color space. Approximately, HLS correspond to the azimuth angle, polar angle, and magnitude of spin density, respectively. the 2D topological insulator model in Equation (12) can also be extended to 3D versions with some generalizations. A sequence of 3D models is constructed on a simple cubic lattice \(\mathbf{r}=\sum_{\mu}n_{\mu}\mathbf{a}_{\mu}\) (\(n_{\mu}\in\mathbb{Z}\)) with an orthogonal set of unit lattice vectors \(\mathbf{a}_{\mu}\) (\(\mu=1,2,3\)). Every site is classified into either A or B sub-lattice as \(\mathbf{r}\in\text{A}(\text{B})\) when \(\sum_{\mu}n_{\mu}=\text{even}(\text{odd})\), in which a sign symbol \(\left(-1\right)^{\mathbf{r}}\) can be introduced as \(\left(-1\right)^{\mathbf{r}}=\left(-1\right)^{\sum_{\mu}n_{\mu}}\). Moreover, each sub-lattice forms a face-centered cubic lattice. The sequence of 3D models is characterized by three types of parameters \(t_{\mu}\), \(t_{\mu\
u}(=t_{\
u\mu})\), and \(v_{s}\), along with SU(2) matrices \(\{U_{\mu}\}\) (\(\mu,\
u=1,2,3\)) representing the spin-precession processes in the \(\mu\)-directional nearest-neighbor hoppings.