# Topological Photonic Media and the Possibility of Toroidal Electromagnetic Wavepackets
## 2 Geometric Phases and Angular Momenta of Electromagnetic Vortices
The degenerate eigenstates of the upper band \(|u_{\mathbf{k}\lambda}\rangle\) in terms of the spherical coordinate of wavevector space \((k,\theta,\phi)\) are expressed as  
\[|u_{\mathbf{k}\pm}\rangle=\frac{1}{\sqrt{2E_{\mathbf{k}}}}\left( \begin{array}{c} \sqrt{E_{\mathbf{k}}+\Delta}\chi_{\mathbf{k}\pm}\\ \pm\sqrt{E_{\mathbf{k}}-\Delta}\chi_{\mathbf{k}\pm}\end{array} \right),\quad\chi_{\mathbf{ k}+}=\left( \begin{array}{c}e^{-i\frac{\theta}{2}}\cos\frac{\theta}{2}\\ e^{i\frac{\theta}{2}}\sin\frac{\theta}{2}\end{array} \right),\quad\chi_{\mathbf{ k}-}=-i\sigma_{2}\overline{\chi}_{\mathbf{k}+}, \tag{5}\]  
where \(E_{\mathbf{k}}=\sqrt{v^{2}k^{2}+\Delta^{2}}\) and the index \(\lambda\) corresponds to the degrees of helicity, i.e., spin component in the direction of \(\mathbf{k}\). Note that the above expressions are well-defined only in regions excluding the points \(\theta=0\) and \(\pi\). However, we shall not step into regularization at these points but only mention that expressions valid at \(\theta=0\) or \(\pi\) are obtained by some gauge transformations of the above expressions. One can easily confirm that the above expressions satisfy the time-independent Schrodinger equation in \(\mathbf{k}\)-subspace,  
\[E_{\mathbf{k}}|u_{\mathbf{k}\lambda}\rangle=\mathcal{H}_{\mathbf{k}}|u_{\mathbf{k}\lambda}\rangle. \tag{6}\]  
At this point, we would like to mention that the electromagnetic polarization vectors \(\mathbf{e}_{\mathbf{k}\lambda}\) can be interpreted as solutions of a similar Schrodinger-type matrix equation, which is a transcription of [PERSON]'s equations. The correspondence relation is confirmed by the replacements, \(E_{\mathbf{k}}=k/(\sqrt{\epsilon}\sqrt{\mu})\), and  
\[\mathcal{H}_{\mathbf{k}}=\left( \begin{array}{cc}0&ie^{-\frac{1}{2}}\mathbf{k}\cdot \mathbf{S}\mu^{-\frac{1}{2}}\\ -i\mu^{-\frac{1}{2}}\mathbf{k}\cdot\mathbf{S}e^{-\frac{1}{2}}&0\end{array} \right), \quad[S_{i}]_{ijk}=-i\epsilon_{ijk},\quad|u_{\mathbf{k}\lambda}\rangle=\frac{1}{ \sqrt{2}}\left( \begin{array}{c}\mathbf{e}_{\mathbf{k}\lambda}\\ \mathbf{e}_{\mathbf{k}}\times\mathbf{e}_{\mathbf{k}\lambda}\end{array} \right), \tag{7}\]where \(\epsilon\), \(\mu\), and \(\epsilon_{ijk}\) are the relative permittivity, relative permeability, and the Levi-Civita symbol, respectively. For simplicity, we assumed a homogeneous and isotropic background medium here. Therefore, by means of the state vectors \(|u_{k\lambda}\rangle\), we can define the Berry connection in a form common in electronic and electromagnetic systems,  
\[\left[\boldsymbol{\Lambda}_{\boldsymbol{k}}\right]_{\alpha\beta}=-i\langle u_{ k\alpha}|\boldsymbol{\
abla}_{\boldsymbol{k}}|u_{k\beta}\rangle. \tag{8}\]  
Now, let us get back to the degenerate two-band model. On the helicity basis, the Berry connection and curvature of the upper band are expressed by  
\[\boldsymbol{\Lambda}_{\boldsymbol{k}}=\tfrac{1}{2k}\left\{\tfrac{\Delta}{E_{ \boldsymbol{k}}}(-\sigma_{2}\boldsymbol{e}_{\theta}+\sigma_{1}\boldsymbol{e}_ {\phi})-\tfrac{\cos\theta}{\sin\theta}\sigma_{3}\boldsymbol{e}_{\phi}\right\}, \quad\boldsymbol{\Omega}_{\boldsymbol{k}}=\tfrac{y^{2}}{2E_{\boldsymbol{k}}^{2 }}\left\{\tfrac{\Delta}{E_{\boldsymbol{k}}}\left(\sigma_{1}\boldsymbol{e}_{ \theta}+\sigma_{2}\boldsymbol{e}_{\phi}\right)+\sigma_{3}\boldsymbol{e}_{ \boldsymbol{k}}\right\}. \tag{9}\]  
In the massless limit \(\Delta\to 0\), this coincides with the Berry curvature of photons except for the overall coefficient due to a different spin magnitude. In the non-relativistic limit, i.e., an increasing \(|\Delta|\) for a fixed \(k\), the Berry curvature decreases in the form \(1/\Delta^{2}\), similar to the scale of the spin-orbit interaction. Next, let us consider the relation between the spin angular momentum and the geometric phase in this spin-\(1/2\) massive fermionic system. The spin operator \(\boldsymbol{s}\) and its projection to the upper band \(\boldsymbol{s}_{\boldsymbol{k}}\) are given as follows:  
\[\boldsymbol{s}=\left( \begin{array}{cc}\boldsymbol{\sigma}&0\\ 0&\boldsymbol{\sigma}\end{array}