## 2 Energy Band Structure and Electronic States in Kek-Y Graphene
\right]\,, \tag{9}\]  
or explicitly,  
\[\mathcal{H}_{Y}(\mathbf{k}\,|\,\Delta_{0})=\hbar v_{F}\left[ \begin{array}{ccc|cc} 0&k_{x}+ik_{y}&\Delta_{0}(k_{x}-ik_{y})&0\\ \dfrac{k_{x}-ik_{y}}{k_{x}-ik_{y}}&0&0&\Delta_{0}(k_{x}-ik_{y})\\ \Delta_{0}(k_{x}+ik_{y})&0&0&k_{x}+ik_{y}\\ 0&\Delta_{0}(k_{x}+ik_{y})&k_{x}-ik_{y}&0\end{array} \right]. \tag{10}\]  
The energy dispersions of Kekule-Y graphene are simply obtained as eigenvalues of the matrix defined in Equation (10), yielding:  
\[\varepsilon_{Y}(s_{1},s_{2}\,|\,k,\Delta_{0})=\hbar v_{F}\left(s_{1}+s_{2}\, \Delta_{0}\right)k\,, \tag{11}\]  
which represent two Dirac cones with different Fermi velocities \(v_{F}(1\pm\Delta_{0})\), as displayed in Figure 2, where \(s_{1}=\pm 1\) is a band index while \(s_{2}=\pm 1\) corresponds to two different Fermi velocities due to strain coupling. These two types of subbands are often referred to as \"fast\" (\(s_{2}=+1\)) and \"slow\" (\(s_{2}=-1\)) Dirac cones, and importantly, photo-induced electronic transitions are allowed between these two. In correspondence with energy dispersions in Equation (11), the normalized wave functions are calculated as:  
\[\Psi_{Y}(s_{1},s_{2}\,|\,\mathbf{k},\Delta_{0})=\dfrac{1}{2}\left[ \begin{array}[] {c}e^{-i\theta_{\mathbf{k}}}\\ s_{1}\\ s_{2}\\ s_{1}s_{2}\,e^{+i\theta_{\mathbf{k}}}\end{array} \right]\,, \tag{12}\]  
where \(\theta_{\mathbf{k}}=\tan^{-1}(k_{y}/k_{x})\) is the angle associated with the electron wave vector \(\mathbf{k}\) with respect to the \(x\) axis. The wave function in Equation (12) consists of two subspinors corresponding to standard graphene eigenstates, namely:\[\Psi_{Y}(s_{1},s_{2}\,|\,\mathbf{k},\Delta_{0})=\frac{1}{2}\bigg{[}\frac{\Psi_{g}(s_{ 1}\,|\,\mathbf{k})}{s_{2}\,\mathsf{e}^{+i\theta_{\mathbf{k}}}\,\Psi_{g}(s_{1}\,|\,\mathbf{k} )}\bigg{]}=\frac{1}{2}\left[\frac{\left[ \begin{array}{c}1\\ s_{1}\,\mathsf{e}^{+i\theta_{\mathbf{k}}}\end{array} \right]}{\right.} \tag{13}\]  
Quantum-mechanically, the velocity operators \(\hat{V}_{x}\) for an electron can be determined by the Heisenberg equation, that is:  
\[\hat{V}_{x}\equiv\frac{d\hat{x}}{dt}=\frac{1}{i\hbar}\left[\hat{x},\hat{H}_{Y} (\mathbf{k}\,|\,\Delta_{0})\right]\,, \tag{14}\]  
which relates to an intraband probability current density \(\hat{f}_{x}(s_{1},s_{2}\,|\,\mathbf{k},\Delta_{0})\), where \(\mathcal{H}_{Y}(\mathbf{k}\,|\,\Delta_{0})\) is presented in Equation (9). Semi-classically, on the other hand, we define another particle velocity \(v_{x}\), determined by:  
\[v_{x}=\frac{1}{\hbar}\,\frac{\partial\langle\mathcal{H}_{Y}(\mathbf{k}\,|\,\Delta _{0})\rangle_{\text{av}}}{\partial k_{x}}\,, \tag{15}\]  
which relates to an intraband transport current density \(j_{x}=n_{0}v_{x}\), where \(n_{0}\) is the areal density of electrons in our system, while \(\langle\cdots\rangle_{\text{av}}\) denotes a quantum-mechanical average with respect to the wave function \(\Psi_{Y}(s_{1},s_{2}\,|\,\mathbf{k},\Delta_{0})\) presented in Equation (13). Generally, from the Schrodinger equation, we know that the intraband probability-current density can be calculated by:  
\[J_{x}(s_{1},s_{2}\,|\,\mathbf{k},\Delta_{0})\approx\left(\frac{v_{F}}{k_{F}} \right)\text{Im}\left[\Psi^{*}\frac{\partial}{\partial x}\Psi\right]=v_{x}| \Psi_{Y}(s_{1},s_{2}\,|\,\mathbf{k},\Delta_{0})|^{2}\,, \tag{16}\]  
where we treat a Kekule particle as a free one. The definition in Equation (15) can be employed for calculating the semi-classical transport current density, yielding:  
\[j_{x}(s_{1},s_{2}\,|\,\mathbf{k},\Delta_{0}) = n_{0}v_{x}=n_{0}v_{F}\Big{[}\hat{\mathcal{T}}_{0}^{(2)}\otimes \hat{\Sigma}_{x}^{(2)}+\Delta_{0})\left(\mathbf{k}\cdot\hat{\mathbf{\Upsilon}}_{x}^{( 2)}\right)\otimes\hat{\Sigma}_{0}^{(2)}\Big{]} \tag{17}\] \[= n_{0}v_{F}\left[ \begin{array}{c|c}0&1&\Delta_{0}&0\\ 1&0&0&\Delta_{0}\\ \hline\Delta_{0}&0&0&1\\ 0&\Delta_{0}&1&0\end{array} \right]\,. \tag{18}\]  
An expression similar to Equation (17) should also be obtained for transport current \(\hat{j}_{y}(s_{1},s_{2}\,|\,\mathbf{k},\Delta_{0})\).