## 4 Inversion Symmetry in 2 LG Systems  
One major intriguing feature of 2 LG is a band structure that can be tuned with external fields, such as electric or magnetic fields. [PERSON] _et al._[67] used angle resolved photoemission to show that 2 LG has a gap that can be tuned from 0 to up to 200 meV. Fundamentally, the gap opening in bilayer graphene originates from the breaking of inversion symmetry, that can be achieved, for example, by applying an electric field perpendicular to the layers. This creates an asymmetry between the two layers that lowers the whole symmetry of the bilayer system, consequently opening an electronic band gap at the K point. For \({}^{12/12}\)C AB-2 LG, at \(q=0\) the E\({}_{2g}\) phonon mode (G band) of monolayer graphene splits into two double degenerated modes, associated with the in-phase (E\({}_{g}\)) and out-of-phase (E\({}_{u}\)) displacements of the atoms in the two layers [68]. The E\({}_{u}\) mode is not Raman active, but, as in 1 LG, the E\({}_{g}\) mode can be seen as one peak in the Raman spectrum, known as G band at \(\sim\)\(1582\) cm\({}^{-1}\). Moreover, since there are two valence (\(\pi_{1}\) and \(\pi_{2}\)) and two conduction (\(\pi_{1}\)* and \(\pi_{2}\)*) bands in this material, phonons can couple with e-h pairs produced by interband or intraband transitions. Interbands transitions are those were the hole is in the valence band and the electron in the conduction band. The intraband transitions occur when both the electron and the hole are in the conduction or in the valence band. [PERSON] [69] calculated the dependence of the self-energy for the in-phase (IP) and for the out-of-phase (OP) phonons as a function of Fermi energy. The phonon renormalization effect in 2 LG is understood by considering the selection rules for the interaction of the IP and OP phonon modes with the interband or intraband e-h pairs. The e-ph interaction can be described by a \(2\times 2\) matrix for each phonon symmetry given by [69]:  
\[\Phi_{jj^{\prime}}^{E_{g}}(k)=\frac{1}{2}\left( \begin{array}{cc}\sin^{2}\psi& \cos^{2}\psi\\ \cos^{2}\psi&\sin^{2}\psi\end{array} \right),\ \ \Phi_{jj^{\prime}}^{E_{u}}(k)=\frac{1}{2} \left( \begin{array}{cc}0&1\\ 1&0\end{array} \right) \tag{19}\]  
where each matrix element gives the contribution of e-h pairs involving different electronic sub-bands \(\pi_{j}\) or \(\pi_{j}\)*. The diagonal terms are responsible for the interband e-ph coupling, while the out-of-diagonal terms give the intraband coupling. For the IP lattice vibration, all matrix elements are different from zero, and this phonon can interact with both interband or intraband e-h pairs (see Figure 12). When \(E_{\mathrm{F}}=0\), there is a couple of the phonon with the interband e-h pair, giving rise to the phonon energy renormalization. However, for the OP phonon mode, the diagonal terms of the matrix are null, showing that there is no e-ph interactions for interband transitions. Therefore, no Kohn anomaly is expected for the antisymmetric phonon mode when the Fermi level is at the K point. The frequency of the out-of-phase vibration is, then, higher then the frequency of the in-phase vibration for \(E_{\mathrm{F}}=0\). If the Fermi energy is now moved away from the K point, the intraband e-h pairs can be produced by phonons (see Figure 12). In this case, the OP phonons also have their energies renormalized, giving rise to the Kohn anomaly and lowering the frequency of the phonons. Figure 13a,b show, respectively, the frequency shift (solid lines) and the line width broadening (dashed lines) of the IP and OP phonon modes as a function of the Fermi energy calculated by [PERSON] [69]. The \(\delta\) parameter is due to disorder in the crystal. The frequency of the IP mode undergoes a hardening with increasing \(E_{\mathrm{F}}\) and  
Figure 12: **(a)** When \(E_{\mathrm{F}}=0\), only interband electron-hole pair creation by the absorption of a phonon is allowed; **(b)** When \(E_{\mathrm{F}}\
eq 0\), intraband electron-hole pair creation by the absorption of a phonon are also allowed [69].