Electron-phonon interactions and Kohn anomalies

Electron-phonon interactions

Derivation of Hamiltonian Electron-Phonon Coupling

The Hamiltonian for the electron-phonon interaction can be described as

$H=H_{el}^{0}+H_{ph}^{0}+H_{coul}+H_{int}$

Where

$H_{el}^{0}=\sum _{k\sigma }E_{k}c_{k\sigma }^{\dagger }c_{k\sigma }$

$H_{ph}^{0}=\sum _{k\lambda }\omega _{k\lambda }a_{k\lambda }^{\dagger }+{\frac {1}{2}}$ $H_{el}^{0}=\sum _{k\sigma }E_{k}c_{k\sigma }^{\dagger }c_{k\sigma }$

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