Electron-phonon interactions and Kohn anomalies

Electron-phonon interactions

The study of interactions between electrons and phonons, is an interesting and classical topic in quantum many body theory as well as condensed matter physics. The electron-phonon interaction leads to many novel properties in metals, for instance, electrical resistance, thermal resistance, superconductivity and the renormalization of linear electronic specific heat. [1]

Free electrons in lattice

In contrary to the independent electron model, where In the tight-binding approximation,

Lattice Vibration and Phonons in 1D

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^0_{ph} = \sum_{k \lambda}\hbar\omega_{k \lambda}( a^\dagger_{k \lambda} + \frac{1}{2} )}

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Acoustical and Optical Phonon in 3D

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^0_{ph} = \sum_{k \lambda}\hbar\omega_{k \lambda}( a^\dagger_{k \lambda} + \frac{1}{2} )}

Derivation of Hamiltonian Electron-Phonon Coupling

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = H^0_{el} + H^0_{ph} + H_{coul} + H_{int}}

Where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^0_{el} = \sum_{k \sigma}E_k c^\dagger_{k \sigma} c_{k \sigma}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^0_{ph} = \sum_{k \lambda}\omega_{k \lambda}( a^\dagger_{k \lambda} + \frac{1}{2} )}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{coul} = \frac{1}{2} \sum_{k k^\prime q \atop \sigma \sigma^{\prime} } V(q)c^\dagger_{k ^\prime + q \sigma^\prime } c^\dagger_{k \sigma} c_{k+q \sigma} c_{k^{\prime} \sigma^{\prime}} }

$H_{int}=\sum _{kk^{\prime }\sigma \lambda }g_{kk^{\prime }c_{k\sigma }^{\dagger }}c_{k^{\prime }\sigma }c_{k^{\prime }\sigma }(a_{-q\lambda }^{\dagger }+a_{q\lambda })$

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