Density matrix of electrons coupled to Einstein phonons and the electron-phonon entanglement content of excited states

Abstract

We derive the exact reduced density matrix for the electrons in an analytically solvable electron-phonon model. Here, the electrons are described as a Luttinger liquid that is coupled to Einstein phonons. We further derive analytical expressions for the electron-phonon entanglement, its spectrum, and mutual information at finite and zero temperature as well as for excited states. The entanglement entropy is additive in momentum for the quasiparticle excited states but not in the electron-phonon coupled eigenmodes of the system.