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

TL;DR

This paper derives an exact reduced density matrix for electrons in a solvable electron-phonon model, analyzing electron-phonon entanglement, spectrum, and mutual information at various temperatures and states.

Contribution

It provides analytical expressions for electron-phonon entanglement and related measures in a Luttinger liquid coupled to phonons, revealing new insights into their quantum correlations.

Findings

Entanglement entropy is additive in momentum for quasi-particle excited states.

Entanglement entropy is not additive in the coupled eigenmodes.

Derived exact formulas for entanglement spectrum and mutual information.

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 quasi-particle excited states, but not in the electron-phonon coupled eigenmodes of the system.