The coupling between electronic and lattice degrees of freedom lies at the core of many important properties of solids. Nevertheless, surprisingly little is known about the entanglement between these degrees of freedom. Here, we calculate the entanglement entropy at zero temperature as well as the mutual information and the logarithmic entanglement negativity at finite temperatures between the electrons and the lattice for a one-dimensional chain. The electrons are described within Luttinger-liquid theory. Our results show that the entanglement entropy diverges when one approaches the limit of stability, the so-called Wentzel-Bardeen singularity. We find that the mutual information and the logarithmic entanglement negativity decrease with temperature. The mutual information reaches a finite value in the infinite-temperature limit, which is a consequence of the infinite linear electron spectrum of Luttinger theory. The logarithmic entanglement negativity becomes exactly zero above a certain temperature; that is, the lattice and the electrons become nonentangled above this temperature. If the electron-electron interaction is unscreened or weakly screened, this characteristic temperature diverges with the system size. However, if the interaction is strongly screened, the characteristic temperature is finite and independent of size, indicating a phase transition in the thermodynamic limit.

• Received 5 February 2021
• Revised 15 June 2021
• Accepted 21 June 2021

DOI:https://doi.org/10.1103/PhysRevB.104.035405