  We study the mode entanglement in the one-dimensional Frenkel-Kontorova
model, and found that behaviors of quantum entanglement are distinct before and
after the transition by breaking of analyticity. We show that the more extended
the electron is, the more entangled the corresponding state. Finally, a
quantitative relation is given between the average square of the concurrence
quantifying the degree of entanglement and the participation ratio
characterizing the degree of localization.
