  To this day, von Neumann definition of entropy remains the most popular
measure of quantum entanglement. Much of the literature on entanglement
entropy, particularly in the context of field theory, has focused on isolating
the UV divergences. Zero-mode divergences of the entanglement entropy are less
studied in this context, and apart from being easier to isolate, they offer an
interesting insight into the physics of the system. To gain a better
understanding of the system in this limit, we develop the free particle
approximation of Harmonic oscillator, with which we investigate the properties
of entropy divergence in continuous bi-partite quantum systems such as the
coupled Harmonic oscillators and the Hydrogen atom. We also show zero-mode
divergence of the entropy of environment-induced entanglement in a tri-partite
oscillator system. We discuss the implications of our result for field theory
and IR structure of gravity.
