  In this paper, we would like to systematically explore the implications of
non-perturbative effects on entanglement in a many body system. Instead of
pursuing the usual path-integral method in a singular space, we attempt to
study the wavefunctions in detail. We begin with a toy model of multiple
particles whose interaction potential admits multiple minima. We study the
entanglement of the true ground state after taking the tunnelling effects into
account and find some simple patterns. Notably, in the case of multiple
particle interactions, entanglement entropy generically decreases with
increasing number of minima. The knowledge of the subsystem actually increases
with the number of minima. The reduced density matrix can also be seen to have
close connections with graph spectra. In a more careful study of the two-well
tunnelling system, we also extract the exponentially suppressed tail
contribution, the analogues of instantons. To understand the effects of
multiple minima in a field theory, it inspires us to inspect wavefunctions in a
toy model of bosonic field describing quasi-particles of two different
condensates related by Bogoliubov transformations. We find that the area law is
naturally preserved. This is probably a useful set of perspectives that promise
wider applications.
