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In this paper, we would like to systematically explore the implications ofnon-perturbative effects on entanglement in a many body system. Instead ofpursuing the usual path-integral method in a singular space, we attempt tostudy the wavefunctions in detail. We begin with a toy model of multipleparticles whose interaction potential admits multiple minima. We study theentanglement of the true ground state after taking the tunnelling effects intoaccount and find some simple patterns. Notably, in the case of multipleparticle interactions, entanglement entropy generically decreases withincreasing number of minima. The knowledge of the subsystem actually increaseswith the number of minima. The reduced density matrix can also be seen to haveclose connections with graph spectra. In a more careful study of the two-welltunnelling system, we also extract the exponentially suppressed tailcontribution, the analogues of instantons. To understand the effects ofmultiple minima in a field theory, it inspires us to inspect wavefunctions in atoy model of bosonic field describing quasi-particles of two differentcondensates related by Bogoliubov transformations. We find that the area law isnaturally preserved. This is probably a useful set of perspectives that promisewider applications.

Inspecting non-perturbative contributions to the Entanglement Entropy via wavefunctions