Green's Function Approach to Entanglement Entropy on Lattices and Fuzzy Spaces

We develop a Green's function approach to compute R\'{e}nyi entanglement entropy on lattices and fuzzy spaces. The R\'{e}nyi entropy resulting from tracing out an arbitrary collection of subsets of coupled harmonic oscillators is written as zero temperature partition function generated by an Euclidean action with $n$-fold step potential. The associated Green's function is explicitly constructed and an alternative new formula for the R\'{e}nyi entropy is obtained. Finally it is outlined how this approach can be used to go beyond the gaussian state and include interaction by writing a perturbative expansion for the entanglement entropy .