Entanglement entropy of coupled conformal field theories and Fermi liquids

In this paper, we calculate the entanglement Rényi entropy of two coupled gapless systems in general spatial dimension $d$. The gapless systems can be either conformal field theories or Fermi liquids. We assume the two systems are coupled uniformly in an $h$-dimensional submanifold of the space, with $0\le h\le d$. We will focus on the scaling of the Rényi entropy with the size of the system, and its scaling with the intersystem coupling constant $g$. Three approaches will be used for our calculation: (1) exact calculation with ground-state wave functional, (2) perturbative calculation with functional path integral, and (3) scaling argument.