  Generic quantum states in the Hilbert space of a many body system are nearly
maximally entangled whereas low energy physical states are not; the so-called
area laws for quantum entanglement are widespread. In this paper we introduce
the novel concept of entanglement susceptibility by expanding the 2-Renyi
entropy in the boundary couplings. We show how this concept leads to the
emergence of area laws for bi-partite quantum entanglement in systems ruled by
local gapped Hamiltonians. Entanglement susceptibility also captures
quantitatively which violations one should expect when the system becomes
gapless. We also discuss an exact series expansion of the 2-Renyi entanglement
entropy in terms of connected correlation functions of a boundary term. This is
obtained by identifying Renyi entropy with ground state fidelity in a doubled
and twisted theory.
