  We study the bipartite von Neumann entropy of two particles interacting via a
long-range scale-free potential $V(r)\sim -g/r^2$ in three dimensions, close to
the unbinding transition. The nature of the quantum phase transition changes
from critical ($-3/4<g<1/4$) to first order ($g<-3/4$) with $g=-3/4$ as a
multicritical point. Here we show that the entanglement entropy has different
behaviours for the critical and the first order regimes. But still there exists
an interesting multicritical scaling behaviour for all $g\in (-2<g<1/4)$
controlled by the $g=-3/4$ case.
