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

Signature of special behaviours of $1/r^2$ interaction in the quantum entanglement entropy