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We study the bipartite von Neumann entropy of two particles interacting via a long-range scale-free potential V(r)∼ -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 scalingdoi:10.1088/1751-8113/45/42/425302 fatcat:pliq73pdynfybllfw6s74efuii