Signature of special behaviours of 1/r super(2) interaction in the quantum entanglement entropy

We study the bipartite von Neumann entropy of two particles interacting via a long-range scale-free potential V(r) ~ -g/r super(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 there still exists an interesting multicritical scaling behaviour for all g [setmembership] (-2 < g < 1/4) controlled by the g = -3/4 case.