Exact and Scaling Form of the Bipartite Fidelity of the Infinite XXZ Chain

We find an exact expression for the bipartite fidelity f=|<vac|vac>‘|^2, where |vac> is the vacuum eigenstate of an infinite-size antiferromagnetic XXZ chain and |vac>’ is the vacuum eigenstate of an infinite-size XXZ chain which is split in two. We consider the quantity -ln(f) which has been put forward as a measure of quantum entanglement, and show that the large correlation length xi behaviour is consistent with a general conjecture -ln(f) ~ c/8 ln(xi), where c is the central charge of the UV conformal field theory (with c=1 for the XXZ chain). This behaviour is a natural extension of the existing conformal field theory prediction of -ln(f) ~ c/8 ln(L) for a length L bipartite system with 0<< L <<xi.