  We study the entanglement between disjoint subregions in quantum critical
systems through the lens of the logarithmic negativity. We work with systems in
arbitrary dimensions, including conformal field theories and their
corresponding lattice Hamiltonians, as well as resonating valence-bond states.
At small separations, the logarithmic negativity is big and displays universal
behavior, but we show non-perturbatively that it decays faster than any power
at large separations. This can already be seen in the minimal setting of
single-spin subregions. The corresponding absence of distillable entanglement
at large separations generalizes the 1d result, and indicates that quantum
critical groundstates do not possess long-range bipartite entanglement, at
least for bosons. For systems with fermions, a more suitable definition of the
logarithmic negativity exists that takes into account fermion parity, and we
show that it decays algebraically. Along the way we obtain general results for
the moments of the partially transposed density matrix.
