  We investigate separability and entanglement of Rokhsar-Kivelson (RK) states
and resonating valence-bond (RVB) states. These states play a prominent role in
condensed matter physics, as they can describe quantum spin liquids and quantum
critical states of matter, depending on their underlying lattices. For dimer RK
states on arbitrary tileable graphs, we prove the exact separability of the
reduced density matrix of $k$ disconnected subsystems, implying the absence of
bipartite and multipartite entanglement between the subsystems. For more
general RK states with local constraints, we argue separability in the
thermodynamic limit, and show that any local RK state has zero logarithmic
negativity, even if the density matrix is not exactly separable. In the case of
adjacent subsystems, we find an exact expression for the logarithmic negativity
in terms of partition functions of the underlying statistical model. For RVB
states, we show separability for disconnected subsystems up to exponentially
small terms in the distance $d$ between the subsystems, and that the
logarithmic negativity is exponentially suppressed with $d$. We argue that
separability does hold in the scaling limit, even for arbitrarily small ratio
$d/L$, where $L$ is the characteristic size of the subsystems. Our results hold
for arbitrary lattices, and encompass a large class of RK and RVB states, which
include certain gapped quantum spin liquids and gapless quantum critical
systems.
