  We study a vast family of continuum Rokhsar-Kivelson (RK) states, which have
their groundstate encoded by a local quantum field theory. These describe
certain quantum magnets, and are also important in quantum information. We
prove the separability of the reduced density matrix of two disconnected
subsystems, implying the absence of entanglement between the two subsystems --
a stronger statement than the vanishing of logarithmic negativity. As a
particular instance, we investigate the case where the groundstate is described
by a relativistic boson, which is relevant for certain magnets or Lifshitz
critical points with dynamical exponent $z=2$, and we propose nontrivial
deformations that preserve their RK structure. Specializing to 1D systems, we
study a deformation that maps the groundstate to the quantum harmonic
oscillator, leading to a gap for the boson. We study the resulting correlation
functions, and find that cluster decomposition is restored. We analytically
compute the $c$-function for the entanglement entropy along a renormalization
group flow for the wavefunction, which is found to be strictly decreasing as in
CFTs. Finally, we comment on the relations to certain stoquastic quantum spin
chains. We show that the Motzkin and Fredkin chains possess unusual
entanglement properties not properly captured by previous studies.
