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We study a vast family of continuum Rokhsar-Kivelson (RK) states, which havetheir groundstate encoded by a local quantum field theory. These describecertain quantum magnets, and are also important in quantum information. Weprove the separability of the reduced density matrix of two disconnectedsubsystems, implying the absence of entanglement between the two subsystems --a stronger statement than the vanishing of logarithmic negativity. As aparticular instance, we investigate the case where the groundstate is describedby a relativistic boson, which is relevant for certain magnets or Lifshitzcritical points with dynamical exponent $z=2$, and we propose nontrivialdeformations that preserve their RK structure. Specializing to 1D systems, westudy a deformation that maps the groundstate to the quantum harmonicoscillator, leading to a gap for the boson. We study the resulting correlationfunctions, and find that cluster decomposition is restored. We analyticallycompute the $c$-function for the entanglement entropy along a renormalizationgroup flow for the wavefunction, which is found to be strictly decreasing as inCFTs. Finally, we comment on the relations to certain stoquastic quantum spinchains. We show that the Motzkin and Fredkin chains possess unusualentanglement properties not properly captured by previous studies.

Entanglement and separability in continuum Rokhsar-Kivelson states