  Renormalization is often described as the removal or "integrating out" of
high energy degrees of freedom. In the context of quantum matter, one might
suspect that quantum entanglement provides a sharp way to characterize such a
loss of degrees of freedom. Indeed, for quantum many-body systems with Lorentz
invariance, such entanglement monotones have been proven to exist in one, two,
and three spatial dimensions. In each dimension d, a certain term in the
entanglement entropy of a d-ball decreases along renormalization group (RG)
flows. Given that most quantum many-body systems available in the laboratory
are not Lorentz invariant, it is important to generalize these results if
possible. In this work we demonstrate the impossibility of a wide variety of
such generalizations. We do this by exhibiting a series of counterexamples with
understood renormalization group flows which violate entanglement RG
monotonicity. We discuss bosons at finite density, fermions at finite density,
and majorization in Lorentz invariant theories, among other results.
