  Two significant consequences of quantum fluctuations are entanglement and
criticality. Entangled states may not be critical but a critical state shows
signatures of universality in entanglement. A surprising result found here is
that the entanglement entropy may become arbitrarily large and negative near
the dissociation of a bound pair of quantum particles. Although apparently
counter-intuitive, it is shown to be consistent and essential for the phase
transition, by mapping to a classical problem of DNA melting. We associate the
entanglement entropy to a subextensive part of the entropy of DNA bubbles,
which is responsible for melting. The absence of any extensivity requirement in
time makes this negative entropy an inevitable consequence of quantum mechanics
in continuum. Our results encompass quantum critical points and first-order
transitions in general dimensions.
