  In quantum statistical mechanics, finite-temperature phase transitions are
typically governed by classical field theories. In this context, the role of
quantum correlations is unclear: recent contributions have shown how
entanglement is typically very short-ranged, and thus uninformative about
long-ranged critical correlations. In this work, we show the existence of
finite-temperature phase transitions where a broader form of quantum
correlation than entanglement, the entropic quantum discord, can display
genuine signatures of critical behavior. We consider integrable bosonic field
theories in both two- and three-dimensional lattices, and show how the two-mode
Gaussian discord decays algebraically with the distance even in cases where the
entanglement negativity vanishes beyond nearest-neighbor separations.
Systematically approaching the zero-temperature limit allows us to connect
discord to entanglement, drawing a generic picture of quantum correlations and
critical behavior that naturally describes the transition between entangled and
discordant quantum matter.
