  Pumping a finite energy density into a quantum system typically leads to
`melted' states characterized by exponentially-decaying correlations, as is the
case for finite-temperature equilibrium situations. An important exception to
this rule are states which, while being at high energy, maintain a low entropy.
Such states can interestingly still display features of quantum criticality,
especially in one dimension. Here, we consider high-energy states in
anisotropic Heisenberg quantum spin chains obtained by splitting the ground
state's magnon Fermi sea into separate pieces. Using methods based on
integrability, we provide a detailed study of static and dynamical spin-spin
correlations. These carry distinctive signatures of the Fermi sea splittings,
which would be observable in eventual experimental realizations. Going further,
we employ a multi-component Tomonaga-Luttinger model in order to predict the
asymptotics of static correlations. For this effective field theory, we fix all
universal exponents from energetics, and all non-universal correlation
prefactors using finite-size scaling of matrix elements. The correlations
obtained directly from integrability and those emerging from the Luttinger
field theory description are shown to be in extremely good correspondence, as
expected, for the large distance asymptotics, but surprisingly also for the
short distance behavior. Finally, we discuss the description of dynamical
correlations from a mobile impurity model, and clarify the relation of the
effective field theory parameters to the Bethe Ansatz solution.
