  We consider two semi-infinite quantum Ising chains initially at thermal
equilibrium at two different temperatures and subsequently joined by an
interaction between their end points. Transport properties such as the heat
current are determined by the dynamics of the left- and right-moving fermionic
quasi-particles which characterize the ensuing unitary dynamics. Within the
so-called semi-classical space-time scaling limit we extend known results by
determining the full space and time dependence of the density and current of
energy and of fermionic quasi-particles. Upon approaching the edge of the
propagating front, these quantities as well as the two-point correlation
function display qualitatively different behaviors depending on the transverse
field of the chain being critical or not. While in the latter case corrections
to the leading behavior are described, as expected, by the Airy kernel, in the
former a novel scaling form emerges with universal features.
