  We consider bosonic transport through one-dimensional spin systems. Transport
is induced by coupling the spin systems to bosonic reservoirs kept at different
temperatures. In the limit of weak-coupling between spins and bosons we apply
the quantum-optical master equation to calculate the energy transmitted from
source to drain reservoirs. At large thermal bias, we find that the current for
longitudinal transport becomes independent of the chain length and is also not
drastically affected by the presence of disorder. In contrast, at small
temperatures, the current scales inversely with the chain length and is further
suppressed in presence of disorder. We also find that the critical behaviour of
the ground state is mapped to critical behaviour of the current -- even in
configurations with infinite thermal bias.
