  Energy transport can reveal information about interacting many-body systems
beyond other transport probes. In particular, in one dimension it has been
shown that the energy current is directly proportional to the central charge,
thus revealing information about the degrees of freedom of critical systems. In
this work, we explicitly verify this result in two cases for translationally
invariant systems based on explicit microscopic calculations. More importantly,
we generalise the result to non-translation invariant setups and use this to
study a composite system of two subsystems possessing different central
charges. We find a bottleneck effect meaning the smaller central charge limits
the energy transport.
