  Translationally invariant finetuned single-particle lattice Hamiltonians host
flat bands only. Suitable short-range many-body interactions result in complete
suppression of particle transport due to local constraints and Many-Body
Flatband Localization. Heat can still flow between spatially locked charges. We
show that heat transport is forbidden in dimension one. In higher dimensions
heat transport can be unlocked by tuning filling fractions across a percolation
transition for suitable lattice geometries. Transport in percolation clusters
is additionally affected by effective bulk disorder and edge scattering induced
by the local constraints, which work in favor of arresting the heat flow. We
discuss explicit examples in one and two dimensions.
