  The phenomenon of many-body localization in disordered quantum many-body
systems occurs when all transport is suppressed despite the fact that the
excitations of the system interact. In this work we report on the numerical
simulation of autonomous quantum dynamics for disordered Heisenberg chains when
the system is prepared with an initial inhomogeneity in the energy density
profile. Using exact diagonalisation and a dynamical code based on Krylov
subspaces we are able to simulate dynamics for up to L = 26 spins. We find,
surprisingly, the breakdown of energy diffusion even before the many-body
localization transition whilst the system is still in the ergodic phase.
Moreover, in the ergodic phase we also find a large region in parameter space
where the energy dynamics remains diffusive but where spin transport has been
previously evidenced to occur only subdiffusively: this is found to be true for
initial states composed of infinitely many hydrodynamic modes (square-wave
energy profile) or just the single longest mode (sinusoidal profile). This
suggestive finding points towards a peculiar ergodic phase where particles are
transported slower than energy, reminiscent of the situation in amorphous
solids and of the gapped phase of the anisotropic Heisenberg model.
