  We apply the hierarchical equations of motion technique to analyzing
nonequilibrium heat transport in a spin-boson type model, whereby heat transfer
through a central spin is mediated by an intermediate pair of coupled harmonic
oscillators. The coupling between each pair of oscillators is shown to
introduce a localized gap into the effective spectral densities characterizing
the system-oscillator-reservoir interactions. Compared to the case of a single
mediating oscillator, we find the heat current to be drastically modified at
weak system-bath coupling. In particular, a second-order treatment fails to
capture the correct steady-state behavior in this regime, which stems from the
$\lambda^4$-scaling of the energy transfer rate to lowest order in the coupling
strength $\lambda$. This leads naturally to a strong suppression in the
steady-state current in the asymptotically weak coupling limit. On the other
hand, the current noise follows the same scaling as in the single oscillator
case in accordance with the fluctuation-dissipation theorem. Additionally, we
find the heat current to be consistent with Fourier's law even at large
temperature bias. Our analysis highlights a novel mechanism for controlling
heat transport in nanoscale systems based on tailoring the spectral properties
of thermal environments.
