  We study the non-adiabatic dynamics of a two-state subsystem in a bath of
independent spins using the non-interacting blip approximation, and derive an
exact analytic expression for the relevant memory kernel. We show that in the
thermodynamic limit, when the subsystem-bath coupling is diluted (uniformly)
over many (infinite) degrees of freedom, our expression reduces to known
results, corresponding to the harmonic bath with an effective,
temperature-dependent, spectral density function. We then proceed and study the
heat current characteristics in the out-of-equilibrium spin-spin-bath model,
with a two-state subsystem bridging two thermal spin-baths of different
temperatures. We compare the behavior of this model to the case of a spin
connecting boson baths, and demonstrate pronounced qualitative differences
between the two models. Specifically, we focus on the development of the
thermal diode effect, and show that the spin-spin-bath model cannot support it
at weak (subsystem-bath) coupling, while in the intermediate-strong coupling
regime its rectifying performance outplays the spin-boson model.
