  We theoretically study the spin current and its dc noise generated between
two spin-1/2 spin chains weakly coupled at a single site in the presence of an
over-population of spin excitations and a temperature elevation in one
subsystem relative to the other, and compare the corresponding transport
quantities across two weakly coupled magnetic insulators hosting magnons. In
the spin chain scenario, we find that applying a temperature bias exclusively
leads to a vanishing spin current and a concomitant divergence in the spin Fano
factor, defined as the spin current noise-to-signal ratio. This divergence is
shown to have an exact analogy to the physics of electron scattering between
fractional quantum Hall edge states and not to arise in the magnon scenario. We
also reveal a suppression in the spin current noise that exclusively arises in
the spin chain scenario due to the fermion nature of the spin-1/2 operators. We
discuss how the spin Fano factor may be extracted experimentally via the
inverse spin Hall effect used extensively in spintronics.
