  We investigate theoretically the spiral staircase Heisenberg spin-$1/2$
ladder in the presence of antiferromagnetic long-range spin interactions and a
uniform magnetic field. As a special case we also consider the Kondo necklace
model. If the magnetizations of the two chains forming the ladder satisfy a
certain resonance condition, involving interchain couplings as perturbations,
the system is in a partially gapped magnetic phase hosting excitations
characterized by fractional spins, whose values can be changed by the magnetic
field. We show that these fractional spin excitations can be probed by spin
currents in a transport setup with a spin conductance that reveals the
fractionalized spin. In some special cases, the spin conductance reaches
universal values in units of $(g\mu_B)^2/h$, where $g$ is the $g$-factor,
$\mu_B$ the Bohr magneton, and $h$ the Planck constant. We obtain our results
with the help of bosonization and numerical density matrix renormalization
group methods.
