 Solution

A pure spin-one-half state on the equator of the Bloch sphere has $⟨Z_k⟩=0$, so $|{\alpha }_k{|}^2=|{\beta }_k{|}^2=1/2$. Its azimuthal phase does not enter the conditional environment overlap, because the interaction is diagonal in $Z_k$. Substitution in the decoherence factor givesThe Zurek spin-bath model now has a real coherence factor: positive and negative values correspond to opposite relative phases, while suppression of coherence depends on $|z|$.