The s-d interaction Hamiltonian is as fellows
$H_I=Js.S$, J is the coupling strength.
We focus on the antiferromagnetic case, where $J>0$.
According Anderson's poor man's scaling, the renormalized coupling strength $J$ increases as the band width $D$ decreases. Anderson originally did the scaling to the lowest order of $J$.
But the situation is different as $D$ decreases and even tend to zero. In this case, higher order terms become more and more important, since $J$ becomes bigger or even divergent. These higher order terms typically are different from s-d type.
To some extent, poor man's scaling loose its effect when $D$ is very small. Is it correct?
Thanks very much.