# Will Anderson's Poor Man's Scaling loose its effect when band width is small?

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.

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