# 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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The poor man's scaling is a perturbation method. The polynomial series that are produced by this method are not convergent (zero radius of convergence). This serie are called asymptotic serie. They give you good results until some small corrections ($\sim D_0 e^{-\frac{1}{\rho J}}$), when the series start to diverge. This is typical in Many-Body Physics and Quantum Field Theory in general.
So, you can only shrink the band and apply the method if the bandwidth is large when compered with this small energy. To probe more smaller scale non-perturbation methods are required or other perturbation expansion around other regime like $J=\infty$, the strong coupling regime. Exist a full non-perturbation method called Numerical Renormalization Group that generalize the idea of poor man's scaling using fully the Renormalization Group.