# What is Resonance Width? Why we use it to distinguish different Regimes of the Anderson Model

The single inpurity Anderson Hamiltonian is $H=\sum_{\sigma}\epsilon_{\sigma}n_{d,\sigma}+Un_{d,\uparrow}n_{d,\downarrow}+\sum_{k,\sigma}\epsilon_{k}c_{k,\sigma}^{+}c_{k,\sigma}+\sum_{k,\sigma}(V_{k}c_{d,\sigma}^+c_{k,\sigma}+h.c.)$

where $n_{d,\sigma}$ is the occupation number of d electron and $\epsilon_d$ is the energy level of d electron, $U$ is the Hubbard interaction.

I know that there are several regimes of parameters $\epsilon_d, U, \Delta$ ($\Delta$ is the resonance width. I do not know why it is called resonance width and this is actually my question), in which the physics of anderson model is different.

For example, there are intermediate valence regime where $\epsilon_d$ and $\epsilon_d+U$ are comparable with $\Delta$. For another example, when $\epsilon_d-\epsilon_F>>\Delta$ and $\epsilon_d+U-\epsilon_F<<\Delta$, they are called empty orbit regime.

My questions are

1. What is the resonance width? What is its physical meaning?

2. Why should we compare $\epsilon_d-\epsilon_F$ and $\epsilon_d+U-\epsilon_F$ with $\Delta$, rather than 0, when we distinguish different regimes of Anderson Model?

Thanks very much!

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