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I have been trying to understand the Kondo physics. Based on Anderson model, at low temperature $T<T_\text{K}$, the local spin gets screened by the itinerant electrons with a formation of the spin singlet. Renormalization techniques have been implemented to show the divergence of Kondo coupling at very low energy scales, which indicates the formation of a spin singlet at low temperature. The singlet ground state has also been examined by the variational approach.

In addition, there is the well-known Kondo resonance, which corresponds to a peak of local (normally from $f$-orbital) electron's spectral function at the Fermi energy, and there is an emergent heavy fermion Fermi liquid (HFFL) phase at low temperature, however, according to what I have seen, these latter phenomena are "best" described by a field theoretic $\textit{slave-boson}$ approach.

Here my question is if there is a unified theory which can simultaneously capture both the formation of a singlet ground state (local spin gets screened) and the formation of Kondo resonance and HFFL phase? Maybe the slave-boson approach already indicates the formation of a singlet state but I was not able to see that, hope some expert can elaborate more details on the Kondo physics and thanks in advance.

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I will take the lattice version as an example to give the relation between the formation of Kondo singlet and heavy fermion phase. And you can "generalized" to Kondo resonance in single impurity system.

First of all, the "naive" phase diagram is:
enter image description here

the left side is the magnetic ordered phase origins from RKKY interaction, and the right side is the heavy fermion which is characterized by larger Fermi surface volume and large effective mass.

The key to understand this phase diagram is to realize there exists two different components electrons: itinerant electrons(s-p orbitals) and local moments(f electrons). Naturally, the magnetic ordered phase is contributed by local moments so that now the Fermi surface totally comes from the itinerant electrons. At the critical point($J\rho_c$), there exists a "jump" of Fermi surface volume, which means "jump" to a larger Fermi surface volumn when enter in the heavy fermion phase. This indicates the number of electrons with charge increase, i.e. more than itinerant electrons take part in forming Fermi surface. But where are these additional electrons from? Local moments! Wait, since they always localized in their sites like Mott insulator and only have spin degree of freedom, why does local moments can obtain the charge degree? In the other words, why it seems like the number of electrons(spin$+$negative charge) are not conserve? This is because the local moments(spin) can "fractionalized" to electrons(spin$+$ negative charge) and Kondo singlets (spinless particle with positive charge):

enter image description here

If someone is not familiar with the picture of "fractionalization", this is actually similar with the BCS theory, i.e. electron can transform to Cooper pairs and hole:enter image description here

And this local moments fractionalization process can be illustrated as:

enter image description here

As the result, local moments transform to electrons which contribute to Fermi surface and leaves Kondo singlet as the background.

Then, next question is why electrons are "heavy"? This is because the above picture actually indicates the hybridization of electrons and local moments(or called "spinon"), which gives the new band dispersion:

enter image description here

Loosely speaking, the "dissipative" itinerant electrons couple with "flat" local moments, resulting in the new bands which is more "flat" than original "dissipative" itinerant electrons near Fermi surface. Also, we can also find the enlarged Fermi surface in this dispersion, which is consistent with the above analysis.

Reference

This discussion is actually the physical picture of "slave particle" as you said in the question, and more details can be obtained in:

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