Simple Explanation of Kondo Effect Does anyone have a simple explanation of the Kondo Effect?
(i.e. a simple physical picture + maybe equations to think of?)
My current understanding is this:
If we consider an electron scattering off of a magnetic impurity, we could have:
$$ |\vec{k},\,\uparrow \downarrow\rangle\rightarrow |\vec{k}',\,\uparrow \downarrow\rangle$$
This is just normal scattering off of an impurity.
We could also have:
$$ |\vec{k},\,\uparrow \downarrow\rangle\rightarrow |\vec{k}',\,\downarrow\uparrow \rangle
$$
Where there is a single spin flip. For some reason, we aren't interested in this.
Additionally, we can have:
$$ |\vec{k},\,\uparrow \downarrow\rangle\rightarrow |\vec{k}'',\,\downarrow\uparrow \rangle \rightarrow |\vec{k}',\,\uparrow \downarrow\rangle
$$
Which has the same final state as the first process, but with an intermediate state.  Doing math on this gives a logarithmic increase to the resistivity.
I don't particularly like this picture, and I don't understand why the second option I listed isn't something important.
Thanks for the help!
 A: The Kondo effect is a phenomenon that occurs when we have a magnetic impurity located in one place of a non-magnetic metal. The magnetic impurity has an residual spin due your electronic configuration. The electrons of the conduction band would interact with this electron via exchange interaction. We can see in equation 10 of the wiki page that the interaction is something like:
$$
\Delta H = - J \vec{S}_{cb}\cdot \vec{S}_{imp}\,,
$$
when the $\vec{S}_{cb}$ is the density of spin of electrons at the point where the impurity is located, and $\vec{S}_{imp}$ is the residual spin of the impurity. The J is the parameter of exchange, or the coupling parameter. We say that the coupling is ferromagnetic when $J>0$, and antiferromagnetic when $J<0$. All this is known as the Kondo model, or s-d interaction model.
The Kondo effect is a quantum phenomenon in the sense that if you think in the model in terms of classical mechanics is impossible to appreciate the phenomenon. Classically, the interaction between the impurity and the electrons of the metal is very simple: a simple scattering of the electron in the impurity by a process like:
$$
|\vec{k},\,\uparrow ,\, \downarrow \rangle \rightarrow |\vec{k}',\,\uparrow ,\, \downarrow \rangle\,,
$$
obeying the conservation of energy. This is the famous Born term of perturbation theory. We are neglecting the non-commutation between the Hamiltonian of the conduction band and the operator $\vec{S}_{cb}$. The total energy (Hamiltonian) of the system do not commute with the spin density operator $\vec{S}_{cb}$ of the conduction band, so we need to appreciate this as well (remember that non-commutation is an quantum feature).
(Here is "hard" math)
By seek of simplicity let take the Hamiltonian of the electrons of the conduction band as:
$$
H_{cd} =\int_{-D}^{D} \frac{|\vec{k}|^2}{2m}\, \left( c_{\vec{k},\,\uparrow}^{\dagger}c_{\vec{k},\,\uparrow} + c_{\vec{k},\,\downarrow}^{\dagger}c_{\vec{k},\,\downarrow} \right) \,d^3\vec{k}
$$
If we assume that the impurity is located at one point $\vec{r}_0$ in space, we have for the operator $\vec{S}_{cb}$:
$$
\vec{S}_{cb} = \int_{-D}^{D} d^3\vec{k} \int_{-D}^{D} d^3\vec{k}'\sum_{\mu,\,\nu - \left(\uparrow,\,\downarrow \right)}\,c_{\vec{k},\,\mu}^{\dagger}\langle \vec{k},\,\mu|\sqrt {2\pi}^3\delta^3(\vec{r}-\vec{r}_0) \frac{\hbar}{2}\vec{\sigma}|\vec{k}',\,\nu \rangle c_{\vec{k}',\,\nu}\,.
$$
The term $\delta^3(\vec{r}-\vec{r}_0)$ means that $\vec{S}_{cb}$ is the density at one point only, the point where the impurity is located. The $\vec{\sigma}$ is the Pauli matrices. All the terms between the bra and ket is operators. So, the Dirac delta function acts as an operator in the state $|\vec{k}\,\mu\rangle$:
$$
\sqrt{2\pi}^3\delta^3(\vec{r}-\vec{r}_0)|\vec{k}\,\mu\rangle \rightarrow\int_{-D}^{D}\frac {d^3\vec{k}}{\sqrt {2\pi}^3} e^{i\vec{k}\cdot(\vec{r}-\vec{r}_0)}|\vec{k}\,\mu\rangle
$$
I'm using Fourier transform.
(Here ends the math)
Ok. Now, we have as an interaction a term like this:
$$
\Delta H = J \left( A_{\uparrow}^{\dagger}A_{\uparrow} - A_{\downarrow}^{\dagger}A_{\downarrow}\right)S_z + J A_{\uparrow}^{\dagger}A_{\downarrow} S_{-} + J A_{\downarrow}^{\dagger}A_{\uparrow} S_{+}\,
$$
where $S_{\pm}$ is the ladder operators of the impurity's spin. The operator $A_{\uparrow \downarrow}$ is the core of the Kondo effect. This operators are obtained by the equation of $\vec{S}_{cb}$, and are written as:
$$
A_{\uparrow \downarrow} = \int_{-D}^{D}\frac{d^3\vec{k}}{\sqrt{2\pi}^3} e^{-i\vec{k}\cdot\vec{r}_0}c_{\vec{k}\,\uparrow \downarrow}
$$
Is a superposition of all energies, from the high energy $D$ to zero. Of course that $A$ do not commute with $H_{cb}$. When we calculates the next correction in the perturbation theory (corrections to the Born term), we found integrals:
$$
J^2\sum_{k\neq n} \frac{\langle n| \Delta H| k \rangle \langle k| \Delta H|n \rangle}{E_n-E_k}\sim J^2\,\int_{0}^{D}\,\frac{dk}{k}\,.
$$
This term is nothing more than a transition $|\vec{k}\uparrow \downarrow \rangle \rightarrow |\vec{k}''  \downarrow \uparrow \rangle \rightarrow |\vec{k}'\uparrow \downarrow \rangle$, and other types of transitions as you suggest give the same thing, but the intermediate transition do not obey conservation of energy. This integrals are divergent. Jun Kondo, in 1964, discover that if you are interested in thermodynamic quantities, you can put a lower bound on the integration:
$$
\sim J^2\,\int_{k_BT}^{D}\,\frac{dk}{k}\,.
$$
This integral is small, justifying the perturbation approach, when:
$$
T\gg D\,e^ \frac{1}{-J^2}
$$
This temperature is known as the Kondo temperature. When we get close to this small temperature the perturbation theory fails and the systems in in a crossover to form a bound state between a few electrons in the metal and the magnetic impurity.
In the last year I work with the Numerical Renormalization Group, a method designed to calculate the the thermodynamic quantities of the Kondo model for every temperature, specially the small ones.
The physics of the phenomenon is that at zero temperature, there is a bound state formed by a lot of electrons around the impurity via exchange interaction. All this electrons are sharing the impurity in a very non-trivial way, after all they are fermions. Fermions do not want to be at the same state. Turn outs that this electrons are all around the impurity with alternating spins is a such a way that they blind the magnetica impurity. Is this bound state the total spin of the system is zero. When we increase the temperature the energy $k_B T$ start to perturb this bound state until reaching the Kondo temperature. To larger temperatures, $k_BT$ destroy completally the bound state and we have a naked impurity. All this phenonmena take place because the operator $A_{\uparrow\downarrow}$ is an integral over suficient small energies (small than $k_BT_K$, where $T_K$ is the Kondo temperature).
