I’ve decided to post several pages about theory before presenting a program, to avoid having long posts that nobody has patience to read. I’ll refer to those posts for details when presenting a program. This post starts a series of posts that will lead to a Numerical Renormalization Group program. A relatively simple one, allowing one to understand it without spending insane amounts of time on it (get Flexible DM-NRG and try to understand it to see what I mean). The serious ones can take years to develop and are not so easy to understand, you end up not seeing the forest because of the trees. There are so many details that you can be overwhelmed.
I’ve written one that is quite simple to understand by comparison, but I still have to add some UI for options and check some constants. I also wrote a charting class that is still not functional enough to be released, I still have some work to do on it. I want to reuse it in more projects. Hopefully by the time I’ll end up writing those posts about the theory I’ll be able to post the program on GitHub.
So, let’s start with the beginnings.
It all started in 1934 when they measured the gold resistivity at a low temperature. Gold is a non magnetic metal, but it had some magnetic impurities in it. They expected a decrease in resistivity as the temperature was lowered, the resistivity having a term because of the lattice vibrations (that is, phonons), one because of the electron-electron interactions and one constant, being proportional with the impurity density. Instead of finding that, they found that the resistivity reached a minimum at some low temperature (about 8K or something like that) then when lowering the temperature further, the resistivity started to increase. Quite a difference from the expected decrease down to 0K towards a resistivity given by the impurity density. It took 30 years to understand the cause.
A Partial Solution
The effect, which is now known as the Kondo Effect was explained by Jun Kondo using the Kondo model. Very shortly, the Kondo model describes the impurity as being a 1/2 spin coupled by a spin-spin interaction to the conduction band electrons, considered as non interacting, the Hamiltonian being:
where is the term for the impurity-conduction electrons interaction and is the term for the conduction electrons, that is:
more detailed for the conduction electrons:
where and and , with the number operator, that is, .
Kondo considered the coupling J as being small and used the perturbation theory to calculate resistivity. The article1 linked at the bottom of this page is the published article on the issue, you can also find some info here. I will link to some thesis pdfs in the post on numerical renormalization group, a lot of info on the subject will also be found there.
Very shortly, for the first order there is no temperature dependency, but for the second order he got a logarithmic dependency. The second order correction has a term of the third order in J and logarithmic in T. The resistivity for this correction looks like this:
where K is some value that depends on the square of J and the impurity concentration and so on.
It indicates that for low temperature there is an increase in scattering close to the Fermi level, known as Kondo resonance. The spin flip scattering are responsible for the temperature dependency, they involve virtual intermediate – with a spin flip – states that occur between two scattering events. Anticipating a little, the Kondo resonance is what you see as the middle narrow peak in the chart displayed at the beginning of the post. I generated it with the nrg program I’m going to release.
A problem still remains and that’s the logarithmic divergence. The resistivity does not really go to infinity as the temperature goes to zero.
As going to a lower temperature J term cannot be considered a small perturbation so the perturbation theory breaks down. The temperature where that happens is the Kondo temperature. A non perturbative approach have to be used to solve this, but on this I’ll give more details in the post about the numerical renormalization group.
The Anderson Model
A model2 that is more general than the Kondo model and it’s also useful to study impurities in metals (or quantum dots!) is the Anderson model. The Hamiltonian has three parts, one for the impurity/quantum dot, , one for the conduction electrons, – similar with the one from the Kondo model, so I won’t detail it – and one for the interaction between the electrons in the impurity/quantum dot with the ones from the conduction band, that is the ones from the metallic host/quantum dot leads, :
The quantum dot part of the Hamiltonian has an ‘on site’ energy term and the Coulomb repulsion terms:
i, j are level indices, the first Coulomb interaction term is the inter-levels term, that is, the Coulomb interaction between electrons on different levels, and the last one is the intra-level term, that is, simply the repulsion between the spin up electron and the spin down electron from the same level. The term that couples the impurity is:
h.c. stands for Hermitian Conjugate.
Having a strong Coulomb interaction in the impurity/quantum dot, energy levels ‘tuned’ in such a way that only an electron – on average – is in the impurity/quantum dot, considering that the tunneling is weak, one can use perturbation theory to find that the Anderson Hamiltonian includes the Kondo Hamiltonian. The Kondo Hamiltonian is obtained by perturbatively eliminating excitations to doubly occupied and empty states, as an effective Hamiltonian.
Spin flips happen through virtual excitations. For example, if a spin down electron is initially in the quantum dot, it can tunnel outside (or if you like this picture, a hole can tunnel in), then a spin-up electron can tunnel in. Or with a spin-up electron inside, a spin-down electron can tunnel in then the spin-up electron tunnels out. Again, the end result is a spin flip. More, here4.
I described very briefly the Kondo effect and models for it, but I also gave some links that should help. To anticipate things a little, here5 is a very useful link. I’ll give more in further posts, but this one you might want to check. You’ll find details not only on the Kondo and Anderson models, Schrieffer Wolff transformation, but also on Quantum Dots and the Numerical Renormalization Group and that’s what I intend to present in the future. If you manage to understand the Michael Sindel dissertation you will have no troubles understanding the program.