‘Connecting the nodes’ in the knowledge graph: Renormalization

One of the things that excites me about learning is the process of creating connections between different things. I believe, this is also the process of learning that is persistent throughout our lives, and in my opinion it can be as powerful as the process of learning in our childhood.

I therefore decided to start writing down notions that I read and which connect to notions I had read before. It’s funny how this can also be converted into a ‘computational’ network, and observe how clusters are formed between its nodes. The difference is that here I will not be connecting words between them, but rather presenting the same word, with the same conceptual meaning, used in different sub-domains in physics. Maybe the most appropriate measure here is when the network would ‘break down’. That is, when the meaning between different uses will change completely.

Here I will talk about ‘renormalization’, which I first encountered when running DFT calculations of a molybdenum, mentioned inside the pseudopotential file:

Max angular momentum in pseudopotentials (lmaxx) = 3

file Mo.pbe-mt_fhi.UPF: wavefunction(s) 4f renormalized
file S.pbe-mt_fhi.UPF: wavefunction(s) 4f renormalized

I knew that the f subshell electron occupancies are hard to capture in DFT. Due to the fact that these electrons are very close to the Fermi surface, and at the same time produce significant amount of screening, their occupancies show dependencies equal to that of valence electrons.

After spending some time creating my own pseudopotential configurations, I learnt that empty states can be added in the calculation – both in the valence and in the core part of the pseudopotential. Including f orbitals in the core, with the correct occupancy (provided that it is known in advance) is a way to approximate the ground state for f-electron strongly correlated materials by avoiding the common inaccuracies of DFT.

But the term ”renormalized’ here refers specifically to the wavefunctions. Generally, renormalization is an elaborate set of techniques used in different domains, including quantum field theory. It is the ‘re-calibration’, or re-set of the values of some parameters used in calculations in order to account for interactions between the constituents of the systems that are examined. The usefulness of this is evident in situations similar to that I mentioned in my previous post on convergence in finite elements method calculations. That is, we can avoid infinite values in parameters, which may be true when looking the solution to part of a system, but are ‘invisible’ to reality (or in other words, non-existent), because of processes like screening.

Another encounter of renormalization is in ferroelectricity. A finite sample (thin film) placed on top of a lattice-mismatched substrate can be treated like a bulk material, with the terms included in the Landau free energy renormalized, resulting in a different temperature required for a paraelectric to ferroelectric phase transition to take place.

There are, of course, many other specific examples of renormalization that could be mentioned here, but what I found interesting is the use of the term in the simple ‘high-school’ notion of integrals in calculus, and specifically the infinitesimal difference (dt) term:

The infinitesimal dt may be viewed as a “renormalizing” factor, weighted so that \int_a^b dt = b-a.

“Calculus for Mathematicians, Computer Scientists, and Physicists”, Andrew D. Hwang

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