# Fermi liquid theory vs Hohenberg-Kohn theorems + Kohn-Sham equations

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## Main Question or Discussion Point

Are the Hohenberg-Kohn theorems insanely more powerful than the Fermi liquid theory?

At first glance it looks like I'm comparing apples to oranges. But here is my reasoning.
The Fermi liquid theory describes well the normal state (i.e. non superconductive and other exotic behaviors) of metals at low temperatures. Out of it, one gets the lifetime of quasiparticles (quasielectrons), their "renormalized" mass and things like that. So essentially the math becomes as easy as in the free electron gas, except that the quasielectrons have different mass than in vacuum. One is thus able to get analytical expressions for the conductivity and other important properties.

On the other hand the Hohenberg-Kohn theorems tackle any N-body problem, including all metals at low temperature. The theorems imply that all the interesting properties (such as the conductivity) are to be found in the ground state of the system. All the information of the ground state can be found in the functional of the electron density. And it is always possible to simplify the N-body problem into N 1-body problems. The difficulty resides in numerically calculating the electronic density functional, but once it is done, the whole problem is solved.

(I am aware that DFT has "problems" to deal with predicting high temperature superconductors and things like that, but I suppose that's due to the approximations made in the models to calculate the electronic density functional, but that's besides the point and I hope the focus of the thread do not switch uselessly in that direction.)

So it would seem that Hohenberg-Kohn theorem + the Kohn-Sham equations can do everything the Fermi liquid theory can, plus a lot more, i.e. tackling semiconductors and much more.

Is my reasoning correct? Or am I missing many things? (I'm 100% sure that I do).

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Henceforth, I will refer to the Hohenberg-Kohn theorems and the Kohn-Sham equations as merely DFT (density functional theory).

One quick thing to note is that a more sophisticated derivation of the DFT equations allows one to compute the equilibrium charge density at any temperature, not just the ground state one. This was a relatively recent (past couple decades?) development. Now we may continue.

In practice, both Fermi Liquid (FL) theory and DFT work in similar situations. FL assumes you have a fermi surface, and thus are working with a metal. DFT works best for metals, and can fail for semiconductors and insulators sometimes. Both formalisms have one construct a set of self-consistent equations for the desired quantities. In FL theory, one solves for the state-resolved (e.g. which excited state) charge density at finite temperatures, whereas in DFT one solves for an averaged sort of charge density, and naively is a zero-temperature theory.

From the above paragraph, you may glean that the discussion is a little more nuanced than what you've suggested. While DFT is a bit more basis-independent, you are stuck with calculating quantities that only rely on the averaged charge density of the system (since that's what you self-consistently solve for). Fermi-Liquid theory doesn't necessarily have that restriction, since it doesn't average over the states of your system.

DFT also has a representability problem. Usually it is taken for granted (and indeed, there are some old representability theorems with some assumptions) that the Kohn-Sham potential we solve for captures perfectly the information we want out of the many-body system. That is not always the case. Suppose we do want to look at fluctuations of the electronic density, because we want to know how quickly electronic thermal energy diffuses through our solid. Since DFT maps the N-body problem into N noninteracting problems, there's actually no interaction, and thus no diffusion. So the naive thing to do is perturb the Kohn-Sham potential (or the charge density), re-compute the slightly perturbed state's energy, and say E_{pert} - E_0 is proportional in some way to the amount of time it will take for the system diffuse the energy. One may say, for example, in an ad-hoc manner that the larger the energy difference, the more rapidly the heat will diffuse. But then one is just doing phenomenology again, whereas one can in principle compute this more accurately with the machinery of Fermi-liquid theory (because it doesn't exclude the idea of interactions; they're just a higher-order process).

I hope the above discussion helps! Basically, both have pros and cons, but DFT solves most problems for most simple materials, so it's widely known.

I think the answer is in the definition of 'powerful'. If we would read the word as 'general', then I agree that DFT is more general than Fermi liquid theory, since DFT does not make any assumption on the strength of the interactions involved. I would still have an objection to your example. You say that we can reduce an ##N##-particle object to a 1-particle one. The only problem is to know the functional. Well, that is exactly the problem. The functional is exactly known: It is given by the solution to the Schrödinger equation. As such, it does not really help per se. This is not something unique to DFT, by the way. I can reduce my ##N##-particle problem to a 2-particle one (density matrix theory or Green's function theory), the point being that practically, the exact solution is as difficult as the original Schrödinger equation.

Furthermore, DFT promises the knowledge of all ground-state and excited state quantities from the ground-state density via functionals, but since these functionals cannot be evaluated, it is not practical.

However, if one would read 'powerful' as 'useful', we have a more interesting situation, which is probably unanswerable in general. It all depends on the systems to consider and the quantities sought after. I am convinced both methods (and all others) have their place and usefulness.