- #1

fluidistic

Gold Member

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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).

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).