Friday, February 26, 2010

When interactions do not seem to matter

At wednesday's meeting of the Quantum many-body theory reading group, we discussed the following points about the chapter, "Non-interacting electron gas".

To get more context on this chapter read, Chapter 2 of Ashcroft and Mermin, Solid State Physics. In particular, they show how the non-interacting fermion model of Sommerfeld can give a good semi-quantitative description of a wide range of properties of elemental metals such as heat capacity, magnetic susceptibility, and bulk compressibility.

Why is this success surprising? A simple estimate suggests that average potential energy due to the interactions of the electrons between each other is 1-10 times larger than the kinetic energy. Yet, the non-interacting fermion model ignores these electron-electron interactions.
So why does the theory work so well? For profound reasons embodied in Landau's Fermi liquid theory, the elementary excitations (quasi-particles) in a three-dimensional electron liquid (chapter 5) have a one-to-one correspondence to those of the non-interacting fermion model.
A key aspect to this is that the electron liquid has sufficiently high density that the Fermi energy is of the order of several eV (1 eV= 11,000 K)...
Consequently, most properties of elemental metals are determined by properties of the Fermi surface, i.e. only states near the Fermi energy.

All of this breaks down in one dimension (chapter 9) .
In two dimensions the existence of non-Fermi liquids is still controversial.

Next week we discuss chapter 2, "The Born-Oppenheimer Approximation."

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