Set of supplementary notes.

Chapter 10
Fermi liquid theory
10.1 The problem with the Fermi gas
Our modelling of electrons in solids so far has been based on a fairly simple-minded approach:
instead of looking for the eigenstates of the many-particle system we are interested in, we
instead calculate the electronic eigenstates of a single-particle Hamiltonian (subject to the
periodic lattice potential). We then fill these eigenstates with electrons according to the Fermi
occupation factor, treating our system as a degenerate Fermi gas. This separation of a manyparticle
problem into single-particle states relies on being able to separate the many-body wave
function into an antisymmetrised product of single-particle wavefunctions. For a two-electron
state, for instance, we could write Ψ(r 1 , r 2 ) = √ 1
2
(ψ a (r 1 )ψ b (r 2 ) − ψ a (r 2 )ψ b (r 1 )). Such product
states, however, implicitly ignore correlations between the electrons. For example, we would find
that expectation values of products such as 〈r 1 r 2 〉 decompose into the products of expectation
values 〈r 1 〉 〈r 2 〉.
In the presence of strong electronic interactions, it is doubtful that the electrons’ motion
remains uncorrelated. We saw last term that in the Thomas-Fermi approximation, the electrons
in a Fermi gas react to the introduction of a charged impurity, in such a way that they screen
the impurity potential at long distances. Taking this idea to the next level, we could say
that for any one electron under consideration, which of course carries a negative charge, the
other electrons in the metal execute a correlated screening motion, which would reduce their
density in the vicinity of the first electron. This reduces the range of the Coulomb potential
due to the electron under consideration, which we might take as justification for ignoring the
Coulomb interaction. However, this also implies that the electrons correlate their motion. Such
correlations are not contained in a single-particle description.
On the other hand, the band structure approach which we have used so far has been remarkably
successful in modelling a wide range of materials and phenomena. It explains electronic
transport and thermodynamic properties such as the heat capacity in metals, we have used it
to understand semiconductors and semiconductor devices, and it is consistent with the Fermi
surface probes such as quantum oscillations in high magnetic fields, as well as other probes
of electronic structure. These successes suggests that it is not altogether wrong. How can we
reconcile the success of the single particle picture with its conceptual difficulties
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