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148 CHAPTER 10. FERMI LIQUID THEORY #$% ' ( & $ ! " ! !"#$%&#'()"*'&#+%,()-&.#/ 0.)-%"%1&23&4 56&*6%/3/'"&#/%,()-&.#&*% 1.5%(#()+$%(7*&'"'&.#/ Figure 10.3: Modelling of a Fermi system in terms of its elementary excitations, electrons or holes close to the Fermi surface. This is the essence of Fermi liquid theory. description works so well in many materials. In many cases, the properties of the electrons making up the Fermi liquid carry over with only slight modification to the properties of the fermionic excitations of the Fermi liquid. 10.2.1 Adiabatic continuity We could approach the interacting electron state in this way: let us begin by imagining an assembly of electrons which do not interact. We know that in this case, the electron system will form a Fermi gas, which means that the ground state is represented by completely filled states inside the Fermi surface and empty states outside the Fermi surface, and that the low energy excitations from the ground state are electrons just outside the Fermi surface and holes just inside the Fermi surface. We then very gradually turn on the interaction between the electrons and follow the evolution of the energy levels of the system. The principle of adiabatic continuity (Fig. 10.4) suggests that we can continue to label the energy eigenstates in the same way as for the non-interacting system: energy eigenstates shift, when the system is tuned, but their labels remain useful. We can therefore assume that the excitations of the interacting Fermi system, the Fermi liquid, follow the same basic rules as those of the Fermi gas. One important consequence of this one-to-one correspondence between the quasiparticle states and the states of the non-interacting system is that the volume of the Fermi surface is unchanged, as the interaction is turned on. This is called Luttinger’s theorem. As usual, there are many hidden pitfalls in this argument. For example, it only holds if the energy levels do not cross as the interaction is turned on. This is not guaranteed, and in fact it is difficult to find a non-trivial example of an interacting system in which the energy levels do not cross on tuning. It is safer to say that for interacting Fermi systems, a Fermi liquid state in the sense discussed above is possible, but that not every Fermi system will necessarily be described in this way.
10.3. TOTAL ENERGY EXPANSION FOR LANDAU FERMI LIQUID 149 Energy N=4 N=4 N=3 N=2 N=1 N=0 0 1 N=3 N=2 N=1 N=0 Figure 10.4: An example of adiabatic continuity (from Schofield, Contemporary Physics 40, 95 (1999)): as the Hamiltonian of a system is changed – here, we gradually change a square well potential into a simple harmonic oscillator potential – the energy levels and details of the eigenstates drift, but we can continue to label the states with the same labels as before. 10.3 Total energy expansion for Landau Fermi liquid If we adopt Landau’s Fermi liquid approach, then we label excited states of the interacting system with quantum numbers of non-interacting system, such as wavevector k, spin, band index, etc. In analogy with the case of lattice vibrations, where excited states are expressed in terms of a new particle called the phonon, we talk of a ’quasiparticle’ at wavevector k, if the system is in an excited state labelled with that wavevector, which would have to lie outside the Fermi surface. Of course, there can also be ’quasiholes’, corresponding to excitations at wavevectors inside the Fermi surface. We can then express the total energy of the interacting system as a functional of the occupation numbers of the various states labelled in this way. At low temperatures, when the number of excitations is small, this functional could be approximated by a Taylor expansion. E[n k ] = ∑ k ɛ(k)n(k) + 1 2 ∑ kk ′ f(k, k ′ )n(k)n(k ′ ) (10.1) The first term on the right-hand side is familiar. It expresses the energy of having quasiparticles in band states of energy ɛ(k). The next term on the right-hand side, which is second-order in occupation number n(k), accounts for interactions between excited state, or quasiparticles. By postulating such a relatively simple expression for the total energy, Landau was able to arrive at a variety of key expressions, which link material properties such as heat capacity, magnetic susceptibility and compressibility to properties of the quasiparticles and their interaction function f.
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