11 Quasiparticles
In many-body quantum mechanics it is convenient to simplify problems because the initial equations
may be too difficult to solve. However it is often possible to calculate the groundstate (at zero tem-
perature). In linearizing the underlying equations, neglecting fast osciallating terms,... one can derive
a handy theory for low-lying excitations which can be built in the language of second quantization
using elementary harmonic oscillators. These quasiparticles will serve as elementary exitations. In
this framework it is possible to excite the system to higher energy by adding quanta of quasiparticles.
Or to lower its energy by removing these quanta until the groundstate is reached again. Because the
concept of quasiparticles is a low energy theory it is not valid for high excitations. For example a
crystal in which the ion lattice undergoes tiny vibrations is said to inhibit phonons. For a crystal
which gets macroscopically distorted it makes no sense to speak of phonons. By investigating the
properties of individual quasiparticles, it is possible to obtain a great deal of information about low-
energy systems, including the flow properties and heat capacity. Most many-body systems possess
two types of elementary excitations. The first type, the quasiparticles, correspond to single particles
whose motions are modified by interactions with the other particles in the system. The second type
of excitation corresponds to a collective motion of the system as a whole. These excitations are called
collective modes.
11.1 Fermi Liquid Theory
The use of the free electron model is a very successful way to describe metals. With the electron
density it is possible to calculate the magnitude of the energy of the electron-electron interaction
(calculation of the Coulomb-energy and sum over all local electrons). The only way to solve the
Schrödinger equation, is to neglect the electron-electron interactions.
Landau was very concerned about this explanation and tried to find another solution for this prob-
lem. He tried to go back and reconstruct the theory with including the electron-electron interaction.
He was interested about the proper normal modes of an electron interacting system which he called
quasiparticles.
One easy example for this would be a system of phonons, in fact, masses which are connected by a
spring. If just one of these masses would be pulled to a side, it would oscillate forward and backward
with some particular frequency. But if you connect them all together and one of them would be pulled
to a side, the energy will spread out and all other masses also will start to oscillate. There won’t be
a periodic solution for this problem anymore.
A model for a crystal would be a lot of springs which connect all phonons and these phonons will
be pulled up and down with a certainkvector. The solution are modes which are eigenmodes of
the system. This model is ideal for phonons because they have a specific wavelength, frequency and
energy.
Landau now includes the electron-electron interaction to find the eigenmodes of the system (these
eigenmodes are called quasiparticles).
If there are just weak interactions (in border case the electron-electron interaction is zero⇒free
electron model) the free electron model is almost right. So there is a correspondence between the free
electrons and the quasiparticles. The number of quasiparticles is the same as the number of the free
electrons. This is also true for phonons: