Thermodynamics, a reminder Spontaneous symmetry breaking Landau theory of phase transitions Phonons Bose systems, Bose condensation Magnetism, magnons Fermi systems, Fermions in a metal, Fermi-liquid Instabilities and phase transitions in electronic systems Superconductivity Strongly correlated electronsPreface to the course
Order and elementary excitations.
A course on the theory of solids should start with a definition of what a solid is. It is sensible to define a solid as a regular array of atoms in the sense to having, to good approximation, translational invariance under one of the space groups. Solids have order. We will see that the presence of order gives rise to the defining property of a solid: when you kick it, it hurts your toe. Order leads to a rigidity that opposes external perturbations.
We will study the physical properties of solids. When one measures a certain property of a material, by definition, the system that was in its groundstate is disturbed. So one does not measure the ordered groundstate itself, but merely the deviations from the ordered state. The groundstate wavefunction of a crystal can be a very complicated object, but it is also inert and in itself not very interesting. In solid state physics the ordered state is a stable (local) minimum of the free energy. We will focus on the low energy excitations around this stable "vacuum" configuration - the elementary excitations of the solid. Every form of order (be it crystalline, magnetic or superconducting long range order) has its own particular set of elementary excitations with a characteristic symmetry, dispersion relation and damping.
In chemistry, the relevant energy scale is set by the typical binding energies of atoms, which is of the order of 1- 0.1 electron volt (eV). The excitations in solids that we will be mostly interested in have energies of about kT, at temperatures typically of the order of room temperature and below, i.e. roughly at an energy scale of 100 - 0.1 meV. Excitations on this energy scale are collective oscillations of the constituents of the solids e.g. phonons, plasmons and magnons. Excitations that are related to the local making and breaking of chemical bonds are much higher in energy.
Thus we wish to study the low energy - long wavelength elementary excitations of solids. It is important to note that a low energy excitation cannot be reduced to an intrinsic property of an individual constituent of the solid. It is senseless to try to establish whether for instance a Pb atom is metallic, has a Fermi-surface or is superconducting. Such properties only emerge when we consider a thermodynamically large set of atoms with interactions between them. Thus, for such a set of many interacting particles completely new collective properties arise. Or, in the words of P.W. Anderson: "More is different". We will try to understand how different more is.
Literature: Lecture notes with exercises are available, see below.
The course is based on the lecture notes of D.I. Khomskii "Quantum theory of solids" (Groningen 2002/2003). Also elements from the lecture notes of J. Zaanen "The Classical Condensates" (Leiden 1996) are used. A lot of material can also be found in the following books:
J.M. Ziman "Principles of the theory of solids"
N.W. Ashcroft and N.D. Mermin "Solid state physics"
C. Kittel "Quantum theory of solids"
L.D. Landau and I.M. Lifshits "Course of theoretical physics: statistical physics"