Set of supplementary notes.

Condensed matter systems have collective modes that are a consequence of their order;
both a solid and a liquid support longitudinal sound waves, but a solid that has a nonzero
shear stiffness has also transverse sound modes. In fact the existence of shear waves we might
choose to define as the characteristic feature distinguishing a solid from a liquid or gas. We can
say that solidity is a broken symmetry (with the symmetry being broken that of translational
invariance); because of the broken symmetry, there is a new collective mode (the shear wave).
Because of quantum mechanics, the waves are necessarily quantised as phonons, and they are a
true quantum particle, with Bose statistics, that interact with each other (due to anharmonicity)
and also with other excitations in the solid. This idea, that a broken symmetry can generate
new particles, is one of the central notions of condensed matter physics – and of course of
particle physics too.
A different example is the behaviour of electrons in a semiconductor. If one adds an electron
into the conduction band of a semiconductor it behaves like a particle of charge −|e|, but a
mass different from the free electron mass due to the interaction with the lattice of positively
charge ions as well as all the other electrons in the solid. But we know that if we remove an
electron from the valence band of the semiconductor, it acts as a hole of charge +|e|; the hole
is in fact a collective excitation of the remaining 10 23 or so electrons in the valence band, but
it is a much more convenient and accurate description to think of it as a new fermionic quasiparticle
as an excitation about the ground state of the solid. The electrons and holes, being
oppositely charged, can even bind together to form an exciton - the analog of the hydrogen
atom (or more directly positronium), which however has a binding energy considerably reduced
from hydrogen, because the Coulomb interaction is screened by the dielectric constant of the
solid, and the electron and hole masses are different from the electron and proton in free space.
The solid is a new “vacuum”, inhabited by quantum particles with properties which may be
renormalised from those in free space (e.g. photons, electrons) or may be entirely new, as in the
case of phonons, plasmons (longitudinal charge oscillations), magnons (waves of spin excitation
in a magnet), etc. In contrast to the physical vacuum, there are different classes of condensed
matter systems which have different kinds of vacua, and different kinds of excitations. Many
of these new excitations arise because of some “broken” symmetry , for example, magnetism
implies the existence of spin waves, and solidity implies the existence of shear waves. Some of
these phenomena – superconductivity, superfluidity, and the quantum Hall effect come to mind
– are remarkable and hardly intuitive. They were discovered by experiment; it seems unlikely
that they would ever have been uncovered by an exercise of pure cerebration starting with the
Schrodinger equation for 10 20 particles.
Solid state systems consist of a hierarchy of processes, moving from high energy to low; on
the scale of electron volts per atom are determined the cohesive energy of the solid, (usually)
the crystal structure, whether the material is transparent or not to visible light, whether the
electrons are (locally) magnetically polarised, and so on. But after this basic landscape is determined,
many further phenomena develop on energy scales measured in meV that correspond to
thermal energies at room temperature and below. The energy scales that determine magnetism,
superconductivity, etc. are usually several orders of magnitude smaller than cohesive energies,
and the accuracy required of an ab initio calculation would be prohibitive to explain them. Although
all condensed matter phenomena are undoubtedly to be found within the Schrödinger
equation, they are not transparently derived from it, and it is of course better to start with
specific models that incorporate the key physics; we shall see many of them. These models will
usually be simply of interactions between excitations of the solid, with sets of parameters to
125