Set of supplementary notes.

Chapter 9
Electronic instabilities
General remarks on theories and models in condensed
matter physics
Solid state physics is concerned with the abundance of properties that arise when atoms are
amalgamated together. Much of what we think of as “core physics” is deliberately reductionist;
we look for the very simplest unified description of a basic phenomenon, and the progress of
much of basic physics has always been a progress toward grander unified theories, each of which
is simpler (at least in concept) than the previous generation.
Condensed matter physics is not like this. The Hamiltonian is not in doubt - it is the
Schrödinger equation for the many particle system:
H elec = − ∑ h¯ 2
2m ∇2 i + ∑ ( )
PI
2 + 1 ∑ Z I e 2
2M
i
I 4πɛ 0 |r i − R I | + 1 ∑ e 2
2 |r i − r j | + 1 ∑ Z I Z J e 2
,
2 |R i − R j |
I
i,I
i≠j
I≠J
(9.1)
where the r i , R I label the coordinates of the electrons and the ions respectively, Z I , M I are
the nuclear charge and mass. The terms in (9.1) represent, in order, the kinetic energy of the
electrons, the kinetic energy of the nuclei, and the Coulomb interaction between electron and
nucleus, electron and electron, and between nucleus and nucleus. In some sense, a complete
theory of solids would be to solve the Schrodinger equation and then apply all the standard
methods of statistical physics to determine thermodynamic and physical properties. From
this point of view, there is no “fundamental” theory to be done, although the calculations
may indeed be complex (and in fact, impossible to perform exactly for solids with macroscopic
numbers of atoms). Because an exact solution for a macroscopic number of atoms is impossible,
we have to treat (9.1) by a sequence of approximations (for example, perhaps fixing the ions in
place, or neglecting electron-electron interactions) that will make the problem tractable.
This view of condensed matter physics as a series of approximations is widely held, but
severely incomplete. Suppose for a moment that we could solve the full Hamiltonian, and we
would then have a wavefunction describing some 10 23 particles that contained all of the physics
of solids. Writing the solution down would be hard enough, but comprehending its meaning
would be beyond us. Condensed matter physics is about phenomena, from the mundane (why
is glass transparent), to the exotic (why does 3 He become a superfluid). There are a host
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