Download Lecture 3: Electronic Band Theory: A Many

Lecture 3: Electronic Band Theory: A Many-Body Warmup
Website with slides and handouts:
http://home.uchicago.edu/ matthewroberts/compton/index.html
No Compton lecture next week (Oct 22) due to the Kadanoff Memorial Symposium
The crystalline structure of solids
Solids are comprised of a lattice of atoms. The nuclei of the atoms bond via covalent (or
other) bonds. For certain atoms there are electrons not used in the covalent bond that can
be “delocalized”, that is, their wavefunction can spread over the entire lattice.
The “tight-binding” approximation
For many materials it is reasonable to assume the electron wavefunctions only have support
at the ions, vanishing away. This means the wave function lives on a lattice, not in the
continuum. In other words, our wave function looks like ψ(nx , ny ) instead of ψ(x, y). The
consequence of this is that momenta for extended electron wave functions live on a torus.
The density of states
If we add up the number of states for each energy (at different momentum) we get the
density of states, N (E). Depending on the number of electrons in our system, we can be
in a conducting or insulating state. The equilibrium state of the system is the one where
we put all our electrons into the lowest energy states possible. A band can be completely
empty, partially filled or completely filled.
Figure 1:
Left: the energy bands for a generic insulator. Right: the density of states for conductor
and insulators.
Topology
We can decompose the electronic
wave
function into the wave and prefactor parts, for inψ↑ (k)
stance with spin ψ(nx , ny ) =
exp[ikx nx + iky ky ]. Then the first factor, which with
ψ↓ (k)
shorthand I’ll call ~u(k), defines an angle everywhere in momentum space.1 If we look at
this angle over momentum space we find it can have vortices and antivortices, points about
which it rotates a full ±2π.
Figure 2:
Left: The winding of the angle θ about vortices in an electric band. Right: multiple bands
with different vortex numbers. The total vortex number, called the Chern number,
determines the Hall conductance.
A famous result by Thouless, Kohmoto, Nightingale (D. J. Thouless, M. Kohmoto,
M. P. Nightingale and M. den Nijs, “Quantized Hall Conductance in a Two-Dimensional
Periodic Potential,” Phys. Rev. Lett. 49 (1982) 405.) and den Nijs proves that the Hall
conductance of an insulator is always given by the signed sum of the vortices in all filled
bands,
e2
σxy =
Nfilled
2π~
where Nfilled is the number of vortices minus the number of antivortices in all filled band.
1
This is a bit of a cheat to make things more visual. What you actually do is define a vector on
∂
momentum space, Ai (k) = u† ∂k
u. I can then define a field strength in analogy with electromagnetism,
i
∂
∂
F = ∂kx Aky − ∂ky Akx , and the integral of this over the momentum torus must be quantized.