If you zoom in on a piece of metal, you’ll see that at small scales it’s a crystal—a triply periodic arrangement of atoms. Being triply periodic means having a symmetry group generated by three translations: \(T_1, T_2, T_3\).
We can start learning how metals conduct electricity by studying how a single electron moves through a crystal. In quantum mechanics, every motion is described by a differential equation \(\psi' = H\psi\), where \(H\) is a linear map. Symmetries, like the translations of our crystal, appear as linear maps that commute with \(H\). Since translations also commute with each other, we can simultaneously diagonalize \(H\) and \(T_1, T_2, T_3\).
The basic rules of quantum mechanics guarantee us eigenvalues of the form \[\begin{align*} H\psi & = E \psi \\ T_1\psi & = e^{ip_1} \psi \\ T_2\psi & = e^{ip_2} \psi \\ T_3\psi & = e^{ip_3} \psi \end{align*}\]
The real numbers \(E, p_1, p_2, p_3\) are the energy and momentum of the electron in the state of motion described by \(\psi\). Their allowed values sweep out a region in energy-momentum space, called the electron’s dispersion relation.
In physically sensible models, you can expect the dispersion relation to be a smooth submanifold of energy-momentum space (maybe with a few singularities). In fact, it's typically a branched cover of momentum space! The sheets of the cover are called energy bands.
Since the eigenvalues that define \(p_1, p_2, p_3\) lie on the unit circle \(\mathbb{T}\), momentum space is the 3-dimensional torus \(\mathbb{T}^3\). In this talk, all the triply periodic solids we get from physics will come from momentum space.
In a 2-dimensional crystal, an electron’s momentum only has two components, so we can visualize the energy-momentum relationship as a surface in (2+1)-dimensional space. Here are some representative examples.