A Hamiltonian $H$ is invariant under the discrete translational symmetry of a Bravais lattice $\Lambda$. This means that there exists a unitary translation operator $T_{\mathbf{r}}$ such that $\left[H, T_{\mathbf{r}}\right]=0$ for all $\mathbf{r} \in \Lambda$. State and prove Bloch's theorem for $H$.

Consider the two-dimensional Bravais lattice $\Lambda$ defined by the basis vectors

$$\mathbf{a}_{1}=\frac{a}{2}(\sqrt{3}, 1), \quad \mathbf{a}_{2}=\frac{a}{2}(\sqrt{3},-1)$$

Find basis vectors $\mathbf{b}_{1}$ and $\mathbf{b}_{2}$ for the reciprocal lattice. Sketch the Brillouin zone. Explain why the Brillouin zone has only two physically distinct corners. Show that the positions of these corners may be taken to be $\mathbf{K}=\frac{1}{3}\left(2 \mathbf{b}_{1}+\mathbf{b}_{2}\right)$ and $\mathbf{K}^{\prime}=\frac{1}{3}\left(\mathbf{b}_{1}+2 \mathbf{b}_{2}\right)$.

The dynamics of a single electron moving on the lattice $\Lambda$ is described by a tightbinding model with Hamiltonian

$$H=\sum_{\mathbf{r} \in \Lambda}\left[E_{0}|\mathbf{r}\rangle\langle\mathbf{r}|-\lambda\left(|\mathbf{r}\rangle\left\langle\mathbf{r}+\mathbf{a}_{1}|+| \mathbf{r}\right\rangle\left\langle\mathbf{r}+\mathbf{a}_{2}|+| \mathbf{r}+\mathbf{a}_{1}\right\rangle\left\langle\mathbf{r}|+| \mathbf{r}+\mathbf{a}_{2}\right\rangle\langle\mathbf{r}|\right)\right]$$

where $E_{0}$ and $\lambda$ are real parameters. What is the energy spectrum as a function of the wave vector $\mathbf{k}$ in the Brillouin zone? How does the energy vary along the boundary of the Brillouin zone between $\mathbf{K}$ and $\mathbf{K}^{\prime}$ ? What is the width of the band?

In a real material, each site of the lattice $\Lambda$ contains an atom with a certain valency. Explain how the conducting properties of the material depend on the valency.

Suppose now that there is a second band, with minimum $E=E_{0}+\Delta$. For what values of $\Delta$ and the valency is the material an insulator?