Momentum matrix-element expansion of band energy

For a real scalar lattice-periodic potential, let $|n⟩$ be a nondegenerate zone-centre periodic eigenstate with energy $E_n$. Conjugation by $e^{i\mathbf{k}\cdot \mathbf{r}}$ gives $H(\mathbf{k})=H(0)+\hslash \mathbf{k}\cdot \mathbf{p}/m+{\hslash }^2{k}^2/(2m)$. The zone-centre state can be chosen real, so $⟨n|\mathbf{p}|n⟩=0$. Second-order nondegenerate perturbation theory then givesTime-reversal symmetry makes this isolated spinless band even in $\mathbf{k}$. In a direction $\mathbf{e}$, the curvature givesA degeneracy requires a matrix treatment before extracting curvatures; a nonlocal potential or spin-orbit terms can also alter the simple momentum vertex.