Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 34E / b

Translation preserves the alternating on-site term only when $(-1{)}^{j+\ell }=(-1{)}^j$, soThe primitive lattice spacing is therefore $2a$.

Using one even and one odd site per unit cell, Bloch theorem reduces the Hamiltonian toIts eigenvalues are

For lattice spacing $2a$, the first Brillouin zone isAt its boundaries $E_{\pm }=\pm t$, while at $k=0$ the energies are $E_{\pm }=\pm \sqrt{5},t$. Thus the lower band runs from $-\sqrt{5},t$ to $-t$, the upper from $t$ to $\sqrt{5},t$, and each has bandwidthThe requested plot consists of these two even branches, separated by a gap of size $2t$ at the zone boundaries.