Past exam of the condensed matter physics major option course of the University of Oxford / 2015 / 6 / a / Solution

Choose orthonormal localized orbitals $|\mathbf{R}⟩$. The on-site energy is $E_0=⟨\mathbf{R}|H|\mathbf{R}⟩$, and the sign convention in the question defines $t(\mathbf{T})=-⟨\mathbf{R}|H|\mathbf{R}+\mathbf{T}⟩$. Bloch state diagonalizes this translationally invariant hopping problem. For real, inversion-symmetric nearest-neighbor amplitudes, $t(\pm a\hat{x})=t_x$ and $t(\pm b\hat{y})=t_y$, so the exponential pairs yieldThe first zone is the rectangle $|k_x|\le \pi /a$, $|k_y|\le \pi /b$, wider in $k_x$ because $a<b$. Use $\Gamma =(0,0)$, $X=(\pi /a,0)$, $Y=(0,\pi /b)$ and $M=(\pi /a,\pi /b)$.

For positive $t_y=t$ and $t_x=2t$, the four corner energies and the requested path areAlong a path parametrized in reduced coordinates $x=k_{x}a$, $y=k_{y}b$, the full curve is simply $E-E_0=-4t\cos x-2t\cos y$. In particular the two appearances of $\Gamma$ and the two diagonal traversals must have consistent endpoint energies.^[ref] A reversed hopping sign changes which point is the band minimum; the usual positive-hopping convention is stated explicitly here.