Past exam of the quantum condensed matter course of the University of Cambridge / 2022 / Section B / 5 / c / Solution

Write the dispersive band as $E_p=A{k}^2$ with $A>0$ and define $k_c=\sqrt{E_{\mathrm{fl}}/A}$. The uncoupled sketch is an upward-opening parabola and a horizontal flat band at positive energy. They cross on the sphere $k=k_c$ in three dimensions, or at $\pm k_c$ in a one-dimensional cut:Without mixing the crossing is allowed. The left panel of the figure shows this case.

For dimensional clarity $A={\hslash }^2/(2m_{\ell })$ in the usual band effective mass convention. The printed product $m_p{\hslash }^2{k}^2$ is consistent only if $m_p$ is a curvature coefficient with inverse-mass units; it cannot simultaneously denote the effective mass itself. Using $A$ avoids relying on that notation.