Zone-face two-plane-wave gap for a cosine potential

For $-V_0\cos Qx$, the Fourier coupling between the two free states at $k=\pm Q/2$ is $-V_0/2$. The split energies are $E_Q/4\mp V_0/2$, with $E_Q={\hslash }^2{Q}^2/(2m)$. The direct face gap is $V_0$, not $2V_0$.

For the separable square-lattice potential $V=-V_0(\cos Qx+\cos Qy)$, four free plane waves with $(k_x,k_y)=(\pm Q/2,\pm Q/2)$ meet at a Brillouin zone corner. Their common unperturbed energy is $E_Q/2$, where $E_Q={\hslash }^2{Q}^2/(2m)$.

Each cosine couples the opposite signs of its own momentum component with matrix element $-V_0/2$. In the four-state space the perturbation is $-(V_0/2)({\sigma }_x\otimes I+I\otimes {\sigma }_x)$, with eigenvalues $-V_0,0,0,V_0$. Thus the corner energies at leading order are $E_Q/2-V_0$, twice-degenerate $E_Q/2$, and $E_Q/2+V_0$. Additional Fourier components or coupling to remote waves can change this result. A local corner splitting alone does not establish a global insulating gap.