Indirect gap of a separable square cosine potential

For $-V_0(\cos Qx+\cos Qy)$, two-dimensional bands are sums of one-dimensional bands. At first order, the first-band maximum is $E_Q/2-V_0$ at the corner, while the second-band minimum is $E_Q/4+V_0/2$ at an edge centre. Hence ${\Delta }_{\mathrm{ind}}\simeq 3V_0/2-E_Q/4$, giving a rough onset $V_0\sim E_Q/6$. Higher plane waves shift this estimate; insulating filling must be checked separately.