= Four-plane-wave splitting at a square-zone corner
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 $E_Q=\hbar^2Q^2/(2m)$.
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.
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