Fermi surface on a 2D square lattice

What you are seeing: the Brillouin zone (kx, ky) in $[-\pi, \pi]^2$ for a tight-binding electron on a square lattice with dispersion $E(k_x, k_y) = -2t (\cos k_x + \cos k_y)$. Below half-filling the Fermi surface is a closed loop around the $\Gamma$ point; at half-filling it is the perfectly nested square diagonals (van Hove singularity in DOS); above half-filling it shifts to closed loops around the (pi, pi) corners.

Sliding the filling fraction $f$ from 0 to 1 sweeps through these three regimes. The right panel shows the density of states with the Fermi energy marked; the van Hove peak at $E = 0$ is the signature of the saddle points at $(\pm \pi, 0)$ and $(0, \pm \pi)$.