Fermi Surface 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 $(0, \pm \pi)$ .

Figure 1. Fermi surface and DOS for a 2D tight-binding square lattice.

filling f0.300

WHAT TO TRY

Slide the filling f: the Fermi surface grows from a small loop around the zone centre to the perfect diamond at half filling, where it touches the zone edges. The shape is a direct cut of the tight-binding dispersion.
Park the filling right at half: the Fermi surface is a nested square and the density of states spikes at a van Hove singularity, the logarithmic divergence visible as the tall peak in the DOS panel.
Read the DOS against the Fermi level marker: states pile up at the band edges and the van Hove energy, the structure that drives instabilities like magnetism and superconductivity in real square-lattice materials.