Minimal two-band model of a nodal-line semimetal (Weng-type):
H(k) = (kx² + ky² + kz² − k0²) σx + m·kz σy
Eigenvalues: E±(k) = ± sqrt{ (kx²+ky²+kz²−k0²)² + (m·kz)² }
The gap E+ vanishes only where both terms are zero at once: kz = 0 and kx² + ky² = k0². That is a whole ring of degeneracies in the 3D Brillouin zone, not an isolated point — the orange torus you see. Away from the ring, the crater-shaped surface is the bulk energy gap |E+|; it opens on both sides (inward and outward), which is why the slice at kz=0 looks like a circular moat.
Cut the crystal into a slab: on the kz=0 surface Brillouin zone, every k-point inside the ring (kx²+ky²<k0²) hosts a topologically protected, nearly dispersionless "drumhead" surface state pinned near zero energy — the translucent disk floating inside the moat, like a drum skin stretched across the ring. The ε slider adds a small surface-hybridization term that domes the drumhead away from perfect flatness, as real finite-thickness slabs do.
Near any point on the ring, writing u = |k∥| − k0 and v = kz, the Hamiltonian linearizes to H ≈ 2k0·u·σx + m·v·σy — an ordinary 2D Dirac Hamiltonian in the (u,v) plane, shown as the small cone. A loop that encircles that local Dirac point picks up a Berry phase of exactly π, computed here as a discretized Wilson loop of the lower-band eigenvectors. That π is quantized and topologically protected: it stays pinned at 180° no matter how you move the k0 or m sliders, which is exactly what guarantees the drumhead band cannot be removed by smooth deformations — only by closing the ring itself.