Tight-Binding Band Structure

Quantum electrons hopping between lattice sites. In 1D: E(k) = ε₀ − 2t·cos(ka). In 2D square lattice: E(kx,ky) = ε₀ − 2t·[cos(kx·a) + cos(ky·a)]. The bandwidth = 4t (1D) or 8t (2D). The Fermi surface separates filled from empty states.

Hopping t: 1.0 ε₀ (site energy): 0.0 Filling: 0.5

1D: bandwidth = 4t = 4.0 | half-filling → Fermi wavevector kF = π/2a

At half-filling the 2D Fermi surface is a perfect square (nesting) — enabling charge-density waves and superconducting instabilities. Adding a next-nearest-neighbor hopping t' warps the square: key for cuprate superconductors.