Tight-Binding Band Structure

The simplest model of a solid: one orbital per site, an on-site energy, and a hopping amplitude $t$ to neighbours. Bloch's theorem turns it into a band $E(k)=\varepsilon_0-2t\cos ka$ for the 1D chain (width $4t$ ), the curvature at the bottom fixing the effective mass $m^*=\hbar^2/2ta^2$ . Dimerize the chain (alternating $t_1,t_2$ ) and a gap $2|t_1-t_2|$ opens at the zone boundary, the SSH insulator. On the 2D square lattice $E=-2t(\cos k_xa+\cos k_ya)$ has a van Hove saddle and a Fermi surface that you can fill by dragging $E_F$ .

Figure 1. Tight-binding dispersion with the filled states and density of states (1D / SSH), or the 2D band and Fermi surface. Method: closed-form Bloch dispersion; SSH 2x2 Hamiltonian; 2D Fermi-surface contouring.

lattice

hopping t1.00

dimerization0.50

Fermi level E_F0.00

WHAT TO TRY

Watch the Bloch state sweep along the real-space chain: near k = 0 every orbital shares one sign (the bonding state at the band bottom), and near k = pi the signs alternate site to site (the antibonding state at the top), with the marker on E(k) = e0 - 2t cos(ka) tracking which state you are seeing.
Raise the hopping t: the band widens to 4t and its curvature at the bottom sharpens, lowering the effective mass; switch to the dimerized SSH chain and the alternating t1, t2 bonds split the band in two, opening the 2|t1-t2| gap that is the one-line topological-insulator toy model.
Read the DOS beside the band on the shared energy axis: it spikes at the band edges where the dispersion flattens (the 1D van Hove singularities), and the Fermi level marker sets the filling.