This is the Benalcazar–Bernevig–Hughes (BBH) quadrupole insulator: a finite square flake of L×L unit cells, 4 sites each, with real nearest-neighbour hoppings whose signs thread a π-flux through every plaquette:
H = Σ v·(sign) c†_i c_j (intracell bonds)
+ Σ w·(sign) c†_i c_j (intercell bonds)
signs: (1↔2)=+, (3↔4)=−, (1↔3)=+, (2↔4)=− per cell,
matching signs for the bonds crossing to the next cell
The Hamiltonian is diagonalized exactly (Jacobi eigenvalue algorithm) on every parameter change. In the bulk this model behaves like two decoupled SSH chains stacked in x and y — but unlike an ordinary edge-state topological insulator, its edges stay gapped. Only when the intercell hopping dominates (w > v, the "dimerized" topological regime) do four states pin close to E = 0 in the middle of the bulk gap, and their wavefunctions localize almost entirely on the four corners of the flake — a genuine higher-order (0-dimensional boundary) topological effect, protected by the lattice's C₄ and mirror symmetries rather than by particle-hole or time-reversal symmetry.
- v, w sliders — intracell vs intercell hopping strength; the ratio w/v drives the topological transition.
- Flake size — number of unit cells per edge (finite open boundaries are required for corner states to exist at all).
- Pillar height / colour — summed probability density of the 4 eigenstates nearest E = 0, so corners light up and rise only in the topological phase.
- Bonds — cylinder colour marks the hopping sign (teal = +, amber = −); thickness marks |amplitude|, showing the SSH-like dimerization directly.
Real-world relevance: this exact lattice has been realized in microwave and mechanical metamaterial arrays and in electrical LC-circuit networks to demonstrate quantized corner charge — the solid-state analogue of higher-order topological insulators now studied in bismuth and other real crystals.