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 — the same routine and sign convention as the 3D version, numerically re-verified for this page: with w > v the four states nearest E = 0 collapse the gap and pile onto the four corners (≈70% of their weight in a 6×6 flake), while with v > w the gap stays large (≈1) and corner weight drops below 3%. Unlike an ordinary edge-state topological insulator, this model's edges stay gapped; only the corners host protected zero modes, a genuine higher-order (0-dimensional boundary) effect protected by the lattice's C₄ and mirror symmetries rather than 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).
- Dot size / glow — summed probability density of the 4 eigenstates nearest E = 0, so corners light up only in the topological phase.
- Bonds — line colour marks the hopping sign (teal = +, amber = −); thickness marks |amplitude|, showing the SSH-like dimerization directly.
- Spectrum strip (bottom panel) — every eigenvalue of the flake plotted on the energy axis; the four states nearest zero are highlighted in accent colour so you can watch the gap close as w/v crosses 1.
- Drag / wheel on the lattice — pan and zoom the top-down view.
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.
Why corners? With w > v the lattice is in the quadrupole (obstructed-atomic-limit) phase: a bulk quadrupole moment qxy = 1/2 forces fractional corner charge, and with chiral symmetry the four corner modes sit at E = 0. For w < v the cells are simply separate molecules (E = ±√2 v at w = 0) and no state enters the gap. At w = v the bulk gap closes.
Honest simplifications: a finite flake splits the four corner modes by roughly (v/w)L, so they sit near but not exactly at zero; the model is spinless with real hoppings and no disorder, interactions or loss; the flux pattern is a gauge-fixed π-flux per plaquette. Real realisations (microwave, acoustic and electric-circuit arrays) add on-site terms and losses.