Fig. 1: The Aharonov–Bohm cages.
a A chain of Aharonov–Bohm cages [cf. Eq. (1)], with three sites an, bn, cn in the nth unit cell and a flux ϕ threading each plaquette. b The energy dispersion E(k) of the chain as a function of the flux ϕ. c The energy dispersion E(k) at ϕ = π consists of three flat band at energies 0 and ±2t. The band at E = 0 has a quantized Zak's phase γ = π while the other two bands show a non-quantized winding of π∕2. At a termination of the chain with site an, two in-gap boundary states appear at \(E=\pm \sqrt{2}t\). d Squaring the Hamiltonian (1) yields a model (2) with one flat band at E = 0 and two degenerate flat bands at E = 4t2. Both bands have a quantized Wilzcek–Zee phase ∣γ∣ = π.
