Fig. 2: Impurity states.
Analytical energy curves (green and yellow curves) obtained from diagonalization of the effective two-state system in (35), as a function of the magnetic flux ϕ for (a) opposite impurities ϵB = − ϵC = ϵ, (b) equal impurities ϵB = ϵC = ϵ, and (c) a single impurity, ϵB = ϵ and ϵC = 0, with ϵ = 0.1t. Blue and red dots correspond to the numerically obtained energies of the two states of the FB that are affected by the impurity, for an open diamond chain with 53 plaquettes pierced by a ϕ flux, with the impurities placed at the B and C sites of the midchain plaquette in (a, b), and at its B site in (c). d–f Probability amplitudes of the eigenstate marked by a blue circle in the corresponding cases above. Only the central part of the chain is shown. Golden bars (red dots) correspond to numerical (analytical) results. The left insets plot the same results in a semi-logarithmic scale and for a smaller portion of the chain. Sites without red dots are A sites where the CLSs forming the analytical subspace have zero weight. The right insets depict the impurities at the midchain plaquette, with green (purple) color indicating an ϵ ( − ϵ) value for the impurity at that site.
