FIG. 22.

Topological pumping with a 1D superlattice described by the potential (50). Left: The phase ϕ of the long lattice [see Eq. (50)] is varied from 0 to 2π from top to bottom. Initially the phase ϕ = 0 and the particle is supposed to be localized on the site A_j of a given lattice cell j. In the limiting case where the energy difference between A and B sites is large compared to the tunnel matrix elements, this state would be stationary if ϕ was kept at the value 0. When ϕ is increased up to π/2, the sites A_j and B_j have the same energy and the particle is adiabatically transferred to B_j. Note that we neglect here the tunneling of the particle from A_j to B_j−1, assuming that it is inhibited by the large barrier between these two sites. The particle then remains in B_j until the phase reaches the value 3π/2, when the particle again undergoes an adiabatic transfer, now from B_j to A_j+1. (Here again we neglect tunneling across the large barrier now present between B_j and A_j.) When the phase ϕ = 2π the potential is back to its initial value and the particle has moved by one lattice site. Note that a motion in the opposite direction occurs if the particle starts for the site B_j when ϕ = 0. Right: In the two-band approximation corresponding to the Rice-Mele model, the system performs a closed loop around the origin in the parameter space (J′ − J, Δ).