Fig. 1: Topological pumping in the interacting Rice–Mele model.
From: Quantization and its breakdown in a Hubbard–Thouless pump
a, Schematic of the dynamical optical superlattice setup. The interfering lattice (yellow) is imbalanced along the x direction, leading to a movement of the ‘long’ lattice with respect to the ‘short’ lattice (non-interfering, red arrows) when ramping the phase φ of incoming light. The running-wave component is due to a rotated polarization of the retro-reflected laser beam (λ/4 waveplate). The standing wave in the y direction is not shown for clarity. Here d = λ = 1,064 nm is the size of one unit cell. b, Resulting lattice structure along x corresponds to the interacting Rice–Mele Hamiltonian (equation (1)). c, Sketch of the pumping trajectory in the parameter space spanned by site offset Δ and dimerization δ. Here Δ0 corresponds to half of the gap in the ionic Hubbard model (δ = 0). d, Measured in situ c.m. position of the fermionic cloud within five pumping cycles for U/Δ0 = {–3.1(2), 0, 3.1(2)} (red, grey and blue data points, respectively). Attractive and non-interacting atoms exhibit quantized pumping (black dashed line), whereas the movement of the repulsive cloud is strongly reduced. The data points and error bars correspond to the mean and standard error of eight individual measurements. e, Measured pumping efficiency (fitted slopes of b, averaged over the pumping direction) as a function of Hubbard U. Nearly quantized pumping efficiency persists for weakly interacting atoms (both attractive and repulsive) and strongly attractive interactions up to |U| = 3.1(2)Δ0 = 9.2(3)t. In the strongly repulsive regime, topological pumping breaks down. The error bars in the y direction correspond to the propagated error estimated from the uncertainty of the fit and those in the x direction, to the propagated error from lattice fluctuations. All the measurements in this figure were taken at a fixed period of T = 41.5(1.5)ℏ/t.
