Quantization of the Hall conductivity in the Harper-Hofstadter model

We study the robustness of the quantization of the Hall conductivity in the Harper-Hofstadter model towards the details of the protocol with which a longitudinal uniform driving force F-x(t) is turned on. In the vector potential gauge, through Peierls substitution, this involves the switching on of complex time-dependent hopping amplitudes e(-i/(h) over bar Ax(t)) in the (x) over cap direction such that partial derivative(t)A(x)(t) = F-x(t). The switching on can be sudden, F-x(t) = theta(t)F, where F is the steady driving force, or more generally smooth F-x(t) = f (t/t(0))F, where f (t/t(0)) is such that f(0) = 0 and f(1) = 1. We investigate how the time-averaged (steady-state) particle current density j(y) in the (y) over cap direction deviates from the quantized value j(y)h/F = n due to the finite value of F and the details of the switching-on protocol. Exploiting the time periodicity of the Hamiltonian (H) over cap (t), we use Floquet techniques to study this problem. In this picture the (Kubo) linear response F -> 0 regime corresponds to the adiabatic limit for (H) over cap (t). In the case of a sudden quench j(y)h/F shows F-2 corrections to the perfectly quantized limit. When the switching on is smooth, the result depends on the switch-on time t(0): For a fixed t(0) we observe a crossover force F* between a quadratic regime for F < F* and a nonanalytic exponential e(-gamma/vertical bar F vertical bar) for F > F*. The crossover F* decreases as t(0) increases, eventually recovering the topological robustness. These effects are in principle amenable to experimental tests in optical lattice cold atomic systems with synthetic gauge fields.