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 Fx(t) is turned on. In the vector potential gauge, through Peierls substitution, this involves the switching on of complex time-dependent hopping amplitudes e-iAx(t) in the ? direction such that ?tAx(t)=Fx(t). The switching on can be sudden, Fx(t)=?(t)F, where F is the steady driving force, or more generally smooth Fx(t)=f(t/t0)F, where f(t/t0) is such that f(0)=0 and f(1)=1. We investigate how the time-averaged (steady-state) particle current density jy in the ? direction deviates from the quantized value jyh/F=n due to the finite value of F and the details of the switching-on protocol. Exploiting the time periodicity of the Hamiltonian ?(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 ?(t). In the case of a sudden quench jyh/F shows F2 corrections to the perfectly quantized limit. When the switching on is smooth, the result depends on the switch-on time t0: For a fixed t0 we observe a crossover force F* between a quadratic regime for FF*. The crossover F* decreases as t0 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.