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 $\mathrm{ e}^{ -\frac{ i}{ \hbar}\mathcal{ A}_x(t)}$ in the $\hat{ \mathbf{ x}}$-direction such that $\partial_t \mathcal{ A}_x(t)=F_x(t)$. 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 $\hat{ \mathbf{ y}}$-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 $\hat{ H}(t)$, we use Floquet techniques to study this problem. In this picture the (Kubo) linear response $F\to 0$ regime corresponds to the adiabatic limit for $\hat{ H}(t)$. 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 { \em non-analytic} exponential $\mathrm{ e}^{ -\gamma/|F|}$ 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.
Categories: cond-mat.quant-gas cond-mat.stat-mech quant-ph