On the quantization of the Hall conductivity in the Harper-Hofstadter model

Abstract

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 ${\mathrm{e}}^{-\frac{i}{\hslash }{\mathcal{A}}_x(t)}$ 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 $F^{\ast }$ between a quadratic regime for $F<F^{\ast }$ and a {\em non-analytic} exponential ${\mathrm{e}}^{-\gamma /|F|}$ for $F>F^{\ast }$. The crossover $F^{\ast }$ 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.

arXiv comment13 pages, 11 figures

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Cited by in corpus (5)

• Localization, topology and quantized transport in disordered Floquet systems
• Topological Lifshitz Transitions, Orbital Currents, and Interactions in Low-dimensional Fermi Gases in Synthetic Gauge Fields
• Exploring helical phases of matter in bosonic ladders
• Dissipation assisted Thouless pumping in the Rice-Mele model
• Thouless pumping in Josephson junction arrays