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

TL;DR

This paper investigates how the quantization of Hall conductivity in the Harper-Hofstadter model is affected by different protocols for turning on a driving force, revealing conditions under which topological robustness is maintained or broken.

Contribution

It analyzes the impact of switching protocols on the quantization of Hall conductivity using Floquet techniques, highlighting the crossover behavior and conditions for topological robustness.

Findings

Sudden switching causes quadratic corrections to quantization.

Smooth switching introduces a crossover force F* with exponential decay behavior.

Topological robustness is recovered for slow or smooth switching protocols.

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 $\mathrm{e}^{-\frac{i}{\hbar}\mathcal{A}_x(t)}$ 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.