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 $F_x\left(t\right)$ is turned on. In the vector potential gauge, through Peierls substitution, this involves the switching on of complex time-dependent hopping amplitudes $e^{-\frac{i}{\hslash }{\mathcal{A}}_x\left(t\right)}$ in the $\widehat{\mathbf{x}}$ direction such that ${\partial }_t{\mathcal{A}}_x\left(t\right)=F_x\left(t\right)$. The switching on can be sudden, $F_x\left(t\right)=\theta \left(t\right)F$, where $F$ is the steady driving force, or more generally smooth $F_x\left(t\right)=f(t/t_0)F$, where $f(t/t_0)$ is such that $f\left(0\right)=0$ and $f\left(1\right)=1$. We investigate how the time-averaged (steady-state) particle current density $j_y$ in the $\widehat{\mathbf{y}}$ 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 $\widehat{H}\left(t\right)$, 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 $\widehat{H}\left(t\right)$. 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 $e^{-\gamma /\left|F\right|}$ 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.

Figure 1

Possible schedules for the switching on of the uniform driving force $F_x\left(t\right)$. The sudden quench case $F_x\left(t\right)=\theta \left(t\right)F$, where $\theta \left(t\right)$ is the Heaviside step function, is recovered for $t_0\to 0$.

Figure 2

Top: Energy bands of the Harper-Hofstadter model for $\alpha =1/3$ vs $k_y$ for $N_x=30$. The dashed red lines represent the energy averaged over the phase $k_x+{\kappa }_x\left(t\right)$. Bottom: Charge pumped in the first period in each magnetic cell, $3a{Q}_1$, as a function of the driving field $aF/J_0$, where $\hslash \omega =aF$, after a sudden switch on of the driving. For $F\to 0$, $3a{Q}_1$ is quantized to the first Chern number, respectively +3, $-6$, +3, of the band in which the system is initially prepared. (The simulation has been repeated by preparing the initial Slater determinant insulating state to be one of the three completely filled bands, in order to compute the Chern numbers.) The first and the third band give exactly the same response. This figure is essentially equivalent to Fig. 1 of Ref. [7], where the abscissa is $1/\omega$.

Figure 3

Adiabatic quasienergies ${\epsilon }_{k_y,\nu }^{\left(0\right)}$ in units of $\hslash \omega$ for the three bands of the Harper-Hofsdtadter model. The frequency is $\hslash \omega =0.1J_0$. ${\epsilon }_{k_y,\nu }^{\left(0\right)}$ is the sum of the $k_x$-averaged band, dashed line in Fig. 2 (upper panel), plus the geometric contribution, giving rise to the loss of periodicity for ${\epsilon }_{k_y,\nu }^{\left(0\right)}$ in the BZ: ${\epsilon }_{\frac{2\pi }{a},\nu }^{\left(0\right)}={\epsilon }_{0,\nu }^{\left(0\right)}+\hslash \omega C_{\nu }$, where $C_{\nu }$ is the Chern number of the $\nu \mathrm{th}$ band.

Figure 4

(a) Floquet spectrum as a function of $k_y$, for $\hslash \omega /J_0=0.1$. The squares signal the Floquet resonance avoided crossings, the circle an ordinary avoided crossing. Both are magnified in the top insets, where the size of the points is proportional to the $k_x$-averaged occupation $n_{k_y,\nu }$, see Eq. (34). (b) Floquet adiabatic quasienergies, Eq. (29), folded in the Floquet BZ. (c) $({\epsilon }_{k_y,2}^{\left(0\right)}-{\epsilon }_{k_y,1}^{\left(0\right)})/\hslash \omega$, the energy difference between the two lowest adiabatic bands in the extended-zone scheme, for $\hslash \omega =0.1J_0$. The vertical lines indicate the Floquet resonances, ${\epsilon }_{k_y,2}^{\left(0\right)}-{\epsilon }_{k_y,1}^{\left(0\right)}=m\hslash \omega$ with $m=29,26,23,20$, giving rise to the avoided crossing gaps of panel (a).

Figure 5

The deviation of $3a{Q}_{\nu }^{\mathrm{F}}$, Eq. (28), from integer quantization, $3-3a{Q}_{\nu }^{\mathrm{F}}$, for the lowest Floquet band, assuming $n_{\mathbf{k},{\nu }^{\prime }}={\delta }_{\nu ,{\nu }^{\prime }}$, showing that the exponentially small gaps in the quasienergy spectrum at finite $\omega$ lead to exponentially small deviations.

Figure 6

Correction to the $\mathbf{k}$-averaged adiabatic Floquet mode occupation $n_{\nu }$, Eq. (37), vs $1/\omega$, showing the good agreement between the numerical data and the perturbation theory prediction from Eq. (35).

Figure 7

Pumped charge vs $\omega$, both for one period (solid line) $3a{Q}_1$, Eq. (25), and in the diagonal ensemble (dashed line) $3a{Q}_{\mathrm{d}}$, Eq. (26). Notice the oscillations in $Q_1$, signaling an essential singularity in $\omega =0$. The inset shows the deviation from the quantized value $3a{Q}_1(\omega \to 0)=3$ vs $1/\omega$. For small frequency this deviation is quadratic in $\omega$.

Figure 8

Correction to the occupation of the lowest energy Floquet band for different switch-on times $t_0$. The inset highlights the crossover from an exponential regime $e^{-1/\omega }$ to the quadratic one ${\omega }^2$ for finite ramp time $t_0$. The solid lines correspond to the functions $0.05e^{-\frac{0.5J_0}{\hslash \omega }}$ and $0.002{(\hslash \omega /J_0)}^2$.

Figure 9

Floquet quasienergies in $\mathbf{k}=0$ as a function of the frequency. The width of the line is proportional to the occupation of the state when the system is in the ground state with filling factor $1/3$. The inset zooms on a level crossing to highlight the presence of gaps, showing also that the adiabatic band is the “excited” state after the avoided crossing. The solid black lines are the boundary of the first Floquet-Brillouin zone.

Figure 10

Correction to the occupation of the $\nu \mathrm{th}$ Floquet mode compared with the prediction of Eq. (43) (solid black lines).

Figure 11

Correction to the occupation number of the adiabatic Floquet state, when the driving force is smoothly turned on with the switching function $f(s=t/t_0)=\frac{1}{2}(1-cos\left(\pi s\right))$. Equation (43), with a slight modification due to the different driving schedule, still gives a good estimate of the quadratic regime for small $\omega$.