Floquet topological phase in a generalized $\mathcal{PT}$-symmetric lattice

We consider a driven, non-Hermitian generalization of the Aubry-Andre-Harper model. We show that the introduction of periodic driving allows us to obtain fully real quasienergy spectra in configurations where the corresponding static system has complex energy eigenvalues for any non-Hermitian degree. We illustrate how generalized parity-time ($\mathcal{PT}$) symmetry can be applied within the Floquet formalism and show that our model's fully real quasienergy spectrum corresponds to an unbroken symmetry. This model exhibits a Floquet topological phase, few examples of have been found in non-Hermitian potentials.

Figure 1

Quasienergy spectra for $\beta =1/2, \lambda =0.4, \gamma =3, m_0=3, \omega =2\pi$, and (a) $N=49$, (b) $N=50$. (c) shows the time-averaged, first Floquet-Brillouin zone midgap ($\epsilon =0$) mode corresponding to the spectrum in (a) with $\mathrm{\Phi }=0$. This mode is localized to the first photon sector, and so only that sector is shown.

Figure 2

$\mathcal{A}$ symmetry-breaking threshold ${\gamma }_c$ as a function of gain site location $m_0$ and tunneling amplitude phase $\mathrm{\Phi }$. Here, $N=59, \omega =2\pi , \lambda =0.5$, and $\beta$ is a rational number of the form $q/p$. (a) has $\beta =1/3$ and (b) has $\beta =1/4$. In both cases, ${\gamma }_c$ is approximately periodic with period $p$, except for defect placement near the edges or the center of the lattice.

Figure 3

$\mathcal{A}$ symmetry-breaking threshold ${\gamma }_c$ for $\beta =1/7, N=48$, and $\lambda =0.5$ (a) has $\omega =\pi$, (b) has $\omega =2\pi$, and $\omega =10\pi$

Figure 4

Quasienergy spectrum for $\beta =\sqrt{2}, N=48, \omega =2\pi , \lambda =0.4, \gamma =3$, and $m_0=12$. As in the static case, lattices with irrational $\beta$ possess fractional numbers of (quasi)energy bands.

Figure 5

Time evolution of a Floquet state over the course of a period. (a) shows a time-averaged Floquet state. (b) shows the expectation value of position for the fully time-dependent Floquet state at each instant of time. The horizontal dashed line shows the value of $\langle x\rangle$ for $\psi (t=0)$. [And, since $\psi (t=T)=\psi (t=0), \langle x\rangle$ is the same at $t=0$ and $t=T$.] The parameters correspond to those of Fig. 3, with $\beta =1/7, N=48, \lambda =0.5, m_0=7, \mathrm{\Phi }=0, \gamma =3$, and $\omega =2\pi$ ($T=1$). This state corresponds to the lowest quasienergy in the zeroth photon sector ($\epsilon =-2.57$) and is localized to the two defect sites over the course of the period (though the magnitude of the state at each site changes over the course of a period).

Figure 6

(a) Quasienergy spectrum with next-nearest-neighbor tunneling [Eq. (22)] with $\beta =1/2, N=49, \omega =2\pi , \lambda =0.4, \gamma =1, m_0=1$, and $t^{\prime }=.2$. (b) Time-averaged, first Floquet-Brillouin zone $\epsilon =0$ quasienergy state for the spectrum in (a) with $\mathrm{\Phi }=0$. Again, this mode is localized to the first Floquet-Brillouin zone, and so only this region is shown.