Parafermion chain with $2\pi /k$ Floquet edge modes

We study parafermion chains with ${\mathbb{Z}}_k$ symmetry subject to a periodic binary drive, focusing on the case $k=3$. We find that the chains support different Floquet edge modes at nontrivial quasienergies, distinct from those for the static system. We map out the corresponding phase diagram by a combination of analytics and numerics, and provide the location of $2\pi /3$ modes in parameter space. We also show that the modes are robust to weak disorder. While the previously studied ${\mathbb{Z}}_2$-invariant Majorana systems possess a transparent weakly interacting limit where the existence of a $\pi$-Majorana mode is manifest, our intrinsically strongly interacting generalization demonstrates that the existence of such a limit is not necessary.

Figure 1

Phase diagram of the periodically driven Majorana chain, showing parameter regimes where edge modes of quasienergy $0,\pi$, or both, are present. Note that the parameters $J$ and $h$ are in units where the time period $T$ of the drive is 2.

Figure 2

Solutions of $z$ for $U(0,z)\propto Q^p$. Filled circles represent the case of $p=1,2$ (locations of $\frac{2\pi }{3}$ modes) and unfilled ones represent $p=0$ (locations of 0 modes). The full lattice of solutions arises from tiling the shaded region along the arrows [40].

Figure 3

Waterfall plots showing $S_1(E_n,\eta )$ as a function of $\eta$ for different “in-states” $E_n$ picked uniformly from the spectrum. $\eta$ corresponds to the change in quasienergy after scattering by ${\sigma }_1$. $S_1(E_n,\eta )$ was averaged over 50 disorder realizations in a system of size $L=7$. Top and bottom panels show results for parameter points $(J=ı,h=0.1)$ and $(J=ı,h=1.2ı)$, respectively. Sharp peaks of strength 1 at $\eta =0$ and $\frac{2\pi }{3}$ indicate a mode localized at the edge. Right panels show the complex $h$ plane in the parameter space defined by $J=ı$, showing the solutions of $z$ (empty circles) given in Eq. (15), and location of the parameters (blue cross) corresponding to the left panels.

Figure 4

Variation of the eigenstate order parameters ${\mathcal{O}}_0$ and ${\mathcal{O}}_{\frac{2\pi }{3}}$ defined in Eqs. (17) and (19) as $h$ is varied along different paths in complex parameter space depicted in the insets. Left and middle panels correspond to two paths along the real and imaginary axis of $h$ in the plane defined by $J=ı$. Right panel corresponds to $h$ varying along the real axis in the plane defined by $J=1$. Filled dots in the insets indicate solutions of $z$. Strong size dependence as well as rapid decay of the order parameters in the case of $J=1$ (right) as $\left|h\right|$ deviates from 0 is indicative of the absence of a well-localized zero mode for $arg\left(J\right)=0$.

Figure 5

Contour plots of ${\mathcal{O}}_0$ and ${\mathcal{O}}_{2\pi /3}$ in the two-dimensional cross-sectional surfaces of parameter space defined by $J=ı$ (left panels), $J=1$ (middle panels), and $\left[\right|J|=1,\mathrm{Re}(h)=0]$ (right panels). In all figures, red and blue shades represent $\mathcal{O}=1$ and 0, respectively. All results are for systems of size $L=6$. The domains supporting edge modes are the largest and smallest when $arg\left(J\right)=\pi /6(\text{mod}\pi /3)$ and $0(\text{mod}\pi /3)$, respectively. Figure 8 shows the relative locations of these planes in a three-dimensional context.

Figure 6

Contour plots of ${\mathcal{O}}_0$ and ${\mathcal{O}}_{2\pi /3}$ in the cross sections of parameter space defined by $J=ı/2$ (left panels), $\mathrm{Re}\left(h\right)=\mathrm{Re}\left(J\right)=0$ (middle panels), and $\left[\mathrm{Re}\right(h)=\pi /3,\mathrm{Re}(J)=0]$ (right panels). All results are for systems of size $L=6$. The figures show that the domains increase in size as $\left|J\right|$ increases from 0. For larger $\left|J\right|$, the domain size varies periodically.

Figure 7

The figures show the three-dimensional section defined by $arg\left(J\right)=\pi /2$. The planes labeled $a,b$, and $c$ show the two-dimensional sections in Fig. 5 (left), Fig. 6 (left), and Fig. 6 (right), respectively.

Figure 8

The three-dimensional cross section of parameter space defined by fixing $\left|J\right|=1$. The planes labeled $A,B$, and $C$ correspond to the two-dimensional cross sections shown in Fig. 5 left, middle, and right, respectively. Left and right panels here show ${\mathcal{O}}_0$ and ${\mathcal{O}}_{2\pi /3}$, respectively.

Figure 9

Contour plot showing disorder-averaged ${\mathcal{O}}_0$, for parameters along the line connecting $(J,h)=(ı,0)$ and $(J,h)=(ı,ı\pi /\sqrt{3})$.