Robustness of symmetry-protected topological states against time-periodic perturbations

The existence of gapless boundary states is a key attribute of any topological insulator. Topological band theory predicts that these states are robust against static perturbations that preserve the relevant symmetries. In this article, using Floquet theory, we examine how chiral symmetry protection extends also to states subject to time-periodic perturbations—in one-dimensional Floquet topological insulators as well as in ordinary one-dimensional time-independent topological insulators. It is found that, in the case of the latter, the edge modes are resistant to a much larger class of time-periodic symmetry-preserving perturbations than in Floquet topological insulators. Notably, boundary states in chiral time-independent topological insulators also exhibit an unexpected resilience against a certain type of symmetry-breaking time-periodic perturbations. We argue that this is a generic property for topological phases protected by chiral symmetry. Implications for experiments are discussed.

Figure 1

A schematic illustration of the harmonically driven SSH chain, with hopping amplitudes ${\gamma }_{1/2}\pm v\left(t\right)$, where $v\left(t\right)\sim cos\left(\mathrm{\Omega }t\right)$.

Figure 2

The topological invariants ${\nu }_0$ and ${\nu }_{\pi }$ calculated for $V_{\text{ac}}=0.2\mathrm{\Omega }$ and different values of ${\gamma }_1$ and ${\gamma }_2$. ${\nu }_{\pi /0}=1\left(0\right)$ corresponds to a topologically nontrivial (trivial) winding number.

Figure 3

Quasienergy spectra of harmonically driven SSH chains subject to time-periodic boundary perturbations, here realized as a spatial disordering of the amplitudes for hopping $(\sim {\gamma }_1,{\gamma }_2)$ or of an added staggered chemical potential $\left(\sim \mathrm{\Delta }\right)$. The chains have 80 sites, with unperturbed ${\gamma }_1=0.15\mathrm{\Omega },V_{\text{ac}}=0.2\mathrm{\Omega }$, and with ${\gamma }_2$ swapped from 0 to $1.2\mathrm{\Omega }$. The disorder is added over 20 sites from one of the boundaries, with the disorder amplitudes varying randomly within the interval $[-0.2\mathrm{\Omega },0.2\mathrm{\Omega }]$. Here we display levels for bulk states (black), perturbed (red), and unperturbed (green) edge states corresponding to single disorder realizations. For each type of perturbation 100 different disorder realizations were considered, with the midgap quasienergies always to be found confined to the corresponding pink regions.

Figure 4

Quasienergy spectra of time-independent SSH chains $\left[v\right(t)=0$ in Eq. (1)] under influence of various time-periodic boundary perturbations. These perturbations were implemented as a spatial disordering of the hopping amplitudes $\left(\sim {\gamma }_1,{\gamma }_2\right)$ or of an added staggered chemical potential $\left(\sim \mathrm{\Delta }\right)$. The chains were taken with ${\gamma }_1=0.15\mathrm{\Omega }$ and ${\gamma }_2$ varying between 0 and $1.2\mathrm{\Omega }$. Each chain consists of 80 sites and is disordered over the first 20 sites from one of the boundaries, with the corresponding disorder amplitudes chosen randomly within the interval $[-0.2\mathrm{\Omega },0.2\mathrm{\Omega }]$. Here we illustrate quasienergies for bulk states in black and edge states in red, respectively (the quasienergies of the perturbed and unperturbed edge modes perfectly match in this case).

Figure 5

Scaling of maximum zero-quasienergy shifts with maximum disorder amplitude for time-independent (panels with subindex 1) and harmonically driven (panels with subindex 2) SSH chains with ${\gamma }_1=0.15\mathrm{\Omega }$ and ${\gamma }_2=0.5\mathrm{\Omega }$. The chains consist of 80 sites and in the driven case the harmonic modulation was fixed to $v\left(t\right)=0.4cos\left(\mathrm{\Omega }t\right)$. The added time-periodic boundary perturbation, extending over 20 sites from one of the edges, was implemented as a spatial disordering of two independent parameters: (a) ${\gamma }_{1,j}cos\left(\mathrm{\Omega }t\right)+{\mathrm{\Delta }}_{j}cos\left(2\mathrm{\Omega }t\right)$, (b) ${\gamma }_{2,j}cos\left(\mathrm{\Omega }t\right)+{\gamma }_{1,j}sin\left(2\mathrm{\Omega }t\right)$, (c) ${\mathrm{\Delta }}_{j}sin\left(\mathrm{\Omega }t\right)+{\gamma }_{2,j}sin\left(2\mathrm{\Omega }t\right)$ with $j=1,2,...,10$. In red we plot the largest zero-quasienergy shift $\varepsilon \left[\mathrm{\Omega }\right]$ maximized over 100 disorder realizations versus the largest allowed disorder site amplitude, denoted by $V\left[\mathrm{\Omega }\right]$. The blue curve represents smoothed data obtained by replacing every 20 points by their average.

Figure 6

Quasienergy spectra obtained from disordering the first 20 sites of 80-site SSH chains with unperturbed ${\gamma }_1=0.15\mathrm{\Omega }$ and ${\gamma }_2$ swapped from 0 to $1.2\mathrm{\Omega }$. The levels for bulk states, perturbed and unperturbed edge states are displayed in black, red, and green, respectively. The panels having subindex 1 or 2 correspond to the time-independent or harmonically driven [with $v\left(t\right)=0.4cos\left(\mathrm{\Omega }t\right)]$ cases, respectively. The disorder is here added as a time-periodic spatial disordering of the amplitudes for hopping $\left(\sim {\gamma }_1,{\gamma }_2\right)$ and/or a staggered chemical potential $\left(\sim \mathrm{\Delta }\right)$: (a) ${\gamma }_{1,j}cos\left(\mathrm{\Omega }t\right)+{\gamma }_{2,j}sin\left(2\mathrm{\Omega }t\right)$, (b) ${\gamma }_{2,j}cos\left(\mathrm{\Omega }t\right)+{\gamma }_{1,j}sin\left(2\mathrm{\Omega }t\right)$, (c) ${\mathrm{\Delta }}_{j}sin\left(\mathrm{\Omega }t\right)+{\gamma }_{1,j}sin\left(2\mathrm{\Omega }t\right)$, (d) ${\gamma }_{1,j}cos\left(\mathrm{\Omega }t\right)+{\mathrm{\Delta }}_{j}cos\left(2\mathrm{\Omega }t\right)$, (e) ${\gamma }_{2,j}cos\left(\mathrm{\Omega }t\right)+{\mathrm{\Delta }}_{j}cos\left(2\mathrm{\Omega }t\right)$, (f) ${\mathrm{\Delta }}_{j}sin\left(\mathrm{\Omega }t\right)+{\gamma }_{2,j}sin\left(2\mathrm{\Omega }t\right)$ with $j=1,...,10$. The maximum disorder strength was taken to be $0.5\mathrm{\Omega }$ in each of the disordering parameters. The pink areas define the regions to which the shifted midgap quasienergies were found to be confined after collecting data from 100 distinct disorder realizations.

Figure 7

Numerical data obtained from time-independent SSH chains $\left[v\right(t)=0$ in Eq. (1)] with 80 sites subject to static and time-periodic boundary perturbations, here realized as a spatial disordering of an added chemical-potential term $\left(\sim \mu \right)$ and extending over 20 sites from one of the boundaries. Panels (a)–(d) illustrate quasienergy spectra for the chains with unperturbed hopping amplitudes ${\gamma }_1=0.15\mathrm{\Omega }$ and ${\gamma }_2$ varying from 0 to $1.2\mathrm{\Omega }$. The boundary perturbations were considered to be of the form (a) ${\mu }_{\sigma ,j}$ (static disorder), (b) ${\mu }_{\sigma ,j}sin\left(\mathrm{\Omega }t\right)$, (c) ${\mu }_{\sigma ,j}cos\left(\mathrm{\Omega }t\right)$, and (d) ${\mu }_{\sigma ,j}[sin\left(\mathrm{\Omega }t\right)+cos\left(\mathrm{\Omega }t\right)]$, with $\sigma =A,B$ and $j=1,...,10$. The disordered site amplitudes ${\mu }_{\sigma ,j}$ vary randomly in the interval $[-0.1\mathrm{\Omega },0.1\mathrm{\Omega }]$ $\left([-1.0\mathrm{\Omega },1.0\mathrm{\Omega }]\right)$ in the static (time-dependent) cases. The levels for the bulk states and perturbed and unperturbed edge states are colored black, red, and green, respectively. 100 disordering configurations were checked, with the midgap quasienergies always found to be within the corresponding pink areas. In panel (e) we present the scaling of the midgap quasienergy shifts in the time-independent SSH chains $({\gamma }_1=0.15\mathrm{\Omega }$ and ${\gamma }_2=0.5\mathrm{\Omega })$ perturbed by time-periodic boundary disorder in chemical potential $\sim \mu [sin(\mathrm{\Omega }t)+cos(\mathrm{\Omega }t\left)\right]$. Here the largest midgap quasienergy shift $\varepsilon \left[\mathrm{\Omega }\right]$ maximized over 500 disorder realizations is plotted in red versus the upper limit of the on-site perturbation amplitude in the chemical potential, denoted by $\mu \left[\mathrm{\Omega }\right]$. The blue curve represents smoothed data obtained by replacing every 20 points by their average.