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# Bimorphic Floquet topological insulators

## Abstract

Topological theories have established a unique set of rules that govern the transport properties in a wide variety of wave-mechanical settings. In a marked departure from the established approaches that induce Floquet topological phases by specifically tailored discrete coupling protocols or helical lattice motions, we introduce a class of bimorphic Floquet topological insulators that leverage connective chains with periodically modulated on-site potentials to reveal rich topological features in the system. In exploring a ‘chain-driven’ generalization of the archetypical Floquet honeycomb lattice, we identify a rich phase structure that can host multiple non-trivial topological phases associated simultaneously with both Chern-type and anomalous chiral states. Experiments carried out in photonic waveguide lattices reveal a strongly confined helical edge state that, owing to its origin in bulk flat bands, can be set into motion in a topologically protected fashion, or halted at will, without compromising its adherence to individual lattice sites.

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## Data availability

The experimental data that support the findings of this study are available from M.H. upon reasonable request (matthias.heinrich@uni-rostock.de).

## Code availability

The MATLAB codes corresponding to the beam-propagation method and band structure algorithms are available from G.G.P. upon reasonable request (pyrialak@knights.ucf.edu).

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## Acknowledgements

We thank C. Otto for preparing the high-quality fused silica samples used for the inscription of all photonic structures employed in the experiments presented here. G.G.P. acknowledges the support of the Bodossaki Foundation. This work was partially supported by Defense Advanced Research Projects Agency (DARPA; D18AP00058), Office of Naval Research (ONR) Multidisciplinary University Research Initiative (MURI; N00014-16-1-2640, N00014-18-1-2347, N00014-19-1-2052, N00014-20-1-2522, N00014-20-1-2789), Air Force Office of Scientific Research (AFOSR MURI; FA9550-20-1-0322, FA9550-21-1-0202), the National Science Foundation (DMR-1420620, EECS-1711230, ECCS-1454531, DMR-1420620, ECCS-1757025, CBET-1805200, ECCS-2000538, ECCS-2011171), Mathematics and Physical Sciences (MPS) Simons collaboration (Simons grant 733682), W. M. Keck Foundation, US–Israel Binational Science Foundation (BSF 2016381) and the US Air Force Research Laboratory (FA9550-14-1-0037, FA9550-20-1-0322, FA9550-21-1-0202, FA86511820019). We furthermore acknowledge funding from the Deutsche Forschungsgemeinschaft (SCHE 612/6-1, SZ 276/12-1, BL 574/13-1, SZ 276/15-1, SZ 276/20-1) and the Alfried Krupp von Bohlen and Halbach Foundation.

## Author information

Authors

### Contributions

G.G.P. initiated the idea, formulated the index-modulated lattice and performed the theoretical calculations and simulations. J.B. developed the experimental implementation, fabricated the samples and conducted the measurements. J.B., L.J.M. and M.H. evaluated the measurements and interpreted the data. M.K., N.V.K., A.S. and D.N.C. supervised the efforts of their respective groups. All authors discussed the results and cowrote the manuscript.

### Corresponding author

Correspondence to Demetrios N. Christodoulides.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature Materials thanks Alexander Khanikaev, Sunil Mittal and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 Comparison of staggered and flat-phased broad excitations.

(a-d): Staggered excitations of the primary waveguides of edge unit cells successfully populate the topological Chern mode near the edge of the Brillouin zone. (e-h) Absent the appropriate phase modulation, the injected wave packets instead represent a superposition of the bulk bands near the center of the Brillouin zone. As a result, the light diffracts freely across the entire lattice instead of being captured in the helical Chern channel.

### Extended Data Fig. 2 Honeycomb helical FTI.

A driven honeycomb lattice with sinusoidally modulated time-periodic coupling terms exhibits a secondary topological phase in response to an increase of the driving period. At the critical driving period $$T_{{{\mathrm{C}}}} = \pi /3$$ the gap at the Floquet zone collapses and reopens with a topologically non-trivial winding number. This corresponds to an anomalous phase with a trivial Chern number, signifying a topological phase transition.

### Extended Data Fig. 3 Chained honeycomb FTI.

Above its critical modulation period $$T_{{{\mathrm{C}}}}^\prime = \pi$$, the chain-driven honeycomb lattice enters a secondary topological phase characterized by the band diagram shown on the right. In this configuration, the bands manifest a non-trivial topological structure characterized by higher order Chern invariants ($${{{\mathcal{C}}}} = 2$$), and, in turn, the Chern gaps host an increased number of unidirectional edge states. Details on the modal amplitudes and propagation dynamics of the Chern states in the $${{{\mathcal{C}}}} = 2$$ phase are provided in Supplementary Fig. 5.

### Extended Data Fig. 4 Losses in index-modulated waveguides.

Waveguide fluorescence40 characterization confirms that even coarsely discretized index modulation (twelve constant-index segments approximating the ideal cosine Floquet cycle) only introduces excess losses of (0.096 ± 0.010)dB/cm relative to a straight constant-index waveguide. With half of the lattice sites being modulated, the mean excess losses of the bimorphic FTI are (0.048 ± 0.005)dB/cm, substantially below the value of 1.7 dB/cm reported for bending losses in conventional FTIs based on helically modulated waveguides4. Note that the apparent oscillations in the normalized intensity of the modulated waveguide are due to the different concentration of color centers formed at different writing speeds, resulting in a modulation of the fluorescence efficiency between the individual segments of each modulation period. Due to the large number of periods, this oscillation does not notably impact the measurement of the loss coefficient γ. The fluorescence itself is a feature of femtosecond laser-written waveguides in fused silica, and does not pose a substantial source of propagation losses40.

## Supplementary information

### Supplementary Information

Supplementary Figs. 1–6 and Discussion.

### Supplementary Video 1

Simulation of propagation dynamics of two isolated compact localized states in a ribbon lattice. Power is injected into the sites indicated by labels A and B. At the 00:06 time mark, the Floquet modulation is switched off in the vicinity of wave packet B (shaded region). As a consequence, wave packet B couples to a static compact localized state and remains stationary, while wave packet A finds a route around the shaded region. At the 00:20 time mark, the Floquet modulation is restored in the entire lattice, and both wave packets resume propagation to, finally, become spatially interchanged.

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Pyrialakos, G.G., Beck, J., Heinrich, M. et al. Bimorphic Floquet topological insulators. Nat. Mater. 21, 634–639 (2022). https://doi.org/10.1038/s41563-022-01238-w

• Accepted:

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• DOI: https://doi.org/10.1038/s41563-022-01238-w

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