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Incommensurability enabled quasi-fractal order in 1D narrow-band moiré systems

Nature Physics volume 20, pages 1933–1940 (2024)Cite this article

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An Author Correction to this article was published on 17 February 2025

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Abstract

A moiré potential—the superposition of two periodic potentials with different wavelengths—will either introduce a new periodicity into a system if the two potentials are commensurate or force the system to be quasiperiodic if they are not. Here we demonstrate that quasiperiodicity can change the ground-state properties of one-dimensional moiré systems with respect to their periodic counterparts. We show that although narrow bands play a role in enhancing interactions, for both commensurate and incommensurate structures, only quasiperiodicity is able to extend the ordered phase down to an infinitesimal interaction strength. In this regime, the state enabled by quasiperiodicity has contributions from electronic states with a very large number of wavevectors. This quasi-fractal regime cannot be stabilized in the commensurate case even in the presence of a narrow band. These findings suggest that quasiperiodicity may be a critical factor in stabilizing non-trivial ordered phases in interacting moiré structures and highlight that multifractal non-interacting phases might be particularly promising parent states.

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Fig. 1: Phase diagrams for IS and CS.
Fig. 2: LL-CDW transition.
Fig. 3: Momentum-space charge fluctuations for V2 < 1.
Fig. 4: Momentum-space charge fluctuations for V2 > 1.

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Data availability

All data presented in the figures are available via Zenodo at https://doi.org/10.5281/zenodo.8082294 (ref. 60). Source data are provided with this paper.

Code availability

The code used in this work is available via GitHub at https://github.com/gmiguel17/Phd_codes/tree/main/Incommensurability_enabled_quasi-fractal_order_in_1D_narrow-band_moire_systems.

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Acknowledgements

M.G. and P.R. acknowledge partial support from Fundação para a Ciência e Tecnologia (FCT-Portugal; Grant No. UID/CTM/04540/2019 and to the Research Unit UID/04540: CeFEMA financed by FCT-Portugal). B.A. and E.V.C. acknowledge partial support from FCT-Portugal (Grant No. UID/04650 - Centro de Física das Universidades do Minho e do Porto). M.G. acknowledges further support from FCT-Portugal (Grant No. SFRH/BD/145152/2019). B.A. acknowledges further support from FCT-Portugal (Grant No. CEECIND/02936/2017). We also acknowledge the Tianhe-2JK cluster at the Beijing Computational Science Research Center, the Bob∣Macc supercomputer (Computational Project CPCA/A1/470243/2021) and the OBLIVION supercomputer (Projects HPCUE/A1/468700/2021, 2022.15834.CPCA.A1 and 2022.15910.CPCA.A1). The OBLIVION supercomputer is at the High Performance Computing Center, University of Évora, and is funded by the ENGAGE SKA Research Infrastructure (Reference POCI-01-0145-FEDER-022217 - COMPETE 2020), the Foundation for Science and Technology, Portugal, the BigData@UE project (Reference ALT20-03-0246-FEDER-000033 - FEDER) and the Alentejo 2020 Regional Operational Program. Computer assistance was provided by the Computational Science Research Center, Bob∣Macc and OBLIVION support teams.

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Authors and Affiliations

  1. CeFEMA-LaPMET, Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

    Miguel Gonçalves, Flavio Riche & Pedro Ribeiro

  2. Centro de Física das Universidades do Minho e do Porto (CF-UM-UP), Laboratório de Física para Materiais e Tecnologias Emergentes (LaPMET), Universidade do Minho, Campus de Gualtar, Braga, Portugal

    Bruno Amorim

  3. Centro de Física das Universidades do Minho e do Porto, LaPMET, Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, Porto, Portugal

    Eduardo V. Castro

  4. Beijing Computational Science Research Center, Beijing, China

    Eduardo V. Castro & Pedro Ribeiro

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  1. Miguel Gonçalves
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M.G., F.R., B.A., E.V.C. and P.R. planned and defined the project, analysed and interpreted the results, and wrote the paper. M.G. performed the numerical calculations.

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Correspondence to Miguel Gonçalves.

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Gonçalves, M., Amorim, B., Riche, F. et al. Incommensurability enabled quasi-fractal order in 1D narrow-band moiré systems. Nat. Phys. 20, 1933–1940 (2024). https://doi.org/10.1038/s41567-024-02662-2

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