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Twist-driven interlayer hybridization and multiple mobility edges in bilayer quasiperiodic lattices

Ankit Mishra* and Kang Hao Cheong†

  • *Contact author: ankitphy0592@gmail.com
  • †Contact author: kanghao.cheong@ntu.edu.sg
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Phys. Rev. Research 8, 033079 – Published 20 July, 2026

DOI: https://doi.org/10.1103/mszt-nzpz

Abstract

We investigate two coupled one-dimensional Aubry-André-Harper rings with N sites, where a site-dependent angular twist in the second ring generates a spatially varying mosaic of distance-modulated couplings. By varying the quasiperiodic potential strength V and twist periodicity k, we find multiple closely spaced mobility edges separating localized, extended, and critical eigenstates within narrow energy intervals. Fractal dimension analysis shows that k strongly controls the spectral distribution of these states, with larger k enhancing the coexistence of different localization characters. In addition, we find that for k=2, most eigenstates remain nearly equally distributed between the two rings, whereas larger k drives pronounced layer polarization. We further examine real-time wave-packet dynamics and show that increasing k accelerates the crossover from nearly ballistic spreading to subdiffusive motion and lowers the long-time saturation of the normalized mean-square displacement. At the same time, an initially layer-balanced wave packet remains strongly hybridized for k=2, whereas the interlayer overlap is reduced for k=3 and k=4. Thus, the site-dependent twist provides a tunable mechanism to control spectral localization, wave-packet transport, and interlayer mixing in quasiperiodic bilayer lattices.

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References (43)

  1. P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
  2. S. Fishman, D. R. Grempel, and R. E. Prange, Chaos, quantum recurrences, and Anderson localization, Phys. Rev. Lett. 49, 509 (1982).
  3. Y. Gefen, D. J. Thouless, and Y. Imry, Localization effects near the percolation threshold, Phys. Rev. B 28, 6677 (1983).
  4. A. Mishra and S. Jalan, Eigenvector localization in hypergraphs: Pairwise versus higher-order links, Phys. Rev. E 107, 034311 (2023).
  5. A. Mishra, J. N. Bandyopadhyay, and S. Jalan, Multifractal analysis of eigenvectors of small-world networks, Chaos Solit. Fractals 144, 110745 (2021).
  6. A. Mishra and K. H. Cheong, Exploring universality of the β-Gaussian ensemble in complex networks via intermediate eigenvalue statistics, Phys. Rev. E 109, 014218 (2024).
  7. A. Mishra, T. Raghav, and S. Jalan, Eigenvalue ratio statistics of complex networks: Disorder versus randomness, Phys. Rev. E 105, 064307 (2022).
  8. G. Semeghini, M. Landini, P. Castilho, S. Roy, G. Spagnolli, A. Trenkwalder, M. Fattori, M. Inguscio, and G. Modugno, Measurement of the mobility edge for 3D Anderson localization, Nat. Phys. 11, 554 (2015).
  9. M. Pasek, G. Orso, and D. Delande, Anderson localization of ultracold atoms: Where is the mobility edge? Phys. Rev. Lett. 118, 170403 (2017).
  10. J. Biddle and S. Das Sarma, Predicted mobility edges in one-dimensional incommensurate optical lattices: An exactly solvable model of Anderson localization, Phys. Rev. Lett. 104, 070601 (2010).
  11. S. Ganeshan, J. H. Pixley, and S. Das Sarma, Nearest neighbor tight binding models with an exact mobility edge in one dimension, Phys. Rev. Lett. 114, 146601 (2015).
  12. Y. Wang, C. Cheng, X.-J. Liu, and D. Yu, Many-body critical phase: Extended and nonthermal, Phys. Rev. Lett. 126, 080602 (2021).
  13. K. Huang, D. Vu, S. Das Sarma, and X. Li, Interaction-enhanced many-body localization in a generalized Aubry-André model, Phys. Rev. Res. 6, L022054 (2024).
  14. A. Štrkalj, E. V. H. Doggen, and C. Castelnovo, Coexistence of localization and transport in many-body two-dimensional Aubry-André models, Phys. Rev. B 106, 184209 (2022).
  15. X. Deng, S. Ray, S. Sinha, G. V. Shlyapnikov, and L. Santos, One-dimensional quasicrystals with power-law hopping, Phys. Rev. Lett. 123, 025301 (2019).
  16. X.-C. Zhou, Y. Wang, T.-F. J. Poon, Q. Zhou, and X.-J. Liu, Exact new mobility edges between critical and localized states, Phys. Rev. Lett. 131, 176401 (2023).
  17. Y. Wang, X. Xia, L. Zhang, H. Yao, S. Chen, J. You, Q. Zhou, and X.-J. Liu, One-dimensional quasiperiodic mosaic lattice with exact mobility edges, Phys. Rev. Lett. 125, 196604 (2020).
  18. S. Longhi, Resonances, mobility edges, and gap-protected Anderson localization in generalized disordered mosaic lattices, Phys. Rev. B 110, 184201 (2024).
  19. Y.-C. Zhang, R. Yuan, S. Song, M. Hu, C. Liu, and Y. Wang, Odd-even effect of the mosaic modulation period of quasiperiodic hopping on the Anderson localization in a one-dimensional lattice model, Phys. Rev. B 111, 064201 (2025).
  20. X. Li, X. Li, and S. Das Sarma, Mobility edges in one-dimensional bichromatic incommensurate potentials, Phys. Rev. B 96, 085119 (2017).
  21. Y.-C. Zhang and Y.-Y. Zhang, Lyapunov exponent, mobility edges, and critical region in the generalized Aubry-André model with an unbounded quasiperiodic potential, Phys. Rev. B 105, 174206 (2022).
  22. D. Dwiputra and F. P. Zen, Single-particle mobility edge without disorder, Phys. Rev. B 105, L081110 (2022).
  23. H. P. Lüschen, S. Scherg, T. Kohlert, M. Schreiber, P. Bordia, X. Li, S. Das Sarma, and I. Bloch, Single-particle mobility edge in a one-dimensional quasiperiodic optical lattice, Phys. Rev. Lett. 120, 160404 (2018).
  24. M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of interacting fermions in a quasirandom optical lattice, Science 349, 842 (2015).
  25. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Anderson localization of light, Nature (London) 446, 52 (2007).
  26. M. Le Ster, T. Maerkl, P. J. Kowalczyk, and S. A. Brown, Moiré patterns in van der Waals heterostructures, Phys. Rev. B 99, 075422 (2019).
  27. J. M. B. L. dos Santos, N. M. R. Peres, and A. H. C. Neto, Continuum model of the twisted graphene bilayer, Phys. Rev. B 86, 155449 (2012).
  28. E. Y. Andrei and A. H. MacDonald, Graphene bilayers with a twist, Nat. Mater. 19, 1265 (2020).
  29. Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Topological states and adiabatic pumping in quasicrystals, Phys. Rev. Lett. 109, 106402 (2012).
  30. H. Cory and C. Zach, Wave propagation in metamaterial multi-layered structures, Microwave Opt. Technol. Lett. 40, 460 (2004).
  31. G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Anderson localization of a non-interacting Bose–Einstein condensate, Nature (London) 453, 895 (2008).
  32. T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, et al., Topological photonics, Rev. Mod. Phys. 91, 015006 (2019).
  33. Y. Wang, L. Zhang, W. Sun, T.-F. J. Poon, and X.-J. Liu, Quantum phase with coexisting localized, extended, and critical zones, Phys. Rev. B 106, L140203 (2022).
  34. C. Guo, Multiple intermediate phases in the interpolating Aubry-André-Fibonacci model, Phys. Rev. B 109, 174203 (2024).
  35. C. W. Duncan, Critical states and anomalous mobility edges in two-dimensional diagonal quasicrystals, Phys. Rev. B 109, 014210 (2024).
  36. M. Rossignolo and L. Dell’Anna, Localization transitions and mobility edges in coupled Aubry-André chains, Phys. Rev. B 99, 054211 (2019).
  37. I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A: Math. Theor. 42, 504003 (2009).
  38. M. O. T. Cunha, J. A. Dunningham, and V. Vedral, Entanglement in single-particle systems, Proc. R. Soc. A 463, 2277 (2007).
  39. A. Bulgac, Entanglement entropy, single-particle occupation probabilities, and short-range correlations, Phys. Rev. C 107, L061602 (2023).
  40. E. Barré, M. Dandu, S. Kundu, A. Sood, F. H. da Jornada, and A. Raja, Engineering interlayer hybridization in van der Waals bilayers, Nat. Rev. Mater. 9, 499 (2024).
  41. D. A. Ruiz-Tijerina and V. I. Fal'ko, Interlayer hybridization and moiré superlattice minibands for electrons and excitons in heterobilayers of transition-metal dichalcogenides, Phys. Rev. B 99, 125424 (2019).
  42. G. Yu, Y. Wang, M. I. Katsnelson, H.-Q. Lin, and S. Yuan, Interlayer hybridization in graphene quasicrystal and other bilayer graphene systems, Phys. Rev. B 105, 125403 (2022).
  43. See Supplemental Material at https://link.aps.org/supplemental/10.1103/mszt-nzpz for additional contour plots showing the robustness of interlayer hybridization under variations of the twist angle and distance-decay exponent.

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Phys. Rev. Research 8, 033079– Published 20 July, 2026
  • Received 24 July 2025
  • Accepted 4 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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