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  • Letter
  • Open Access

Distinct reentrant transitions in a quasiperiodic Raman lattice

Xingbo Wei1,*, Zhentian Xie1, Tong Liu2, Gao Xianlong3, and Yunbo Zhang1,†

  • *Contact author: weixingbo@zstu.edu.cn
  • †Contact author: ybzhang@zstu.edu.cn
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Phys. Rev. Research 8, L022044 – Published 5 June, 2026

DOI: https://doi.org/10.1103/g15k-q96k

Abstract

We investigate a one-dimensional lattice with spin-orbit coupling (SOC) and a Zeeman potential containing uniform and quasiperiodic components. By tuning SOC, anomalous mobility edges emerge that separate critical from noncritical states, yielding a reentrant transition between two mixed phases, M1→M2→M1, where M1 (M2) lacks (hosts) anomalous mobility edges. A reentrant criticality transition, defined as multiple entries into the critical phase, is identified. As a counterpart to reentrant delocalization/localization transitions, it completes the basic framework of reentrant phenomena across extended, localized, and critical states. The uniform Zeeman potential drives a reentrant delocalization transition, arising from the splitting of the localized region induced by the shift of mobility edges. This reveals a distinct pathway for reentrant phenomena beyond hybridization mechanisms.

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References (54)

  1. P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
  2. X.-C. Zhou, B.-C. Yao, Y. Wang, Y. Wang, Y. Wei, Q. Zhou, and X.-J. Liu, The fundamental localization phases in quasiperiodic systems: A unified framework and exact results, Sci. Bull. 71, 1654 (2026).
  3. 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).
  4. M. Pasek, G. Orso, and D. Delande, Anderson localization of ultracold atoms: Where is the mobility edge? Phys. Rev. Lett. 118, 170403 (2017).
  5. S. Longhi, Dephasing-induced mobility edges in quasicrystals, Phys. Rev. Lett. 132, 236301 (2024).
  6. 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).
  7. 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).
  8. 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).
  9. H. Yao, A. Khoudli, L. Bresque, and L. Sanchez-Palencia, Critical behavior and fractality in shallow one-dimensional quasiperiodic potentials, Phys. Rev. Lett. 123, 070405 (2019).
  10. T. Liu, X. Xia, S. Longhi, and L. Sanchez-Palencia, Anomalous mobility edges in one-dimensional quasiperiodic models, SciPost Phys. 12, 027 (2022).
  11. Z. Wang, Y. Zhang, L. Wang, and S. Chen, Engineering mobility in quasiperiodic lattices with exact mobility edges, Phys. Rev. B 108, 174202 (2023).
  12. 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).
  13. Y. Wang and Q. Zhou, Exact new mobility edges, arXiv:2501.17523.
  14. N. Mott, Electrons in disordered structures, Adv. Phys. 16, 49 (1967).
  15. 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).
  16. F. A. An, K. Padavić, E. J. Meier, S. Hegde, S. Ganeshan, J. H. Pixley, S. Vishveshwara, and B. Gadway, Interactions and mobility edges: Observing the generalized Aubry-André model, Phys. Rev. Lett. 126, 040603 (2021).
  17. Y. Wang, J.-H. Zhang, Y. Li, J. Wu, W. Liu, F. Mei, Y. Hu, L. Xiao, J. Ma, C. Chin, and S. Jia, Observation of interaction-induced mobility edge in an atomic Aubry-André wire, Phys. Rev. Lett. 129, 103401 (2022).
  18. Y.-J. Chang, J.-H. Zhang, Y.-H. Lu, Y.-Y. Yang, F. Mei, J. Ma, S. Jia, and X.-M. Jin, Observation of photonic mobility edge phases, Phys. Rev. Lett. 134, 053601 (2025).
  19. T. Xiao, D. Xie, Z. Dong, T. Chen, W. Yi, and B. Yan, Observation of topological phase with critical localization in a quasi-periodic lattice, Sci. Bull. 66, 2175 (2021).
  20. V. Goblot, A. Štrkalj, N. Pernet, J. L. Lado, C. Dorow, A. Lemaître, L. Le Gratiet, A. Harouri, I. Sagnes, S. Ravets, et al., Emergence of criticality through a cascade of delocalization transitions in quasiperiodic chains, Nat. Phys. 16, 832 (2020).
  21. S. Roy, T. Mishra, B. Tanatar, and S. Basu, Reentrant localization transition in a quasiperiodic chain, Phys. Rev. Lett. 126, 106803 (2021).
  22. A. Padhan, M. K. Giri, S. Mondal, and T. Mishra, Emergence of multiple localization transitions in a one-dimensional quasiperiodic lattice, Phys. Rev. B 105, L220201 (2022).
  23. A. Štrkalj, Elmer V. H. Doggen, I. V. Gornyi, and O. Zilberberg, Many-body localization in the interpolating Aubry-André-Fibonacci model, Phys. Rev. Res. 3, 033257 (2021).
  24. X. Wei, Reentrant localization transition induced by a composite potential, Phys. Rev. B 111, 134204 (2025).
  25. Z.-S. Xu, J. Gao, A. Iovan, I. M. Khaymovich, V. Zwiller, and A. W. Elshaari, Observation of reentrant metal-insulator transition in a random-dimer disordered SSH lattice, npj Nanophotonics 1, 8 (2024).
  26. S. Vaidya, C. Jörg, K. Linn, M. Goh, and M. C. Rechtsman, Reentrant delocalization transition in one-dimensional photonic quasicrystals, Phys. Rev. Res. 5, 033170 (2023).
  27. Q. Guo, C. Cheng, Z.-H. Sun, Z. Song, H. Li, Z. Wang, W. Ren, H. Dong, D. Zheng, Y.-R. Zhang, et al., Observation of energy-resolved many-body localization, Nat. Phys. 17, 234 (2021).
  28. Y.-H. Shi, Z.-H. Sun, Y.-Y. Wang, Z.-A. Wang, Y.-R. Zhang, W.-G. Ma, H.-T. Liu, K. Zhao, J.-C. Song, G.-H. Liang, et al., Probing spin hydrodynamics on a superconducting quantum simulator, Nat. Commun. 15, 7573 (2024).
  29. X. Li, H. Xu, J. Wang, L.-Z. Tang, D.-W. Zhang, C. Yang, T. Su, C. Wang, Z. Mi, W. Sun, et al., Mapping the topology-localization phase diagram with quasiperiodic disorder using a programmable superconducting simulator, Phys. Rev. Res. 6, L042038 (2024).
  30. Y. Wang, L. Zhang, S. Niu, D. Yu, and X.-J. Liu, Realization and detection of nonergodic critical phases in an optical Raman lattice, Phys. Rev. Lett. 125, 073204 (2020).
  31. Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Realization of two-dimensional spin-orbit coupling for Bose-Einstein condensates, Science 354, 83 (2016).
  32. B. Song, C. He, S. Niu, L. Zhang, Z. Ren, X.-J. Liu, and G.-B. Jo, Observation of nodal-line semimetal with ultracold fermions in an optical lattice, Nat. Phys. 15, 911 (2019).
  33. See Supplemental Material at http://link.aps.org/supplemental/10.1103/g15k-q96k for details on (1) matrix form of the Hamiltonian, (2) multifractal analysis of different states, (3) transition points in the phase diagram and (4) in the limiting cases, (5) reentrant phenomenon and pure phase criterion, (6) the effect of W on the purely critical phase, (7) the influence of disorder and interactions on phase diagrams, (8) experimental detection, (9) destruction of the reemerged extended states, (10) the transition in the case of W/t=0, (11) spectrum for Δ/t=1 and typical states, (12) Δ-induced transition for large W, (13) the phase diagram for V/t=2.5, and (14) reentrant delocalization transitions, which includes Refs. [2, 15, 20, 30, 38, 52, 53, 54].
  34. X.-J. Liu, Z.-X. Liu, and M. Cheng, Manipulating topological edge spins in a one-dimensional optical lattice, Phys. Rev. Lett. 110, 076401 (2013).
  35. M. Kohmoto, Metal-insulator transition and scaling for incommensurate systems, Phys. Rev. Lett. 51, 1198 (1983).
  36. J. Zhao, Y. Zhao, J.-G. Wang, Y. Li, and X.-D. Bai, Phase transition of a non-Abelian quasiperiodic mosaic lattice model with p-wave superfluidity, Phys. Rev. B 108, 054204 (2023).
  37. S. Aubry and G. André, Analyticity breaking and Anderson localization in incommensurate lattices, Ann. Isr. Phys. Soc. 3, 133 (1980).
  38. J. Wang, X.-J. Liu, G. Xianlong, and H. Hu, Phase diagram of a non-Abelian Aubry-André-Harper model with p-wave superfluidity, Phys. Rev. B 93, 104504 (2016).
  39. M. Yahyavi, B. Hetényi, and B. Tanatar, Generalized Aubry-André-Harper model with modulated hopping and p-wave pairing, Phys. Rev. B 100, 064202 (2019).
  40. Q.-B. Zeng, S. Chen, and R. Lü, Generalized Aubry-André-Harper model with p-wave superconducting pairing, Phys. Rev. B 94, 125408 (2016).
  41. X. Li, X. Li, and S. Das Sarma, Mobility edges in one-dimensional bichromatic incommensurate potentials, Phys. Rev. B 96, 085119 (2017).
  42. R. Ghosh, M. Sarkar, and I. M. Khaymovich, Reentrant localization induced by short-range hopping in the fractal Rosenzweig-Porter model, Phys. Rev. B 111, L220102 (2025).
  43. D. J. Luitz, F. Alet, and N. Laflorencie, Universal behavior beyond multifractality in quantum many-body systems, Phys. Rev. Lett. 112, 057203 (2014).
  44. G. De Tomasi and I. M. Khaymovich, Multifractality meets entanglement: Relation for nonergodic extended states, Phys. Rev. Lett. 124, 200602 (2020).
  45. R. Ji and Z. Xu, Fibonacci-modulation-induced multiple topological Anderson insulators, Commun. Phys. 8, 336 (2025).
  46. G.-Q. Zhang, L.-Z. Tang, L. Quezada, S.-H. Dong, and D.-W. Zhang, Reentrant topological phases and spin density wave induced by 1D moiré potentials, Commun. Phys. 8, 275 (2025).
  47. X. Li and S. Das Sarma, Mobility edge and intermediate phase in one-dimensional incommensurate lattice potentials, Phys. Rev. B 101, 064203 (2020).
  48. R. Qi, J. Cao, and X.-P. Jiang, Multiple localization transitions and novel quantum phases induced by a staggered on-site potential, Phys. Rev. B 107, 224201 (2023).
  49. E. Guan, G. Wang, X.-W. Guan, and X. Cai, Reentrant localization and mobility edges in a spinful Aubry-André-Harper model with a non-Abelian potential, Phys. Rev. A 108, 033305 (2023).
  50. T. Lv, T.-C. Yi, L. Li, G. Sun, and W.-L. You, Quantum criticality and universality in the p-wave-paired Aubry-André-Harper model, Phys. Rev. A 105, 013315 (2022).
  51. T. L. G. X. Xingbo Wei, Z. Xie and Y. Zhang, Distinct reentrant transitions in a quasi-periodic Raman lattice, Zenodo, 2025, 10.5281/zenodo.17082777.
  52. F. Pietracaprina, N. Macé, D. J. Luitz, and F. Alet, Shift-invert diagonalization of large many-body localizing spin chains, SciPost Phys. 5, 045 (2018).
  53. X. Wei, C. Cheng, G. Xianlong, and R. Mondaini, Investigating many-body mobility edges in isolated quantum systems, Phys. Rev. B 99, 165137 (2019).
  54. H. Tabanelli, C. Castelnovo, and A. Štrkalj, Reentrant localization transitions and anomalous spectral properties in off-diagonal quasiperiodic systems, Phys. Rev. B 110, 184208 (2024).

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Phys. Rev. Research 8, L022044– Published 5 June, 2026
  • Received 9 September 2025
  • Accepted 22 May 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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