- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Screening and localization in the nonlinear Anderson problem

Alexander V. Milovanov1,2 and Alexander Iomin2,3

PDF

Phys. Rev. E 112, 034206 – Published 5 September, 2025

DOI: https://doi.org/10.1103/svxt-wsk7

Abstract

We study the spreading dynamics of an initially localized wave packet of finite norm in one-dimensional nonlinear Schrödinger lattices with random potential. The problem has gained considerable interest in the literature, and it continues to attract attention due to its connection with the general properties of behavior of systems with competition between nonlinearity, nonlocality, and randomness. It is shown that adding small dielectric coupling to the ambient random medium leads to asymptotic localization of the nonlinear field regardless of the Kerr nonlinearity strength. If the electric susceptibility is zero, then the nonlinear field undergoes sharp localization-delocalization transition above a certain critical value of the nonlinearity parameter. The nonlinear localization length is found to be Λloc≃exp[(π/ɛrtanδ)lnβ], where tanδ is the dielectric loss tangent, ɛr is the relative permeability of the medium, and β characterizes the Kerr nonlinearity. The model predicts a possibility of self-induced localization when the “medium” to which the wave field is dielectrically coupled is the wave function itself. The mathematical methods, stipulated here, pave the way towards understanding the wave processes in complex media with competition between randomness, dispersiveness, and nonlinearity, such as Anderson localization of a wave packet interacting with the environment.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (69)

  1. P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
  2. R. Abou-Chacra, P. W. Anderson, and D. J. Thouless, A selfconsistent theory of localization, J. Phys. C 6, 1734 (1973).
  3. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979).
  4. A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, Fifty years of Anderson localization, Phys. Today 62(8), 24 (2009).
  5. D. J. Thouless, Electrons in disordered systems and the theory of localization, Phys. Rep. 13, 93 (1974).
  6. L. Fleishman and P. W. Anderson, Interactions and the Anderson transition, Phys. Rev. B 21, 2366 (1980).
  7. B. L. Altshuler, Y. Gefen, A. Kamenev, and L. S. Levitov, Quasiparticle lifetime in a finite system: A nonperturbative approach, Phys. Rev. Lett. 78, 2803 (1997).
  8. D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states, Ann. Phys. (NY) 321, 1126 (2006).
  9. D. A. Abanin and Z. Papic, Recent progress in many-body localization, Ann. Phys. (NY) 529, 1700169 (2017).
  10. F. Alet and N. Laflorencie, Many-body localization: An introduction and selected topics, C. R. Physique 19, 498 (2018).
  11. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  12. S. Roy and D. E. Logan, The Fock-space landscape of many-body localization, J. Phys.: Condens. Matter 37, 073003 (2025).
  13. A. S. Pikovsky and D. L. Shepelyansky, Destruction of Anderson localization by a weak nonlinearity, Phys. Rev. Lett. 100, 094101 (2008).
  14. S. Flach, D. O. Krimer, and Ch. Skokos, Universal spreading of wave packets in disordered nonlinear systems, Phys. Rev. Lett. 102, 024101 (2009).
  15. D. L. Shepelyansky, Localization of diffusive excitation in multi-level systems, Physica D 28, 103 (1987).
  16. D. L. Shepelyansky, Delocalization of quantum chaos by weak nonlinearity, Phys. Rev. Lett. 70, 1787 (1993).
  17. Ch. Skokos, D. O. Krimer, S. Komineas, and S. Flach, Delocalization of wave packets in disordered nonlinear chains, Phys. Rev. E 79, 030702 (2009).
  18. W.-M. Wang and Z. Zhang, Long time Anderson localization for the nonlinear random Schrödinger equation, J. Stat. Phys. 134, 953 (2009).
  19. Y. Krivolapov, S. Fishman, and A. Soffer, A numerical and symbolic approximation of the nonlinear Anderson problem, New J. Phys. 12, 063035 (2010).
  20. A. Iomin, Subdiffusion in the nonlinear Schrödinger equation with disorder, Phys. Rev. E 81, 017601 (2010).
  21. B. Senyange, B. M. Manda, and Ch. Skokos, Characteristics of chaos evolution in one-dimensional disordered nonlinear lattices, Phys. Rev. E 98, 052229 (2018).
  22. D. M. Basko, Weak chaos in the disordered nonlinear Schrödinger chain: Destruction of Anderson localization by Arnold diffusion, Ann. Phys. (NY) 326, 1577 (2011).
  23. A. V. Milovanov and A. Iomin, Localization-delocalization transition on a separatrix system of nonlinear Schrödinger equation with disorder, Europhys. Lett. 100, 10006 (2012).
  24. A. V. Milovanov and A. Iomin, Topological approximation of the nonlinear Anderson model, Phys. Rev. E 89, 062921 (2014).
  25. M. V. Ivanchenko, T. V. Laptyeva, and S. Flach, Quantum chaotic subdiffusion in random potentials, Phys. Rev. B 89, 060301(R) (2014).
  26. A. V. Milovanov and A. Iomin, Destruction of Anderson localization in quantum nonlinear Schrödinger lattices, Phys. Rev. E 95, 042142 (2017).
  27. I. Vakulchyk, M. V. Fistul, and S. Flach, Wave packet spreading with disordered nonlinear discrete-time quantum walks, Phys. Rev. Lett. 122, 040501 (2019).
  28. A. V. Milovanov and A. Iomin, Dynamical chaos in nonlinear Schrödinger models with subquadratic power nonlinearity, Phys. Rev. E 107, 034203 (2023).
  29. B. Shapiro, Expansion of a Bose-Einstein condensate in the presence of disorder, Phys. Rev. Lett. 99, 060602 (2007).
  30. B. Shapiro, Cold atoms in the presence of disorder, J. Phys. A 45, 143001 (2012).
  31. I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Interacting electrons in disordered wires: Anderson localization at low-T transport, Phys. Rev. Lett. 95, 206603 (2005).
  32. V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B 75, 155111 (2007).
  33. V. Rosenhaus and G. Falkovich, Interaction renormalization and validity of kinetic equations for turbulent states, Phys. Rev. Lett. 133, 244002 (2024).
  34. A. I. Akhiezer and V. B. Berestetskii, Quantum Electrodynamics (Wiley, New York, 1965).
  35. Conceptual Foundations of Quantum Field Theory, edited by T. Yu. Cao (Cambridge University, Cambridge, England, 2004).
  36. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, A Course of Theoretical Physics Vol. 8 (Pergamon, New York, 1960).
  37. S. R. Elliott, On the super-linear frequency dependent conductivity of amorphous semiconductors, Solid State Commun. 28, 939 (1978).
  38. S. R. Elliott, A.c. conduction in amorphous chalcogenide and pnictide semiconductors, Adv. Phys. 36, 135 (1987).
  39. M. Pollak, On the frequency dependence of conductivity in amorphous solids, Philos. Mag. 23, 519 (1971).
  40. S. R. Elliott, A theory of A.C. conduction in chalcogenide glasses, Philos. Mag. 36, 1291 (1977).
  41. H. Scher and M. Lax, Stochastic transport in a disordered solid. I. Theory, Phys. Rev. B 7, 4491 (1973).
  42. A. K. Jonscher, The “universal” dielectric response, Nature (London) 267, 673 (1977).
  43. J. C. Dyre and T. B. Schrøder, Universality of ac conduction in disordered solids, Rev. Mod. Phys. 72, 873 (2000).
  44. J. C. Dyre, Universal low-temperature ac conductivity of macroscopically disordered nonmetals, Phys. Rev. B 48, 12511 (1993).
  45. A. V. Milovanov and J. J. Rasmussen, Critical conducting networks in disordered solids: Ac universality from topological arguments, Phys. Rev. B 64, 212203 (2001).
  46. M. M. Fangary and M. A. O. Ahmed, The influence of frequency and temperature on the AC-conductivity in TlInTe2 semiconductor single crystal, Sci. Rep. 15, 5162 (2025).
  47. A. V. Milovanov and A. Iomin, Destruction of Anderson localization by subquadratic nonlinearity, Europhys. Lett. 141, 61002 (2023).
  48. Z. Mehri and A. Boudjemaa, Diffusive expansion of a dipolar Bose-Einstein condensate in three-dimensional disorder potentials, Eur. Phys. J. B 97, 39 (2024).
  49. The situation may be different in three dimensions, where the inclusion of dipole-dipole interactions introduces spatial anisotropy into the dynamics (see Ref. [48] and references therein). In the limit of strong anisotropy, one may speculate the existence of a preferred direction, along which one reduces to one dimension the more general, three-dimensional localization model. We ought to note that in three and higher dimensions, the Anderson localization is characterized by a mobility edge separating the localized regime at low energy and diffusive states at high energy [3], and in this sense is fundamentally different from the 1D localization case, the subject matter of the present study.
  50. R. Penrose, On gravity's role in quantum state reduction, Gen. Relativ. Gravit. 28, 581 (1996).
  51. R. Penrose, Quantum computation, entanglement and state reduction, Phil. Trans. R. Soc. A 356, 1927 (1998).
  52. R. Penrose, On the gravitization of quantum mechanics 1: Quantum state reduction, Found. Phys. 44, 557 (2014).
  53. I. M. Moroz, R. Penrose, and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Class. Quantum Grav. 15, 2733 (1998).
  54. M. Bahrami, A. Großardt, S. Donadi, and A. Bassi, The Schrödinger-Newton equation and its foundations, New J. Phys. 16, 115007 (2014).
  55. Y. Sharabi, H. Herzig Sheinfux, Y. Sagi, G. Eisenstein, and M. Segev, Self-induced diffusion in disordered nonlinear photonic media, Phys. Rev. Lett. 121, 233901 (2018).
  56. A. V. Milovanov and A. Iomin, Topology of delocalization in the nonlinear Anderson model and anomalous diffusion on finite clusters, Discont. Nonlinear. Complex. 4, 151 (2015).
  57. G. M. Zaslavsky and B. V. Chirikov, Stochastic instability of non-linear oscillators, Sov. Phys. Usp. 14, 549 (1972).
  58. G. M. Zaslavsky and R. Z. Sagdeev, Introduction to the Nonlinear Physics: From Pendulum to Turbulence and Chaos (Nauka, Moscow, 1988).
  59. P. Bak, C. Tang, and K. Wiesenfeld, Self-organized criticality: An explanation of 1/f noise, Phys. Rev. Lett. 59, 381 (1987).
  60. P. Bak, C. Tang, and K. Wiesenfeld, Self-organized criticality, Phys. Rev. A 38, 364 (1988).
  61. Y.-C. Zhang, Scaling theory of self-organized criticality, Phys. Rev. Lett. 63, 470 (1989).
  62. M. R. Schroeder, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise (Freeman, New York, 1991).
  63. The coordination number of a Cayley tree is the number of bonds at each node. The triad couplings in Eq. (14) dictate z=3.
  64. Y. Gefen, A. Aharony, and S. Alexander, Anomalous diffusion on percolating clusters, Phys. Rev. Lett. 50, 77 (1983).
  65. This power law incorporates the ac responses from all clusters at percolation, both infinite and finite. Gefen et al. [64] write the η value in terms of the percolation critical exponents β, ν, and μ, which we do not introduce. The η value, cited here, is obtained straightforwardly [45] with use of the hyperscaling df=d−β/ν, where d is the topological dimension of the ambient space, and θ=(μ−β)/ν. These definitions are further discussed in D. Stauffer, Scaling theory of percolation clusters, Phys. Rep. 54, 1 (1979).
  66. T. Nakayama, K. Yakubo, and R. L. Orbach, Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations, Rev. Mod. Phys. 66, 381 (1994).
  67. S. Havlin and D. Ben-Avraham, Diffusion in disordered media, Adv. Phys. 51, 187 (2002).
  68. A. V. Milovanov and J. J. Rasmussen, Turbulence spreading by resonant wave-wave interactions: A fractional kinetics approach, Phys. Rev. E 109, 045105 (2024).
  69. A. V. Milovanov, A. Iomin, and J. J. Rasmussen, Turbulence spreading and anomalous diffusion on combs, Phys. Rev. E 111, 064217 (2025).

Outline

Information

Phys. Rev. E 112, 034206– Published 5 September, 2025
  • Received 16 December 2024
  • Accepted 18 August 2025
Crossmark - Check for updates button
Creative Commons Logo

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. Open access publication funded by Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation