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Nonlocal adiabatic response of a localized system to local manipulations

Nature Physics volume 11, pages 560–565 (2015)Cite this article

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Abstract

We examine the response of a system localized by disorder to a time-dependent local perturbation that varies smoothly with a characteristic timescale τ. We find that such a perturbation induces a nonlocal response, involving a rearrangement of conserved quantities over a length scale ∼ln τ. This effect lies beyond linear response, is absent in undisordered insulators and highlights the remarkable subtlety of localized phases. The effect is common to both single-particle and many-body localized phases. Our results have implications for numerous fields, including topological quantum computation in quantum Hall systems, quantum control in disordered environments, and time-dependent localized systems. For example, they indicate that attempts to braid quasiparticles in quantum Hall systems or Majorana nanowires will not succeed if the manipulations are performed asymptotically slowly, and thus using such platforms for topological quantum computation will require considerable engineering. They also establish that disorder-localized insulators suffer from a statistical orthogonality catastrophe.

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Figure 1: Schematic illustration of our protocol.
Figure 2: Charge transfer and the orthogonality catastrophe in an Anderson insulator.
Figure 3: Single-particle spectrum of a ten-site Anderson insulator, equation (3), in a given disorder realization as a function of time.
Figure 4: Exact (green, squares) and linear-response (blue, circle) answers for the ground-state charge density difference (equation (1)), δρ(r) for a metal and an Anderson insulator.
Figure 5: Schematic illustration of a proposal for topological quantum computation outlined in ref. 48.

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References

  1. Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958).

    Article  ADS  Google Scholar 

  2. Fleishman, L. & Anderson, P. W. Interactions and the Anderson transition. Phys. Rev. B 21, 2366–2377 (1980).

    Article  ADS  Google Scholar 

  3. Basko, D. M., Aleiner, I. L. & Altshuler, B. L. Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states. Ann. Phys. 321, 1126–1205 (2006).

    Article  ADS  Google Scholar 

  4. Altshuler, B. L., Gefen, Y., Kamenev, A. & Levitov, L. S. Quasiparticle lifetime in a finite system: A nonperturbative approach. Phys. Rev. Lett. 78, 2803–2806 (1997).

    Article  ADS  Google Scholar 

  5. Gornyi, I. V., Mirlin, A. D. & Polyakov, D. G. Interacting electrons in disordered wires: Anderson localization and low-T transport. Phys. Rev. Lett. 95, 206603 (2005).

    Article  ADS  Google Scholar 

  6. Imbrie, J. Z. On Many-Body Localization for Quantum Spin Chains. Preprint at http://arxiv.org/abs/1403.7837 (2014).

  7. Oganesyan, V. & Huse, D. A. Localization of interacting fermions at high temperature. Phys. Rev. B 75, 155111 (2007).

    Article  ADS  Google Scholar 

  8. Pal, A. & Huse, D. A. Many-body localization phase transition. Phys. Rev. B 82, 174411 (2010).

    Article  ADS  Google Scholar 

  9. Znidaric, M., Prosen, T. & Prelovsek, P. Many-body localization in the Heisenberg XXZ magnet in a random field. Phys. Rev. B 77, 064426 (2008).

    Article  ADS  Google Scholar 

  10. Nandkishore, R. & Huse, D. A. Many body localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys. 6, 15–38 (2015).

    Article  ADS  Google Scholar 

  11. Huse, D. A., Nandkishore, R. & Oganesyan, V. Phenomenology of fully many-body-localized systems. Phys. Rev. B 90, 174202 (2014).

    Article  ADS  Google Scholar 

  12. Serbyn, M., Papic, Z. & Abanin, D. A. Local conservation laws and the structure of the many-body localized states. Phys. Rev. Lett. 111, 127201 (2013).

    Article  ADS  Google Scholar 

  13. Nandkishore, R., Gopalakrishnan, S. & Huse, D. A. Spectral features of a many-body-localized system weakly coupled to a bath. Phys. Rev. B 90, 064203 (2014).

    Article  ADS  Google Scholar 

  14. Johri, S., Nandkishore, R. & Bhatt, R. N. Many-body localization in imperfectly isolated quantum systems. Phys. Rev. Lett. 114, 117401 (2015).

    Article  ADS  Google Scholar 

  15. Huse, D. A., Nandkishore, R., Oganesyan, V., Pal, A. & Sondhi, S. L. Localization-protected quantum order. Phys. Rev. B 88, 014206 (2013).

    Article  ADS  Google Scholar 

  16. Pekker, D., Refael, G., Altman, E., Demler, E. & Oganesyan, V. Hilbert-glass transition: New universality of temperature-tuned many-body dynamical quantum criticality. Phys. Rev. X 4, 011052 (2014).

    Google Scholar 

  17. Vosk, R. & Altman, E. Dynamical quantum phase transitions in random spin chains. Phys. Rev. Lett. 112, 217204 (2014).

    Article  ADS  Google Scholar 

  18. Kjall, J. A., Bardarson, J. H. & Pollmann, F. Many-body localization in a disordered quantum Ising chain. Phys. Rev. Lett. 113, 107204 (2014).

    Article  ADS  Google Scholar 

  19. Bauer, B. & Nayak, C. Area laws in a many-body localized state and its implications for topological order. J. Stat. Mech. 09, P09005 (2013).

    Article  MathSciNet  Google Scholar 

  20. Bahri, Y., Vosk, R., Altman, E. & Vishwanath, A. Localization and topology protected quantum coherence at the edge of ‘hot’ matter. Preprint at http://arxiv.org/abs/1307.4092 (2013).

  21. Chandran, A., Khemani, V., Laumann, C. R. & Sondhi, S. L. Many-body localization and symmetry-protected topological order. Phys. Rev. B 89, 144201 (2014).

    Article  ADS  Google Scholar 

  22. Nandkishore, R. & Potter, A. C. Marginal Anderson localization and many-body delocalization. Phys. Rev. B 90, 195115 (2014).

    Article  ADS  Google Scholar 

  23. Serbyn, M. et al. Interferometric probes of many-body localization. Phys. Rev. Lett. 113, 147204 (2014).

    Article  ADS  Google Scholar 

  24. Vasseur, R., Parameswaran, S. A. & Moore, J. E. Quantum revivals and many-body localization. Phys. Rev. B 91, 140202 (2015).

    Article  ADS  Google Scholar 

  25. Bardarson, J. H., Pollman, F. & Moore, J. E. Unbounded growth of entanglement in models of many-body localization. Phys. Rev. Lett. 109, 017202 (2012).

    Article  ADS  Google Scholar 

  26. Alessio, L. D. & Polkovnikov, A. Many-body energy localization transition in periodically driven systems. Ann. Phys. 333, 19–33 (2013).

    Article  ADS  MathSciNet  Google Scholar 

  27. Ponte, P., Chandran, A., Papic, Z. & Abanin, D. A. Periodically driven ergodic and many-body localized quantum systems. Ann. Phys. 353, 196–204 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  28. Anderson, P. W. Infrared catastrophe in Fermi gases with local scattering potentials. Phys. Rev. Lett. 18, 1049–1051 (1967).

    Article  ADS  Google Scholar 

  29. Gefen, Y., Berkovits, R., Lerner, I. V. & Altshuler, B. L. Anderson orthogonality catastrophe in disordered systems. Phys. Rev. B 65, 081106 (2002).

    Article  ADS  Google Scholar 

  30. Bliokh, K. Y., Bliokh, Y., P, Freilikher, V., Genack, A. Z. & Sebbah, P. Coupling and level repulsion in the localized regime: From isolated to quasiextended modes. Phys. Rev. Lett 101, 133901 (2008).

    Article  ADS  Google Scholar 

  31. Labonte, L., Vanneste, C. & Sebbah, P. Localized mode hybridization by fine tuning of two-dimensional random media. Opt. Lett. 37, 1946–1948 (2012).

    Article  ADS  Google Scholar 

  32. Altshuler, B. H., Krovi, H. & Roland, J. Anderson localization makes adiabatic quantum optimization fail. Proc. Natl Acad. Sci. USA 107, 12446–12450 (2010).

    Article  ADS  Google Scholar 

  33. Mahan, G. D. Many-Particle Physics (Springer Science, 2000).

    Book  Google Scholar 

  34. Vollhardt, D. & Wölfle, P. Diagrammatic, self-consistent treatment of the Anderson localization problem in d ≤ 2 dimensions. Phys. Rev. B 22, 4666–4679 (1980).

    Article  ADS  Google Scholar 

  35. Mott, N. Conduction in non-crystalline systems I. Localized electronic states in disordered systems. Phil. Mag. 17, 1259–1268 (1968).

    Article  ADS  Google Scholar 

  36. Aspect, A. & Inguscio, M. Anderson localization of ultracold atoms. Phys. Today 62 (8), 30–35 (2009).

    Article  Google Scholar 

  37. Billy, J. et al. Direct observation of Anderson localization of matter waves in a controlled disorder. Nature 453, 891–894 (2008).

    Article  ADS  Google Scholar 

  38. Kondov, S. S., McGehee, W. R., Zirbel, J. J. & DeMarco, B. Three-dimensional Anderson localization of ultracold matter. Science 334, 66–68 (2011).

    Article  ADS  Google Scholar 

  39. Roati, G. et al. Anderson localization of a non-interacting Bose–Einstein condensate. Nature 453, 895–898 (2008).

    Article  ADS  Google Scholar 

  40. Kondov, S. S., McGehee, W. R., Xu, W. & DeMarco, B. Evidence for many-body localization in an ultracold Fermi–Hubbard gas. Phys. Rev. Lett. 114, 083002 (2015).

    Article  ADS  Google Scholar 

  41. Ovadia, M. et al. Evidence for a finite temperature insulator. Preprint at http://arxiv.org/abs/1406.7510 (2014).

  42. Knap, M. et al. Time-dependent impurity in ultracold fermions: Orthogonality catastrophe and beyond. Phys. Rev. X 2, 041020 (2012).

    Google Scholar 

  43. Azbel, M. Ya. Eigenstates and properties of random systems in one dimension at zero temperature. Phys. Rev. B 28, 4106–4125 (1983).

    Article  ADS  Google Scholar 

  44. Pendry, J. B. Quasi-extended electron states in strongly disordered systems. J. Phys. C: Solid State Phys. 20, 733–742 (1987).

    ADS  Google Scholar 

  45. Lifshitz, I. M. & Kirpichenkov, V. Y. Tunnel transparency of disordered systems. Zh. Eksp. Teor. Fiz. 77, 989–1016 (1979); Sov. Phys. JETP 50, 499–511 (1979)

    Google Scholar 

  46. Iyer, S., Oganesyan, V., Refael, G. & Huse, D. A. Many-body localization in a quasiperiodic system. Phys. Rev. B. 87, 134202 (2013).

    Article  ADS  Google Scholar 

  47. Aubry, S. & André, G. Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Isr. Phys. Soc. 3, 133–164 (1980).

    MathSciNet  MATH  Google Scholar 

  48. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Sarma, S. D. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1051 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  49. Sau, J. D., Clarke, D. J. & Tewari, S. Controlling non-Abelian statistics of Majorana fermions in semiconductor nanowires. Phys. Rev. B 84, 094505 (2011).

    Article  ADS  Google Scholar 

  50. Alicea, J., Oreg, Y., Refael, G., Oppen, F. V. & Fisher, M. P. A. Non-Abelian statistics and topological quantum information processing in 1D wire networks. Nature Phys. 7, 412–417 (2011).

    Article  ADS  Google Scholar 

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Acknowledgements

We acknowledge useful conversations with B. L. Altshuler, P. W. Anderson, J. E. Avron, R. Bhatt, A. Elgart, M. S. Rudner and, especially, J. Chalker. We thank J. Chalker and D. Huse for comments on a draft. This work was supported by NSF Grant Numbers DMR 1006608, 1311781 and PHY-1005429 and the John Templeton Foundation (V.K. and S.L.S.) and by a PCTS fellowship (R.N.).

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Authors and Affiliations

  1. Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

    Vedika Khemani & S. L. Sondhi

  2. Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany

    Vedika Khemani & S. L. Sondhi

  3. Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA

    Rahul Nandkishore

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All authors contributed extensively to the work presented in the paper and the writing of the manuscript.

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Correspondence to S. L. Sondhi.

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Khemani, V., Nandkishore, R. & Sondhi, S. Nonlocal adiabatic response of a localized system to local manipulations. Nature Phys 11, 560–565 (2015). https://doi.org/10.1038/nphys3344

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