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A Rigorous Theory of Many-Body Prethermalization for Periodically Driven and Closed Quantum Systems

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Abstract

Prethermalization refers to the transient phenomenon where a system thermalizes according to a Hamiltonian that is not the generator of its evolution. We provide here a rigorous framework for quantum spin systems where prethermalization is exhibited for very long times. First, we consider quantum spin systems under periodic driving at high frequency \({\nu}\). We prove that up to a quasi-exponential time \({\tau_* \sim {\rm e}^{c \frac{\nu}{\log^3 \nu}}}\), the system barely absorbs energy. Instead, there is an effective local Hamiltonian \({\widehat D}\) that governs the time evolution up to \({\tau_*}\), and hence this effective Hamiltonian is a conserved quantity up to \({\tau_*}\). Next, we consider systems without driving, but with a separation of energy scales in the Hamiltonian. A prime example is the Fermi–Hubbard model where the interaction U is much larger than the hopping J. Also here we prove the emergence of an effective conserved quantity, different from the Hamiltonian, up to a time \({\tau_*}\) that is (almost) exponential in \({U/J}\).

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References

  1. Abanin D.A., De Roeck W., Ho W.W., Huveneers F.: Effective hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems. Phys. Rev. B 95, 014112 (2017)

    Article  ADS  Google Scholar 

  2. Abanin D.A., De Roeck W., Huveneers F.: Exponentially slow heating in periodically driven many-body systems. Phys. Rev. Lett. 115(25), 256803 (2015)

    Article  ADS  Google Scholar 

  3. Abanin D.A., De Roeck W., Huveneers F.: Theory of many-body localization in periodically driven systems. Ann. Phys. 372, 1–11 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  4. Aidelsburger M., Lohse M., Schweizer C., Atala M., Barreiro Julio T., Nascimbène S., Cooper N.R., Bloch I., Goldman N.: Measuring the chern number of hofstadter bands with ultracold bosonic atoms. Nat. Phys. 11(2), 162–166 (2015)

    Article  Google Scholar 

  5. Bambusi D., Graffi S.: Time quasi-periodic unbounded perturbations of schrödinger operators and kam methods. Commun. Math. Phys. 219(2), 465–480 (2001)

    Article  ADS  MATH  Google Scholar 

  6. Bourgain J., Wang W.-M.: Anderson localization for time quasi-periodic random schrödinger and wave equations. Commun. Math. Phys. 248(3), 429–466 (2004)

    Article  ADS  MATH  Google Scholar 

  7. Bru J.-B., Pedra W.: Lieb-robinson bounds for multi-commutators and applications to response theory. Springer Briefs in Math. Phys. 13, 1–110 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bukov M., D’Alessio L., Polkovnikov A.: Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to floquet engineering. Adv. Phys. 64(2), 139–226 (2015)

    Article  ADS  Google Scholar 

  9. Bukov M., Gopalakrishnan S., Knap M., Demler E.: Prethermal floquet steady states and instabilities in the periodically driven, weakly interacting bose-hubbard model. Phys. Rev. Lett. 115(20), 205301 (2015)

    Article  ADS  Google Scholar 

  10. Carati A., Galgani L., Maiocchi A., Gangemi F., Gangemi R.: Persistence of regular motions for nearly integrable hamiltonian systems in the thermodynamic limit. Regul. Chaot. Dyn. 21(6), 660–664 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Chudnovskiy A.L., Gangardt D.M., Kamenev A.: Doublon relaxation in the bose-hubbard model. Phys. Rev. Lett. 108(8), 085302 (2012)

    Article  ADS  Google Scholar 

  12. Combescure M.: Spectral properties of a periodically kicked quantum hamiltonian. J. Stat. Phys. 59(3-4), 679–690 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. D’Alessio Luca, Rigol Marcos: Long-time behavior of periodically driven isolated interacting lattice systems. Phys. Rev. X 4, 041048 (2014)

    Google Scholar 

  14. De Roeck W., Huveneers F.: Asymptotic localization of energy in nondisordered oscillator chains. Commun. Pure Appl. Math. 68(9), 1532–1568 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ducatez, R., Huveneers, F.: Anderson localisation for periodically driven systems. (2016). arXiv preprint arXiv:1607.07353

  16. Eckardt A., Anisimovas E.: High-frequency approximation for periodically driven quantum systems from a floquet-space perspective. New J. Phys. 17(9), 093039 (2015)

    Article  ADS  Google Scholar 

  17. Eliasson H.L., Kuksin S.B.: On reducibility of schrödinger equations with quasiperiodic in time potentials. Commun. Math. Phys. 286(1), 125–135 (2009)

    Article  ADS  MATH  Google Scholar 

  18. Fishman S., Grempel D.R., Prange R.E.: Chaos, quantum recurrences, and anderson localization. Phys. Rev. Lett. 49(8), 509 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  19. Goldman N., Dalibard J.: Periodically driven quantum systems: effective hamiltonians and engineered gauge fields. Phys. Rev. X 4(3), 031027 (2014)

    Google Scholar 

  20. Howland J.S.: Stationary scattering theory for time-dependent hamiltonians. Math. Ann. 207(4), 315–335 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  21. Imbrie J.Z.: On many-body localization for quantum spin chains. J. Stat. Phys. 163(5), 998–1048 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Kapitza P.L.: Dynamic stability of a pendulum with an oscillating point of suspension. J. Exp. Theor. Phys. 21(5), 588–597 (1951)

    Google Scholar 

  23. Kitagawa T., Berg E., Rudner M., Demler E.: Topological characterization of periodically driven quantum systems. Phys. Rev. B 82(23), 235114 (2010)

    Article  ADS  Google Scholar 

  24. Kuwahara T., Mori T., Saito K.: Floquet–Magnus theory and generic transient dynamics in periodically driven many-body quantum systems. Ann. Phys. 367, 96–124 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  25. Lazarides A., Das A., Moessner R.: Equilibrium states of generic quantum systems subject to periodic driving. Phys. Rev. E 90(1), 012110 (2014)

    Article  ADS  Google Scholar 

  26. Lazarides A., Das A., Moessner R.: Fate of many-body localization under periodic driving. Phys. Rev. Lett. 115(3), 030402 (2015)

    Article  ADS  Google Scholar 

  27. Lieb, E.H., Robinson, D.W.: The finite group velocity of quantum spin systems. Commun. Math. Phys. 28(3), 251–257 (1972). doi:10.1007/BF01645779

  28. Lindner N.H., Refael G., Galitski V.: Floquet topological insulator in semiconductor quantum wells. Nat. Phys. 7(6), 490–495 (2011)

    Article  Google Scholar 

  29. Magnus W.: On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7(4), 649–673 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mori T., Kuwahara T., Saito K.: Rigorous bound on energy absorption and generic relaxation in periodically driven quantum systems. Phys. Rev. Lett. 116(12), 120401 (2016)

    Article  ADS  Google Scholar 

  31. Nachtergaele B., Ogata Y., Sims R.: Propagation of correlations in quantum lattice systems. J. Stat. Phys. 124(1), 1–13 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Ponte P., Papić Z., Huveneers F., Abanin D.A.: Many-body localization in periodically driven systems. Phys. Rev. Lett. 114(14), 140401 (2015)

    Article  ADS  Google Scholar 

  34. Rahav S., Gilary I., Fishman Shmuel: Effective hamiltonians for periodically driven systems. Phys. Rev. A 68(1), 013820 (2003)

    Article  ADS  Google Scholar 

  35. Sensarma R., Pekker D., Altman E., Demler E., Strohmaier N., Greif D., Jördens R., Tarruell L., Moritz H., Esslinger T.: Lifetime of double occupancies in the Fermi–Hubbard model. Phys. Rev. B 82(22), 224302 (2010)

    Article  ADS  Google Scholar 

  36. Simon B.: The Statistical Mechanics of Lattice Gases, vol. 1. Princeton University Press, Princeton (2014)

    Google Scholar 

  37. Soffer, A., Wang, W.-M.: Anderson localization for time periodic random Schrödinger operators. Commun. Partial. Differ. Equ. 28(1–2), 333–347 (2003). doi:10.1081/PDE-120019385

  38. Ueltschi D.: Cluster expansions and correlation functions. Mosc. Math. J. 4(2), 511–522 (2004)

    MathSciNet  MATH  Google Scholar 

  39. Verdeny A., Mielke A., Mintert F.: Accurate effective hamiltonians via unitary flow in floquet space. Phys. Rev. Lett. 111(17), 175301 (2013)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Theoretical Physics, University of Geneva, Geneva, Switzerland

    Dmitry Abanin & Wen Wei Ho

  2. Instituut voor Theoretische Fysica, KU Leuven, Leuven, Belgium

    Wojciech De Roeck

  3. Université Paris-Dauphine, PSL Research University, CNRS, CEREMADE, 75016, Paris, France

    François Huveneers

Authors
  1. Dmitry Abanin
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  2. Wojciech De Roeck
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  3. Wen Wei Ho
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  4. François Huveneers
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Corresponding author

Correspondence to Wojciech De Roeck.

Additional information

Communicated by H. Spohn

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Abanin, D., De Roeck, W., Ho, W.W. et al. A Rigorous Theory of Many-Body Prethermalization for Periodically Driven and Closed Quantum Systems. Commun. Math. Phys. 354, 809–827 (2017). https://doi.org/10.1007/s00220-017-2930-x

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  • DOI: https://doi.org/10.1007/s00220-017-2930-x

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