Abstract
We consider a quantum lattice system with infinite-dimensional on-site Hilbert space, very similar to the Bose–Hubbard model. We investigate many-body localization in this model, induced by thermal fluctuations rather than disorder in the Hamiltonian. We provide evidence that the Green–Kubo conductivity κ(β), defined as the time-integrated current autocorrelation function, decays faster than any polynomial in the inverse temperature β as \({\beta \to 0}\). More precisely, we define approximations \({\kappa_{\tau}(\beta)}\) to κ(β) by integrating the current-current autocorrelation function up to a large but finite time \({\tau}\) and we rigorously show that \({\beta^{-n}\kappa_{\beta^{-m}}(\beta)}\) vanishes as \({\beta \to 0}\), for any \({n,m \in \mathbb{N}}\) such that m−n is sufficiently large.
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References
Anderson P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109(5), 1492 (1958)
Fröhlich J., Spencer T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88(2), 151–184 (1983)
Basko D., Aleiner I., Altshuler B.: Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states. Ann. Phys. 321(5), 1126–1205 (2006)
Vosk R., Altman E.: Many-body localization in one dimension as a dynamical renormalization group fixed point. Phys. Rev. Lett. 110(6), 067204 (2013)
Pal A., Huse D.A.: Many-body localization phase transition. Phys. Rev. B 82(17), 174411 (2010)
Oganesyan V., Huse D.A.: Localization of interacting fermions at high temperature. Phys. Rev. B 75(15), 155111 (2007)
Imbrie, J., Spencer, T.: Personal communication and research talks (2012)
Linden N., Popescu S., Short A.J., Winter A.: Quantum mechanical evolution towards thermal equilibrium. Phys. Rev. E 79, 061103 (2009)
Lebowitz J.L., Goldstein S., Mastrodonato C., Tumulka R., Zanghi N.: On the approach to thermal equilibrium of macroscopic quantum systems. Phys. Rev. E 81, 011109 (2010)
Aizenman M., Warzel S.: Extended states in a Lifshitz tail regime for random Schrödinger operators on trees. Phys. Rev. Lett. 106(13), 136804 (2011)
Imbrie, J., Spencer, T.: Unpublished note (2013)
Oganesyan V., Pal A., Huse D.A.: Energy transport in disordered classical spin chains. Phys. Rev. B 80(11), 115104 (2009)
Basko D.: Weak chaos in the disordered nonlinear Schrödinger chain: destruction of Anderson localization by Arnold diffusion. Ann. Phys. 326(7), 1577–1655 (2011)
Fishman S., Krivolapov Y., Soffer A.: Perturbation theory for the nonlinear Schrödinger equation with a random potential. Nonlinearity 22(12), 2861 (2009)
Huveneers F.: Drastic fall-off of the thermal conductivity for disordered lattices in the limit of weak anharmonic interactions. Nonlinearity 26(3), 837–854 (2013)
Aubry S., André G.: Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc. 3(133), 18 (1980)
Iyer S., Oganesyan V., Refael G., Huse D.A.: Many-body localization in a quasiperiodic system. Phys. Rev. B 87, 134202 (2013)
Schiulaz, M., Müller, M.: Ideal quantum glass transitions: many-body localization without quenched disorder. arXiv:1309.1082 (2013)
De Roeck, W., Huveneers, F.: Search for many-body localization in translation-invariant systems (2014). arXiv:1405.3279
De Roeck, W., Huveneers, F.: Asymptotic localization of energy in non-disordered oscillator chains (2013). arXiv:1305.5127
Rumpf B.: Simple statistical explanation for the localization of energy in nonlinear lattices with two conserved quantities. Phys. Rev. E 69(1), 016618 (2004)
Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier’s law: a challenge to theorists. In: Fokas, A., Grigoryan, A., Kibble, T., Zegarlinsky, B. (eds) Mathematical Physics 2000. Imperial College, London, p. 128 (2000)
Nachtergaele, B., Sims, R.: Lieb–Robinson bounds in quantum many-body physics, “Entropy and the Quantum”, Contemporary Mathematics, vol. 529. Amer. Math. Soc., Providence, pp. 141–176 (2010). arXiv:1004.2086
Abdesselam A., Procacci A., Scoppola B.: Clustering bounds on n-point correlations for unbounded spin systems. J. Stat. Phys. 136(3), 405–452 (2009)
Cipriani, A., Pra, P.D.: Decay of correlations for quantum spin systems with a transverse field: a dynamic approach (2010). arXiv:1005.3547
Bodineau T., Helffer B.: The log-Sobolev inequality for unbounded spin systems. J. Funct. Anal. 166(1), 168–178 (1999)
Netočný, K., Redig, F.: Large deviations for quantum spin systems. J. Stat. Phys. 117(521) (2004)
Ueltschi D.: Cluster expansions and correlation functions. Moscow Math. J. 4, 511–522 (2004)
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De Roeck, W., Huveneers, F. Asymptotic Quantum Many-Body Localization from Thermal Disorder. Commun. Math. Phys. 332, 1017–1082 (2014). https://doi.org/10.1007/s00220-014-2116-8
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DOI: https://doi.org/10.1007/s00220-014-2116-8