Abstract
We study a one-dimensional quantum system with an arbitrary number of hard-core particles on the lattice, which are subject to a deterministic attractive interaction as well as a random potential. Our choice of interaction is suggested by the spectral analysis of the XXZ quantum spin chain. The main result concerns a version of high-disorder Fock-space localization expressed here in the configuration space of hard-core particles. The proof relies on an energetically motivated Combes–Thomas estimate and an effective one-particle analysis. As an application, we show the exponential decay of the two-point function in the infinite system uniformly in the particle number.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abdul-Rahman, H., Nachtergaele, B., Sims, R., Stolz, G.: Entanglement dynamics of disordered quantum XY chains. Lett. Math. Phys. 106(5), 649–674 (2016)
Abdul-Rahman, H., Nachtergaele, B., Sims, R., Stolz, G.: Localization properties of the XY spin chain. A review of mathematical results with an eye toward many-body localization. To appear in: Annalen der Physik (2017). arXiv:1610.01939 [math-ph]
Aizenman, M.: Localization at weak disorder: Some elementary bounds. Rev. Math. Phys. 6(5A), 1163–1182 (1994)
Aizenman, M., Graf, G.M.: Localization bounds for an electron gas. J. Phys. A 31(32), 6783 (1998)
Aizenman, M., Warzel, S.: Localization bounds for multiparticle systems. Commun. Math. Phys. 290, 903–934 (2009)
Aizenman, M., Warzel, S.: Random operators: Disorder effects on quantum spectra and dynamics. Volume 168 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2015)
Altman, E., Vosk, R.: Universal dynamics and renormalization in many-body-localized systems. Ann. Rev. Condens. Matter Phys. 6(1), 383–409 (2015)
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)
Bols, A., De Roeck, W.: Asymptotic localization in the Bose–Hubbard model (2016, preprint). arXiv:1612.04731
Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Probability and Its Applications. Birkhäuser Boston Inc, Boston(1990)
Chulaevsky, V., Suhov, Y.: Multi-particle Anderson localisation: Induction on the number of particles. Math. Phys. Anal. Geom. 12, 117–139 (2009)
Chulaevsky, V., Suhov, Y.: Multi-scale analysis for random quantum systems with interaction. Progress in Mathematical Physics vol. 65. Birkhäuser/Springer, New York (2014)
Ducatez, R.: Anderson localisation for infinitely many interacting particles in Hartree-Fock theory. J. Spectr. Theory (2017, to appear). arXiv:1602.02896
Elgart, A., Klein, A., Stolz, G.: Many-body localization in the droplet spectrum of the random XXZ quantum spin chain (2017, preprint). arXiv:1703.07483
Fischbacher, C.: On the spectrum of the XXZ spin chain. TMP Master Thesis (2013)
Fischbacher, C., Stolz, G.: The infinite XXZ quantum spin chain revisited: structure of low lying spectral bands and gaps. Math. Model. Nat. Phenom. 9(5), 44–72 (2014)
Hamza, E., Sims, R., Stolz, G.: Dynamical localization in disordered quantum spin systems. Commun. Math. Phys. 315(1), 215–239 (2012)
Imbrie, J.Z.: Diagonalization and many-body localization for a disordered quantum spin chain. Phys. Rev. Lett. 117, 027201 (2016)
Imbrie, J.Z.: On many-body localization for quantum spin chains. J. Stat. Phys. 163(5), 998–1048 (2016)
Imbrie, J. Z., Ros, V., Scardicchio, A.: Review: local integrals of motion in many-body localized systems. Annalen der Physik (2017, to appear). arXiv:1609.08076
Klein, A., Perez, J.F.: Localization in the ground-state of the one-dimensional \(X\)–\(Y\) model with a random transverse field. Commun. Math. Phys. 128(1), 99–108 (1990)
Mastropietro, V.: Localization of interacting Fermions in the Aubry–André model. Phys. Rev. Lett. 115, 180401 (2015)
Mastropietro, V.: Localization in interacting Fermionic chains with quasi-random disorder. Commun. Math. Phys. 351(1), 283–309 (2017)
Nachtergaele, B., Spitzer, W., Starr, S.: Droplet excitations for the spin-\(1/2\) \(XXZ\) chain with kink boundary conditions. Ann. Henri Poincaré 8(1), 165–201 (2007)
Nachtergaele, B., Starr, S.: Droplet states in the \(XXZ\) Heisenberg chain. Commun. Math. Phys. 218(3), 569–607 (2001)
Nandkishore, R., Huse, D.A.: Many-body localization and thermalization in quantum statistical mechanics. Ann. Rev. Condens. Matter Phys. 6(1), 15–38 (2015)
Seiringer, R., Warzel, S.: Decay of correlations and absence of superfluidity in the disordered Tonks–Girardeau gas. New J. Phys. 18(3), 035002 (2016)
Sims, R., Warzel, S.: Decay of determinantal and Pfaffian correlation functionals in one-dimensional lattices. Commun. Math. Phys. 347(3), 903–931 (2016)
Žnidarič, M., Prosen, T., Prelovšek, P.: Many-body localization in the Heisenberg XXZ magnet in a random field. Phys. Rev. B 77, 064426 (2008)
Acknowledgements
We thank B. Nachtergaele for proposing the problem and for illuminating discussions during his stay at TUM as a John von Neumann Fellow. We are indebted to the referee for valuable comments. V.B. was supported by grant P2EZP2_162235 of the Swiss National Science Foundation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Vieri Mastropietro.
Rights and permissions
About this article
Cite this article
Beaud, V., Warzel, S. Low-Energy Fock-Space Localization for Attractive Hard-Core Particles in Disorder. Ann. Henri Poincaré 18, 3143–3166 (2017). https://doi.org/10.1007/s00023-017-0591-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-017-0591-0