Abstract
We consider a one dimensional many body fermionic system with a large incommensurate external potential and a weak short range interaction. We prove, for chemical potentials in a gap of the non interacting spectrum, that the zero temperature thermodynamical correlations are exponentially decaying for large distances, with a decay rate much larger than the gap; this indicates the persistence of localization in the interacting ground state. The analysis is based on the renormalization group, and convergence of the renormalized expansion is achieved using fermionic cancellations and controlling the small divisor problem assuming a Diophantine condition for the frequency.
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References
Anderson P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492 (1958)
Froehlich J., Spencer T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88, 151 (1983)
Aizenman M., Molchanov S.: Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157, 245 (1993)
Aubry S., André G.: Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Isr. Phys. Soc 3, 133 (1980)
Dinaburg E., Sinai Y.: The one-dimensional Schrdinger equation with a quasiperiodic potential. Funct. Anal. Appl. 9, 279 (1975)
Pastur L.: Spectra of random self-adjoint operators. Russ. Math. Surveys 28, 1 (1973)
Eliasson L.H.: Floquet solutions for the 1-dimensional quasi-periodic Schroedinger equation. Commun. Math. Phys 146, 447 (1992)
Bellissard J., Lima R., Testard D.: A metal-insulator transition for the almost Mathieu model. Commun. Math. Phys. 88, 207 (1983)
Sinai Ya.: Anderson Localization for one dimensional difference Schroedinger operator with quasiperiodic potential. J. Stat. Phys. 46, 861 (1987)
Froehlich J., Spencer T., Wittvwer T.: Localization for a class of one-dimensional quasi-periodic Schrdinger operators. Commun. Math. Phys. 132(1), 5 (1990)
Avila, A., Jitomirskaya, S.: The ten Martin problem. Ann. Math. 170, 303 (2009)
Fleishmann L., Anderson P.W.: Interactions and the Anderson transition. Phys. Rev. B 21, 2366 (1980)
Giamarchi T., Schulz H.J.: Localization and interaction in onedimensional quantum fluids. Europhys. Lett. 3, 1287 (1987)
Gornyi I., Mirlin A., Polyakov D.: Interacting electrons in disordered wires. Phys. Rev. Lett. 9, 206603 (2005)
Vidal J., Mouhanna D., Giamarchi T.: Correlated Fermions in a one-dimensional quasiperiodic potential. Phys. Rev. Lett. 83, 3908 (1999)
Basko D.M., Alteiner I., Altshuler B.L.: Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states. Ann. Phys. 321, 1126 (2006)
Oganesyan V., Huse D.A.: Localization of interacting fermions at high temperature. Phys. Rev. B 75, 155111 (2007)
Znidaric M., Prosen T., Prelovek P.: Localization in the Heisenberg Magnet in a Random Field. Phys. Rev. B 77, 064426 (2008)
Pal A., Huse D.A.: Many-body localization phase transition. Phys. Rev. B 82, 174411 (2010)
Ros, V., Mueller, M., Scardicchio, A.: Integrals of motion in the many-body localized phase. Nucl. Phys. B 420 (2015)
Nandkshore R., Huse D.A.: Many-body localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys. 6, 15 (2015)
Iyer S., Oganesyan V., Refael G., Huse D.A.: Many-body localization in a quasiperiodic system. Phys. Rev. B 87, 134202 (2013)
Klein A., Perez J.F.: Localization in the ground state of a disordered array of quantum rotators. Commun. Math. Phys. 147, 241 (1992)
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. 147(1), 99 (1990)
Fauser M., Warzel S.: Multiparticle localization for disordered systems on continuous space via the fractional moment method. Rev. Math. Phys. 27, 1550010 (2015)
Imbrie, J.: On many-body localization for quantum spin chains. arXiv:1403.7837
Spencer, T.: Talk at a conference in Rome (2012)
De Roeck W., Huveneers F.: Asymptotic quantum many-body localization from thermal disorder. Commun. Math. Phys. 332(3), 1017 (2014)
Gallavott G.: Twistless KAM tori. Commun. Math. Phys. 164(1), 145 (1994)
Gentile G., Mastropietro V.: Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications. Rev. Math. Phys. 8(3), 393 (1996)
Benfatto, G., Gentile, G., Mastropietro, V.: Electrons in a lattice with an incommensurate potential. J. Stat. Phys. 89, 655 (1997)
Gentile G., Mastropietro V.: Anderson localization for the Holstein model. Commun. Math. Phys. 215, 69 (2000)
Mastropietro V.: Small denominators and anomalous behaviour in the incommensurate Hubbard–Holstein model. Commun. Math. Phys. 201, 81 (1999)
Roati G., D’Errico C., Fallani L., Fattori M., Fort C., Zaccanti M., Modugno G., Modugno M., Inguscio M.: Anderson localization of a non-interacting Bose–Einstein condensate. Nature 453, 895 (2008)
Benfatto G., Falco P., Mastropietro V.: Universality of one-dimensional Fermi systems, I. Response functions and critical exponents. Commun. Math. Phys. 330, 153 (2014)
Benfatto G., Mastropietro V.: Renormalization group, hidden symmetries and approximate Ward identities in the XYZ model. Rev. Math. Phys. 13, 1323 (2001)
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Communicated by R. Seiringer
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Mastropietro, V. Localization in the Ground State of an Interacting Quasi-Periodic Fermionic Chain. Commun. Math. Phys. 342, 217–250 (2016). https://doi.org/10.1007/s00220-015-2498-2
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DOI: https://doi.org/10.1007/s00220-015-2498-2