Abstract
We evaluate the localization length of the wave solution of a random potential characterized by an arbitrary autocorrelation function. We go beyond the Born approximation to evaluate the localization length using a non-linear approximation and calculate all the correlators needed for the localization length expression. We compare our results with numerical results for the special case, where the autocorrelation decays quadratically with distance. We look at disorder ranging from weak to strong disorder, which shows excellent agreement. For the numerical simulation, we introduce a generic method to obtain a random potential with an arbitrary autocorrelation function. The correlated potential is obtained in terms of the convolution between a Wiener stochastic potential and a function of the correlation.
Graphical abstract
Similar content being viewed by others
References
M. Cutler, N. Mott, Phys. Rev. 181, 1336 (1969)
D. Thouless,Ill-Condensed Matter (North-Holland, Amsterdam, 1979)
L. Berthier, G. Biroli, Rev. Mod. Phys. 83, 587 (2011)
J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clément, L. Sanchez-Palencia, P. Bouyer, A. Aspect, Nature 453, 891 (2008)
G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, M. Inguscio, Nature 453, 895 (2008)
J. Elson, J. Rahn, J. Bennett, Appl. Opt. 22, 3207 (1983)
S. John, Phys. Rev. Lett. 58, 2486 (1987)
J. Topolancik, B. Ilic, F. Vollmer, Phys. Rev. Lett. 99, 253901 (2007)
M. Hilke, Phys. Rev. A 80, 063820 (2009)
S. Karbasi, R.J. Frazier, K.W. Koch, T. Hawkins, A.M. John Ballato, Nat. Commun. 5, 3362 (2014)
S.E. Skipetrov, Nat. Nanotechnol. 9, 335 (2014)
R. Weaver, Wave Motion 12, 129 (1990)
D. Basko, I. Aleiner, B. Altshuler, Ann. Phys. 321, 1126 (2006)
J.H. Bardarson, F. Pollmann, J.E. Moore, Phys. Rev. Lett. 109, 017202 (2012)
S. Fishman, D. Grempel, R. Prange, Phys. Rev. Lett. 49, 509 (1982)
R. Brandenberger, W. Craig, Eur. Phys. J. C 72, 1881 (2012)
M. Hilke, Phys. Rev. B 78, 012204 (2008)
B. Huckestein, Rev. Mod. Phys. 67, 357 (1995)
P.W. Anderson, Phys. Rev. 109, 1492 (1958)
E. Abrahams, P. Anderson, D. Licciardello, T. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979)
H. Kunz, B. Souillard, Commun. Math. Phys. 78, 201 (1980)
J. Pendry, Adv. Phys. 43, 461 (1994)
H. Cao, Waves Random Media 13, R1 (2003)
L. Sanchez-Palencia, D. Clément, P. Lugan, P. Bouyer, G.V. Shlyapnikov, A. Aspect, Phys. Rev. Lett. 98, 210401 (2007)
M. Hilke, H. Eleuch, Ann. Phys. 529, 1600347 (2017)
F.A. de Moura, M.L. Lyra, Phys. Rev. Lett. 81, 3735 (1998)
F. Izrailev, A. Krokhin, Phys. Rev. Lett. 82, 4062 (1999)
H. Shima, T. Nomura, T. Nakayama, Phys. Rev. B 70, 075116 (2004)
F.M. Izrailev, A.A. Krokhin, N. Makarov, Phys. Rep. 512, 125 (2012)
H. Eleuch, M. Hilke, New J. Phys. 17, 083061 (2015)
P. Erdös, R. Herndon, Adv. Phys. 31, 65 (1982)
J.C. Flores, J. Phys.: Condens. Matter 1, 8471 (1989)
D. Dunlap, H.L. Wu, P. Phillips, Phys. Rev. Lett. 65, 88 (1990)
J. Flores, M. Hilke, J. Phys. A: Math. General 26, L1255 (1993)
A. Sánchez, E. Maciá, F. Domínguez-Adame, Phys. Rev. B 49, 147 (1994)
M. Hilke, J. Flores, Phys. Rev. B 55, 10625 (1997)
V. Bellani, E. Diez, R. Hey, L. Toni, L. Tarricone, G. Parravicini, Domínguez-Adame, R. Gómez-Alcalá, Phys. Rev. Lett. 82, 2159 (1999)
M. Hilke, J. Phys. A: Math. General 27, 4773 (1994)
M. Hilke, Phys. Rev. Lett. 91, 226403 (2003)
L. Tessieri, J. Phys. A: Math. General 35, 9585 (2002)
H. Eleuch, Y. Rostovtsev, M. Scully, Europhys. Lett. 89, 50004 (2010)
H. Eleuch, Y.V. Rostovtsev, J. Mod. Opt. 57, 1877 (2010)
A. Moroz, Europhys. Lett. 117, 40001 (2017)
H. Eleuch, Y. Rostovtsev, Europhys. Lett. 117, 40002 (2017)
H. Eleuch, M. Hilke, Results Phys. 11, 1044 (2018)
H. Makse, S. Havlin, H.E. Stanley, M. Schwartz, Chaos Solitons Fractals 6, 295 (1995)
O. Usatenko, S. Melnik, S. Apostolov, N. Makarov, A. Krokhin, Phys. Rev. E 90, 053305 (2014)
F. Li, H. Zheng, S.Y. Zhu, J. Mod. Opt. 63, 1340 (2016)
Author information
Authors and Affiliations
Corresponding author
Additional information
Contribution to the Topical Issue “Recent Advances in the Theory of Disordered Systems”, edited by Ferenc Iglói and Heiko Rieger.
Rights and permissions
About this article
Cite this article
Eleuch, H., Hilke, M. One dimensional localization for arbitrary disorder correlations. Eur. Phys. J. B 92, 269 (2019). https://doi.org/10.1140/epjb/e2019-100474-5
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2019-100474-5