[en] We study the Anderson transition on a generic model of random graphs with a tunable branching parameter 1 < K < 2, through large scale numerical simulations and finite-size scaling analysis. We find that a single transition separates a localized phase from an unusual delocalized phase that is ergodic at large scales but strongly nonergodic at smaller scales. In the critical regime, multifractal wave functions are located on a few branches of the graph. Different scaling laws apply on both sides of the transition: a scaling with the linear size of the system on the localized side, and an unusual volumic scaling on the delocalized side. The critical scalings and exponents are independent of the branching parameter, which strongly supports the universality of our results.
Research Center/Unit :
CESAM - Complex and Entangled Systems from Atoms to Materials - ULiège
Disciplines :
Physics
Author, co-author :
Garcia-Mata, Ignacio
Giraud, Olivier
Georgeot, Bertrand
Martin, John ; Université de Liège > Département de physique > Optique quantique
Dubertrand, Rémy ; Université de Liège > Département de physique > Optique quantique
Lemarié, Gabriel
Language :
English
Title :
Scaling Theory of the Anderson Transition in Random Graphs: Ergodicity and Universality
Publication date :
17 April 2017
Journal title :
Physical Review Letters
ISSN :
0031-9007
eISSN :
1079-7114
Publisher :
American Physical Society, Ridge, United States - New York
Volume :
118
Pages :
166801
Peer reviewed :
Peer Reviewed verified by ORBi
Tags :
CÉCI : Consortium des Équipements de Calcul Intensif
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Bibliography
R. V. Jensen and R. Shankar, Phys. Rev. Lett. 54, 1879 (1985); PRLTAO 0031-9007 10.1103/PhysRevLett.54.1879
J. M. Deutsch, Phys. Rev. A 43, 2046 (1991); PLRAAN 1050-2947 10.1103/PhysRevA.43.2046
M. Srednicki, Phys. Rev. E 50, 888 (1994); PLEEE8 1063-651X 10.1103/PhysRevE.50.888
P. Calabrese and J. Cardy, Phys. Rev. Lett. 96, 136801 (2006); PRLTAO 0031-9007 10.1103/PhysRevLett.96.136801
M. Rigol, V. Dunjko, and M. Olshanii, Nature (London) 452, 854 (2008). NATUAS 0028-0836 10.1038/nature06838
P. W. Anderson, Phys. Rev. 109, 1492 (1958). PHRVAO 0031-899X 10.1103/PhysRev.109.1492
E. Abrahams, 50 Years of Anderson Localization (World Scientific, Singapore, 2010), Vol. 24.
F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008). RMPHAT 0034-6861 10.1103/RevModPhys.80.1355
P. Jacquod and D. L. Shepelyansky, Phys. Rev. Lett. 79, 1837 (1997). PRLTAO 0031-9007 10.1103/PhysRevLett.79.1837
D. Basko, I. Aleiner, and B. Altshuler, Ann. Phys. (Amsterdam) 321, 1126 (2006). APNYA6 0003-4916 10.1016/j.aop.2005.11.014
A. Pal and D. A. Huse, Phys. Rev. B 82, 174411 (2010). PRBMDO 1098-0121 10.1103/PhysRevB.82.174411
R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015). ARCMCX 1947-5454 10.1146/annurev-conmatphys-031214-014726
E. Altman and R. Vosk, Annu. Rev. Condens. Matter Phys. 6, 383 (2015). ARCMCX 1947-5454 10.1146/annurev-conmatphys-031214-014701
R. Abou-Chacra, D. Thouless, and P. Anderson, J. Phys. C 6, 1734 (1973). JPSOAW 0022-3719 10.1088/0022-3719/6/10/009
K. B. Efetov, Sov. Phys. JETP 61, 606 (1985). SPHJAR 0038-5646
M. R. Zirnbauer, Phys. Rev. B 34, 6394 (1986); PRBMDO 0163-1829 10.1103/PhysRevB.34.6394
M. R. Zirnbauer Nucl. Phys. B265, 375 (1986). NUPBBO 0550-3213 10.1016/0550-3213(86)90316-0
C. Castellani, C. Di Castro, and L. Peliti, J. Phys. A 19, L1099 (1986). JPHAC5 0305-4470 10.1088/0305-4470/19/17/009
A. D. Mirlin and Y. V. Fyodorov, J. Phys. A 24, 2273 (1991); JPHAC5 0305-4470 10.1088/0305-4470/24/10/016
Y. V. Fyodorov and A. D. Mirlin, Phys. Rev. Lett. 67, 2049 (1991); PRLTAO 0031-9007 10.1103/PhysRevLett.67.2049
A. D. Mirlin and Y. V. Fyodorov J. Phys. I (France) 4, 655 (1994).
A. D. Mirlin and Y. V. Fyodorov, Phys. Rev. Lett. 72, 526 (1994). PRLTAO 0031-9007 10.1103/PhysRevLett.72.526
C. Monthus and T. Garel, J. Phys. A 42, 075002 (2009). JPAMB5 1751-8113 10.1088/1751-8113/42/7/075002
C. Monthus and T. Garel, J. Phys. A 44, 145001 (2011). JPAMB5 1751-8113 10.1088/1751-8113/44/14/145001
G. Biroli, A. Ribeiro-Teixeira, and M. Tarzia, arXiv:1211.7334.
A. De Luca, B. L. Altshuler, V. E. Kravtsov, and A. Scardicchio, Phys. Rev. Lett. 113, 046806 (2014). PRLTAO 0031-9007 10.1103/PhysRevLett.113.046806
V. Kravtsov, I. Khaymovich, E. Cuevas, and M. Amini, New J. Phys. 17, 122002 (2015). NJOPFM 1367-2630 10.1088/1367-2630/17/12/122002
B. Altshuler, E. Cuevas, L. Ioffe, and V. Kravtsov, Phys. Rev. Lett. 117, 156601 (2016). PRLTAO 0031-9007 10.1103/PhysRevLett.117.156601
K. Tikhonov, A. Mirlin, and M. Skvortsov, Phys. Rev. B 94, 220203 (2016). PRBMDO 2469-9950 10.1103/PhysRevB.94.220203
E. Tarquini, G. Biroli, and M. Tarzia, Phys. Rev. Lett. 116, 010601 (2016). PRLTAO 0031-9007 10.1103/PhysRevLett.116.010601
C. Monthus, J. Stat. Mech. (2016) 093304. JSMTC6 1742-5468 10.1088/1742-5468/2016/09/093304
D. Facoetti, P. Vivo, and G. Biroli, Europhys. Lett. 115, 47003 (2016). EULEEJ 0295-5075 10.1209/0295-5075/115/47003
K. Tikhonov and A. Mirlin, Phys. Rev. B 94, 184203 (2016). PRBMDO 2469-9950 10.1103/PhysRevB.94.184203
M. Mézard, G. Parisi, and M.-A. Virasoro, Spin Glass Theory and Beyond. (World Scientific Publishing Co., Inc., Pergamon Press, Singapore, 1990).
B. Derrida and H. Spohn, J. Stat. Phys. 51, 817 (1988). JSTPBS 0022-4715 10.1007/BF01014886
J. D. Miller and B. Derrida, J. Stat. Phys. 75, 357 (1994). JSTPBS 0022-4715 10.1007/BF02186867
A. M. Somoza, M. Ortuno, and J. Prior, Phys. Rev. Lett. 99, 116602 (2007). PRLTAO 0031-9007 10.1103/PhysRevLett.99.116602
C. Monthus and T. Garel, Phys. Rev. B 80, 024203 (2009). PRBMDO 1098-0121 10.1103/PhysRevB.80.024203
G. Biroli (private communication).
C.-P. Z. Zhu and S.-J. Xiong, Phys. Rev. B 62, 14780 (2000). PRBMDO 0163-1829 10.1103/PhysRevB.62.14780
C.-P. Z. Zhu and S.-J. Xiong, Phys. Rev. B 63, 193405 (2001). PRBMDO 0163-1829 10.1103/PhysRevB.63.193405
O. Giraud, B. Georgeot, and D. L. Shepelyansky, Phys. Rev. E 72, 036203 (2005). PRESCM 1539-3755 10.1103/PhysRevE.72.036203
E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979). PRLTAO 0031-9007 10.1103/PhysRevLett.42.673
J. Pichard and G. Sarma, J. Phys. C 14, L127 (1981). JPSOAW 0022-3719 10.1088/0022-3719/14/6/003
A. MacKinnon and B. Kramer, Phys. Rev. Lett. 47, 1546 (1981). PRLTAO 0031-9007 10.1103/PhysRevLett.47.1546
K. Slevin and T. Ohtsuki, Phys. Rev. Lett. 82, 382 (1999). PRLTAO 0031-9007 10.1103/PhysRevLett.82.382
J. Chabé, G. Lemarié, B. Grémaud, D. Delande, P. Szriftgiser, and J. C. Garreau, Phys. Rev. Lett. 101, 255702 (2008). PRLTAO 0031-9007 10.1103/PhysRevLett.101.255702
D. J. Watts and S. H. Strogatz, Nature (London) 393, 440 (1998). NATUAS 0028-0836 10.1038/30918
See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.118.166801 for more information about the topological properties of the random graphs considered, the recursion equations used in the glassy physics approach, and the scaling analysis performed in this work, which includes Refs. [43,44].
B. Bollobás and W. Fernandez de la Vega, Combinatorica 2, 125 (1982). COMBDI 0209-9683 10.1007/BF02579310
E. Bogomolny and O. Giraud, Phys. Rev. E 88, 062811 (2013). PRESCM 1539-3755 10.1103/PhysRevE.88.062811
M. V. Feigelman, L. B. Ioffe, and M. Mézard, Phys. Rev. B 82, 184534 (2010). PRBMDO 1098-0121 10.1103/PhysRevB.82.184534
B. Derrida and D. Simon, Europhys. Lett. 78, 60006 (2007). EULEEJ 0295-5075 10.1209/0295-5075/78/60006
Note that in the scaling law (1), the scaling parameter (Equation presented) is to be distinguished from the localization length (Equation presented). In particular, they have different critical exponents (Equation presented) and (Equation presented), respectively (see also Ref. [16]).
M. Bollhöfer and Y. Notay, Comput. Phys. Commun. 177, 951 (2007). CPHCBZ 0010-4655 10.1016/j.cpc.2007.08.004
S. Dorogovtsev, Lectures on Complex Networks (Oxford University Press, New York, 2010).
A. Rodriguez, L. J. Vasquez, K. Slevin, and R. A. Römer, Phys. Rev. B 84, 134209 (2011). PRBMDO 1098-0121 10.1103/PhysRevB.84.134209
R. Dubertrand, I. García-Mata, B. Georgeot, O. Giraud, G. Lemarié, and J. Martin, Phys. Rev. Lett. 112, 234101 (2014); PRLTAO 0031-9007 10.1103/PhysRevLett.112.234101
R. Dubertrand, I. García-Mata, B. Georgeot, O. Giraud, G. Lemarié, and J. Martin Phys. Rev. E 92, 032914 (2015). PRESCM 1539-3755 10.1103/PhysRevE.92.032914
B. L. Altshuler, L. B. Ioffe, and V. E. Kravtsov, arXiv:1610.00758.
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