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1 1 2 abstract: 1 [1] 1970 Furstenberg [2] 1 [3] 1979 Anderson [4] H = NX n=1 jn >V n <nj + NX n (jn ><n+1j + jn +1>< nj); fv n g HjΨ >= EΨ > (ffi n =<njψ >), ffi n+1 + ffi n 1 + V n ffi n = Effi n, M n ψ! ψ! ψ ffin +1 =Π N n=1 ffi M ffi1 E Vn 1 n ; M n = N ffi hyamada@cc.niigata-u.ac.jp! 1
2 2.1 Harper V n V n =2 cos(2ßffn + ) (ff ; 2 Q) [5] self-duality Herbert-Jones-Thouless ( )[6] =1 [7] Harper Soukoulis-Economou [8], V n =1:9[cos(Qn)+ 1 cos(2qn)](q =0:7), 3 V n = W j cos(2ßffn)j[9] Harper V n =2 cos(ßffn ν )(0 <ν<1)[] 2.2 :C n < V n 0 V n >= ffi n 0;n (C n ο e cn ) [11] c 1 random-dimer [12] fv A ;V B g V A! V A V A ;V B! V B. [13, 14] 1 (C n ο n s (s>0)) [15, 16] fv n g S(f) ο f ff (Fourier Filtering Method) ff ff >ff c ff 2 (E =0) ff 1: [17, 18] 0» ff» 1 2
3 Furstenberg 1» ff» 1:5 ff ' 1:5 ff 1 fv n g Furstenberg 3 X n+1 = X n +2 B 1 (1 2b)Xn B + b (0» X n < 1=2) X n 2 B 1 (1 2b)(1 X n ) B + b (1=2» X n» 1): [19] B b B 2 b = 13 i b =(B 1) 1 (2b) (1 B)=B [19]. W V n =(2X n 1)W fv n g 0» Xn < 1=2! V n = W 1=2» X n < 1! V n = W C(n) ο n B 1 2 B (n >>1) [19]. B < 2 B 2 b =0, S(f) ο f 2B 3 B 1 ff =2 B!1 B 2 2 B B 1 W! W( W! W ) m P (m) ο m B 1 B B <2 (< m><1) B 2 [20] 4 (B» 2), fl =< ln(jffi N+1j 2 +jffin j 2 ) 2N > < ::: > N!1 N [17, 18, 21] 3
4 Furstenberg [17] B <3=2 3=2 < B < 2 fx n g V n [22, 23] 5 (B 2) 2» B» 3 [21, 24] [25] 0x γ ξ W= W= E E 1: Lyapunov exponents fl as a function of energy, for several potential strength W within a range [0.01, 0.8]. The system and ensemble size are 2 20 and 50 respetively. (B =2:0.) 2: Localization length ο as a function of energy, for several potential strength W = 0:1; 0:3. The system and ensemble size are 2 22 and 50 respetively. (B =3:0.) B =2:0 fl E (W << 1) (E =0) fl B = 3:0 ( ) E =0 4
5 B= W^ γ γ 'B=1.1' '1.8' '2.3' '3.0' W^2/3 W~1/ W W 3: Lyapunov exponents at band center (E = 0) as a function of potential strength W for several bifurcation parameter B in symbolic cases. 4: Lyapunov exponents at band edge (E = 2:0) as a function of potential strength W for several bifurcation parameter B in symbolic cases. (E =0) fl W fl ο W 2 [26, 27] W (E =2:0) fl W (fl ο W 2=3 ) fl ο W 1=2 ( [28, 27] ) Van-Hope [24] fl B B = 2:0 fl ( 6) fl B W fl = C(W )B + D(W )(C; D ) fl(b Λ )=0 B Λ 7 B 2 B =2 fl>0 B =2 5
6 20x γ 15 5 W= x x γ 0.5 W= B B : Lyapunov exponents as a function of bifurcation parameter B, for several potential strength W in symbolic cases (E =0). The inset is expansion of a case, W =0:01. 6: Lyapunovexponents as a function of bifurcation parameter B, for several potential strength W in non-symbolic cases (E =0) B * W : Bifurcation parameter B Λ estimated by fl(b Λ )=0:0 for several potential strength W in symbolic (2) and non-symbolic cases (fl) ate =0. N W B fl N [17] [16] N, N 8 9 W N ο = N B =2:8 N ο / N ν ν<1 6
7 W= W= ln ξ ln ξ ln N ln N : Localization length ο as a function of system size N for several potential strength W at B=2.8. The line shows ο = N. 9: Localization length ο as a function of system size N for several potential strength W at B=3.0. The line shows ο = N. 9(B =3:0) W ο / N ν (ν ο 1) (W << 1) S(f) ο f 1:5 6 B» 2:8 (W << 1) S(f) ο f ff (ff ' 1:5) ff c 2( B!1 ) [15] [16] ff c =1:9 B fl hopping( 1 (off-diagonal model) stretched fl =0 [29] hopping 7
8 Long-tail r 1 [30, 31] 2 2N 2N [32, 33, 34] 4 [35] DNA A,T,G,C [36] [37] [38] well-define (W << 1) E c fl οje E c j ff ff [38] [39] [40] [1] P.W.Anderson, Phys. Rev. 9, 1492(1958). [2] H. Furstenberg, Trans. Am. Math. Soc. 8, 377(1963). [3] K. Ishii, Prog.Theor. Phys. Suppl. 53, 77(1973). [4] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673(1979). [5] P.G.Harper, Proc.Phys.Soc.London, Sect. A 68, 874(1955). [6] D.C.Herbert and R.Jones, J. Phys. C 4, 1145(1971); D.J.Thouless, ibid. 5, 77. [7] S.Aubry and G.Andre, Ann.Isr.Phys.Soc.3, 133(1980). 8
9 [8] C.M.Soukoulis and E.N.Economou, Phys. Rev. B 24, 5698(1981). [9] J.F.Weisz, M. Johansson and R. Riklund, Phys. Rev. B 41, 6032(1990); M. Johansson and Rolf Riklund, ibid, 42, 8244(1990). [] S.D.Sarma, S.He and X.C.Xie, Phys. Rev. Lett. 61, 2144(1988). [11] M. J. de Oliver and A. Petri, Phys. Rev. E 53, 2960(1996) ; ibid, 56, 215(1997). [12] D.H.Dunlap. H.-L.Wu and P.W.Phillips, Phys. Rev. Lett. 65, 88(1990). [13] H.-L.Wu and P.W.Phillips, Phys. Rev. Lett. 66, 1366(1991). [14] V. Bellani, E. Diez, R. Hey, L. Toni, L. Tarricone, G.B. Parravicini, F. Dominguez- Adame, and R. Gomez-Alcala, Phys. Rev. Lett. 82, 2159(1999). [15] F.A.B.F. de Moura and M.L.Lyra Phys. Rev. Lett. 81, 3735(1998); Physica A 266, 465(1999). [16] P. Carpena, P.B.Galvan, P.Ch.Ivanov and H.E.Stanly, Nature 418, 955(2002). [17] H. Yamada, M. Goda and Y. Aizawa, J. Phys.:Condens. Matter 3, 043(1991). [18] H. Yamada, M. Goda and Y. Aizawa, J. Phys. Soc. Jpn. 60, 3501(1991). [19] Y. Aizawa, C. Murakami and T. Kohyama, Prog. Theor. Phys. Suppl. 79, 96(1984). [20] K.Tanaka and Y. Aizawa, Prog. Theor. Phys. 90, 547(1993). [21] H.Yamada, J.Phys.Soc.Jpn.Suppl., to appear (2003). [22] A.K.Mohanty and A.V.S.S. Narayana Rao, Phys. Rev. Lett. 84, 1832(2000). [23] N.Scafetta, V.Latora, and P.Grigolini, Phys. Rev. E 66, (2002). [24] H. Yamada, in preparation. [25] S.Russ, J.W.Kantelhardt, A.Bunde and S.Havlin, Phys. Rev. B 64, (2001). [26] M.Kappus and F.J.Wegner, Z. Phys. B 45, 15(1981). [27] M.Yamanaka, Y.Avishai and M.Kohmoto, Phys. Rev. B 54, 228(1996). [28] B.Derrida and E.Gardner, J.Phys.(France),45, 1283(1984). [29] C.M.Soukoulis and E.N.Economou, Phys. Rev. Lett. 48, 43(1982). [30] B.L.Altshuler and L.S.Levitov, Phys. Rep. 288, 487(1997). 9
10 [31] L.Borland, L.G.Menchero and C.Tsallis, Phys. Rev. B 61, 1650(2000). [32] A. Crisanti, G. Paladin, and A. Vulpiani, Products of Random Matrices in Statistical Physics (Springer-Verlag, Berlin, 1993), and references therein. [33] H. Yamada and T. Okabe, Phys. Rev. E63, 26203(2001). [34] J. Heinrichs, Phys. Rev. B 66, (2002). [35] D.Porath, A.Bezryain,S.de Vaies and Cees Dekker, Nature 403, 635(2000). [36] D.Holste, I.Grosse and H.Herzel, Phys. Rev. E 64, (2001). [37] K. Iguchi, J.Phys.Soc.Jpn. 70, 593(2001). [38] F.M. Izrailev and A.A. Krokhin, Phys. Rev. Lett. 82, 4062(1999). [39] E.Abrahams, S.V.Kravchenko and M.P.Sarachik, Rev. Mod. Phys. 73, 251(2001). [40] V.N. Kuzovkov, W. Niessen, V.Kashcheyevs and O.Hein, J. Phys.:Condens. Matter 14, 13777(2002).
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