Multifractal statistics of eigenstates of 2D disordered conductors

surface science ELSEVIER

Surface Science 361/362 (1996) 735-738

Multifractal statistics of eigenstates of 2D disordered conductors V l a d i m i r I. F a l ' k o

a,d,*, K.B.

E f e t o v b,c

• Max-Planck-lnsatutfar FestloJrperforschungHelsenbergstr. 1, 70569 Stuttgart, Germany b Max-Planck-lnstttutfar Physik Komplexer Systeme, Helsenbergstr. 1, 70569 Stuttgart, Germany * Landau htstitutefor Theoretical Physics, Chernogokmka, Russia d Institute of Solid State Physics, RAS, Chernogoiovka, Russia Received 21 June 1995; ~zepted for publication 25 September 1995

AImract We have studied the manifestation of pro-localized states in the distribution of local ampfitudes of wave functions of a 2D disordered metal. Although the distribution of comparatively small amplitudes obeys the universal laws known from the random matrix theory, its large-amplitude tails are non-universal and have a logarithmlcully-normal dependence. The inverse participation numbers calculated on the basis of the exact form of the distribution function in the weak localiTation regime indicate multifractal behavior. Our calculation is based on the derivation of the non-trivial saddle-point of the reduced supersymmetric ~-modeL

In recent years, the use of non-pertarbative approaches to the problem of quantum chaos and localization has resulted in a detailed understanding of quantum states in classically chaotic systems. In particular, it has been realized that the energy spectra and wave functions both in chaotic ballistic and weakly disordered systems obey universal statistical laws which are the same as those known in the random matrix theory [ 1] and which do not depend on the level of disorder and physical d/mensionality of the object I-2.3]. On the other hand, the properties of quantum states of disordered conductors where chaotic motion of a wave has diffusive character are affected by Anderson localization and have shown deviations from the above universality. That is to say that universal statistics is applicable to met* Corresponding author. Fax: +49 711 689 1010; e-mail: falko~daor.mpi-stuttgartanpg.de.

aUic-type states which equally test the details of a random potential all over the sample, whereas the development of localization has to be described beyond the random matrix theory approacll Below, we report the results of studies of wave functions in disordered 2D conductors that are smaller than the localization length (whatever it is) and discuss the states which are the precursors of localization at long distances. The statistical a n a l y s i s of wave functions consists in the calculation of the distribution function f ( 0 of local amplitudes t-I~(ro)l 2 and of the set of inverse participation numbers (IPN),

f(t) =(Vv) -x (~ $(t-,~k.(ro),2)$(e-e.))

(1)

t, = j? e'f(t) dt. Here ~k,(r) and e, are the eigenfunctions and eigen-

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V.L Fal'ko, K R Efetov/Surface Science 361/362 (1996) 735-738

736

f.(tv)

values of a confined system, < > means averaging over disorder, v is the density of states, Voc L2 is the system area. In a ballistic chaotic cavity where the wave functions extend over the whole system, one has Vt, oc V-", and the universality means that the distribution function scales as f(tV) and depends only on the fundamental symmetry of the system: orthogonal (with time-reversal symmetry), unitary (subjected to a magnetic field) or symplectic (with sufficient spin-orbit coupling). The goal of our analysis is to detect deviations of the distribution function f(t) from the generalized universal (Porter-Thomas) form [1], especially at the tails where tV>>1. Thus, we find statistically averaged envelopes of the states which are responsible for that deviation (pre-localized states). The latter information is implicit in the correlation function

R(t, r)

=

Our theory is based on the use of the suporsymmetry technique [2]. Analytical study of wave functions using this method makes it possible to calculate the distribution function of local amplitudes, f(t), as a whole [4] rather than having to reconstruct it from the full set of IPN's (as has been done before [3,13]). At the same time, the supersymmetric a-model can be solved 'using the saddle-point method [5]. To study the statisticsof quantum states in an isolated system, we have developed a reduced version of the a-model [6] which, on the one hand, has a very clear saddlepoint all over the metallic r e , me and, on the other hand, enables us to assess a statistically averaged form of pre-localized states (i.e. R(t,r)/f(t) for t>> V -t) directly from the form of solution of the saddle-point equation [6,7]. The length of this report does not allow us to enter into the details of calculations. The latter are available in Refs. [6,7]. Below, we sketch out the results and try to interpret them. The use of the saddie-point method enables us to determine the distribution function f(t) for all amplitudes t < (2vl) -t. A typical form of the distribution function (in the logarithmic scale) is shown in Fig. 1 which illustrates the results for the orthogonal

SYSTEM

nol s y m m e t r y

0.01 0.001 0.0001

L,A- 1oo; 2 ~ o - a L/I-5~, 2~vo-16

O.COO01

L,/I-25; 2 ~ 0 - 3 2 0.000001

x

0.0000001

Pocter-Thomo~l X "~

lb

2b

3b

tV

Fig. 1. Distribution function of the local dentfity of the wave

function, f(t), for the orthogonal symmetry class. The input parameters are given in the picture: these are the ratio of the sample size to the length of the mean free path and the conductance per square, 2narD, meamLrod in quantum units.

(Vv) -t

x <~ [~=(ro+r)128(t-l~=(ro)12)8(e-e,)).

2D

0.1

.~

symmetry class, since it models chaotic microwaves in a thin resonator of a slab geometry which has been studied experimentally [8]. The dashed curve in the figure demonstrates the universal PorterThomas (PT) statistics, foa~r)~(t)=e-'V/2~, whereas the solid lines are drawn for diffusive systems. Fig. 1 shows that, close to the origin, the PT statistics give a satisfactory approximation for all samples, but fails for large values of t. This can be better described in the asymptotic limits: of small, t < , and large t > 2n'2vD/(Vln(L/l)) amplitudes. In the asymptotic re#rues and with the exponential accuracy, all three fundamental symmetry classes can be described simultaneously, and

rexp(-flvt[1- T/2 +...]), f ,,, V { / fln'2vD 2 'X T-

T<
tV in(L/O 2~zvD '

where the parameter fl distinguishes between unitary (flffi~i= 1), orthogonal (flort= 1/2) and symplectic (fl~= = 2) ensembles. On the basis of the expressions in Eq. (2), one can conclude that disorder makes the appearance of high-amplitude splashes of wave functions much more probable than one would expect on the basis of the PT formulae. Moreover, the similarity between the large-t asymptotes in Eq. (2) and the

V.l.. Fal'ko, KB. Efetov/Surface Science 361/362 (1996) 735-738

log-normal tails of the distribution of fluctuations of the local density of states discovered in Ref. [9] reveals their deep relationship. It seems that they are both due to localization effects. But the tails of the states which are responsible for these rare events do not decay exponentially, as do tails of a wave function confined in the well or of loealiTexl states in a one-dimensional wire. Even in the asymptotic region T>> 1, the size L of the system influences the distribution The long-range structure of pre-localiTed states (under the localization-length scale) can be anticipated after one has calculated the cross-correlation function R(t, r). It turns out that the statistically averaged envelope of the tail of a state associated with a large-amplitude splash Id/(ro)la=t>>l/V follows the form of the optimal solution (saddiepoint) of the supersymmetric field theory and its form can be approximated by [~(r) 120c e -°'~., ~. (l/r) 2~, where [6,7]

z(T) # - - 2 ln(L/l)'

e" z

= T-

tV ln(L/O 2;r2vD

The exponents # = #(t)< 1 in this expression are individual for each amplitude t, and I¢,(r)] 2 tends to approach the r -2 dependence when t increases up to tm~ = (12F)-l/in(L/1). On the other hand, some information about the short-range structure can be anticipated from the value of the maximal density t~,.~ which can be described in the framework of our theory. The heart of the matter is that the stronger localization of a state is considered, the shorter scale of distances is involved in the formation of the optimal configuration of the a-model, i.e. of the pre-loealized state. The conventional formulation of the a-model allows us to study the system at distances not shorter than the mean free path I. If the prelocalized states we are discussing were isotropic, the maximal density which could be described must be smaller than 1-2, whereas we can show [6] that our scheme of calculus leads to t~x = (2e)-l/ln(L/l) which is approximately the density of a state bound to the forward-and-backward scattered trajectory within the mean free path length. This is only possible if the states concerned axe locally anlsotropic. On the fine-scale of dis-

737

tances of ca. 1 they typically have a snake-like structure. In some sense, the anomalous events of high-amplitude splashes of ¢(r) we predict in this paper are analogous to the scars of wave functions of chaotic ballistic billiards found by Heller [ 10] and by other workers [ 11 ], although in our case of disordered conductors they have a stochastic nature. Also, the statistics of local amplitudes of wave functions which we obtained cont~in.~ somewhat more detailed information about the structure of quantum states in classically diflhsive media. That is, using the full form of the distribution function f(t) at any value of the variable t, we can show that these states represent the class of physical objects which are multifractal [12]. The ideas of fractal and multifractal analysis have already penetrated the theory of disordered systems which characterize the complexity of wave functions in the vicinity of metal-insulator transitions [13]. Approaching the transition from the metallic side, one deals with the system where the localization length is always infinite, so that the inverse participation numbers t. should obey a power-law dependence on the size L of the system, t,(e) oc L -'t")-2,

x(n) = (n -- 1)d*(n).

(3)

In a 3D metal far from the critical conditions, the fractal dimension d* coincides with real physical dimensions: d* = d = 3. The criticality shows up as the set of dimensions d* which differ from the physical one and, as observed in numerical simulations [14], depend on n. According to common belief [13], the dimension d = 2 is critical for the Anderson transition model, and from this point of view our approach allows us to calculate the full spectrum of d*(n). To find the moments t, accurately enough, we have to take into account that tmiversal statistics fail unless the condition tV<<2V~£T--I) is satisfied [see Eq. 2 ~ . Hence, only the first few ratios of t,, 2
El. Fal'ko, K R Efetov/Surface Science 361/362 (1996) 735-738

738

ways are in g o o d agreement with each other and, in the order cited, take the f o r m

[3] ICB. Efetov and V.N. Prigodin, Phys. Rev. Lett. 70 (1993) 1315. [4.1 VJ. FaI'ko and K.B. Efetov, Phy~ Rev. B 50 (1994) 11267.

[5.1 B~. Muzykantsk/i and D.E. Khmelnitskii, Phys. Rev. B 51 (1995) 5480.

tn ~,

1(, - 1X2- J*)

L - is- 1)d*- 2,

fl - I n d*(n) ~, 2 - 4T~.~v----~,

(4)

which indicates multifractal behavior. Note: the multifractality seems to be a generic p r o p e r t y of 2 D disordered systems, since this result is valid for all fundamental s y m m e t r y classes: the p a r a m eter fl distinguishess between unitary (fl,~i = 1), o r t h o g o n a l (flo~t = 1/2) a n d symplectic ( f t , = = 2) ensembles.

Acknowledgements The authors t h a n k P. Fulde for continuous encouragement, a n d V.F. acknowledges partial s u p p o r t f r o m N A T O C R G 921333.

References [1] T.A. Brody et aL Rev. Mod. Phys. 53 (1981) 385. [2"1 K.B. Efetov, Adv. Phys. 32 (1983) 53.

[6] V1 FaTko and ICB. Efetov, Cond-mat/9503096, submitted to PRL. [7.1 V.I. Fal'ko and K.B. Efetov, Cond-mat/9507091, submitted to PRB. [8] A. Kudrolli, V. Kidambi and S. Sridhar, Phys. Rev. I.ett. 75 (1995) 822. [9.1 B.L. Altshuler, V.E. Kravtsov and LV. Lerner, in: Mesoscopic Phenomena in Solids, Eds. B. Altshuler et al~ (Elsevier, Amsterdam 1991) p. 449. [10.1 E.J. Heller, Phys. Rev. Lett. 53 (1984) 1515; Phys. Rev. A 35 (1987) 1360, E3. Heller, P.W. O'Connor and J. Gehlen, Phys. Scr. 40 (1989) 354. [11] E.G. Bogomolnyi, Physica D 31 (1988) 169;, M.V. Berry, Proc. Roy. Soc. London A 423 (1989) 219. [12] B.B. Mandelbrot, The Fractal Geometry of Nature, (W.H. Freeman, San Francisco 1983). [13] F. Wegner, Z. Phys. B 36 (1980) 209; C. Castellani and L. Pelifi, J. Phys. A 19 (1986) L429. [14] H. Aoki, J. Phys. C 16 (1983) L205; Phys. Rev. B 33 (1985) 7310, M. Schreiber, J. Phys. C 18 (1985) 2493; B. Kramer et al., Sur~ Sci. 196 (1988) 127. [15] Y.V. Fyodorov and A.D. Mirlin, Phys. Rev. B 51 (1995) 13403.