Download pdf - Delocalization transition of two-dimensional disordered systems in a strong magnetic field

Z. Phys. B 90. 427440 (1993) Condensed

Zeitschrift M a t t e r f~r Physik B �9 Springer-Verlag 1993

Delocalization transition of two-dimensional disordered systems in a strong magnetic field B. Mieck* Max-Planck-Institut f/ir Kernphysik, Postfach 103980, W-6900 Heidelberg, Germany

Received: 7 August 1992

Abstract. The delocalization transition in two-dimensional systems and a strong magnetic field is investigated with respect to its dependence on the Landau band index j and on the type of disorder. The generation of random potentials according to a given correlation funct ionfand for a chosen correlation length d is described. The spectral properties of random eigenvalue sequences are examined as measures for the extension of wavefunctions and indicate a nonuniversal delocalization behaviour in higher Landau bands for short ranged correlated potentials. The critical exponents of the localization length of wavefunctions are determined for rapidly varying potentials in the second lowest Landau band (j = 1) and depend on the correlation length d of the disorder. This different critical behaviour compared to that in the lowest band is confirmed by calculations for the density-density correlations of wavefunctions at the centers of the Landau levels. Calculations in different geometries also show that the critical systems of delocalized states are conformal invariant in the case of the nonuniversal delocalization transition (d ~< lo), whereas such local rescaling properties cannot be expected for slowly varying potentials.

1. Introduction

Conventional Anderson localization theory allows only strong or weak localization in two-dimensional disordered systems [ 1,2] whereas the presence of a sufficiently strong magnetic field causes a diverging localization length ~ of the wavefunctions at the centers Ej = h ~ ( j + �89 of disorder broadened Landau bands (for a review see [3]). This delocalization phenomenon has been established both experimentally [4-7] and theo- retically [10-24] by a variety of methods and has been examined with respect to a phase transition and a universal critical behaviour of the energy dependent localization length ~ (E) oc [ E - Ej ] - v. In the numerical sim-

* Present address: Department of Physics, University of Maryland, College Park, MD 20742, USA

ulations of this paper we consider very different kinds of disordered potentials and therefore test to what extent universality does hold in the presence of a strong magnetic field. Furthermore, the critical systems at the centers of the bands are investigated with respect to the conformal transformation which maps a cylinder into a disk geometry.

The first experiments for the determination of the exponent v used a temperature-driven scaling of effective sample size [4-6]. They gave evidence for a delocalization transition, but could not clarify the question of a universal behaviour (independence of Landau band index j and type of disorder). In these experiments one can measure only a combined exponent ~ c = p / ( 2 v ) so that, in order to obtain the exponent v, the knowledge of the exponent p for the inelastic scattering time is involved. Since the exponent p may even depend on the strength of disorder [8, 9], values for v are difficult to estimate from these experimental data [6]. Recently, direct mea- surements of the exponent v, including only elastic effects, have become available by studying the size dependent width of the delocalization transition (sharpness of the step-like behaviour of axy or the width of axx around the centers Ej of the Landau bands versus the sample size) [7]. The results indicate a universal exponent v~2.3 in the three lowest Landau bands, except for a value of v ~ 6.5 which has been attributed to a non-spin- split Landau band ( j = 2) in the case of a low mobility.

Various numerical simulations [15-24] as well as a semiclassical approximation [13] have confirmed a value of v ~2.3 or v = 7/3. However, we have pointed out recently [23, 24] that such a universal behaviour has to be expected in the lowest Landau band, as well as in higher Landau bands if the random potential varies only on a larger scale than the magnetic length. In this case the modified percolation picture [ 17, 18] and the semiclassical approach [13, 14] can be applied because the fast cyclotron motion can be separated from the drifting motion along the equipotential lines of the hills and valleys of the impurity potential. In the opposite limit of a rapidly varying potential and higher Landau bands the delocal-

428

ization transition appears to be nonuniversal and the sep- aration of the cyclotron and drifitng motion is no longer valid. Furthermore, the extended states show peculiar multifractal properties [26] which have not been observed in the semiclassical limit. Experimental hints for such a nonuniversal behaviour have only been obtained from measuring the combined exponent a:=p/(2v) [5] in- volving the unknown exponent p.

The numerical simulations in this paper are therefore used to test whether such a nonuniversal behaviour is possible for certain kinds of disorder. Since a proper comparison of delocalization properties relies on an exact and systematic projection of various random potentials onto a single Landau band, we describe in Sect. 2 how to generate various kinds of disorder for single bands, which are characterized by a correlation function f, and study the second moments of their probability distribution. This is of crucial importance in order to classify and discern relevant cases where a different delocalization behaviour may occur. The results of the numerical simulations for these cases are presented in Sect. 3. We show that the different critical behaviour of the localizaion length in the first Landau band ( j -- 1) can be recognized by com- paring fluctuation properties of eigenvalues with the lowest Landau level (j = 0). The density-density correlation function of the wavefunctions is calculated in the first band ( j = 1) and yields new values for the critical exponent v which depend on the ratio of the correlation length d to the magnetic length l 0. Computations at the centers of the bands support the point of view of a nonuniversal delocalization transition in the case of a small correlation length d because the delocalized states are characterized by different exponents r/ of the density- density correlation function depending on the Landau band under consideration. These calculations at the centers of the bands are also performed in different geometries and show the conformal invariance of the critical systems for short ranged correlated potentials. In the case of a larger correlation length (d>lo) the extended wavefunctions connect distant saddle points of the disordered potential along equipotential lines so that a local scale invariance cannot be expected because of the nonlocal properties of the critical systems at the centers of the bands.

2. Generation of disorder in a strong magnetic field

We consider a one particle Hamiltonian with potential A in Landau gauge and a random, distributed, zero-centered potential V(r)

the vector Gaussian-

H = ( p - e A ) 2 / 2 m + V(r) A = B ( 0 , x , 0) r. (2.1)

The second moment of V(r) is defined by a correlation function so that various kinds of disorder can be specified

V(r0 V(r2 ) = 22f (irl _ r21 IN). (2.2)

Apart from the functional form of f, there are two pa- rameters, the strength 3~ of the random potential and the

correlation length d. We restrict ourselves to the case of a very strong magnetic field so that the condition ~. 10 1 ~hO) is satisfied (102 = h/eB, o) = eB/m); furthermore we require that the integral over the correlation function in (2.2) is normalized to unity. The spatial variation of V(r) is characterized by the correlation length d. Weak disorder can be represented by a slowly varying potential with a long-ranged correlation function or a large correlation length d whereas strong disorder can be imagined as a rapidly varying potential with a sufficiently small correlation length d. The last case is usually preferred in field theoretical models because such a potential can be realized by the white-noise distribution which yields independent, Gaussian distributed potential values at each space point

1 ~ [ V ] = J U e x p I - ~ d 2 r [ V ( r ) ] 2 1 . (2.3)

In general the probability distribution is not accessible because one has to construct the inverse of the correlation function

I 1 ~,~[ V] =JKexp - 2 ~ X j" d2rld2r2 g(rl)

X ~ ( [r I -- r 2 [/d) g(r2) 1 (2.4)

(rt - r2) -= f d2rf ( [r, - r I/d) 57( [r - r 21/d). (2.5)

The explicit construction of the probability distribution can be avoided by taking the Fourier transform of the random potentials V(r0, V(r2) in the second moment (2.2)

1 V ( r ) = ~ 5 d ike- ikr lT(k) 17*(k)= 17(-k) (2.6)

V(rO V(r2)

1 -- (27r)2 5 d2kl d2k2e-lkl~-ik2'~il~(kl) 17(k2)

~2 -2zr ~ d2kf ( lk ld / l~ ik (,,-,2) (2.7)

(In the remainder length scales, wave vectors and momenta are expressed with respect to the magnetic length /0; energies and the strength parameter ~ of the potential are scaled by the cyclotron energy hog, ~ = (2 .lol)/ho).) Due to the translation invariance of the correlation function f, the second moment of the Fourier transform 17(k) is independently distributed with a weight given by the Fourier transform f of the correlation function

J,~(kl) l~(k2) = 27/7 ~'27( I k 1 [d/lo) a (k 1 -{- k2). (2.8)

We introduce a new scaled Fourier transform l?(k)= (2 7~)1/2~ [f(lk[ d/lo)]l/2O(k) so that the potential V(r), distributed according to (2.2), can be generated from in-

dependent random variables/7(k)

v(r)

_ ( 2 ~ ) ~ J" d2k[f(Ikl d/l~

~* (k l ) /7 (k2) = d (k I - - k2 ) .

(2.9)

(2.1o)

This simple relation is very useful for numerical simulations without a magnetic field because the Fourier transform requires a negligible amount of operations compared to the calculations of measures for the degree of localization. Thus various kinds of correlation functions can be investigated with respect to a universal or nonuniversal critical behaviour of the localization length.

In the case of a strong magnetic field we have to pro- ject the Hamiltonian (2.1) and the relation (2.9) onto a single or a few Landau bands. In Landau gauge the wave functions are given by

1 ~gw (x, y) - ,_/7- e 'wyz; (x - w)

Xj- (x) = (2Jj !~21/2) - 1/2e-X2/2Hj (X), (2.11)

where we have introduced dimensionless, continuous wave vectors w, labelling the degenerate, internal states of single Landau bands j. A cylinder geometry with periodic boundary conditions in y-directions is chosen so that a periodicity of the circumference Ly = N o �9 l o restricts w to the discrete values w~ = 2 ~- n / N o, n ~ Z. The quantum numbers w, shift the wavefunction in x-direction so that in a numerical simulation for a single Landau band, including up to M internal states, a cylinder of width Ly = N o �9 l o and length L~ = 2 ~. M / N o �9 l o is investigated. The subdivision of a length unit in x-direction becomes smaller with increasing length in y-direction so that one needs No2/2 ~ states in order to simulate a square sample of size Ly • Ly.

The transformation of the Hamiltonian (2. l) and the relation (2.9) into Landau band states yields

( j w l Hl j " w" ) = ( j + l /2),Sji, ,5 ( w - w' )

+<jw I V l j ' w ' > (2.12)

_ ~" ( jw] V l j ' w ' ) (2~),/2 j ' d2k f f ( l k ld /10 ) ] ~/2

• ( j w I exp ( - ik- r) I J ' w' ) ~ (k), (2.13)

so that the matrix elements ( jw] V ] j ' w ' ) for a given subspace j , j" can be generated from the random variables /7(k) if one knows in (2.13) the matrix elements of the plane wave operator ( jw [ exp (ik. r) [j ' w' ) . In [ 11,24] it has been shown how to separate the matrix elements ( j w I exp (ik- r) [j ' w' ) by the introduction of magnetic translation operators ~/, ~ i n t o two matrix polynomials depending on the internal states w, w' and the subspace j , j "

429

( j w I exp (ik- r) I J ' w' ) = U~. (k) Wjj. (k) (2.14)

U,~, (k) = exp {ikx. (w + w') /2}

• (2.15)

Wjj" (k)=iAa (2 +Aa) ! e - Itl2/4 ( I k [/]~-)A~

TAX (1 e-iaz .~o ~z ,~[kl 2) (2.16)

2 = M i n ( j , j ' ) a x = j ' - j (2.17)

A ~ = I ~ I t a n ~ = ~ / k x .

Due to the delta function in the magnetic translation operator Uw~, (k) for the internal states, there are correlations along the diagonals kyo = w' - w of the matrix elements ( j , w] V I j ' , w + kyo) whereas elements of different diagonal fields kyo, k',o are independently distributed. Substituting (2.15, 2.16) into (2.14), the relation (2.13) for generating matrix elements within a single Lan- dau band becomes

( j , wl VI j , w + k y o )

- ~ ~ dk~ (2 x)1-7~ _

x e-i~'(w+k'o/2)Lj (�89 2) e-1/4, ikl ?

X [ f ( [k [ d/lo)] 1/2/7 (kx, ky0)" (2.18)

A further Fourier transformation of/7(k) shows that it suffices to generate for each diagonal or correlation field ky o a single Gaussian distributed random vector w (px, k~0)

1 /7(kx, kyo)=(2r01/~ 2- ~ dpxeikxpxw(p~,k~) (2.19)

(2.20) w* (px, k,o) = w (px, - IC~o)

w* (Px~, kyl) w (Px2, ky2)

=fi (Pxl--Px2) fi (ky,-- ky2) (2.21)

(j,w[ Vlj, w§

_

- (2 701/5 ~ dPx C ( p x - w - kyo/2, kyo)

• w (Px, kyo) (2.22)

c (px, ky)

_ 1 �88 [k1'- - - ( 2 7 0 1 / 2 ~ dkxei~x'Pxe

• Lj (�89 2) [ f ( I k l d / l o ) ] 1/~ . (2.23)

Thus the exact generateion of the random potential according to f (2.2) and to the projection onto subspaces

430

can be performed by a matrix multiplication of the random vector w(p~,ky) with the correlation matrix C(p~, ky) in (2.22). In the following we choose the functional form of the correlation function to be a Gaussian

1 120 -~lo2/d21rl-r212 f(Ir~-r21l~ 2n d 2 e (2.24)

1 -�89 Ik/ f ( l k l d / l o = ~ e

so that the variation of the correlation length d between 0 ~ oc allows the simulation of models ranging from the white-noise to a slowly varying potential. The projection of the Gaussian (2.24) onto higher Landau bands (j > 1) implies as well a test of a set of correlation functions gj (Ir~- r2l ) for the lowest Landau band which have a different functional form in coordinate space. This can be seen from the integrand in (2.23) containing the La- guerre polynomial Lj( l lk I ~) which can be introduced by the projection onto Landau band j (as described above) or by the Fourier transform fi If we choose the Fourier transform in (2.23) to be of the form

f (Ikldllo)=gj(lkl) 1 2 2 - ~ d l/d-Ikl =

=[Lj(�89 Ikl:)] 2 e , (2.25)

and if we take as the subspace the lowest Landau band Lo(�89 2) = 1 in (2.23), then we perform in the lowest Landau band numerical simulations for a different spatial correlation function f=gi , but whose corresponding correlation matrix C(p~, ky) in (2.23) is identical to the case of Gaussian correlations (2.24) in coordinate space, projected onto Landau level with index j. By similar pro- jections one can obtain identical correlation matrices C(p~,ky) from different correlation functions in coordinate space. Such modified correlation functions & ( [ r l - r2l ) for the lowest Landau band have a Gaussian structure but with extra nodes in their spatial dependence. The functional form follows from the Fourier transform of ~j. ( [k l ) in (2.25)

oo

gj-(lrl--r2 ) = ~ dkkg, j(k)Jo(k. ] r~-r2]) o

(2.26)

1 1 l g 1/2.lUd2.r 2 gj (r) = 2 n 22j d 2 e

( 2 : ) ( 2 j - 2 n ) (1-2lg/d2)Zn • n=O J - - n

Io 'r • (2a2(1 _ 1/2.d2/12)]. (2.27)

1.0 ~

0.8 '

- 0 .2 h ', ;J .~ ~ - 0.2

- 0.4 t I I I I - 0.4 0.0 0.1 0.2 0.3 0.4 0.5 0.0

~" -------4

M~)/Mo) g~(~)/Mo) 1 . 0 ~ ~ 1 . 0 ,

0.6 --~,! e 0.6

0 . 4 - ~'~ 0.4

0.2 ~ 0.2

o o 0 0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0

Fig. 1 a -d . Modif ied correlat ion functions gj ( I rl - r21 )- The correlation functions gj (r)/gj (0) are displayed for three values of j and four values o f the correlat ion length d/l o (see case a to d, solid line: j = 0 corresponding to the Gaussian correlation function (2.24), dashed dot ted line: j = 1, dashed line: j = 2). In the case j ~ 1 and

g,(T)/M0) 1.0 -,~\ .....

) 0.2 , x

0.0 :---=--

I"" I L. I 0.5 1.0 1.5 2.0 2.5

,'%

"\ "\ d ' \ \ \

- 0 \ " \ \ \

2.0 4.0 6.0 8.0 10.0 T ----+

d/lo<, 1 the spatial correlat ion funct ion takes also negative values, a new feature which cannot be realized in the pure Gaussian case (2.24) and which may be the cause of the different delocalization behaviour repor ted in Sect. 3

ta.) j = O, d / l o = 0.0!

(mtll'lwz} (we+ ~.wlllw~ - l - ~ w ~

o : o ~ g - ~ ~ o.o

' - - ' 3 : o ' ~ ~ ~ 3.o 4.0 ~ 4 .0

[c.) j = 1, d/la = 0.0l

(,<lVIw,}

2.0 �84

1.5 �84

O 1.0 X

0 . 5

0.0

- 0 . 5

0

'e.) j = I, d/lo = 0.5i

{~dI

x

.'4

Fig. 2a-L Second moments of matrix elements for single Landau bands. The distribution of second moments in the lowest Landau band has a Gaussian form for the overall size as well as for the correlations among matrix elements in a diagonal (a). A change of the correlation length just causes a different scale in the distributions (compare a and b). The next Landau band j = 1 shows significant nodes for a small correlation length and a much broader

431

b.) j = o, d/lo = 2,0~

(uql-~-rw:

1.2

�9 ~ 1.0

c, 0.8

x 0.6

0.4

0.2

0.0

(

a.) j = 1, a/to = 2.0]

(w,I

0 .0

f.) j = 2, d / lo = o.oi

(w~illw~)

1.6

1.2

c~ 0.8 •

0.4

0.0

-0.4

C

b.U u,v

).0

distribution which indicates more nonlocal couplings than in the lowest band (see e and e). This different behaviour becomes even more pronounced in higher Landau bands (f). But in the case of a large correlation length the distributions tend to a Gaussian in all Landau bands and yield a universal delocalization behaviour (compare b and d)

In the case o f a small correlation length d / l o , these modified correlation functions gj force the random potential to alter its sign on spatially short-ranged scales (see Fig. l a - d ) . Numerica l s imulat ions for higher Landau bands and a small correlation length are therefore an

important test o f universality in the case o f the lowest Landau band and spatial correlation functions which have a different form than just a pure Gaussian.

432

Distribution of matrix elements in single Landau bands

In the following we only consider the Gaussian correlation function (2.24), but keep in mind that numerical simulations for higher Landau bands allow for conclu- sions of the critical behaviour in the lowest Landau band and different types of correlation functions as well. The second moments of single Landau bands are displayed in the Fig. 2a- f for various bands index j and various correlation lengths d/I o in order to extract cases of interest for our simulations. The displayed distributions of second moments follow from the relations (2.13, 2.14) and (2.10) which combine to yield

(Z w~ I Vlj; w~ y (A w21 Vlj~ w;)

- 2 ~ I d2kf(Ikld/lo)

• Wj l j i (k)Uw2wl(-k)Wj2A(-k) . (2.28)

Restricting to a single Landau band, the dky integration in (2.28) can be performed easily and gives a delta-function 6 ((% - w~) - (w~ - w2)) which causes correlations among the matrix elements of a diagonal. The qualitative influence of such correlations on the delocalization transition has been investigated separately [24], in this paper we are interested in the actual distributions which are given by two exponentials and a sum over Laguerre polynomials (ky o = W~ -- W 2 Aw = w 2 -- w[)

<jwl I V l jw; ) (jw21 V l jw; ) ~2

- (2 rc)37f fi ((W 1 -- W;) -- (W; -- W2) )

• + 2 2 2 d /lo).kyo/2}

• e x p { - (awy/(2 (1 + d2/lg))}

VII + d2/8

(2.29)

j jMin(2k,2m) 1 ) l ( 2 ~ ) ( 2 7 ) • Z ( - - - k=0 m=0 1=0

l ' ( k @ m - l ) ' ( d2/l 2 )rn+k+, x ~m~. 7 +a~ll~

• L5--~12),--,o (ka --,/~-" -.,r!-- 1/2)m (/<:2o/2) r(- , /2) // 1 CAw) 2

• 2 4 7 ~ - - 2 (d2/lX Ti 2/lg ) '

The exponential with the argument (1 + d2/lg)k2o/2 de- termines the overall size of the matrix elements whereas the second exponential with the argument l(Aw)2/(1 + d2/lg) gives the strength of the correlations. The exponentials dominate over the polynomials for sufficiently large d/l o (see Fig. 2d) so that a universal Gaussian distribution and a universal delocalization behaviour of all Landau bands is obtained because similar distributions of matrix elements result necessarily in the same delocalization properties and the same critical exponent v

(compare Fig. 2b and d). The generic distribution for large d/l o is always realized in the lowest Landau band

(jwa I Vljw[ > (jw21 Wljw;> ~2

- ( 2 ~)375 a ( (W 1 - - W~) - - (W~ - - W2) )

• 2 2 2 d /l~).kyo/2 }

• exp { - (Aw)2/(2 (1 + d2/12o))}

[/1 + d2/lg

(2.30)

Various numerical simulations [15-23] have shown that the distribution (2.30) yields a critical exponent v ~2.3. According to these simulations, a postulated universality can only refer to a parameter independence on the correlation length d. Different situations arise for a short correlation length d~ l o and higher Landau bands. In Fig. 2c, e, f one can notice how in the case of a higher Landau band index j > 1 the Laguerre polynomials introduce significant nodes into the second moments which then depend on the index j of the Landau level under consideration. These are the cases where numerical simulations become of interest in order to test for a deviation from the universal delocalization with exponent v ,-~2.3.

3. Nonuniversal delocalization in higher Landau bands

Numerical simulations have been performed for the Gaussian correlation function (2.24) and two values of the correlation length (d/Io=O.O, d/lo=0.5 ) for the Landau band with j = 1. The distributions of second moments (w 11 V I w2) (w2 + Aw I V[ w 1 + A w) for these cases show considerable deviations from the situations realized for a large correlation length or for the lowest Landau band (compare Fig. 2c, e with 2a, b). This choice is a compromise because computations in higher Landau levels become much more time consuming, due to the broader distribution of the squares of matrix elements. Different kinds of measures are used to detect the delocalization transition around the centers of the bands. The first part of our simulations involves only the diagonali- zation of random matrices, generated according to (2.22), and the calculation of fluctuation properties of random eigenvalue sequences which contain also information about the extension of the corresponding wavefunctions. In the second part of this section the average localization length is obtained from the calculation of the density- density correlation function by employing finite size scaling methods. The simulations at the centers of the bands are presented in the third part and are investigated with respect to a conformal invariance.

3.1. Spectral properties and the delocalization transition

In order to compare given eigenvalue sequences E i of different kinds of random matrices, it is necessary to perform a transformation so that the energy dependence of the mean level density 13 (E) is removed and the average level spacing of neighbouring eigenvalues becomes con-

stant. This is accomplished by convolution of Ez with 9 (E) which is either known analytically or obtained by averaging the spectrum generated numerically

Ei

X~= ~ d E p ( E ) . (3.1) --oo

There are two specific nearest-neighbour spacing distributions p (S), S = Xi+ 1 - Xi of the transformed eigenvalues Xi which indicate localized or delocalized states, respectively [24]. For the eigenvalue sequence of extended states one expects p (S) to have the form of the Wigner distribution for the Gaussian unitary ensemble because the magnetic field breaks time-reversal invariance p ( S ) o c S 2 exp ( - 4/zr- $2). This is because extended states 'talk' to each other and repel each other such that the probability for an accidental degeneracy becomes small. The spectrum appears to be rather rigid. In the case of localized states the spacings of neighbouring eigenvalues tend to a Poisson distribution p (S)oc exp ( - S). The spectra of localized states show large fluctuations and usually a high probability for an approximated degeneracy of the transformed eigenvalues X~ because the spectra can be viewed as being a superposition of several independent spectra corresponding to states that are localized in different regions of space. Another measure for the strength of fluctuations or the rigidity of a spectrum is given by the quantity Z "2 (L). Due to the convolution

433

(3.1), a chosen interval of length L contains on the average L transformed eigenvalues X~. The quantity 272 (L) measures just the standard deviation from this mean value

M X2(L) = ~, ( N ~ - L ) Z / M , (3.2)

i=l

where N~ counts the number of eigenvalues in the interval i and an average over M random samples is implied. For localized states corresponding to a Poisson type of distribution, there are strong fluctuations so that a linear increase of X 2 (L) versus L can be expected. In the case of extended states or a rather rigid spectrum, a weaker dependence of ~ 2 (L) versus L is realized which may be approximated by a logarithmic dependence as for the classical ensembles like the GUE, GOE or the GSE.

In Figs. 4 a - d and 6a -d we display the nearest neighbour distribution p (X, S) and 222 (X, L) for j = 1 and a vanishing correlation length d / l o = 0.0. In comparison the same quantities have been calculated for the lowest Lan- dau band j = 0 (see Figs. 3a -d and 5a-d). According to the considerations above, one can recognize the localization transition around the centers of the band by a change from a Poisson to a Wigner type distribution and by a change from a linear to a logarithmic dependence of 222 (L) so that the nearest neighbour distribution and the standard deviation have to be calculated locally around an interval of the transformed eigenvalue X~. The

p(x,s) [2) j=o, L~=5o.Olo I p(x,s) [ b . ) j = U ~ : - ~

9• ~ 4 . 0 40

3 26255 t 2 0 " 3 O 5 ~ H , , ~ X~=287

6 96 ~ 0 O

p(x,s) ~a00.)J = ~ 7 , T = ~

4 '3 I 0 1

- , NNIIIIIIIIIIImmmD" x =.6 5 .U!!lllllmm v0 s _ �9 qgr I --.l ~ (1 {I 0

Fig. 3a-d. Nearest neighbour distribution in the lowest Landau bandj = 0 for d/l o = 0.0. p (X, S) has been calculated for four system sizes (see cases a to fl) Ly = 50.0, 60.0, 70.0, 100.0 l 0 corresponding to matrices of dimension 398, 573, 780, 1592 which is also the range of the transformed eigenvalues X i. The ensemble average has been

performed a 1000, 400, 160 and 80 times for the cases a to d, respectively. Note the gradual change from the Poisson to the Wig- ner distribution as one approaches the center of the band which is given by half the value of the matrix dhnension

434

p(X, S) [a.) j = 1, L u = 50.0/o] p(X, S) Ib.) j = 1, Lu = 60.0/o1

4 0 4 0 39 5

7, 3S2 ~ ~ l i i l l l ~ ~176 ,,i ~ r ~ l ~ P ~ i l F - , ~ 2 ~ ~ I I l ~ ~ W ' ~ 2 o

9 6 " ; I )~

p(X, S) [c.) j = 1, L~, = 70.0/o1

090 ~ - - x , 65o ~ l l l l i A, 520 - ~ J l ~ l l J I I mm "

130 " N ~ 0 , 0

0.0

[ , 0

1592

-~, io6~ " ~ i ~ l l l l ~ i v ~ 2o 531 ~ ' ~ S-----~

t v o ~ - 9 u

0

Fig. 4a-d. Nearest neighbour distribution in the first Landau band j = 1 for d/l o = 0.0. The same system sizes and number of averages have been used as in the calculations for the lowest Landau band.

The Wigner distribution can be noticed for a much larger range of the X i than in the lowest band. This difference remains as one increases the system size (compare cases a to d in Figs. 3 and 4)

computat ions have been performed for square samples of different size Ly • Ly in order to exclude possible finite size effects. The range of the transformed eigenvalues for squares with Ly = N o. l o lies between zero :and the corresponding matrix dimension Mo=No2/2~z where the centers X C of the bands are situated at half the value of the matrix dimension. As can be recognized in Fig. 3 to 6, the Wigner type distribution or logarithmic increase of 272 (X, L) near the center of the first Landau band is realized for a much larger range of the X i than in the lowest band. In general the spectra of the first Landau band are more rigid than that for j = 0. This difference remains as we increase the sample size or matrix dimension, indicating a much stronger tendency towards delocalization for j = 1. Therefore, the significant differences of the nearest neighbour distributions p (X, S) and 272 (X, L) between the lowest and first Landau band for a very small correlation length allow the conclusion for a nonuniversal delocalization transition.

3.2. Density-density correlation function and critical exponents

Quantitative information about the extension of wavefunctions is contained in the averaged density-density correlation function F(E; [ri - r21 ) which decays expo- nentially for large distances and yields the definition for the averaged localization length ~ (E) at energy E

F(E; I r ] - - r 2 l )

= Z I q/- (r,) 12 [ w'. (rz) 12 0 ( E - E/7) / 7

ocexp ( -- 21rl - r 2 1 / ~ (E)) . (3.3)

F (E; I rl -- r21 ) can be obtained by averaging the densities of the wavefunctions with respect to a reference point r 1 or can be expressed in terms o f an retarded Greensfunc- tion [251

1 (E) = 1 / ~ (E) = lim lim

Irl--r~l . . . . 0 26r]- - r2[

• (3.4)

A recursive technique [27] for the inverse of large band matrices has been used to calculate the inverse localization length e (E, Ly) from (3.4) in cylinder geometry versus energy E and circumference Ly. The system becomes one-dimensional for a large length Lx>>Ly and allows only localized states. One has to extrapolate the circumference Zy--*oo in order to estimate the asymptotic values ~ ( E ) f rom which the critical exponent

~ (E) = ~0" [ E - - E] I v of the localization length follows. Recently, there have been indications for a stronger energy dependence of e (E, Zy) in the first Landau band ( j = 1) for d<l o, but an asymptotic value could not be

r (X,L) g .

6 .

4 .

2 .

O .

0 .

a.) j = 0, L u = 50.01o I I L I I I i l l I I I I I I t I i

2. 4. 6. 8. i0. 12.

L >

~(X,L) c.) j = O, Ly = 70.0/o] S .

'+. " ' , , , I, !

0 . I I I I I [ I I I t -

0 . 2 . 4 . 6 . 8 . i0 . 12 .

L

Fig. 5a-d. 272 (X, L) distribution for j = 0 and d/l o = 0.0. The fluctuations around the average number of eigenvalues in an interval of length L decreases towards the center X c of the band, indicating the transition to extended states (r 272(X, L) for X= 0.2.X~ (tail

435

N(X,L) 1 0 .

8.

6.

4.

2.

O.

O.

Ib.) j = 0, Ly = 60.010

1 ! 1 I I I I I I I I [ I I I I t I

2 . 4 . 6 . 8 . i0 . 12 .

L-

N(X,L) [d.) j = 0, L u = 100.0/01 1 0 .

8 .

6 .

4 .

2 .

0 .

O. 2 . 4 . S . 8 . I0 . 12 .

L > of the band); o: X= 0.4.Xc; zx: X= 0.6-Xc; n: X= 0.8.Xc; • : center of the band X= X~). The system sizes used in cases a to d correspond to the calculations of the nearest neighbour distribution in Figs. 3 and 4

evaluated due to the small system sizes [23, 24]. In the range o f the critical regime one expects a one parameter scalling relation which has been derived for second order phase transitions [28] and which has been verified numerically for sufficiently large system sizes a round the centers o f the Landau bands [21, 22]. Such a scaling relation can be used to check whether the critical regime is reached and whether asymptot ic critical exponents can be extracted f rom the simulations.

In the case o f the first Landau band ( j = 1) and for the correlat ion lengths d / l o = 0.0 and rill o = 0.5, we could perform computa t ions o f the density-density correlat ion function according to (2.22) for systems up to the size L y = 3 0 0 l o and L x ~ 1 0 5 / o in the energy ranges E = 0.9 - 1.6 ~'/2 ~r (d / l o = 0.0) and E = 0.8 - 1.4 ~'/2 rc (d / l o --0.5) relative to the center o f the band (see Fig. 7a, b

for the density o f states). Closer to the center o f the band, the localization length exceeds rapidly the available system sizes o f Ly ~ 300 l o so that the results o f the simulations become un impor tan t for the determinat ion o f v. Nevertheless, a one parameter scaling relation o f the form

~z (E, L y ) . Ly = h (~ + (E) . Ly ) (3.5)

seems to be approximately satisfied for both cases d / l o = 0.0 and d/lo = 0.5 in the specified energy ranges above (see Fig. 8a, b). Therefore, the inverse localization length c ~ ( E ) = c % . E ~ of these energy values can be regarded as relevant for the calculation o f an asymptot ic critical exponent v. Fo r the considered correlat ion lengths, we obtain according to our simulations the following result for ~ + (E) (see also Fig. 9)

436

3

3

2

2

1 5

1 0

0 5

0.0

2(X,L) a.) j = 1, Lu= 50.0/o 5

- I t I I I t I t t I I L I I I , I I - -- I I i I I L ~ _ -

~ i 5

0

O . 2 . 4 . 6 . 8 . i0 , 12 .

L

2 ( X , L ) .[c.) y = 1, = 7o.o;o 5

4

3

2

1

0

0 . 2 . 4 . 6 . 8 . i0 . 12 .

L Fig. 6a-d. 22(X,L) distribution for j = 1 and d/lo=O.O. The no- tations and matrix dimensions are equivalent to the ones for the lowest Landau band in Fig. 5. The fluctuations are strongly reduced

Z(X,L) 4 .

3 .

2 .

1 .

O .

O .

b.) j = l , L u=60.01o

I I I L I I I [ i ~

2 �9 4 . 6 . 8 . lO . 12 �9

L >

2(X,L) [d,) j = 1, L u = lO0.Olo 1 5

2 - - I

0

0 . 2 . 4 , 6 . 8 . lO . 12 .

L compared to that with the lowest Landau band (note the smaller range of Z "2 (X, L)). This difference is also independent from the system size

d/lo=O.O: c% = (4.5_+ 0.05)-10-3 /0 -1

v=6.2• d / ~ = 0 . 5 : ~ o = ( 8 . 6 • 10 2 ~

(3.6)

v = 3 . 7 •

The dependence of v on the correlation length d/ l o shows the nonuniversal critical behaviour and excludes the ex- istence of another simple universality class as the one in the lowest band. According to further simulations in the first Landau band j = 1, the change f rom the nonuniversal to the universal behaviour with exponent v,-~2.3 may occur already at values of d/ l o = 0 . 7 5 - 1.0. The experimental observation of these nonuniversal exponents has to be expected only for special cases where the strong disorder has very short ranged correlations.

3.3. Conformal invariance of the delocalization transition

One may still argue that the critical behaviour with the large exponent v = 6.2 may change for energies very close (e.g. E ~ ~/(2 zr)) to the center of the first Landau band ( j = 1) to that with the exponent v =2 .3 or v = 7 / 3 . A change near the center of the band can be hardly detected by numerical simulations as reported in the previous section, even if a one-parameter scaling relation holds within an energy range where the extrapolated localization length

~ (E) does not exceed the sample size considerably. The possibility of such a change can be taken into account by performing simulations for the density-density correlation function at the center of the first Landau band j = 1 for d/l o = 0.0 and in comparison in the lowest band. At the critical point one expects a power law behaviour with an exponent t/

~(E) 160.0

120.0

~.) d/Io = o.o!

_ / \ /

_ /

80.0

40.0 - -

0.0 - 4 . 0

/

l ?

]

- 2 . 0 0.0 2.0 4.0 z. ~/~/(

200.0

150.0

100.(

50.(

b.) d/~o = 0.~

o.o _ ~ ; I I K/ - 3 . 0 - 2 . 0 - 1 . 0 0.0 1.0 2.0 3.0

E. , /~/~ Fig. 7a, b. Density of states p (E) for d/ l o = 0.0 and d/ l o = 0.5 in the first Landau band j = 1. The shaded region indicate the energies where the inverse localization lengths a (E, Ly) have been calculated

437

log(c~(E, Ly).Ly) 1.2[ ,_ ~ : ~ , ,

~.0 t I~) a/z0 x--0.0!

0 . 8 ~

0.6

0.4 ao

O +

0.2-- + +

0.0 + ,! , i i ~ , l

10 -1 i0 ~

i i i i ~ l J l [ i , i I l l l ~ I

g

A

,A Q

I ! I l I J l l ] , i r F l , l ' i

I01 10 2 log(a~(E).Lv)

log(a(E, L~).Ly)

~b.) ~/lo : 05} 1 . 6 - -

1 . 2 - -

0.8

0.4--

100

A +

+

§

I ' ~ ' I I t i !

10 ~

C L~

u r

I I I ' I I I I

10 2 log(a~(E)-Lv)

Fig. 8a, b. One parameter scaling relation. For the cases d/l o = 0.0 and d/ l o =0.5 a fit of the data log(~(E)-Ly) versus log(e (E, Ly) .Ly) has been performed (,: E=0.8, + : E= 1.0, o: E = 1.2, A: E = 1.3, • : E = 1.4, rn: E = 1.5, ~: E = 1.6)

n

1 Or

I r l - - r21" ' (3.7)

whose value is very different f rom the exponent v for the localization length and which describes the strength o f density correlations t ~" (r) 12 for the critical system. Nu- merical simulations at the centers o f the bands test whether the significant nodes in the second moments (see Fig. 2c, e, f) for d/lo<,% 1 and j > 1 m a y lead to different critical systems so that a universal delocalization behaviour with an exponent v ~ 2 . 3 near the critical point for d / l o = 0.0 becomes less probable if the exponent r/ de- pends on the Landau band index j .

The corresponding calculations for Y C ( [ r 1 - - r 2 ] ) have been performed in a disk geometry whose center has been chosen as the reference point r I for the averages in (3.7). R a n d o m potentials in a disk geometry which are sampled according to a given correlat ion funct ion f (2.2) and which are projected onto single subspaces can be generated in a similar manner as in the cylinder geometry. One

log(~(E).10) 100

10 1 d/lo, = 0.5! /~/ ~"

10 -2 /

7~}~i~7-~176 *z j ~ = 6.2 • o.1

j - . . / .

10-a /

- 0110 ' 0'.0 ' 0110 ' 0120 Iog(E.~2~ , / r

Fig. 9. Scaling plot for the calculation of the exponent v from the extrapolated inverse localization lengths ~ (E)

has to substitute the symmetric gauge for the Landau gauge in the expressions o f Sect. 2 because the wavefunctions o f the kinetic energy in syrrmaetric gauge possess an azimuthal symmetry and are degenerate with respect to a discrete label m = 0, 1, 2 .. . . which is related to the

438

log G ( f ~ - ~2])

- 5 . 1

- 5.2

- 5 . 3

- 5.4

- 5 . 5

- 5 . 6

- 5 . ~

a.) Rdi,~ = 14.0l~

,['j = O; d / lo = 0 .0

0.2 0.4 0.6 0.8 1.0

- 6 . 0

- 6 . 1

- 6 . 2

- 6 . 3

- 6 . 4

-6 .5

- 6 . 6

- 6 . 7

- 6 . 8

- 6 . 9

b.) P ~ = 28.Otol

- i i i I i I I I I 0 . 2 0.4 0.6 0.8 1.0 1.2

log( I r l - r

- 6 . 8

- 6 . 9

- 7 . 0

7.1

- 7 . 2

- 7 . 3

- 7 . 4

- 7 . 5

7.6

~ R disk --~ 42.010~

0.3 0.6 0.9 ~.2 L 5 ~og(l~l - ~ 1 )

Fig. 10a-c. Density correlation function f~ ( ] r I - r 2 [ ) in the lowest Landau band j = 0 for d/l o = 0.0 and for three different system sizes

log G(16 - ~1) - 5.90

- 5.95

- 6.00

a.) Rdl, k = 14.010~

._~; = .

0.7 0.8 0.9 1.0 l o g ( f 1 - ~=1)

log rc(1r - ~2F) - 7.00

7.05

7.10

7.15

0.7

b.) P~i,k = 28.010q

- I I I I I 0.s 0.9 1.0 t.1 1.2 1.3

log(lr - ~21)

log rc(1r - ~21) c ) Rai,k = 42.0z~

- 7 . 6 [ j = 1 ; d/ lo = 0.0

- 7.7 - - ~ - 7.8

-7.9 I I I I t I I I 0.6 0.8 1.0 1.2 1.4

log(lf'l - ~'21)

Fig. l l a - e . Density correlation function Fc( [r~-r2[ ) in the first Landau band j = 1 for d/l o = 0.0 and for three different system sizes

angu la r m o m e n t u m pe rpend icu la r to the p lane (see [24] for a de ta i led descr ip t ion) . The densi ty cor re la t ion func- t ion F c ( I r l - r21 ) has been ca lcu la ted for three system sizes wi th radi i R = 14.0, 28.0, 42.0 l o using inverse iter- a t ions to de te rmine the wavefunct ions at the centers o f the bands (see Figs. 1 0 a - c and l l a - c ) . W e ob ta in f rom these s imula t ions the exponents r/

r / ( j = 0) = 0.56 :J: 0.03 (3.8)

,~ ( j = 1 )= 0.28 +0 .03 , (3.9)

which obvious ly depend on the index j under consider- a t i on (for d / t o = 0 . 0 ) . The differences observed for the

exponen t v between the first and lowest L a n d a u band wi thin a specified energy range o f the local ized regime cont inue to the cri t ical poin ts which have to be charac- ter ized by different exponents r/ for d / l o = 0.0. The sig- nif icant nodes and b r o a d e r d i s t r ibu t ion o f the squares o f ma t r ix e lements for j = 1 and d / l o = 0.0 obvious ly have a s t rong influence on the de loca l iza t ion behaviour .

The poss ib i l i ty to calcula te F c ( I r l - r2[ ) in different geometr ies by choos ing an a p p r o p r i a t e gauge can be ap- p l ied for a test o f con fo rma l invar iance at the centers o f the L a n d a u bands . C o n f o r m a l invar iance is an extens ion o f the g loba l scale invar iance o f cor re la t ion funct ions at the cri t ical p o i n t wi th a cons t an t fac tor q to a local scale

439

invariance with a space dependent factor q (r). If we as- sume a field S(r ) (e.g. spins), this local invariance of a correlation function means

r = q (r) r ' (3.10)

1 (S (rl) S(r2) ) oc

Ir --r l"

= q - " / 2 ( r ~ ) q - ~ / 2 ( r 2 ) ( S ( r ; ) S ( r ; ) ) (3.11)

that one can relate by suitable transformations q (r) correlation functions in different geometries to one another (for a review and rigorous definitions of conformal invariance see Cardy [29]). This concept has been established for various critical systems both numerically and in simpler cases analytically. Since extended states show a strong sensitivity to a change of boundary conditions according to the idea of the Thouless number [30] g - f i E / W ~ - h / e 2. 1 / R (fiE: energy change due to the boundary conditions, W: mean level spacing), one should also expect a maximal sensitivity to the geometrical shape and to finite size effects in the energy ranges of delocalized states of disordered systems. Let us consider the mapping w = L y / 2 7r. In z which transforms a disk into a cylinder geometry where the complex number z = x + iy describes the coordinates of the disk and w = u + iv refers to the cylinder with the periodic coordinate v. I f a system at the critical point is invariant under conformal transformations or local rescalings, one can show for the mapping w -- L y / 2 ~z. In z that the exponent r / for the disk is related to the localization length by [31]

r /= r/oy~ r/oy~ = 1/re . ( L y / ~ ) . (3.12)

In the case of disordered systems it is not a priori obvious how to represent the field S(r ) and how to define the localization length ~. In [18] the conformal invariance of two-dimensional disordered systems in a strong magnetic field has been checked for the correlation function

F ( r ; E, co)

= ~, d ( E - co/2 - E~) ~ (E + co/2 - Eo) ~u (0) q/~ (0) ~,* (r) q/o (r),

~'~ (3.13)

where one of the delta functions has to be removed by an integration and the limit co ~ 0 is implied. The field S (r) is replaced in (3.13) by the matrix q/* (r) ~u~ (r) whose labels ~,/~ are restricted to the extended wavefunction(s) by the summation over the degenerate state(s) at the critical energy E~. The results of [ 18] have shown the validity

Table 1. Results for the critical exponent r/cy j in cylinder geometry for the two lowest Landau bands (j = 0, 1) and for different circumference Ly. The listed values r/~y I have been calculated from the localization length ~ of the densities of wavefunctions according to the relation t&yl= l/re . (Ly /~)

Ly/lo-~ 100.0 150.0 200.0 300.0

j = 0 0.57• 0.57__+0.01 0.56:k0.01 0.57•

j = 1 0.32_+0.02 0.34_+0.025 0.31• 0.28_+0.02

of the relation (3.12)within 30%. In the calculations of this paper the density [ ~ (r) [a of wavefunctions, its correlations (3.7) and its associated localization length, given by ~/2 in (3.4), have been chosen as the corresponding quantities to test the relation (3.12).

In Table 1 we list the values r/cyl, which have been calculated from the localization length in cylinder geometry according to (3.12), for the lowest and first Lan- dau band in the case of a vanishing correlation length d / l o = 0.0 and for different circumference Ly. The differences between the lowest two bands for d / l o = 0.0 are remarkable, as well as the agreement of the critical exponent r / in the disk geometry and the exponent ~/oyl for the cylinder so that the critical systems can be regarded as conformal invariant, despite the complication that there is a dependence on the Landau band index for a small correlation length. Further investigations, including results for a finite correlation length d ~ 10, will be published elsewhere.

4. S u m m a r y and discussion

One of the questions we have considered in this paper is: to what extent does a universal delocalization transition hold in the presence of a strong magnetic field? The in- vestigation of the second moments for a spatial Gaussian correlation function in Sect. 2 shows that one has to expect for the lowest Landau band j = 0 and for a large correlation length d~> l o in higher Landau bands a universal critical behaviour with an exponent v ~-2.3. This is because the distributions of the squares of matrix elements are similar to a Gaussian (see Fig. 2a, b and d). According to the distribution of second moments, deviations from this transition can only occur for short ranged correlated potentials and higher Landau bands j > 1. In this case the second moments are characterized by significant nodes depending on the Landau band under consideration (see Fig. 2c, e and f). Furthermore, the hopping matrix, given by the nondiagonal elements, has more nonlocal couplings, due to the broader distribution of the squares of matrix elements. It has been verified by numerical simulations for j = 1 that this causes different spectral properties (see Fig. 3 to 6) and different critical exponents v of the localization length (see Fig. 9), depending on the ratio d / l o < ~ 1.0. Moreover, the considerations at the end of Sect. 2 prove that this different, nonuniversal delocalization transition for j > 1 and d~< l o can also be realized in the lowest Landau band ( j = 1) by choosing a spatial correlation function gj ( I r~ - r 2 [ ) which contains suitable nodes and forces the random potential to alter its sign steadily (see Fig. 1). Therefore, a few properties, as the dimension and symmetries of disordered systems, are not sufficient to classify delocalization phenomena; other features, as the Landau band index, the correlation length d, the type of correlation function f and the type of coupling (off-site versus on- site disorder) also determine the presence and kind of delocalization transition.

The numerical simulations at the centers of the bands for a disk geometry have confirmed the differences of the

440

Landau bands for d/ lo~ 1.0 and shown that one obtains different critical systems with density correlation functions F c ( [ r 1 - r 21 ) whose exponents 1/differ between the two lowest Landau bands by a factor of about two (for d/ l o = 0.0 and a Gaussian correlation function). Similar calculations in cylinder geometry have verified conformal invariant properties of the disordered systems at the center of the bands where the fields S (r) have been associated to the density of wavefunctions. It has to be noted that the conformal invariance of delocalized states at the centers is observed despite the fact that the delocalization transition around the centers of the Landau bands is nonuniversal. Therefore, the conformal invariance of extended states, which we have tested in this paper for the mapping w = L y / 2 it. In z in the case of short ranged correlated potentials (d / l o = 0.0), is a new interesting aspect for delocalization phenomena. Since conformal invariance allows local rescalings, it is violated in the case of nonlocal interactions so that one should mainly investi- gate local observables as the density correlation function F c ( [ r I -- r 2 [ ) and systems whose disorder has only short ranged correlations d < l o. However, there is no unique assignment under conformal invariance between critical exponents ~/at the centers of the bands and Landau band index j because of the modified correlation functions & which, properly chosen, yield identical random matrices in different Landau bands (see Sect. 2). The exponents r/ should therefore be related to certain types of random matrices with specific second moments as in Fig. 2 for d/10r

The conformal invariance may also be observed in other disordered, two-dimensional systems with a delocalization transition. In previous calculations [24] we have pointed out that disordered systems in a strong magnetic field have strong rlondiagonal hopping matrix elements. Therefore, conformal invariant properties can also be expected in other two-dimensional systems with nondiagonal disorder [34, 35]. According to recent simulations, conformal invariance is also satisfied for the extended states of random matrices, representing two-dimensional systems with pure off-site disorder between nearest lattice points. In the case of complex coupling matrix elements the calculations in the cylinder and disk geometry have shown values of about r/~-0.28 whereas real matrix elements have given a much larger value of about 0.7. Fur- ther results of these singular matrices may be considered and published elsewhere.

B. Micck would like to acknowledge the financial support of the Heidelberger Akademie der Wissenschaften, the Alexander yon Humboldt-Foundation and the advice of Prof. Weidenmiiller and Prof. Zirnbauer, as well as the hospitality of the Condensed Matter theory group at the University of Maryland. Furthermore, I would like to thank for the possibility to perform calculations at the HLRZ- computation center in Jfilich which yielded most of the numerical results presented in this paper.

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