Eur. Phys. J. B (2013) 86: 48DOI: 10.1140/epjb/e2012-30410-x

Regular Article

THE EUROPEANPHYSICAL JOURNAL B

Partitioning schemes and non-integer box sizesfor the box-counting algorithm in multifractal analysis

Stefanie Thiema and Michael Schreiber

Institut fur Physik, Technische Universitat Chemnitz, 09107 Chemnitz, Germany

Received 22 May 2012 / Received in final form 2 October 2012Published online 11 February 2013 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2013

Abstract. We compare different partitioning schemes for the box-counting algorithm in the multifractalanalysis by computing the singularity spectrum and the distribution of the box probabilities. As modelsystem, we use the Anderson model of localization in two and three dimensions. We show that a partitioningscheme which includes unrestricted values of the box size and an average over all box origins leads to smallererror bounds than the standard method using only integer ratios of the linear system size and the box sizewhich was found by Rodriguez et al. [Eur. Phys. J. B 67, 77 (2009)] to yield the most reliable results.

1 Introduction

Systems with multifractal properties are often found innature, e.g. in biological systems, in meteorology and ge-ology, or in economics, e.g. stock market time series. Fora detailed study of these properties, multifractal analysisplays an important role. Also the electronic wave func-tions Φ of physical models like the Anderson model of lo-calization at the mobility edge, quasiperiodic tilings andchaotic systems show multifractal properties [1–4]. Onecan distinguish two methods for the multifractal analysis:the system-size scaling approach and the box-size scalingapproach [5]. In both approaches the system is partitionedinto boxes and one measures the probability to find theelectron in a particular box, i.e., we study the spatial dis-tribution of the wave functions [6]. While for the system-size scaling the box size is kept constant and the systemsize is varied, for box-size scaling one studies how the prob-ability distribution changes with the box size. The lattermethod is more common because it requires less comput-ing time and storage.

For box-size scaling, different partitioning schemes areknown, where the scheme has a crucial influence on thefit results and the statistical analysis. Rodriguez et al.compared several approaches for partitioning the systeminto boxes including different box shapes and box sizes aswell as an averaging over different box origins [5,7]. Theyfound that a partitioning of the system into cubic boxeswith integer ratios of the linear system size N and thelinear box size L is numerically most reliable in compar-ison to the other studied methods. However, a previouswork by one of us showed that the fit quality can be im-proved by adapting the multifractal analysis to work also

a e-mail: stefanie.thiem@physik.tu-chemnitz.de

for non-integer ratios N/L and by averaging over differ-ent origins of the boxes [2,8,9]. Unfortunately, this methodwas not considered in the comparative study by Rodriguezet al. Therefore, we compare in this paper the fit qualityof the best method proposed in [7] with the participationscheme allowing non-integer ratios N/L [2]. We show thatthe latter partitioning scheme yields smaller errors in theleast squares fits than the standard scheme which allowsonly integer ratios of N/L.

The paper is structured in the following way: first, webriefly introduce the Anderson model in Section 2 and themultifractal analysis in Section 3. We then discuss differ-ent partitioning schemes for the box-counting algorithmin Section 4 and compare their performances in Section 5.This is followed by a brief conclusion in Section 6.

2 Anderson model of localization

As model system for studying the influence of the parti-tioning in the multifractal analysis we use the eigenstatesof the tight-binding Anderson Hamiltonian

H =n.n.∑

r,r′|r〉〈r′| +

∑

r

εr|r〉〈r|, (1)

represented in the orthogonal basis of states |r〉 associatedto a vertex r. The onsite potential εr is chosen accord-ing to a uniform distribution in the interval [−w/2, w/2]for a given disorder strength w. Each site r is connectedto the nearest neighbors in a regular square/cubic lat-tice with periodic boundary conditions. For a given re-alization of disorder we consider only the eigenstate Φr

of the Anderson Hamiltonian closest to the band center

Page 2 of 10 Eur. Phys. J. B (2013) 86: 48

Fig. 1. Probability amplitude |Φr| of a wave function on asquare lattice of N2 = 360 × 360 sites. Every site r with anamplitude larger than the average is shown, i.e. |Φr| > 1/N andfour different grey levels distinguish whether |Φr|2 > 2j/N2 forj = 0, 1, 2, 3.

at E = 0. The wave functions are obtained using the pro-gram JADAMILU, which is based on the Jacobi-Davidsonmethod [10] and allows one to compute selected eigenval-ues and eigenfunctions of large sparse matrices with highefficiency. In Figure 1 we visualize a two-dimensional wavefunction for the disorder strength w = 4 for the linear sys-tem size N = 360.

The investigations in this paper are restricted tomoderate system sizes with at most N = 360 in two di-mensions and N = 60 in three dimensions which are suffi-ciently large for the presented purpose. We have also stud-ied larger system sizes up to N = 720 in two dimensionsand up to N = 90 in three dimensions but we have notfound any significant differences for the fit results for thesingle wave functions. Of course, the system size has a cru-cial influence on the ensemble average, and the below dis-cussed measures are not yet completely size-independentfor these system sizes. Therefore, large system sizes shouldbe preferred. As we are only interested in the optimiza-tion of the partitioning scheme, we mainly consider singlewave functions.

3 Multifractal analysis

The multifractal analysis is based on a standard box-counting algorithm, i.e., in d dimensions the system ofsize V = Nd is divided into B = Nd/Ld boxes of linearsize L [1,4,11,12]. For each box b we measure the proba-bility to find the electron in this particular box. Hence,

for an eigenstate Φr and the box size L we determine theprobability for the bth box [4]

μb(Φ, L) =∑

r∈ box b

|Φr|2 (2)

and construct for each parameter q a measure by normal-izing the qth moments of this probability

μb(q, Φ, L) =μq

b(Φ, L)P (q, Φ, L)

, (3)

where

P (q, Φ, L) =B∑

b=1

μqb(Φ, L). (4)

The mass exponent of a wave function

τq(Φ) = limε→0

ln P (q, Φ, L)ln ε

(5)

can be obtained by a least squares fit of ln P versus ln ε,where ε = L/N . The generalized dimensions are easilyobtained as Dq = τq/(q − 1).

According to equation (5), small boxes should be con-sidered for the least squares fit. Further, the power-lawbehavior defined in equation (5) requires the absence ofa typical length scale in the distribution of the proba-bility density of the wave functions. Therefore, it shouldbe computed for box sizes l � L � N to properly dis-play the multifractal properties, where l corresponds tomicroscopic length scales of the system like e.g. the latticespacing [13]. Rodriguez et al. used box sizes from L = 10to N/2 [7]. For reasons of comparison we have performedmost of the calculations also for this range, but we alsoshow that the range of considered box sizes has a crucialinfluence on the observed results.

Another quantity often used for the multifractal anal-ysis is the singularity spectrum f(αq), which denotes thefractal dimension of the set of points where the wave-function intensity behaves according to |Φr|2 ∝ N−αq ,which means that in a discrete system the number ofsuch points scales as Nf(αq) [14]. The singularity spec-trum f(αq) follows from a parametric representation ofthe singularity strength

αq(Φ) = limε→0

A(q, Φ, L)ln ε

(6)

with

A(q, Φ, L) =B∑

b=1

μb(q, Φ, L) ln μb(1, Φ, L) (7)

and the fractal dimension

fq(Φ) = limε→0

F (q, Φ, L)ln ε

(8)

with

F (q, Φ, L) =B∑

b=1

μb(q, Φ, L) ln μb(q, Φ, L). (9)

Eur. Phys. J. B (2013) 86: 48 Page 3 of 10

L

N

L

N

Fig. 2. (Color online) Partitioning scheme with integer ratioN/L for a system of linear size N = 6 and box sizes L = 2(left) and L = 3 (right). The red dots indicate the consideredbox origins.

Both quantities can be determined by least squares fitsof A(q, Φ, L) or F (q, Φ, L) versus ln ε considering differ-ent box sizes L. The mass exponents τq and the gener-alized dimensions Dq can also be obtained from the sin-gularity spectrum f(αq) via the Legendre transformationτq = Dq(q − 1) = qαq − f(αq) [11].

Regarding the partitioning of the system into boxes,several options are possible, which are discussed in moredetail in the following two sections.

4 Partitioning schemes

In this section we briefly introduce the considered meth-ods for the optimization of the partitioning in multifractalanalysis. In particular, we discuss the following three par-titioning schemes for the box-counting algorithm:

A) cubic boxes with integer ratios N/L;B) cubic boxes with unrestricted values for L; andC) cubic boxes with integer ratios N/L including an av-

erage over all box origins.

To be specific, we define the left upper corners of a box asbox origin in the figures below.

4.1 Scheme A: cubic boxes with integer ratios N/L

This quite simple partitioning scheme is the most straight-forward procedure in which only such box sizes are consid-ered for which the boxes cover the system gapless with-out overlap as visualized in Figure 2. Hence, the ratio1/ε = N/L is restricted to integer values. Obviously thiscondition is a severe limitation of the possible choicesfor N and L. For example, in the case of N = 6, thetwo partitionings with L = 2 and L = 3 shown in Fig-ure 2 are the only possibilities. For N = 5 or N = 7 theredoes not exist any such way of subdividing the system.

Exploiting periodic boundary conditions, one couldutilize different box origins. But usually no additional av-erage over different box origins is considered. Rodriguezet al. studied several partitioning schemes and found that

L

N

LCM(L,N)

Fig. 3. (Color online) Partitioning scheme for unrestrictedvalues of the box size L for a system of linear size N = 4and the box size L = 3. The original system is repeatedLCM(L, N)/N = 3 times in each direction so that the result-ing system of linear size LCM(L, N) = 12 can be filled gaplessby the boxes. For coprime L and N this method intrinsicallyincludes an averaging over all box origins as indicated by thered box origins. The arrows visualize this for the sites in thethird column of the original system.

this one shows the most reliable performance of the con-sidered schemes [7]. However, we show in this paper that amethod considering unrestricted values of box sizes showsa better performance. This scheme was already success-fully applied for the multifractal analysis [2,8]. Unfortu-nately, it was not taken into account in the comparison byRodriguez et al. [7].

4.2 Scheme B: cubic boxes with unrestricted valuesfor L

The fitting of the scaling exponents in the box-size scalingcan be enhanced if all possible box sizes L as well as an av-erage over all box origins is considered [2,8,9]. This methodhas a rather simple geometrical interpretation, which isdisplayed in Figure 3. First, the original system is peri-odically repeated LCM(L, N)/N times (LCM labels theleast common multiple) in each direction. The new sys-tem can then be covered gapless for any linear box size L.If N and L are coprime, then this method automaticallyincludes an average over all different box origins as shownin Figure 3. For the other cases we explicitly consider anaverage over all different box origins.

Page 4 of 10 Eur. Phys. J. B (2013) 86: 48

From the computational perspective, it is easier notto look at the periodically repeated system but rather toimagine that all boxes with origins outside the originalsystem are folded back so that their origins fall into theoriginal system. One can easily see from Figure 3 thatif N and L are not coprime, such a folding back wouldnot yield all sites of the original system, but it is of coursestraightforward to generalize the idea and to place a boxof linear size L on each site of the original system in thiscase, too. Hence, in all cases, each of the N2 sites is con-sidered L2 times and the proper average for a quantity Xis obtained by

〈X〉 =1L2

∑

box origins

X. (10)

This average then replaces the simple sum over all Bboxes.

Rodriguez et al. also studied a method which allowsone to utilize all box sizes L [7]. In contrast to our method,they did not periodically repeat the system LCM(L, N)/Ntimes. Hence, they faced the problem that different effec-tive system sizes occur, and they had to introduce an ad-ditional average over these different effective system sizes.Our method does not require such an additional averagingbecause the boxes fill the new system gapless, so that noeffective system sizes appear.

4.3 Scheme C: cubic boxes with integer ratios N/Lincluding an average over all box origins

This method is a mixture of the two former partitioningschemes. Now we only consider integer ratios but includethe average over all box origins as in the previous method.Rodriguez et al. also studied the average over all initialbox origins but pointed out that for an ensemble aver-age over many wave functions, this method reduces thestandard deviation almost to zero [7]. The reason for thisapparently negative side effect is that they counted eachbox origin as a different disorder configuration, which isclearly not the case.

Hence, a proper ensemble average over different disor-der configurations can still be obtained by computing theaverage quantities P (q, Φ, L), A(q, Φ, L) and F (q, Φ, L) forall box origins for each wave function according to equa-tion (10) but without considering each box origin as anew disorder configuration. In this case, no unwanted re-duction of the standard deviation of the ensemble averageoccurs. However, the average over the box origins in equa-tion (10) means that the data for each wave functions areevaluated with a better accuracy, so that a reasonable re-duction of the standard deviation of the ensemble averagecan be expected.

To avoid a misunderstanding, we point out that thereplicas in Figure 3 are used for an easy explanation ofour scheme only. They are not necessary for our compu-tation. One can also utilize the following interpretation:due to periodic boundary conditions, no site of the sys-tem can or should be distinguished or preferred over any

other site. But the choice of the system origin makes sucha distinction. This bias can be avoided by treating everyother site equivalently, by considering every site as a pos-sible system origin. Then the generic placement of boxesstarting at all the possible system origins is equivalent toour above placement of box origins on each site of thesystem.

5 Comparison of results

For a comparison of the three partitioning schemes westudy the singularity spectrum of wave functions closest tothe band center E = 0 for certain disorder configurations.We consider wave functions in different dimensions andstudy also the influence of the considered range of boxsizes.

5.1 Two-dimensional wave functions

Figures 4 and 5 show the scaling behavior of the quantitiesA(q, Φ, L) and F (q, Φ, L) with the box size L for a two-dimensional wave function for the system sizes N = 350and N = 360. The results can be well described by apower law. The stronger fluctuations for negative valuesof q are well known and can be explained by the less ac-curate values of the wave functions at vertices with smallamplitudes, which are enhanced for negative powers of μb

in equation (3). Apart from fluctuations for very smallbox sizes and negative q, the data are sufficiently closeto straight lines as visualized in Figures 4 and 5 so thatwe can determine the singularity strengths αq(Φ) and thefractal dimensions fq(Φ) from the slopes. If not stated dif-ferently, we consider box sizes in the range from L = 10to N/2 for the computation of the singularity spectrum inaccordance to the setup used by Rodriguez et al. [7,14].

The corresponding singularity spectrum f(αq) isshown in Figure 6. From literature it is known that f(αq)is a convex function of αq, where the maximum of f(αq)corresponds to q = 0 with f(α0) = d being the supportof the measure [11,12]. This is verified in Figure 6 withf(α0) ≈ 2. Further, for q = 1 we have f(α1) = α1 and thelimits q → ±∞ yield the minimal/maximal value of αq,which corresponds to the singularity associated to the boxcontaining the largest/smallest measure [2]. For the pe-riodic system, i.e. for vanishing disorder, the wave func-tions are Bloch waves extending over the entire system andtherefore the singularity spectrum is expected to convergeto the point f(d) = d. In the limit of a strongly disorderedsystem with localized wave functions, the spectrum con-verges to the points f(0) = 0 and f(∞) = d. The wavefunctions of the two-dimensional Anderson Hamiltonianshow large spatial fluctuations, which increase with in-creasing disorder strength [15]. However, for moderate dis-order strengths, the wave functions can be approximatelydescribed by the multifractal analysis.

Comparing the results for the three different partition-ing schemes we observe that the least squares fit of thequantities A(q, Φ, L) and F (q, Φ, L) includes many more

Eur. Phys. J. B (2013) 86: 48 Page 5 of 10

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-8

-6

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-2

0

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1

A(q

,Φ, L

)

ln(ε)

q = -2q = -1q = 0q = 1q = 2

-10

-8

-6

-4

-2

0

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1

F(q

,Φ,L

)

ln(ε)

q = -2q = -1q = 0q = 1q = 2

Fig. 4. (Color online) Scaling behavior of the quanti-ties A(q, Φ, L) (top) and F (q, Φ, L) (bottom) with the ratioε = L/N of the box size L and the linear system size N for thetwo-dimensional wave function of the Anderson Hamiltonianshown in Figure 1 with N = 360 and disorder strength w = 4.The results are shown for the scheme A with integer ratiosN/L (squares) and scheme B with unrestricted values of Land a box-origin average (circles) for 2 ≤ L ≤ N/2. The dot-ted lines correspond to least squares fits considering all boxsizes in the range 5 ≤ L ≤ N/10 marked by the two verticaldashed lines.

points for the scheme B as for the other two methods,which makes the fitting much more reliable for scheme B.For the other two schemes, the actual number of consid-ered data points of course strongly depends on the num-ber of factors of the system size N . Further, schemes Aand B yield different values for the quantities A(q, Φ, L)and F (q, Φ, L) because scheme B considers all possiblebox origins in contrast to scheme A. The biggest differ-ences are observed for large box sizes L because in thiscase only very few box origins are taken into account inscheme A as visualized in Figure 2. The reason is that thenumber of considered box origins has a crucial influenceon the distribution of the box probabilities μb(Φ, L). Thisbecomes clearly visible by plotting the distribution withand without an explicit average in Figure 7. While thedistributions for small box sizes L are quite similar, thereare huge differences already for medium-sized boxes. Even

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-6

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-2

0

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1

A(q

,Φ, L

)

ln(ε)

q = -2q = -1q = 0q = 1q = 2

-10

-8

-6

-4

-2

0

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1

F(q

,Φ,L

)

ln(ε)

q = -2q = -1q = 0q = 1q = 2

Fig. 5. (Color online) Same as Figure 4, but for the systemsize N = 350.

for the small box sizes the distribution is not as smoothas the results averaged over all box origins. As we applyperiodic boundaries there is no preferable choice for thebox origins. Hence, the average over all box origins is abetter estimate for the real box probability distribution.For the method C, which also includes an average over allbox origins, the data points are identical to those of thepartitioning scheme B but only box sizes for integer val-ues of 1/ε are taken into account. Hence, the results formethod B and method C are often rather close, but theaccuracy of the fit for C is not as good as for B.

Having a look at the singularity spectra f(αq), weobserve that the results are rather close for all threeschemes. However, the error bars are significantly largerfor scheme A than for scheme B especially for the systemsize N = 360. This is not surprising because in B manymore data points are considered for the least squares fitand the average over all box origins leads to a better es-timate of the box probabilities μb. Comparing schemes Band C, we observe that the errors are rather similar, i.e.,most of the improvement of the fits is due to the aver-aging over all different box origins. For the system withN = 350 the error bars in Figure 6 are rather close for allthree schemes especially for negative values of q and onlya slight advantage of scheme B can be found. At first view

Page 6 of 10 Eur. Phys. J. B (2013) 86: 48

0

0.5

1

1.5

2

1 1.5 2 2.5 3 3.5

sing

ular

ity s

pect

rum

f(α q

)

singularity strength αq

scheme Ascheme Bscheme C

0

0.5

1

1.5

2

1 1.5 2 2.5 3 3.5

sing

ular

ity s

pect

rum

f(α q

)

singularity strength αq

scheme Ascheme Bscheme C

Fig. 6. (Color online) Singularity spectrum f(αq) for thethree different methods for the systems shown in Figure 4 withN = 360 (top) and in Figure 5 with N = 350 (bottom) withconsidered box sizes from L = 10 to N/2. The plot shows 1σerror bars for integer q from −3 to 3.

this seems surprising because for this system even fewerbox sizes are taken into account for schemes A and C. Thereason is that by considering box sizes from L = 10 to N/2in scheme B, large boxes are strongly overrepresented inthe least squares fits. This leads to a strong contributionof the non-linear behavior of A(q, Φ, L) and F (q, Φ, L) forlarge L and negative q to the fit in B (see Fig. 5) whilefor A and C almost no data points are considered in thisrange.

5.2 Influence of range of box sizes L

Further, we also investigated the influence of the consid-ered range of box sizes. As mentioned in Section 3, theexponents should be computed from the scaling behaviorof preferably small boxes and by considering the range1 � L � N to properly measure the multifractal proper-ties. The first point is not properly considered yet because,especially in scheme B, larger box sizes L dominate the fitas mentioned above. In order to neglect the influence dueto the different number of considered box sizes for the fits,we study the singularity spectrum for an identical numberof box sizes for the different schemes. This means that we

0

0.02

0.04

0 0.002 0.004 0.006 0.008 0.01

perc

enta

ge o

f box

es

box probabilities μ(Φ,L)

scheme Ascheme B

0

0.02

0.04

0.06

0.08 scheme Ascheme B

Fig. 7. (Color online) Distribution of box probabilities forthe scheme A and scheme B for the wave function consideredin Figure 4 with linear box sizes L = 15 (top) and L = 30(bottom).

compute the singularity spectrum for the two-dimensionalwave function in Figure 1 by using exactly 15 different boxsizes, i.e., for scheme A we have taken all box sizes withinteger ratios N/L in the range from L = 10 to N/2 intoaccount and for scheme B we used on the one hand L = 10to 24 and on the other hand L = 5 to 19. In the lattercase we do not start with Lmin = 2 to assure L 1and, therefore, to omit the deviations from the power-lawbehavior which can be observed in Figures 4 and 5 forsmall box sizes and negative values of q. This deviationsare most likely caused by numerical inaccuracies which areenhanced for negative powers of μb and by summing overmany boxes. The results in Figure 8 again show that theerror bars are significantly smaller for scheme B. We canalso observe clear differences in the singularity spectrumespecially for negative values of q for all three consideredsetups.

We also plot in Figure 8 the squares r2 of the correla-tion coefficients obtained from the least squares fit, whichis a measure for the amount of variability in the data thatis accounted for by the fitting curve. Values of r2 = 1 meanthat the data can be fitted perfectly by a straight line, i.e.,in this case we can identify a power-law behavior. Whilethe numerical results in Figure 8 show that for the singu-larity strength αq the quantity r2 ≈ 1 for −5 ≤ q ≤ 5, weobserve smaller values of r2 for large positive and nega-tive q for the least squares fits of the fractal dimension fq

in agreement with the increase of the fit errors for theseq values. The results also depend on the used scheme andrange of box sizes, but again scheme B yields the best re-sults. Using scheme B with box sizes from L = 5 to 19,the scaling behavior can be well described by a power-lawfor −4 ≤ q ≤ 3 and 1 ≤ αq ≤ 2.8, respectively.

5.3 Three-dimensional wave functions

Finally, we have performed the same calculations also forthe three-dimensional Anderson model for the disorder

Eur. Phys. J. B (2013) 86: 48 Page 7 of 10

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0.5

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1.5

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1 1.5 2 2.5 3 3.5

sing

ular

ity s

pect

rum

f(α q

)

singularity strength αq

scheme A, L = 10 ... 180scheme B, L = 5 ... 19scheme B, L = 10 ... 24

0

0.5

1

1 1.5 2 2.5 3 3.5

r α2

singularity strength αq

scheme A, L = 10 ... 180scheme B, L = 5 ... 19scheme B, L = 10 ... 24

0

0.5

1

r f(α)

2 scheme A, L = 10 ... 180scheme B, L = 5 ... 19scheme B, L = 10 ... 24

0

0.5

1

-4 -2 0 2 4

r α2

parameter q

scheme A, L = 10 ... 180scheme B, L = 5 ... 19scheme B, L = 10 ... 24

0

0.5

1

r f(α)

2 scheme A, L = 10 ... 180scheme B, L = 5 ... 19scheme B, L = 10 ... 24

Fig. 8. (Color online) Singularity spectrum f(αq) obtainedusing scheme A or scheme B with different ranges of box sizes Lfor the wave function in Figure 4 with −5 ≤ q ≤ 5 and 1σ errorbars for integer q from −4 to 4 (top) corresponding squares r2

of the correlation coefficients for the singularity strength αq

and the fractal dimension f(αq) (center and bottom).

strength w = 16.5 at the mobility edge and a linear sys-tem size N = 60. At the mobility edge in three dimen-sions, the wave functions are known to be multifractals [2].The results for the scaling of the quantities A(q, Φ, L) andF (q, Φ, L) are shown in Figure 9 and the correspondingsingularity spectrum is shown in Figure 10. Again bothquantities, A(q, Φ, L) and F (q, Φ, L), show a power-law be-havior and yield qualitatively similar results as in two di-mensions for the partitioning schemes A and B. However,for the singularity spectrum we can observe a clear differ-ence. While the results for the schemes B and C are nearlyidentical, the singularity spectrum obtained by the parti-tioning scheme A shows significant differences for negativevalues of q (large αq). In Figure 9 we see that the averagevalues of A(q, Φ, L) and F (q, Φ, L) are already quite dif-ferent for negative values of q. This is probably related tothe rather small system size. Hence, the distribution of the

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A(q

,Φ, L

)

ln(ε)

q = -2q = -1q = 0q = 1q = 2

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0

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F(q

,Φ,L

)

ln(ε)

q = -2q = -1q = 0q = 1q = 2

Fig. 9. (Color online) Same as Figure 4, but for a wavefunction of the three-dimensional Anderson Hamiltonian forthe linear system size N = 60 and the disorder strengthw = 16.5. The least squares fits consider all box sizes in therange 5 ≤ L ≤ N/4 marked by the two vertical dashed lines.

box probabilities is already rather different for relativelysmall box sizes. Further, due to the identical distributionof box probabilities for the partitioning schemes B andC, the only differences are the different number of consid-ered data points for the least squares fit and hence thesingularity spectra for both schemes are very similar.

All results show that scheme B significantly reducesthe fit errors. This can be expected to ultimately lead toa better convergence of the ensemble average. To demon-strate this, we also show the singularity spectrum for wavefunctions closest to E = 0 for ten different disorder real-izations and the corresponding fit errors for the schemesA and B in Figure 11. For moderate values of q, scheme Bshows very small fluctuations of the scaling exponents,which are in all cases smaller than for scheme A. We alsopresent the corresponding ensemble average from these tenwave functions in Figure 12, which is obtained from [7]

αq = limε→0

1ln ε

1〈Pq〉

⟨B∑

b=1

μqb(Φ, L) ln μb(Φ, L)

⟩

fq = limε→0

1ln ε

[q

〈Pq〉

⟨B∑

b=1

μqb(Φ, L) ln μb(Φ, L)

⟩−ln〈Pq〉

].

(11)

Page 8 of 10 Eur. Phys. J. B (2013) 86: 48

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1

1.5

2

2.5

3

0 1 2 3 4 5 6 7

sing

ular

ity s

pect

rum

f(α q

)

singularity strength αq

scheme Ascheme Bscheme C

Fig. 10. (Color online) Singularity spectrum f(αq) for thethree different methods for the system in Figure 9 with con-sidered box sizes from L = 10 to N/2. The plot shows 1σ errorbars for integer q from −3 to 3.

The error bars indicate the standard deviation due to theleast squares fit and are not due to the fluctuations ofthe points for different disorder realizations. We also showthe squares r2

f(α) of the correlation coefficients of the frac-tal dimension for both methods. The standard deviationsare quite small for both methods although we averageonly over ten wave functions. However, method B showsa significantly better performance for large positive andnegative q.

We also like to point out that in literature differentapproaches for the determination of averages and errorsin multifractal analysis can be found. On the one handthe exponents αq and fq were determined from a leastsquares fit according to equation (11) [7,16], and on theother hand they were computed from the average of αq(Φ)and fq(Φ) obtained for the single wave functions [17,18].The first method yields the ensemble average and the sec-ond method corresponds to the so-called typical averageof the multifractal properties of the wave functions. Forreasons of comparison, we show in Figure 13 the results forthe typical average of the ten disorder realizations in Fig-ure 11. The results for schemes A and B are quite closebut the error bars for scheme B are again significantlysmaller. The main differences between the ensemble andthe typical average occur for large positive and negative qbecause for these cases the typical average is dominatedby the behavior of a single wave function [14].

Further, in literature, the error bars denote often dif-ferent errors of the method like e.g. the standard deviationof the least squares fit [4,19] or the standard deviationfrom averaging over the exponents obtained for differ-ent disorder realizations [17,20]. More advanced analysesuse error propagation and consider correlations for thedata points in the least squares fits according to equa-tion (11) [16].

In addition, we compare the resulting singularity spec-trum of the ensemble average in Figure 12 and of the typ-ical average in Figure 13 to the symmetry relation

f(αq) = f(2d − αq) − d + αq (12)

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8

sing

ular

tiy s

pect

rum

f(α q

)

singularity strength αq

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8

sing

ular

tiy s

pect

rum

f(α q

)

singularity strength αq

Fig. 11. (Color online) Singularity spectrum f(αq) for wavefunctions at E = 0 for 10 different disorder realisations ofthe three-dimensional Anderson Hamiltonian for the systemsize N = 60 and the disorder strength w = 16.5 computedwith scheme B with 5 ≤ L ≤ 15 (top) and scheme A with5 ≤ L ≤ 30 (bottom). The plot shows 1σ error bars for inte-ger q from −2 to 2, and identical disorder realizations/wavefunctions are used for schemes A and B.

proposed by Mirlin et al. [21]. We observe that this sym-metry relation is fulfilled well for the ensemble andthe typical average for positive q. However, for large αq,the ensemble averaged data show some differencesfrom the symmetry relations, and we find considerabledeviations for the typical average. Further, the proposedupper bound αq < 2d is satisfied neither for the ensembleaverage nor for the typical average [21] although the re-sults of scheme B for the ensemble average in Figure 12 arequite close to this upper bound within the error margins.These singularity strengths correspond to large negativevalues of q and, therefore, small errors are strongly en-hanced in this region and the fit quality is decreasing (seealso Fig. 8). This has been found before in many othernumerical studies. For instance, the results by Rodriguezet al. show that the bound αq < 2d is approached forlarger system sizes [16,22]. However, the comparison be-tween the numerical results for scheme A with this sym-metry relation shows bigger differences. The observationsfor the symmetry relation and for the ensemble average inFigure 12 support the finding that scheme B can help toimprove the ensemble average in the multifractal analysis.

Eur. Phys. J. B (2013) 86: 48 Page 9 of 10

0

0.5

1

1.5

2

2.5

3

sing

ular

tiy s

pect

rum

f(α q

)

symmetry relation for scheme Asymmetry relation for scheme B

scheme Ascheme B

0

0.5

1

0 1 2 3 4 5 6 7

r f(α)

2

singularity strength αq

scheme Ascheme B

Fig. 12. (Color online) Ensemble average of the singularityspectrum (solid lines) and the correlation coefficients r2

f(α) ofthe fractal dimensions for the wave functions in Figure 11.The error bars reflect only the accuracy of the least squares fitfor q from −1.5 to 3. We check whether the results fulfill thesymmetry relation in equation (12) by plotting the quantityf(2d − αq) − d + αq (dashed lines).

0

0.5

1

1.5

2

2.5

3

sing

ular

tiy s

pect

rum

⟨f(⟨

α q⟩)

⟩

symmetry relation for scheme Asymmetry relation for scheme B

scheme Ascheme B

0

0.5

1

0 1 2 3 4 5 6 7

⟨rf(

α)2

⟩

average singularity strength ⟨αq⟩

scheme Ascheme B

Fig. 13. (Color online) Typical average of the singularity spec-tra (solid lines) and the averaged correlation coefficients 〈r2

f(α)〉of the fractal dimensions for the wave functions in Figure 11.The error bars reflect the standard deviation from the aver-aged exponents 〈αq〉 and 〈fq〉 for q from −2 to 2. We againcheck whether the results fulfill the symmetry relation in equa-tion (12) (dashed lines).

6 Conclusion

We have compared different partitioning schemes in thebox-counting algorithm in order to improve the evaluationof equations (7) and (9) for a single wave function. We be-lieve that the partitioning scheme considering unrestricted

ratios 1/ε of the system size and the box size as well asan average over all box origins is better suited than thebest method proposed by Rodriguez et al. [7]. This con-clusion is based in particular on the observation that ourmethod leads to smaller error bounds in the least squaresfits and on the fact that it is not restricted to system sizeswith preferable large numbers of factors. Our scheme isnot meant to improve the ensemble average procedure andtherefore it does not interfere with the established errorestimates of this average. We improve the evaluation ofequations (7) and (9) by avoiding any bias in the choice ofthe origin of the boxes and by consequently averaging inequation (10) over all possible choices. We found that theaverage over different box origins yields a better estimateof the distribution for the box probability already for mod-erately large box sizes L. Due to the thus improved valuesof A(q, Φ, L) or F (q, Φ, L), a better fit of the singularitystrength αq and the fractal dimension fq becomes feasi-ble. Further, we showed that only small box sizes shouldbe considered for the fit. Additionally, our scheme allowsus to consider many more box sizes L, what also helps toimprove the fits of αq and fq.

Rodriguez et al. also studied the multifractal prop-erties of the Anderson Hamiltonian with system-sizescaling although it is computationally more demand-ing [5,14]. They found that system-size scaling yieldsbetter fit results than the box-scaling approach. How-ever, they compared their results with that of partitioningscheme A. Hence, a comparison with the better partition-ing scheme B is necessary to validate the better perfor-mance of the system-size scaling approach. But we expectthat also for the system-size scaling, scheme B is superiorin comparison with scheme A, not only because of the av-eraging over the box origins, which yields more accuratevalues of A(q, Φ, L) or F (q, Φ, L), but also because thereis no restriction for the system sizes: in A again one needsa value of N which factorizes into many factors while in Bevery N can be included in the evaluation.

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