# Scattering and transport statistics at the metal-insulator transition: A numerical study of the power-law banded random matrix model

###### Abstract

We study numerically scattering and transport statistical properties of the one-dimensional Anderson model at the metal-insulator transition described by the Power-law Banded Random Matrix (PBRM) model at criticality. Within a scattering approach to electronic transport, we concentrate on the case of a small number of single-channel attached leads. We observe a smooth crossover from localized to delocalized behavior in the average scattering matrix elements, the conductance probability distribution, the variance of the conductance, and the shot noise power by varying (the effective bandwidth of the PBRM model) from small () to large () values. We contrast our results with analytic random matrix theory predictions which are expected to be recovered in the limit . We also compare our results for the PBRM model with those for the three-dimensional (3D) Anderson model at criticality, finding that the PBRM model with reproduces well the scattering and transport properties of the 3D Anderson model.

###### pacs:

03.65.Nk, 71.30.+h, 73.23.-b## I Introduction

The study of systems at the Anderson metal-insulator transition (MIT) has been a subject of intensive research activity for several decades. A58 ; AMPJ95 ; AKL91 ; EM08 In particular, much interest has been focused on the scattering properties of critical systems by analysing the probability distribution functions of the resonance widths and Wigner delay times , MK05 ; OF05 ; MFME06 ; OKG03 ; F03 ; TC99 ; KW02 ; KW02b ; WMK06 ; MI06 as well as the transmission or dimensionless conductance . S90 ; M94 ; M99 ; SO97 ; SOK00 ; SMO00 ; WLS98 ; RS01 ; RMS001 ; TM02 ; SM05 ; JMZ99 The distribution functions of and have been shown to be related to the properties of the corresponding closed systems, i.e., the fractality of the eigenstates and the critical features of the MIT. On the other hand, at the MIT, the distribution of conductances has been found to be universal, i.e., size independent, but dependent on the adopted model, dimensionality, symmetry, and even boundary conditions of the system. has been studied for systems in two and more dimensions with a large number of attached single-channel leads. S90 ; M94 ; M99 ; SO97 ; SOK00 ; SMO00 ; WLS98 ; RS01 ; RMS001 ; TM02 ; SM05 ; JMZ99 In fact, concerning the conductance of one-dimensional (1D) systems and its statistical distribution, the regime of small number of leads has been left almost unexplored. JMZ99 ; SZ02 ; MG09a ; MG09b

In the present work we study numerically several statistical properties of the scattering matrix and the electronic transport across disordered systems in one and three dimensions described by the Power-law Banded Random Matrix (PBRM) model at criticality and the three-dimensional (3D) Anderson model at the MIT, respectively. We stress that we concentrate on the case of a small number of attached leads each of them supporting one open channel.

The organization of this paper is as follows. In the next subsection we define the PBRM model, the 3D Anderson model, as well as the scattering setup. We also define the scattering quantities under investigation and provide the corresponding analytical predictions from random scattering-matrix theory (RMT) for systems with time-reversal symmetry. These analytical results will be used as a reference along the paper. In Section II we analyse the average scattering matrix elements, the conductance probability distribution, the variance of the conductance, and the shot noise power for the PBRM model as a function of its effective bandwidth . In Section III we compare the results of the PBRM model at criticality with the scattering and transport properties of the 3D Anderson model. Finally, Section IV is left for conclusions.

### i.1 The PBRM and the 3D Anderson models

As we have mentioned above, in the present study we adopt two models, namely, the PBRM model at criticality and the 3D Anderson model at the MIT.

The PBRM model EM08 ; MFDQS96 describes 1D samples of length with random long-range hopping. This model is represented by () real symmetric matrices whose elements are statistically independent random variables drawn from a normal distribution with zero mean, , and a variance decaying as a power law , where and are parameters. There are two prescriptions for the variance of the PBRM model: the so-called non-periodic,

(1) |

where the 1D sample is in a line geometry; and the periodic,

(2) |

where the sample is in a ring geometry. Field-theoretical considerations EM08 ; MFDQS96 ; KT00 and detailed numerical investigations EM08 ; EM00b ; V03 have verified that the PBRM model undergoes a transition at from localized states for to delocalized states for . This transition shows all the key features of the Anderson MIT, including multifractality of eigenfunctions and non-trivial spectral statistics at the critical point. Thus the PBRM model possesses a line of critical points . We set in our study, i.e., we work with the PBRM model at criticality.

The 3D Anderson model with diagonal disorder is described by the tight-binding Hamiltonian (TBH)

(3) |

where labels all the sites of a cubic lattice with linear size , while the second sum is taken over all nearest-neighbour pairs on the lattice. The on-site potentials for are independent random variables. When are Gaussian distributed, with zero mean and variance , the MIT at energy occurs for . See Refs. [M99, ; RMS01, ; MW99, ]. Then, for () the system is in the metallic (insulating) regime. We set in our study.

We open the isolated samples, defined above by the PBRM model and the 3D Anderson model, by attaching semi-infinite single channel leads. Each lead is described by the 1D semi-infinite TBH

(4) |

Using standard methods MW69 one can write the scattering matrix (-matrix) in the form SOKG00 ; KW02

(5) |

where is the unit matrix, is the wave vector supported in the leads, and is an effective non-hermitian Hamiltonian given by

(6) |

Here, is an matrix that specifies the positions of the attached leads to the sample. Its elements are equal to zero or , where is the coupling strength. Moreover, assuming that the wave vector do not change significantly in the centre of the band, we set and neglect the energy dependence of and .

### i.2 RMT predictions for the Circular Orthogonal Ensemble

Notice from Eqs. (1-2) that in the limit the PBRM model reproduces the Gaussian Orthogonal Ensemble. Therefore, in that limit we expect the statistics of the scattering matrix, Eq. (5), to be determined by the Circular Orthogonal Ensemble (COE) which is the appropriate ensemble for systems with time reversal symmetry. Thus, below, we provide the statistical results for the average -matrix and the transport quantities to be analysed in the following sections, assuming the orthogonal symmetry. In all cases, we also assume the absence of direct processes, i.e., .

We start with the average of the -matrix elements. It is known that

(7) |

where means ensemble average over the COE.

Within a scattering approach to the electronic transport, once the scattering matrix is known one can compute the dimensionless conductance and its distribution . For , i.e., considering two single-channels leads attached to the sample, is given by

(8) |

while for ,

(9) |

For arbitrary , the predictions for the average value of and its variance are

(10) |

and

(11) |

respectively. For the derivation of the expressions above see for example Ref. [MK04, ]. Another transport quantity of interest is the shot noise power , which as a function of reads evgeny ; HMBH06 ; savin

(12) |

In the following sections we focus on , , , and for the PBRM model and the 3D Anderson model, both at the MIT. In all cases we set the coupling strength such that in order to compare our results, in the proper limits, with the RMT predictions introduced above.

## Ii PBRM model

We attach the leads to the first sites of the 1D sample described by the PBRM model. That is, in the non-periodic version of the PBRM model, Eq. (1), we attach the leads at the boundary of the system. See Fig. 1(a). While in the periodic version, Eq. (2), we attach them to the bulk. See Fig. 1(b). In the latter case, finite size effects are considerably reduced. However, the quantities we analyse below are -independent once is much larger than the number of attached leads for both versions of the PBRM model.

We point out that our setup is significantly different from the one used in Refs. [MG09a, ; MG09b, ], where the conductance has also been studied using the PBRM model. There, for example, the leads in the case are attached to sites which are separated a distance of and in the periodic and non-periodic versions of the PBRM model, respectively. In such situation the scattering quantities are strongly -dependent.

### ii.1 Average scattering matrix elements

First we consider the case , where the -matrix is a matrix. In Fig. 2(a) we plot the ensemble average of the elements and as a function of the bandwidth parameter , and , for the periodic (black symbols) and the non-periodic (red symbols) PBRM model. We concentrate on these two matrix elements since the other two elements give no additional information: and . Notice a strong -dependence of the average -matrix elements driving them from a localized-like regime [ and ; i.e. the average conductance is close to zero] for , to a delocalized-like or ballistic-like regime [ and ; i.e., RMT results are already recovered] for .

Moreover, we have found that and , as a function of the bandwidth , are well described by

(13) | |||||

(14) |

where is a fitting parameter. Eq. (13) is a consequence of the unitarity of the scattering matrix, , while the factor 1/3 in Eq. (14) comes from Eq. (7) with . In Fig. 2(a) we plot Eqs. (13) and (14) (dashed lines) and compare them with the corresponding results from the PBRM model (symbols) in the periodic and non-periodic setups. In the same figure Eq. (7) is also plotted (dot-dashed lines).

On the other hand, it is well known that in systems at the disorder driven MIT both the energy spectra and the eigenstates exhibit multifractal characteristics. EM08 The PBRM model is characterized by the effective bandwidth that drives the system from strong () to weak () multifractality. Multifractality can be quantified by the generalized dimensions which describe the fluctuations of the eigenfunctions. The multifractal dimensions of the -th eigenfunction (given as a linear combination of the basis states in a system with linear size , ) are defined through the so-called inverse participation numbers, , by the scaling EM08 ; EM00a ; V03

(15) |

where is the typical value of . However, among all dimensions, the correlation dimension plays a prominent role. HP83 As transits from zero to infinity, takes values from zero to one.

Using numerical diagonalization of matrices with sizes , 256, 512, and 1024, we extracted by the use of Eq. (15). For each system size , we used a number of disorder realizations in order to have at least data for its analysis. For each disorder realization we used of the states at the centre of the spectral band (where the density of states is almost constant) in order to avoid boundary effects. We found good agreement between the numerically obtained and the analytical estimation EM08 ; EM00a

Moreover, in Ref. [MKC05, ] the following phenomenological analytical expression for as a function of was proposed:

(16) |

where is a fitting parameter. See also Ref. [MI06, ]. Eq. (16) describes well our numerical results for with and for the periodic and non-periodic versions of the PBRM model, respectively.

Then, by substituting Eq. (16) into Eqs. (13) and (14) we obtain the following expressions for the averages of and as a function of :

(17) | |||||

(18) |

In Fig. 2(b) we compare Eqs. (17) and (18) (dashed lines) with the ensemble-average numerical results at different values of . The agreement is excellent. In addition, it is interesting to note that

which might be relevant for systems at the MIT where can be tuned.

Finally we want to remark that concerning for the PBRM model, the RMT limit, expected for , is already recovered for . See Fig. 2(a).

50 | |||
---|---|---|---|

100 | |||

200 | |||

400 | |||

800 |

### ii.2 Conductance probability distribution

Now we turn to the conductance statistics. For the conductance distribution is highly concentrated close to . So it is more convenient to analyse the distribution of the transmission logarithm, , instead. Then, in Fig. 3(a) we show for several values of in the case , for the periodic and non-periodic versions of the PBRM model. Notice that the distribution functions do not change their shape by increasing , mainly for , thus being scale invariant. In fact, for clearly displays a linear behavior when plotted as a function of as shown in Fig. 3(b). Then, all distributions functions fall one on top of the other when shifting them along the -axis by the typical value of ,

(19) |

as shown in Fig. 3(c).

Note that for , is proportional to , see Figs. 3(a) and 3(c). Moreover, we found that such behavior extends from small to large values of , . Also, notice that the behavior for small coincides with the RMT prediction , see Eq. (8), since the change of variable leads to .

In Fig. 4 we show for large (). In the limit , is expected to approach the RMT prediction of Eq. (8). However, once , is already well described by .

We want to stress that our results do not depend on the system size , as can be seen in Table 1 where we report for the periodic PBRM model at criticality for some values of and in the case . In fact, does not depend on out of criticality ( and ) either. This makes an important difference with respect to the setup where the leads are attached to opposite sides of the sample where increases (decreases) as a function of for ( and ).

Now, in Figs. 5 and 6 we explore in the case . As well as in the case , studied above, here: (i) for small , is scale invariant with as scaling factor, see Fig. 5; (ii) for , is well described by Eq. (9), the corresponding RMT prediction, see Fig. 6. However, although for small and for large are practically the same for the periodic and non-periodic versions of the PBRM model, they show differences for intermediate values, , as can be seen in Fig. 5(a) and the upper panels of Fig. 6.

We point out that for small , , is proportional to , as shown in Fig. 5(c). This behavior is universal for the PBRM model with ; i.e. it is -independent and valid for the periodic and non-periodic versions of the model. Again, as for the case, here for , the dependence in the limit is consistent with the RMT prediction for ; see Eq. (9).

From the results above and since MK04

(20) |

in the region , we conclude that for the PBRM model

(21) |

for . We argue that the behavior dictated by Eq. (21) can be interpreted in the following way: Since small values of conductance mean very strong reflection, the scattering process is almost direct, i.e. an incident electron is scattered out the system mainly by the interaction with the sites at which the leads are attached. Then the electron does not explore the complete sample, not even part of it, and as a consequence it does not realize that the sample is at criticality. That is, for , the incident electron does not distinguish between a critical system and a random system represented by a full random matrix if the number of attached leads is not much larger than ; i.e., when the leads are attached to sites interconnected by Hamiltonian matrix elements located within the bandwidth of the matrix where all elements have almost the same variance: . So, we can use Eq. (20) and as a consequence Eq. (21) too. In fact, we have verified that Eq. (21) is valid up to for , while for it describes perfectly the left tail of for all the values of used in this work (up to ). Our result, given by Eq. (21), is the generalization of the result shown in Ref. [MG09a, ], where the behavior was reported for the PBRM model in the localized () and delocalized regimes () for .

### ii.3 Mean and variance of the conductance and the shot noise power

We now increase further the number of attached
leads.^{1}^{1}1All quantities reported in Figs. 2
to 6 were obtained for using ensemble
realizations. In Subsection IIC we used and
ensemble realizations. We have verified that in both cases
our results are invariant by further increasing . See
also Table 1.
In Figs. 7(a), 8(a), and 9(a) we plot
the average conductance , the variance of the
conductance , and the shot noise power for
the non-periodic version of the PBRM model for several values
of with .^{2}^{2}2
We recall that for , ten single-channel leads are attached
to the sample.
It is clear from these three plots that changing
from small () to large () values produces a
transition from localized- to delocalized-like behavior in the
scattering properties of the PBRM model. That is, (i) for
, , ,
and ; and (ii) for , ,
, and are well described by the corresponding
RMT predictions given by Eqs. (10),
(11), and (12), respectively.
Similar plots are obtained (not shown here) for the periodic
PBRM model.

Moreover, we have observed that , , and behave (for all ) as does. See Eq. (14). Thus, we can write

(22) |

where represents , , or and is a fitting parameter. Then, in Figs. 7(b), 8(b), and 9(b) we plot , , and normalized to their respective COE average values, now for the periodic PBRM model as a function of for . Also, similar plots are obtained (not shown here) for the non-periodic PBRM model.

## Iii 3D Anderson model

Since the most prominent realization of systems that undergo a MIT is the 3D Anderson model, it is of relevance to analyse its scattering and transport properties taking as a reference the results shown in the previous section for the PBRM model.

We attach the leads to sites at one of the edges of the cubic lattice described by the 3D Anderson model. In this way we make a line contact as we did in the case of the PBRM model. So, we can compare the scattering properties of both models. We use Gaussian distributed on-site potentials and system sizes from to 10 (we have verified that our results do not change by increasing further) with to disorder realizations.

We start our analysis by looking at the average scattering matrix elements for . For the 3D Anderson model at criticality we found and , see Table 2, which are close to those of the non-periodic PBRM model with .

In Fig. 10 we show conductance probability distributions for the 3D Anderson model at criticality in the cases and . We found and for and , respectively. See Table 3. These values of for the 3D Anderson model are close to those of the PBRM model with and , respectively. Notice also that, as well as for the PBRM model, for the 3D Anderson model in the region of is proportional to and for and , respectively (dashed lines in Fig. 10).

Additionally, in Fig. 11 we plot for the 3D Anderson model in the cases and . In the same figure we have also plotted (dashed lines) the conductance distributions from the PBRM model with . We can see that the conductance distributions of both models are similar at this bandwidth value.

6 | ||
---|---|---|

8 | ||

10 |

6 | ||
---|---|---|

8 | ||

10 |

We also compute , , and for the 3D Anderson model and plot them in Figs. 7(a), 8(a), and 9(a), respectively (red curves labeled as 3DAM). We observe that for the 3D Anderson model , , and behave as the corresponding quantities for the PBRM model with close to 0.2.

We want to stress that our results for the 3D Anderson model at criticality do not seem to depend on the on-site potential distribution. Here we used Gaussian distributed potentials. However, we observe practically the same results by the use of box distributed on-site potentials (not shown here).

## Iv Conclusions

We study the scattering and transport properties of the Power-law Banded Random Matrix (PBRM) model and of the three-dimensional (3D) Anderson model, both at criticality.

We observed a smooth crossover from localized- to delocalized-like (ballistic-like) behavior in the scattering properties of the PBRM model by varying from small () to large () values. For this crossover region we proposed heuristic analytical expressions for and . For small , we have shown that is scale invariant with the typical value of , , as scaling factor. We realized that RMT results, expected in the limit , are already recovered for relatively small values of the bandwidth: . However, this fact is closely related to the small number of attached leads we used in this work: the larger the number of leads the larger the value of needed to approach RMT behavior.

Our conclusions are valid for leads attached to the bulk of the system as well as for leads attached at the boundary. In this work we assumed that time-reversal symmetry () is present in our disordered systems. Moreover, the case of broken time-reversal symmetry () has been preliminarily studied in Ref. [AMV09, ] where similar conclusions to this work were made.

We have also shown that the scattering properties of the 3D Anderson model are similar to those for the PBRM model with . This result is in agreement with previous studies EM00b were it was shown that several critical quantities related to the spectrum and eigenstates of the PBRM model with are practically the same as for the 3D Anderson model. This makes the PBRM model an excellent candidate to explore the properties of the 3D Anderson model at criticality at a low computational cost.

We want to stress that for both models at criticality, the PBRM model and the 3D Anderson model, we found the universal behavior for , that we tested here for and . Moreover by the use of the PBRM model with (not shown here) we have already verified that the more general expression , derived from MK04 with , holds.

We emphasize that, even though we used a scattering setup where the leads are attached perfectly to sites at one side of the sample, our conclusions are not restricted to this topology.unpublished

Finally, we recall that we concentrate here on the case of
a small number of single-channel attached leads (up to ten).
The study of the scattering and transport properties of the
PBRM and the 3D Anderson models in the regime of
will be the subject of a forthcoming
contribution.^{3}^{3}3In Ref. [MG09b, ] some
properties of the transmission for the PBRM model were
already studied for .

###### Acknowledgements.

We thank one of the Referees for valuable comments. This work was partially supported by the Hungarian-Mexican Intergovernmental S & T Cooperation Program under grants MX-16/2007 (NKTH) and I0110/127/08 (CONACyT). We also acknowledge support form CONACyT Mexico grant CB-2006-01-60879; the European Social Fund; The MICINN (Spain) under Project No. FIS 2009-16450 and through the Ramón y Cajal Program; the Alexander von Humboldt Foundation; and the Hungarian Research Fund (OTKA) grants K73361 and K75529.## References

- (1) P. W. Anderson, Phys. Rev. 109, 1492 (1958); A. MacKinnon and B. Kramer, Rep. Prog. Phys. 56, 1469 (1993); E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979); L. P. Gorkov, A. I. Larkin, and D. E. Khmelnitskii, JETP Lett. 30 228 (1979).
- (2) Mesoscopic Quantum Physics, Proceedings of the Les Houches Session LXI, ed. E. Akkermans, G. Montambaux, J.-L. Pichard, and J. Zinn-Justin (North-Holland, Amsterdam, 1995).
- (3) B. L. Altshuler, V. E. Kravtsov, and I. V. Lerner, in Mesoscopic Phenomena in Solids, edited by B. L. Altshuler, P. A. Lee, and R. A. Webb (North Holland, Amsterdam, 1991).
- (4) F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008).
- (5) J. A. Méndez-Bermúdez and T. Kottos, Phys. Rev. B 72, 064108 (2005).
- (6) A. Ossipov and Y. V. Fyodorov, Phys. Rev. B 71, 125133 (2005).
- (7) A. D. Mirlin, Y. V. Fyodorov, A. Mildenberger, and F. Evers, Phys. Rev. Lett. 97, 046803 (2006).
- (8) A. Ossipov, T. Kottos, and T. Geisel, Europhys. Lett. 62, 719 (2003).
- (9) Y. V. Fyodorov, JETP Letters 78, 250 (2003).
- (10) C. Texier and A. Comtet, Phys. Rev. Lett. 82, 4220 (1999); A. Ossipov, T. Kottos, and T. Geisel, Phys. Rev. B 61, R11411 (2000).
- (11) T. Kottos and M. Weiss, Phys. Rev. Lett. 89 056401, (2002).
- (12) F. Steinbach, A. Ossipov, T. Kottos, and T. Geisel, Phys. Rev. Lett. 85, 4426, (2000).
- (13) M. Weiss, J. A. Méndez-Bermúdez, and T. Kottos, Phys. Rev. B 73, 045103, (2006).
- (14) J. A. Méndez-Bermúdez and I. Varga, Phys. Rev. B 74, 125114 (2006).
- (15) B. Shapiro, Phys. Rev. Lett. 65, 1510 (1990).
- (16) P. Markoš, Europhys. Lett. 26, 431 (1994).
- (17) P. Markoš, Phys. Rev. Lett. 83, 588 (1999).
- (18) K. Slevin and T. Ohtsuki, Phys. Rev. Lett. 78, 4083 (1997).
- (19) K. Slevin, T. Ohtsuki, and T. Kawarabayashi, Phys. Rev. Lett. 84, 3915 (2000).
- (20) K. Slevin, P. Markoš, and T. Ohtsuki, Phys. Rev. Lett. 86, 3594 (2001).
- (21) X. Wang, Q. Li, and C. M. K. Soukoulis, Phys. Rev. B 58, 3576 (1998).
- (22) M. Rühländer and C. M. Soukoulis, Physica B 296, 32 (2001).
- (23) M. Rühländer, P. Markoš, and C. M. Soukoulis, Phys. Rev. B 64, 172202, 212202 (2001).
- (24) I. Travěnec and P. Markoš, Phys. Rev. B 65, 113109 (2002).
- (25) L. Schweitzer and P. Markoš, Phys. Rev. Lett. 95, 256805 (2005); J. Phys. A: Math. Gen. 39, 3221 (2006).
- (26) M. Jansen, M. Metzler, and M. R. Zirnbauer, Phys. Rev. B 59, 15836 (1999).
- (27) K. Senouci and N. Zekri, Phys. Rev. B 66, 212201 (2002).
- (28) C. Monthus and T. Garel, Phys. Rev. B 79, 205120 (2009).
- (29) C. Monthus and T. Garel, J. Stat. Mech. P07033 (2009).
- (30) A. D. Mirlin, Y. V. Fyodorov, F.-M. Dittes, J. Quezada, and T. H. Seligman, Phys. Rev. E 54, 3221 (1996).
- (31) V. E. Kravtsov and K. A. Muttalib, Phys. Rev. Lett. 79, 1913 (1997); V. E. Kravtsov and A. M. Tsvelik, Phys. Rev. B 62, 9888 (2000).
- (32) E. Cuevas, M. Ortuno, V. Gasparian, and A. Perez-Garrido, Phys. Rev. Lett. 88, 016401 (2001).
- (33) I. Varga, Phys. Rev. B 66, 094201 (2002); I. Varga and D. Braun, ibid. 61, R11859 (2000).
- (34) A. Cohen, Y. Roth, and B. Shapiro, Phys. Rev. B 38, 12125 (1988); C. M. Soukoulis, X. S. Wang, Q. M. Li, and M. M. Sigalas, Phys. Rev. Lett. 82, 668 (1999); K. Slevin and T. Ohtsuki, ibid. 82, 382 (1999).
- (35) K. A. Muttalib and P. Wölfle, Phys. Rev. Lett. 83, 3013 (1999).
- (36) C. Mahaux and H. A Weidenmüller, Shell Model Approach in Nuclear Reactions, (North-Holland, Amsterdam,1969); J. J. M. Verbaarschot, H. A. Weidenmüller, and M. R. Zirnbauer, Phys. Rep. 129, 367 (1985); I. Rotter, Rep. Prog. Phys. 54, 635 (1991).
- (37) F. Steinbach, A. Ossipov, T. Kottos, and T. Geisel, Phys. Rev. Lett. 85 4426, (2000).
- (38) P. A. Mello and N. Kumar, Quantum Transport in Mesoscopic Systems (Oxford University Press, Oxford, 2004).
- (39) E. N. Bulgakov, V. A. Gopar, P. A. Mello, and I. Rotter, Phys. Rev. B 73, 155302 (2006).
- (40) S. Heusler, S. Muller, P. Braun, and F. Haake, Phys. Rev. Lett. 96, 066804 (2006); J. Phys. A: Math. Gen. 39, L159 (2006).
- (41) D. V. Savin and H.-J. Sommers, Phys. Rev. B 73, 081307(R) (2006).
- (42) F. Evers and A. D. Mirlin, Phys. Rev. Lett. 84, 3690 (2000).
- (43) H. G. E. Hentschel and I. Procaccia, Physica D 8, 435 (1983); I. Varga and J. Pipek, J. Phys.: Condens. Matter 10, 305 (1998).
- (44) J. A. Méndez-Bermúdez, T. Kottos, and D. Cohen, Phys. Rev. E 73, 036204 (2006).
- (45) A. Alcazar-López, J. A. Méndez-Bermúdez, and I. Varga, Ann. Phys. (Berlin) 18, 896 (2009).
- (46) J. A. Méndez-Bermúdez, V. A. Gopar, and I. Varga, unpublished.