VOLUME 78, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 26 MAY 1997

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The Anderson Transition: Time Reversal Symmetry and Universality

Keith SlevinThe Institute of Physical and Chemical Research, Hirosawa 2-1, Wako-shi, Saitama 351-01, Jap

Tomi OhtsukiDepartment of Physics, Sophia University, Kioi-cho 7-1, Chiyoda-ku, Tokyo 102, Japan

(Received 13 December 1996)

We report a finite size scaling study of the Anderson transition. Different scaling functions adifferent values for the critical exponent have been found, consistent with the existence of the orthogand unitary universality classes which occur in the field theory description of the transition. The critconductance distribution at the Anderson transition has also been investigated and different distribufor the orthogonal and unitary classes obtained. [S0031-9007(97)03259-6]

PACS numbers: 71.30.+h, 71.23.–k, 72.15.Rn

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It is now widely accepted that the metal-insulator transition for noninteracting electrons, the so-called Andersotransition [1], is a continuous phase transition in whicstatic disorder plays a role analogous to temperaturethermal phase transitions. The field theoretical formultion of the problem [2], though not making reliable predictions for the critical exponent [3,4], does indicate thatshould be possible to describe the critical behavior witha framework of three universality classes: orthogonal, untary, and symplectic. However, recent work put this idein question. It was found that the scaling functions fosystems with orthogonal and unitary symmetry could nbe distinguished at an accuracy of a few percent [5]. Ncould the values of the critical exponent for the three unversality classes be reliably distinguished [5–9]. This, together with recent work on the statistics of energy levein the vicinity of transition, has prompted the suggestio[10] that the universality classes predicted by the field thory do not correctly describe the Anderson transition.

The two important symmetries in the field theory artime reversal symmetry (TRS) and spin rotation symmet(SRS). The system is said to be in the orthogonuniversality class if it has both SRS and TRS, in thunitary class if TRS is broken, and in the symplecticlass if the system has TRS but SRS is broken. Trelevant terms in the Hamiltonian are a coupling to aapplied magnetic field, which breaks TRS, and the sporbit interaction, which breaks SRS.

Here we focus on the breaking of TRS by a constaapplied magnetic field. We report the results of MontCarlo studies which indicate that the critical behavioat least as far as orthogonal and unitary symmetries aconcerned, is in accord with the conventional universaliclasses. By carrying out a precise study of the finisize scaling of the electron localization length, whichanalogous to the correlation length in thermal transitionwe have clearly differentiated the scaling functions for thorthogonal and unitary universality classes. We have aestimated the critical exponents and calculated confiden

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intervals for these estimates. We find a statisticasignificant difference of about12% between the values ofthe critical exponent in the orthogonal and unitary class

To reinforce the above conclusion we have also simlated the critical conductance distribution of a disordermesoscopic conductor. Following the discovery of unversal conductance fluctuations, it was realized thatconductance of a phase coherent system is not saveraging. Extrapolating from the metallic regime,seems that the conductance fluctuations at the critipoint should be of the same order of magnitude asmean conductance and that the full conductance distrition should be a more useful characteristic of the criticpoint. As we approach the critical point, the localizatiolength diverges and the system becomes effectively ssimilar; we then expect the conductance distributionbecome independent of system size and depend onlythe universality class. This expectation was borne outour study, where we found different critical conductancdistributions, depending on whether or not TRS is broke

The model Hamiltonian used in this study describnoninteracting electrons on a simple cubic lattice. Winearest neighbor interactions only, we have

k$rjHj$rl ­ V s$rd ,

k$rjHj$r 2 xl ­ 1 ,

k$rjHj$r 2 yl ­ 1 ,(1)

k$rjHj$r 2 zl ­ exps2i2pfxd ,

where x, y, and z are the basis vectors of the latticeThe electrons are subject to an external magnetic fiapplied in they direction whose strength is parametrizeby the flux f, measured in units of the flux quantumhye, threading a lattice cell. The on-site energies of telectronshV s$rdj are assumed to be independently anidentically distributed with probabilitypsV ddV , where

psV d ­ 1yW jV j # Wy2

­ 0 otherwise.

© 1997 The American Physical Society 4083

VOLUME 78, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 26 MAY 1997

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The critical point, scaling function, and value of thcritical exponent for several values off and Fermi energyEf (see Table I) are determined by examining the finisize scaling [11] of the localization lengthl for electronson a quasi-1d-dimensional bar of cross sectionL 3 L.The localization lengthl ; lsEf , f, W , Ld, defined by

l21 ­ limLz!`

k2 ln gsLzdl2Lz

, (2)

whereLz is the length of the bar andgsLzd is the conduc-tance of the bar measured in units ofse2yhd, can be evalu-ated by rewriting the Schrödinger equation as a producttransfer matrices;l21 can then be determined to within aspecified accuracy using a standard technique [12]. Taccuracy used here ranges between0.1% and0.2%.

On intuitive grounds it has been argued [13] that wshould observe orthogonal scaling whenL ø LH andunitary scaling whenL ¿ LH , whereLH is the magneticlength. For the lattice model (1),LH ­

p1y2pf. For

the smallest system sizeL ­ 6 used here, this criterionyields a crossover fluxfc . 1y226. We thus expect tosee clear unitary scaling behavior for cases Ua andlisted in Table I.

When the dimensionless quantityL ­ lyL is plottedagainst disorderW for different cross sectionsL, thecurves are found to have a common point of intersetion; this indicates the occurrence of the metal-insulattransition at a critical disorderWc in the3d system whichwould be obtained by lettingL ! `. Detailed analysis ofthe data is based on the following assumptions: first, thfor L finite L is a smooth function ofW and L; second,that the data obey a one parameter scaling law

LsW , Ld ­ f6sssLyjswdddd , (3)

wherew ­ sWc 2 WdyWc and the subscript6 refers tow . 0 andw , 0; and, third, that the lengthj appearingin the scaling law has a power law divergence close toWc

of the form jswd ­ j6jwj2n . This relation defines thecritical exponentn and introduces two arbitrary constantj1 sw . 0d andj2 sw , 0d. According to the Wegnerscaling law [14],n is related to the critical exponentsassociated with the conductivity bys ­ sd 2 2dn so thatn ­ s in d ­ 3. The above assumptions imply that ishould be possible to fit the data to

LfitsW , Ld ­ Lc 1Xn­1

AnLnynwn. (4)

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TABLE I. The parameters used in the numerical study: Fermi energyEf , flux f, systemsizesL, the range of disorder, the number of data pointsM, the number of fitting parametersN , the value ofx2 for the best fitx2

min, and the goodness of fitQ as estimated from thex2

distribution.

Label Ef f L fWmin, Wmaxg M N x2min Q

Oa 0 0 h6, 8, 10, 12j f15, 18g 86 6 79 0.50Ob 0.5 0 h6, 8, 10j f15.6, 17.6g 73 6 49 0.96Ua 0 1y3 h6, 9, 12j f17, 20g 55 6 35 0.94Ub 0 1y4 h8, 10, 12, 14j f17.5, 19.5g 66 6 65 0.31

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In practice, we have truncated this series atn ­ 3. Therelation between (3) and (4) can be made apparentwriting

f1 ­ Lc 1Xn­1

ansLyjdnyn, an ­ Ansj1dnyn , (5)

f2 ­ Lc 1Xn­1

bnsLyjdnyn ,

bn ­ s21dnAnsj2dnyn.(6)

In principle, j1 and j2 should depend on energyand flux, though the “amplitude ratio”j1yj2 may beuniversal [15] so thatj1 andj2 may not be independent.Their absolute values cannot be determined using tpresent method. No relative variation as a functioof energy and flux, which would be apparent in thsimulation, was detected. Therefore for convenience wset j1 ­ j2 ­ 1. The most likely fit is determined byminimizing thex2 statistic

x2 ­Xm

µLm 2 LfitsWm, Lmd

sm

∂2

, (7)

where the summationm is over all data points andsm isthe error (standard deviation) in the determination of thmth data point. After being fitted the data are replotteagainstLyj to check that they obey the scaling law (3).

We also need to determine the goodness of fitQand confidence intervals for the fitting parameters. Thgoodness of fit measures the credibility of the fit;Q .

0.001 is often regarded as acceptable in other applicatio[16]. We have checked that the numerical procedure usto estimate the localization lengths does so with an errwhich is approximately normally distributed. If we ignorethe presence of any systematic corrections to scaling in tdata, this permits the use of thex2 likelihood function todetermine the “best fit” and the estimation ofQ from thex2 distribution withM 2 N degrees of freedom.

The confidence intervals for the fitted parameters weestimated in two independent ways: first, from the Hessiamatrix obtained in the least squares fitting proceduand, second, using the bootstrap procedure described[16]. In the latter method the original data are repeatedrandomly sampled (with replacement) and fitted. Thprovides an independent check of the distribution of thfitted parameters. Both methods gave approximately t

VOLUME 78, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 26 MAY 1997

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TABLE II. The best fit values of the critical parametetogether with their95.4% confidence intervals as estimateusing the bootstrap method.

Label Wc Lc n

Oa 16.448 6 .014 0.5857 6 .0012 1.59 6 .03Ob 16.442 6 .018 0.5862 6 .0015 1.60 6 .06Ua 18.316 6 .015 0.5683 6 .0013 1.43 6 .04Ub 18.375 6 .017 0.5662 6 .0016 1.43 6 .06

same results. We chose to present the errors as95.4%marginal confidence intervals as given by the bootstmethod.

The results are summarized in Tables I and II. Givthe confidence intervals, the probability that the valuesthe critical exponent for the orthogonal and unitary unversality classes are the same seems to be negligible.ferent values ofLc can also be distinguished, confirminwhat is clearly evident in Fig. 1 that the orthogonal aunitary data scale differently. We conclude that the scing function is sensitive to the breaking of time reverssymmetry.

We now turn to the conductance distribution. Wconsider a cubic “sample” of sideL attached to semi-infinite leads on two opposite faces. The disorderW isset to zero in the leads, the Fermi energy and magnfield are constant throughout. The zero temperature linconductanceG ­ se2yhdg can be obtained fromg ­trtty, where t is the transmission matrix of the samplThe t matrix can be related to a Green’s function [17,1which can be determined iteratively.

For each set of parameters we have calculated the cductances for an ensemble of100 000 realizations of therandom potential. Some typical results are presented

FIG. 1. The scaling of the data for the parameter sets lisin Table I. The lines are the scaling functions given by (5) a(6). Different scaling functions for the orthogonal and unitauniversality classes can be clearly distinguished.

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FIG. 2. The critical conductance distributions for parameteset Oa listed in Table I.

Fig. 2. Our analysis of these results is based on the geeralization of (3)ps gd ­ ps g, Lyjd. At the critical pointj diverges, and we should obtain a universal critical conductance distributionpcs gd which is independent ofL.The scale invariance ofpcs gd is clearly demonstrated, atleast for the range of system sizes studied, in Fig. 2. Wfound a similar scale invariance for all the cases listedTable I. In Fig. 3 we have plotted the critical conductancdistributions obtained and, in Table III, tabulated somaverages of these distributions. The results are consistwith the existence of distinct orthogonal and unitary critical conductance distributions.

We now discuss the general features ofpcs gd, focus-ing on the orthogonal universality class and making

FIG. 3. The distribution ofg at the critical point. Orthogonaland unitary distributions can be clearly distinguished. Thcritical conductance distribution obtained in thee expansionis also shown.

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VOLUME 78, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 26 MAY 1997

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TABLE III. The means and standard deviations of the criticadistribution ofg and lng.

Label k gl sg kln gl sln g

Oa L ­ 10 0.579 0.598 21.206 1.319Ob L ­ 10 0.578 0.598 21.207 1.318Ua L ­ 12 0.395 0.499 21.869 1.590Ub L ­ 10 0.395 0.499 21.864 1.583

comparison with the critical distribution obtained in thee expansion in the field theory. Thenth cumulantcn ofpcs gd for a d-dimensional cube of linear dimensionLis [19]

cn ­

µp

2

∂nΩen22, n & n0 ,sLylden222n, n * n0 ,

(8)

where e ­ d 2 2, n0 is an integer of order1ye, and lis the elastic mean free path. As described in [20] itpossible, under certain assumptions, to derivepcs gd fromthis. Extrapolating tod ­ 3, we find

pes gd ­1e

ds gd 1p

2e

3

∑u

µpg2

∂1

12!

fu p ugµ

pg2

∂1 · · ·

∏, (9)

whereusxd ­ 4x23 exps22yxd, p denotes the convolution,and e ­ 2.71828, ... . The series (9) is easily handlednumerically. The result is shown in Fig. 3.

The most obvious feature is thatpcs gd is not peakedabout its mean valuek gl. The conductance fluctuationsas measured by the standard deviationsg, are of thesame order of magnitude ask gl. If we compare withpes gd, we find that we have a good approximation tthe central region of the distribution function but a rathebad approximation to its tails. The largeg tail of pes gddecays as1yg3, which means that all cumulants highe

FIG. 4. The critical distribution after changing variables.

4086

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than c1 diverge. This is reflected in (8), where thesecumulants are not universal but depend onl and L. Wecould find no evidence of this behavior, however, in thsimulation; as seen in Fig. 3 there is sharp decay ofpcs gdaboveg . 2, and the higher cumulants, at least as far an ­ 4, seem to have universal values.

Another way to look at the critical distribution isto change variables to lng (see Fig. 4). While thedistribution is certainly not of Gaussian form, it does showa central tendency.

In conclusion, we have presented numerical evidenwhich, we think, confirms that the critical behavior athe Anderson transition is sensitive to perturbations whicbreak time reversal symmetry.

T. O. would like to thank Yoshiyuki Ono for importantdiscussions at the outset of the present work. Some of thwork was carried out on supercomputer facilities at thISSP, University of Tokyo, and the Institute of Physicaand Chemical Research.

[1] For a review, see B. Kramer and A. MacKinnon, RepProg. Phys.56, 1469 (1993).

[2] K. B. Efetov, Adv. Phys.32, 53 (1983).[3] W. Bernreuther and F. Wegner, Phys. Rev. Lett.57, 1383

(1986).[4] S. Hikami, Prog. Theor. Phys. Suppl.107, 213 (1992).[5] M. Henneke, B. Kramer, and T. Ohtsuki, Europhys. Lett

27, 389 (1994).[6] A. MacKinnon, J. Phys. Condens. Matter6, 2511 (1994).[7] T. Ohtsuki, B. Kramer, and Y. Ono, J. Phys. Soc. Jpn.62,

224 (1993).[8] J. T. Chalker and A. Dohmen, Phys. Rev. Lett.75, 4496

(1995).[9] T. Kawarabayashi, T. Ohtsuki, K. Slevin, and Y. Ono,

Phys. Rev. Lett.77, 3593 (1996).[10] E. Hofstetter, Report No. cond-maty9611060 (to be

published).[11] Lectures on Phase Transitions and the Renormalizatio

Group, edited by N. Goldenfeld (Addison-Wesley, Read-ing, MA, 1992), Chap. 9.

[12] A. MacKinnon and B. Kramer, Z. Phys. B53, 1 (1983).[13] I. Lerner and Y. Imry, Europhys. Lett.29, 49 (1995).[14] F. Wegner, Z. Phys. B25, 327 (1976).[15] Quantum Field Theory and Critical Phenomena,edited

by J. Zinn-Justin (Oxford University, New York, 1996),Chap. 28.

[16] Numerical Recipes in Fortran,edited by W. Press,B. Flannery, and S. Teukolsky (Cambridge UniversityPress, Cambridge, England, 1992), Chap. 15.

[17] Electronic Transport in Mesoscopic Systems,edited byS. Datta (Cambridge University Press, Cambridge, England, 1995), Chap. 3.

[18] T. Ando, Phys. Rev. B44, 8017 (1991).[19] B. L. Altshuler, V. Kravtsov, and I. Lerner, Sov. Phys.

JETP64, 1352 (1986); Phys. Lett. A134, 488 (1989).[20] B. Shapiro, Phys. Rev. Lett.65, 1510 (1990); A. Cohen

and B. Shapiro, Int. J. Mod. Phys. B6, 1243 (1992).