Table Of ContentCritical parameters for the disorder-induced metal-insulator transition in FCC and
BCC lattices
Andrzej Eilmes1, Andrea M. Fischer2, Rudolf A. Ro¨mer2
1Department of Computational Methods in Chemistry,
Jagiellonian University, Ingardena 3, 30-060 Krak´ow, Poland
2Department of Physics and Centre for Scientific Computing,
University of Warwick, Coventry, CV4 7AL, United Kingdom
(Dated: February 2, 2008)
8 We use a transfer-matrix method to study the disorder-induced metal-insulator transition. We
0 takeisotropicnearest-neighborhoppingandanonsitepotentialwithuniformlydistributeddisorder.
0 Followingpreviousworkdoneonthesimplecubiclattice,weperformnumericalcalculationsforthe
2 bodycenteredcubicandfacecenteredcubiclattices,whicharemorecommoninnature. Weobtain
the localization length from calculated Lyapunov exponents for different system sizes. This data
n
is analyzed using finite-size scaling to find the critical parameters. We create an energy-disorder
a
phase diagram for both lattice types, noting that it is symmetric about the band center for the
J
body centered cubic lattice, but not for theface centered cubic lattice. We finda critical exponent
2
of approximately 1.5-1.6 for both lattice typesfor transitions occurring either at fixed energy or at
2
fixed disorder, agreeing with results previously obtained for other systems belonging to the same
orthogonal universality class. Wenotice an increase in critical disorder with the numberof nearest
]
n neighbors, which agrees with intuition.
n
-
s
I. INTRODUCTION calculatethecriticalparametersfortheMITintheBCC
i
d andFCClattices. Ofcourse,oneshouldexpectnochange
.
t Thedisorder-inducedmetal-insulatortransition(MIT) in the critical exponent, as all systems considered here
a
m and the concept of Anderson localization1,2,3,4,5 have belongto the sameorthogonaluniversalityclass. Onthe
other hand, because of the different numbers of nearest
been studied extensively for more than forty years. The
d- scaling theory of localization6 provides a very success- neighborsinSC(Z =6),BCC(8)andFCC(12)lattices,
n ful approach for non-interacting electrons. According to the values of the criticaldisorderWc andenergy Ec may
o its predictions, a disorder driven MIT occurs in three- be different. Hence our study tests and reconfirms uni-
c versality while at the same time allowing to see how the
dimensionalsystems,i.e.beyondacriticalamountofdis-
[ non-universal parameters of the transition change with
order W all eigenstates localize. For smaller disorder,
1 extendedcstates exist in the system. increasing coordination number.
v
For the simple cubic (SC) lattice and uniform dis-
8
order distribution, the critical disorder and the crit-
3
II. NUMERICAL APPROACH
3 ical exponent have been successfully calculated using
3 the transfer-matrix method (TMM).7,8 Highly accu-
. rate recent studies9,10,11,12 report W = 16.54 ± 0.02 A. The Transfer-Matrix Approach to
1 c
and the critical exponent ν = 1.57 ± 0.02.9 How- the Anderson Model of Localization
0
8 ever, direct diagonalizationresults based on energy-level
0 statistics13,14,15,16 and multi-fractal analysis17,18 give a To model the MIT in the 3D system we use the stan-
: smallerν =1.44±0.2. Furthermore,experimentalresults dard Anderson Hamiltonian
v
report yet smaller values of ν &1.0.19,20,21,22,23 Many of
i
X these observeddiscrepancies can be explained by the at- H= ǫ |iihi|− t |iihj|. (1)
i ij
ar tainablelimits onsystemsizes,temperatureandstatisti- Xi Xi6=j
cal averages in the above results, nevertheless, the quest
for an accurate determination of the critical parameters The orthonormal states |ii correspond to electrons lo-
at the Anderson transition is not yet complete. cated at sites i = (x,y,z) of a cubic lattice with pe-
The SC lattice is, in addition, not very common24 in riodic boundary conditions. The hopping integrals t
ij
nature. The only element knownto adoptit is the alpha arenon-zeroonlyfori,j beingnearest-neighborsandthe
phase of polonium; most metals exhibit body centered energy scale is set by choosing t = 1. The disorder
ij
(BCC) or face centered (FCC) cubic lattices. Although in the model is incorporated into the diagonal energies
the localization properties for an FCC lattice have been ǫ ∈[−W/2,W/2],randomlydistributedaccordingtothe
i
studied recently25 for a vibrational problem, to the best uniform distribution with width W.
of our knowledge no critical parameters have been re- In order to compute the localization length λ of the
portedforanelectronicAndersontransitioninBCCand wave function, we use the TMM for quasi-1D bars of
FCC lattices. In order to fill this gap, we use in the cross section M × M and length L ≫ M.7,8,9,10 The
present paper the TMM and finite-size scaling (FSS) to Schr¨odinger equation Hψ = Eψ for the Hamiltonian
2
given by Eq. (1) is written in the TMM form:
ψ ψ
l+1 =T l ,
(cid:18) ψl (cid:19) l(cid:18)ψl−1 (cid:19)
−C−1(E1−H ) −C−1C ψ
= l+1 l l+1 l l , (2)
(cid:18) 1 0 (cid:19)(cid:18)ψl−1 (cid:19)
where ψl,Hl and Tl denote the wave function, Hamil- (a) (b)
tonian matrix and transfer matrix of the lth slice of the
bar, respectively. 1 and 0 denote unit and zero matri- FIG. 1: (a) 3 layers of the 3D BCC lattice along a h100i
ces. The localization length λ(M,W) = 1/γ at en- latticevector. Lightgrayspheresmarkthe1stand3rdlayer,
min
ergy E is determined by the smallest Lyapunov expo- darkgrayonesindicatethecentrallayer. Linesbetweenlayers
denotetheconnectionsbetweenlatticesites. Theconnections
nent γ > 0 obtained as an eigenvalue of the product
min
of transfer matrices τ = T T ...T T , where L is to the upper-left sphere in the central layer are emphasized
L L L−1 2 1 by broad lines, illustrating its 8 neighbors. The 4 thick light
increased until the desired accuracy is achieved.26 The
graylinesconnect1standthecentrallayer,theblackonesgo
reduced localization length may then be calculated as
from thecentral tothe3rd layer. (b) Structureof 3layers of
ΛM(W)=λ(M,W)/M. the 3D FCC lattice along a h111i lattice vector. The broken
Cl and Cl+1 are the connectivity matrices describing lines mark the cubic unit cell of the lattice. Sites in the 1st
the connections of the lthslice to slices l−1 andl+1.27 and 3rd layer are dark gray; sites in the central layer are
Element c of the connectivity matrix equals 1 if the light gray. The thick light gray lines represent connections
jk
site j in one slice is connected to the site k in the other; betweenlattice sitesinthesamelayerforoneparticularsite.
otherwisec =0. Inthecaseofthe SC lattice,eachsite The thick black lines represent connections between lattice
jk
sites in neighboring layers for another site. The thin lines
has only one connection to the succeeding (preceding)
layer; therefore all C are unit matrices and the transfer indicate connections to other sites. Some sites in the upper-
matrix T reduces tolthe most often used form28 right corner are removed for clarity.
l
−(E1−H ) −1
Tl = 1 l 0 . (3) h111ivector,sothesubsequentlayersofthebarareclose
(cid:18) (cid:19)
packed. Within the layer each site has six connections
ForBCCandFCClatticestheconnectivitymatricestake to nearest neighbors. In addition there are three con-
amorecomplicatedform,butwithpurelydiagonaldisor- nections to the preceding and three connections to the
der,i.e. no disorderinthe hopping integralst , they are succeeding layer. The resulting connectivity matrix can
ij
constant so that the inverse C−1 needs to be calculated be inverted for each size of the M ×M TMM slice but
l only when we use a mix of periodic boundary conditions
onlyonceatthebeginningoftheTMMcalculationsfora
in one direction and helical boundary conditions in the
givensizeM. Nevertheless,theadditionalneedtomulti-
other. Seeappendixforexamplesoftheconnectivityma-
ply all states at eachstep of the TMM with a dense ma-
trix C−1 reduces the speed of the calculation and hence trices for system size M =3.
restricts the attainable system sizes.
C. Finite-Size Scaling
B. The Lattice Structures
The MIT is characterized by a divergent correlation
The structure of the BCC lattice is displayed in Fig. length, so that at fixed energy E, ξ(W) ∝ |W −Wc|−ν
1(a). The construction of the TMM quasi-1D bar pro- and at fixed disorder W, ξ(E) ∝|E−Ec|−ν, where ν is
ceeds along a h100i vector. In this case each site within thecriticalexponentandWc,Ec arethecriticaldisorder
the slice is connected to four sites in the preceding slice and energy, respectively, at which the MIT occurs.7 In
and to four sites in the succeeding one. There are no thefollowingdiscussionweshallassumethecaseoffixed
connections between sites within the slice, which means energy and varying disorder; the converse case of fixed
that the Hamiltonian matrix H is a diagonal matrix of disorder and varying energy proceeds analogously.
l
energiesǫ . Weuseperiodicboundaryconditionsinboth In order to extract the critical parameters from the
i
transversal directions, which results in the connectivity calculated values of Λ (W), one applies the FSS proce-
M
matrix for a slice of M ×M sites being singular for all dure outlined in Ref. 30. The correlation length for the
even M, thus restricting the system sizes we can use. infinite system ξ may be obtained from the localization
Usingahelicalboundarycondition29 inoneortwodirec- lengths for finite system sizes Λ (W) by using the one-
M
tion provides the same singularities and hence offers no parameter scaling law Λ = f(M/ξ).31 The FSS can
M
advantage. be performed numerically by minimizing the deviations
Fig. 1(b) shows the structure of the FCC lattice. It of the data from a common scaling curve. The criti-
proved convenient to construct the TMM bar along a cal parameters are then obtained by fitting the ξ values
3
as obtainedfromFSS. Better numericalaccuracyfor the
25 localized
FSSprocedurecanbeachievedbyfittingdirectlytheraw
data from TMM calculations using the method applied
previouslytotheTMMdataforthe3DSClattice.9,10We 20
introduce a set of fit functions which include two kinds high precision FSS results
of corrections to scaling, (i) nonlinearities of the W de- 15 E=-E symmetry points
pendence of the scaling variables and (ii) an irrelevant W W=0 (analytical)
extended
scalingvariablewhichaccountsforashiftofthe pointat
which the Λ (W) curves cross. We use10 10
M
ΛM =f˜(χrM1/ν,χiMy) , (4) 5
where χ and χ are the relevant and irrelevant scaling
r i
variablesrespectively. ThefunctionΛM(W)isthenTay- 0-12 -8 -4 0 4 8 12
lor expanded E
ni FIG.2: PhasediagramfortheBCClattice. Thedarkgrayre-
ΛM = χni Mnyf˜n(χrM1/ν) , (5) gionrepresentstheapproximatelocationofthephasebound-
nX=0 ary. Itsedges(thesolidblacklines)weredeterminedbycom-
nr paringlocalizationlengthswitherrors≤10%forsystemsizes
f˜n = ankχkrMk/ν . (6) M = 7 and M = 9 in the (E,W) plane. The solid squares
kX=0 ((cid:3))arepointscalculatedbyperforminghigh-precisionFSSon
localization data with an error ≤ 0.1%. The dashed squares
Nonlinearities are taken into account by expanding χ
r are reflections of the solid squares in the E = 0 axis. The
and χi in terms of w = (Wc −W)/Wc up to order mr diamonds ((cid:7)) denote the band edges at W = 0. They have
and mi, respectively, beenjoinedtothephaseboundaryedgescalculatedforhigher
disordersasaguidetotheeye. Thedashedlinesarethethe-
mr mi
oreticalbandedges±(Z+W/2),whereZ isthecoordination
χ (w)= b wm, χ(w)= c wm , (7)
r m i m number. The light gray, shaded area in the center contains
mX=1 mX=0 extendedstates; statesoutsidethephaseboundaryarelocal-
withb =c =1. The expansionsinthe fitfunctions are ized. Error bars are within symbol size for ((cid:3)).
1 0
carriedoutuptoordersn,n ,m ,m whichareadjusted
i r r i
tothespecificdataandshouldbekeptaslowaspossible, 35
while giving the best fit to the data and FSS plot and localized
minimizing the errors for critical parameters W and ν. 30
c
The Levenberg-Marquardt method was used to perform
extended
the non-linear fit.10,32. 25
We emphasize that this FSS procedure assures the di- high precision FSS results
20
vergenceofξ andhenceitisnotthedivergenceitselfbut W=0 (analytical)
W
rather the quality of how the model fits the computed
15
reduced localization lengths Λ which determines the
M
validity of the scaling hypothesis.
10
5
III. CALCULATIONS AND RESULTS
0
-16 -12 -8 -4 0 4 8
A. Phase Diagrams E
Fig. 2 and Fig. 3 show the phase diagrams for the FIG. 3: Phase diagram for the FCC lattice. Symbols, lines
andshadedareashavethesamemeaningasinFig.2withthe
BCC and FCC lattices, respectively. Originally a grid
diamonds((cid:7))representingthebandedges−12and4 atzero
of W versus E values was created with separation ∆E,
disorder24. Error bars are within symbol size.
∆W = 0.5. At each point the nature of the elec-
tronic wavefunction was determined by comparing the
reduced localization lengths Λ calculated for system
M
sizes M = 7 and M = 9 with error ≤ 10%. If Λ > Λ connecting such averages using a spline fit. We do not
9 7
(<) at the same values of E and W then we identify the obtain data points for lower disorder values, as the fluc-
point (E,W) in the phase diagram as extended (local- tuations in the Lyapunov exponents, due to the small
ized). The edges of the phase boundary were obtained system sizes, become too big; higher values of disorder
by averaging separately over the three extended and lo- smooth out these fluctuations.
calized points (E,W) nearest to the boundary and then A striking difference between the phase diagrams is
4
0.1
M
Λ
0.08
1.2
)M0.06
Λ
(0
g1 10
1.1 o 0.04
l ξ
0.02
5
20.5 21 21.5
1.0 0 W
2 3 4
20.4 20.8 21.2 21.6
W log (ξ/M)
10
FIG. 4: Reducedlocalization lengthsΛM versusdisorder W FIG.5: Scalingfunction(solidline)andscaleddatapointsfor
for BCC lattice. System sizes M are 3(•),5((cid:3)),...,15(×). theBCC lattice and nr =3,ni =2,mr =3,mi =1. Symbols
Error bars are within symbol size. Lines are fits to the data denotethesame valuesof M as inFig. 4. Inset: Dependence
given byEqs. (4) – (7) with nr =3,ni =2,mr =3,mi=1. ofthescalingparameterξ onthedisorderstrengthW forthe
13W valuesshowninFig.4. Inallcaseserrorbarsarewithin
symbol size.
that for the BCC lattice the phase boundary is symmet-
ric about the line E = 0, whereas for the FCC lattice it
is not. This is due to the bipartiteness of the BCC lat-
0.7
tice which consists of two SC sublattices, one displaced
half the distance along a body diagonal of the other. M
Λ
Hence for any site in one sublattice, its nearest neigh-
bors are in the other sublattice. Such connections result
instatescoupledbyabipartitesymmetrytransformation
0.65
— which is exact for the case of no diagonal disorder —
with eigenenergies of the same magnitude but opposite
signhavingapproximatelythe same localizationlengths;
this producesasymmetricphasediagram. TheFCC lat-
tice is non-bipartite, so such a symmetry in its phase 0.6
diagram is not observed.
26 26.5 27 27.5
W
B. Critical Parameters at E =0
FIG.6: Reduced localization lengthsΛM versusdisorder W
for FCC lattice. Symbol sizes M are 3(•),4(▽),...,15(×).
TheTMMcalculationswereperformedforsystemsizes Error bars are within symbol size. Lines are fits to the data
up to M = 15. In order to examine the localization given by Eqs. (4) – (7) with nr = 2,mr = 2. Lines for even
properties at the band center for the BCC lattice and M havebeen removed for clarity.
the barycenter33 for the FCC lattice, we set E = 0 in
Eq. (2). A value of the critical disorder W was approx-
c
imated using the phase diagrams described above and and (7). Fig. 5 shows the results of the scaling proce-
then localizationlengthsλ werecalculatedfora rangeof dure for n = 3,n = 2,m = 3,m = 1. The scaling
r i r i
W closetothisapproximatevaluewithaccuracyranging curve exhibits localized and extended branches as ex-
from 0.1% for small system sizes M to about 0.14% for pected for the MIT. Divergence of the scaling parameter
the largest. Let us remark that we use the term critical ξ at W ≈ 20.75 indicates the critical value of the disor-
disorder to indicate that there are no further extended der. Table I(a) gives some examples of models providing
states at E = 0 for disorders W > W ; extended states the best fits and the resulting critical parameters. The
c
maystillexistforW >W atotherenergiesE,asshown values of the critical disorder and critical exponent ob-
c
in Fig. 3. tained by averaging over all the best fit models are also
The reduced localization lengths for the BCC lattice given.
are displayed in Fig. 4. Note how the crossing point of Results of the TMM calculations for the FCC lattice
the curves shifts with changing M. In most cases this areshowninFig.6. Inthis casethelinesforconstantM
indicates the need for an irrelevant scaling variable in- cross at the same point — at least within the accuracy
troduced via non zero values of n and m in Eqs. (5) of the calculated Λ — indicating that the use of the
i i M
5
-0.14
TABLE I: Critical parameters for the MIT in the BCC lat-
tice. Allerrors quotedarestandard errors. (a)3examplesof
FSSresultswithvaryingnr,ni,mr,mi atfixedenergyE =0. -0.16
We use 91 data points, equally spaced in the indicated in-
tervals (cp. Fig. 4), for each set of nr,ni,mr,mi. Varying )M-0.18
nr,ni,mr,mi, we obtain 77 best fit models in order to pro- Λ
ducetheindicatedaverages. (b)SimilarFSSresultsobtained (10
g
for3outof19bestfitmodelsfrom82non-equallyspaceddata lo -0.2 10
pointsatfixedW =15fortheindicatedenergyintervals. (c) ξ
Results at W =17.5 (cp. Fig. 8) for 3 out of 8 best fit mod-
elswith108non-equallyspaceddatapointsusedineachFSS -0.22 5
procedure. The numerical fitting procedure continued in all 26 26.5 27 27.5
W
cases until convergence was reached or (a) 5000, (b,c) 1000
iterations had been completed. 2 3 4
log (ξ/M)
10
(a)
FIG. 7: Scaling function (solid line) and scaled data points
∆M E ∆W nr ni mr mi Wc ν y for the FCC lattice and nr = 2,mr = 2. Symbols denote
the same values of M as in Fig. 6. Inset: Dependence of the
3 - 15 0 20.3 - 21.5 2 0 1 0 20.95(1) 1.67(5) -
scaling parameter ξ on the disorder strength W for the 13
3 - 15 0 20.3 - 21.5 3 1 1 4 20.92(2) 1.51(9) 1.7(5)
W values shown in Fig. 6. In all cases error bars are within
3 - 15 0 20.3 - 21.5 3 2 3 1 20.75(3) 1.70(9) 3.0(5) symbol size.
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
averages: 20.85(1) 1.61(1)
TABLE II: Critical parameters for the MIT in the FCC lat-
tice. The fitting procedure was continued until convergence
(b) was reachedor until1000 iterations had beencompleted. (a)
At E = 0 (cp. Fig. 6), we use 91 data points for each FSS
∆M ∆E W nr ni mr mi Ec ν y andtheobtained 31bestfitmodelsaverageasindicated. (b)
9 - 13 9.9 - 10.9 15 2 0 1 0 10.38(1) 1.32(5) - 83 data points (cp. Fig. 9) are used to perform the FSS at
W = 18 and 8 best fit models are used to obtain the aver-
9 - 13 9.9 - 10.9 15 2 0 2 0 10.38(1) 1.22(5) -
ages.
9 - 13 9.9 - 10.9 15 3 0 4 0 10.40(1) 1.03(3) -
. . . . . . . . . .
. . . . . . . . . . (a)
. . . . . . . . . .
averages: 10.39(1) 1.27(1)
∆M E ∆W nr ni mr mi Wc ν y
3 - 15 0 26 - 27.5 1 4 3 4 26.72(1) 1.49(4) 6(2)
(c)
3 - 15 0 26 - 27.5 2 0 2 0 26.73(1) 1.58(3) -
∆M ∆E W nr ni mr mi Ec ν y 3 - 15 0 26 - 27.5 3 4 2 4 26.72(1) 1.51(10) 4(2)
. . . . . . . . . .
7 - 15 10.5 - 11.5 17.5 2 0 1 0 10.98(1) 1.55(6) - .. .. .. .. .. .. .. .. .. ..
7 - 15 10.5 - 11.5 17.5 3 0 2 0 10.99(1) 1.48(6) -
averages: 26.72(1) 1.53(1)
7 - 15 10.5 - 11.5 17.5 3 0 4 0 10.99(1) 1.36(7) -
. . . . . . . . . .
. . . . . . . . . . (b)
. . . . . . . . . .
averages: 10.96(3) 1.47(9)
∆M ∆E W nr ni mr mi Ec ν y
9 - 15 8.52 - 8.88 18 1 0 2 0 8.683(3) 1.63(5) -
9 - 15 8.52 - 8.88 18 2 0 2 0 8.687(4) 1.65(5) -
irrelevant variables in Eqs. (5) and (7) is not necessary
9 - 15 8.52 - 8.88 18 3 0 1 0 8.685(3) 1.62(5) -
in most cases and n = m = 0. Results of the fit for
i i . . . . . . . . . .
nr =2,mr =2 are displayed in Fig. 7. The transition at .. .. .. .. .. .. .. .. .. ..
W ≈26.73isclearlyindicated. Moreexamplesofbestfit
averages: 8.684(2) 1.63(2)
modelscanbefoundinTableII(a)aswellastheaverage
values of the critical parameters.
the transition. We remark that it is know that such in-
vestigations are numerically more difficult due to the in-
C. Critical Parameters away from the Band Centre fluenceofdensityofstateseffects.34 ResultsoftheTMM
and FSS calculations for the BCC lattice with W =17.5
Wealsoperformcalculationswherewefixthedisorder can be seen in Fig. 8. Although the lines for constant
andallowtheenergytovaryacrossacriticalvalueE for M do not all cross at the same point, the data is best
c
6
0.4 -0.4
M
Λ
1.2
Λ)M 0 ΛM )M 0.65
g(10 Λ(10-0.45
lo-0.4 og
l
0.8
-0.5
-0.8
0.6
0 2 4 6 8 10.8 11.2
log (ξ/M) E 5 108.5 8.6 8.7 8.8 8.9
10 log (ξ/M) E
10
FIG.8: LocalizationdatafortheBCClatticewithW =17.5.
System sizes M are 7(N),9(+),...,15(×) as in Fig. 4. Error FIG. 9: Data for the FCC lattice with W = 18. System
barsarewithinsymbolsize. Left: Scalingfunction(solidline) sizes M are 9(+),11(♦),...,15(×) as in Fig. 7. Error bars
andscaled datapointsusingnr =2,mr =1. Right: Reduced arewithinsymbolsize. Left: Scalingfunction(solidline)and
localization lengths ΛM versus disorder W. Lines are fits to scaled data points using nr = 1,mr = 2. Right: Reduced
thedata given byEqs. (4) – (7) with nr =2,mr =1. localization lengths ΛM versus disorder W. Lines are fits to
thedata given byEqs. (4) – (7) with nr =1,mr =2.
fitted by models containing no irrelevant scaling. Also
Morenearestneighborsconnectedtoagivensiteprovide
evident is the poorer quality of the fit compared to the
more paths for electronic transport, so stronger disor-
calculations where the energy was fixed at zero. This is
der is needed to localize eigenstates of the system. The
not due to using lower accuracy data, as the maximum
universal localization properties of a 3D system and the
raw-data error remained at 0.1%. Hence we attribute it
presence of an MIT are however not affected in accor-
to complications arising from a varying density of states
dance to the scaling theory oflocalization6 andin agree-
closetoE attheattainablevaluesofM. Resultsforthe
c
ment with results27 showing that they depend only on
critical parameters are shown in Table I (b) and (c) for
the dimensionalityofthe system,but notonthe number
W = 15 and W = 17.5 respectively. The low value of ν
of nearest neighbors in the lattice.
forthecaseW =15canbeattributedtotheuseoffewer
datapointsinthe FSSandonlyusingthreevaluesofM. Ourresultsandtheirinterpretationareconsistentwith
We note that this is consistent with the lower values of investigationsofclassicalbondandsitepercolationmod-
ν obtained in the diagonalization studies as mentioned elsonSC,BCCandFCClattices. InRef.35itwasfound
in the Introduction. It appears that the FSS procedure thatthepercolationthresholdsfortheselatticesdecrease
systematicallyreducesthevaluesofthecriticalexponent withincreasingnumberofnearestneighbors;moreneigh-
for data from smaller systems or of lower accuracy. bors allow for easier formation of a percolating cluster,
or, as in our case, the formation of extended states.
ResultsfortheTMMandFSScalculationsfortheFCC
lattice with W = 18 can be seen in Fig. 9. Table II (b)
gives examples of the best fit models and shows the re-
sulting average critical parameters. Note that both esti- Acknowledgments
mates of ν are consistent with the result 1.57(2) for the
SC lattice.9 We thankfully acknowledge discussions with C. Hoo-
ley, J. Knoester and I. Plyushchay. We are grateful to
T. Wright for producing the M = 7 and 9 data for the
IV. CONCLUSIONS phasediagrams. WethankS.Wellsforcarefulreadingof
the manuscript. AMF and RAR gratefully acknowledge
EPSRC for financial support.
Usingthe transfer-matrixapproachandFSSwedeter-
mined the criticalparametersofthe Andersontransition
for the BCC and FCC lattices. The values of the crit-
ical exponent ν are in good agreement with the results APPENDIX A: CONNECTIVITY MATRICES
obtained previously for other systems belonging to the
orthogonal universality class. The increase of the crit- Forcompleteness,letusgivethe connectivitymatrices
ical disorder W from 16.54 for the SC lattice to 20.85 for BCC and FCC lattices with M = 3. Recall that
c
for BCC and 26.72 for FCC lattice may be attributed C is the connectivity matrix describing the connections
l
to an increasing number of nearest neighbors which for of the lth slice to the l−1th slice. Element c of the
jk
the above structures equals 6, 8 and 12, respectively. connectivity matrix equals 1 if site j in the lth slice is
7
connectedtositek inthel−1thslice;otherwisec =0. For the FCC lattice for odd and even layers
jk
Theboundarytermsareindicatedinitalics. FortheBCC
lattice for odd layers,
1 0 1 0 0 0 1 0 1
1 1 0 0 0 0 1 1 0
0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0
1 0 1 1 0 1 0 0 0 0 1 1 0 0 0 0 0 1
C2l−1 = 1 1 0 1 1 0 0 0 0 . (A1) 1 0 1 1 0 0 0 0 0
0 1 1 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0
0 0 0 1 0 1 1 0 1 Cl = 0 0 1 0 1 1 0 0 0 . (A3)
0 0 0 1 1 0 1 1 0 0 0 0 1 0 1 1 0 0
0 0 0 0 1 1 0 1 1 0 0 0 0 1 0 1 1 0
0 0 0 0 0 1 0 1 1
For even layers 1 0 0 0 0 0 1 0 1
1 1 0 1 1 0 0 0 0
0 1 1 0 1 1 0 0 0
1 0 1 1 0 1 0 0 0
0 0 0 1 1 0 1 1 0
C2l = 0 0 0 0 1 1 0 1 1. (A2) In all cases, l =1,2,... .
0 0 0 1 0 1 1 0 1
1 1 0 0 0 0 1 1 0
0 1 1 0 0 0 0 1 1
1 P. W. Anderson,P1hy0s. R1ev0. 1009,0141920(11958). Phys. Rev. Lett. 83, 3005 (1999), ArXiv:
2 E. N. Economou and M. H. Cohen, Phys. Rev. Lett. 25, cond-mat/9905297.
1445 (1970). 21 K. M. Itoh, M. Watanabe, Y. Ootuka, and E. E. Haller,
3 E. N. Economou and M. H. Cohen, Phys. Rev. B 5, 2931 Ann.Phys.(Leipzig) 8, 631 (1999).
(1972). 22 K. M. Itoh et al., Phys.Rev.Lett. 77, 4058 (1996).
4 E. N. Economou, Journale de PhysiqueC 3, 145 (1972). 23 M. Watanabe, Y. Ootuka, K. M. Itoh, and E. E. Haller,
5 D. C. Licciardello and E. N.Economou, Solid StateCom- Phys. Rev.B 58, 9851 (1998).
mun.15, 969 (1974). 24 N. W. Ashcroft and N. D. Mermin, Solid State Physics
6 E. Abrahams, P. W. Anderson, D. C. Licciardello, and (SaundersCollege, New York,1976).
T. V.Ramakrishnan, Phys. Rev.Lett. 42, 673 (1979). 25 J. J. Ludlam, S. N. Taraskin, S. R. Elliot, and D. A.
7 B.KramerandA.MacKinnon,Rep.Prog.Phys.56,1469 Drabold, Journal of Physics: Condensed Matter 17, L321
(1993). (2005).
8 J.-L. Pichard and G. Sarma, J. Phys. C 14, L127 (1981); 26 V.I.Oseledec, Trans.Moscow Math.Soc. 19, 197 (1968).
L617 (1981). 27 M.SchreiberandM.Ottomeier,J.Phys.: Condens.Matter
9 K.SlevinandT.Ohtsuki,Phys.Rev.Lett.82,382(1999), 4, 1959 (1992).
ArXiv: cond-mat/9812065. 28 Note the minus in the top-left entry of the matrix in Eq.
10 F. Milde, R. A. R¨omer, M. Schreiber, and V. Uski, Eur. (3); this is due to the minus before the hopping term in
Phys. J. B 15, 685 (2000), ArXiv: cond-mat/9911029. Eq. (1).
11 T. Ohtsuki,K.Slevin,andT. Kawarabayashi, Ann.Phys. 29 I. K. Zharekeshev and B. Kramer, Comp. Phys. Comm.
(Leipzig) 8, 655 (1999), ArXiv: cond-mat/9911213. 121–122, 502 (1999).
12 A.MacKinnon,J.Phys.: Condens.Matter6,2511(1994). 30 A. MacKinnon and B. Kramer, Z. Phys. B 53, 1 (1983).
13 I.K.ZharekeshevandB.Kramer,Phys.Rev.B51,17239 31 D. J. Thouless, Phys. Rep. 13, 93 (1974).
(1995). 32 W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T.
14 I.K.ZharekeshevandB.Kramer,Phys.Rev.Lett.79,717 Vetterling, Numerical Recipes in FORTRAN, 2nd ed.
(1997), ArXiv: cond-mat/9706255. (Cambridge University Press, Cambridge, 1992).
15 F. Milde and R. A. R¨omer, Ann. Phys. (Leipzig) 7, 452 33 LetHbesimilartosomediagonalmatrixH0,sothatH=
(1998). UH0U−1, where U is a unitary matrix. Let {Ei} denote
16 F. Milde, R. A. R¨omer, and M. Schreiber, Phys. Rev. B the eigenvalues of H0, which are equal to the eigenvalues
61, 6028 (2000), ArXiv: cond-mat/9909210. of H. Then PiEi = Tr(H0) = Tr(H) = Piǫi ≈ 0 and
17 F. Milde, R. A. R¨omer, and M. Schreiber, Phys. Rev. B theaverage eigenvalue of H is approximately equal to0.
55, 9463 (1997). 34 P. Cain, R. A. R¨omer, and M. Schreiber, Ann. Phys.
18 F. Milde, (2000), Dissertation, Technische Universit¨at (Leipzig) 8, SI33 (1999), ArXiv: cond-mat/9908255.
Chemnitz. 35 S. Galam and A.Mauger, Phys. Rev.E 53, 2177 (1996).
19 H. Stuppet al.,Phys. Rev.Lett. 71, 2634 (1993).
20 S. Waffenschmidt, C. Pfleiderer, and H. v. L¨ohneysen,
