Table Of ContentNumerical verification of universality for the Anderson transition
Keith Slevin
Department of Physics, Graduate School of Science, Osaka University,
1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan
Tomi Ohtsuki
1
Department of Physics, Sophia University, Kioi-cho 7-1, Chiyoda-ku, Tokyo 102-8554, Japan
0
0 (Journal Ref: Physical Review B 63 45108 (2001))
2
We analyze the scaling behavior of the higher Lyapunov exponents at the Anderson transition.
n We estimate the critical exponent and verify its universality and that of the critical conductance
a distribution for box, Gaussian and Lorentzian distributions of therandom potential.
J
8 I. INTRODUCTION est Lyapunov exponent is the inverse of the localization
1
length.)
We also present an estimation of the critical con-
] Universality is a concept that is central to the the-
n ductance distribution p (g). This is expected to
ory of critical phenomena and continuous phase transi- c
n
take a size independent universal form at the critical
- tions. The idea is that critical phenomena are indepen-
s dent of the “details” of the physical system depending point7. The size independence of pc(g) has been verified
i previously8–10. Here, we verify that the form of p (g) is
d only on its symmetry and dimensionality. All systems c
. in the same “universality class” should exhibit the same independent of the distribution of the random potential
t
a critical phenomena1. for the three distributions mentioned.
m
The Anderson transition is a continuous phase transi-
- tion and so the concept of universality should apply. In
d II. SCALING OF THE HIGHER LYAPUNOV
this case,the importantsymmetriesarethoseoftime re-
n EXPONENTS
o versal and spin rotation. Three universality classes are
c expected: orthogonal, unitary, and symplectic. In this
[ paper we are concerned with the orthogonal universal- A. Model and method
ity class, which is comprised of systems that have both
1
v time reversal and spin rotation symmetry. This corre- ThemotionoftheelectronsisdescribedbytheAnder-
2 sponds to an electron in a random potential in the ab- son model
7 sence of a magnetic field and with negligible spin-orbit
2 interaction2,3. H =V C†C + W C†C , (1)
1 X i j X i i i
Oneofthesimplestwaystotestthevalidityofthecon- <i,j> i
0
cept of universality is to change the distribution of the
1
0 randompotentialinwhichtheelectronmoves. Sincethis where Ci†(Ci) denotes the creation (annihilation) opera-
/ does not change the universality class, the critical expo- torofanelectronatthesiteiofathreedimensionalcubic
t
a nents for different distributions of the random potential lattice. The value of the potential at site i is W . Hop-
i
m should be the same4. In a recent paper5 we presented ping is restricted to nearest neighbors and its amplitude
- numerical estimates of the critical exponent for the An- is taken as the unit of energy, V =1.
d derson transition for three different distributions of the We have examined three different distributions of the
n
random potential: box, Gaussian and Lorentzian. The random potential.
o
estimates were made by analyzing the finite size scaling
c
: of the localization length of electrons on long wires, and 1. A box distribution with each Wi uniformly dis-
v the resultssupportthe contentionthat the criticalexpo- tributed on the interval [−W/2,W/2].
i
X nenthasauniversalvalueindependentofthedistribution
2. A Gaussian distribution with mean zero and vari-
of the random potential.
r ance W2/12.
a Inthispaper,weanalyzethescalingofthehigherLya-
punov exponents6 for the box, Gaussian and Lorentzian
3. A Lorentzian distribution of the form
distributions of the random potential. We give a
clear demonstrationof single parameterscaling for these W
p(W )= , (2)
higher Lyapunovs,a clear verificationof the universality i π(W2+W2)
i
of this scaling with respect to these three distributions
of random potential, and show that this scaling yields
for which the variance diverges.
estimates of the critical exponent that are in almost ex-
act agreement with our previous estimates based on the The standard method used in transfer matrix stud-
scalingofonlythelowestLyapunovexponent5. (Thelow- ies of the Anderson transition11 involves rewriting the
1
Schr¨oedinger equation for electrons on a long bar with here. The decay of the irrelevant scaling variable with
dimensions L×L×L as a product of transfer matrices system size is described by an exponent y <0.
z
The functions F(n) and F(n) were expanded either to
Lz 0 1
second or third order in w. Also, when possible, the
M = M , (3)
Y n expansion for ψ was truncated at the first order in w.
n=1
Thedecisionatwhichordertotruncate thesevariousse-
where M is the transfer matrix. From M we can define ries was based on the criteria of obtaining an acceptable
n
a matrix Ω by goodness of fit with as few parameters as possible.
Ω=lnMM†, (4)
B. Results
with eigenvalues {−ν ,...,−ν ,+ν ,...,+ν } where
N 1 1 N
N = L2 and the ν′s are arranged in increasing order.
The eigenvalues occur in pairs of opposite sign as a con- We simulated systems with sizes L =
sequence of current conservation12,13 and therefore we 4,5,6,8,10,12,14 and 16 and Fermi energy EF = 0
need only consider the positive ν′s. The Lyapunov ex- calculating data to 0.1% accuracy with the exception
ponents {α } for the random matrix product (3) can be of some data very close to the critical point that were
n
defined as calculated to 0.05% accuracy. We found that the zn’s
obey single parameter scaling provided that the system
ν
n
αn = lim = . (5) sizes considered are large enough. The minimum system
Lz→∞ 2Lz size required increased with n so that, for example, we
These were calculated to within a specified accuracy us- could fit only the data with L ≥ 10 when analyzing z10
ingthe methoddescribedin11. The localizationlengthλ for the box distribution. In practice, we could demon-
for electrons on the bar is the reciprocal of the smallest strate scaling for the first ten exponents for the box and
positive Lyapunov exponent λ = 1/α . In our previ- Gaussian distributions. For the Lorentzian distribution
1
ous work, we examined only the scaling of the quantity the requiredsystemsizes seemto be largerandwe could
Λ = λ/L = 1/α L. Here we have examined the scaling demonstrate scaling only for the first three exponents.
1
of the quantities z defined by Theresultsofthe scalinganalysisarepresentedinTa-
n
bles I -III and some of the data are plotted in Figs. 1-4.
zn =αnL, (6) Theestimatesofthe criticaldisorderobtainedbyscaling
different Lyapunov exponents for a given distribution of
for highervalues of n. Inperforming the scaling analysis
the random potential are in very close agreement. The
corrections to scaling were taken into account using the
same can also be said for the critical values of the pa-
methoddescribedin(Ref. 5). Weassumedthatthedata
rameter z for various n between different distributions
n
obey the following scaling law of the random potential. The critical values z(n) for a
c
given n is defined by
z =F(n) ψL1/ν,φLy , (7)
n (cid:16) (cid:17)
z(n) =F(n)(0), (11)
where ψ is a relevant scaling variable and φ is an irrele- c 0
vantscalingvariable. Weapproximatedthisscalingfunc-
and is expected to be universal if the scaling function
tion by its first order expansion in the irrelevant scaling
for each Lyapunov is universal. The data clearly sup-
variable and fitted the data for each z to the form
n port this idea. Turning to the critical exponent ν, we
z =F(n)(ψL1/ν)+φLyF(n)(ψL1/ν). (8) find estimates that are remarkably consistent across all
n 0 1 the Lyapunov exponents and disorder distributions we
The scaling variables were approximated by expansions considered.
intermsofthedimensionlessdisorderw =(W −W)/W
c c
whereW isthecriticaldisorderseparatingtheinsulating
c
III. THE ZERO TEMPERATURE
and metallic phases,
CONDUCTANCE DISTRIBUTION
ψ =ψ w+ψ w2, φ=φ . (9)
1 2 0
Next we consideredthe zerotemperature conductance
Thecriticalexponentνdescribesthedivergenceofthelo-
distribution when W =W , i.e. the critical conductance
calizationlength(correlationlengthonthemetallicside) c
distributionattheAndersontransition. WesettheFermi
as the transition is approached
energy E = 0 and, based on the results of Section II,
F
ξ =ξ0|ψ|−ν. (10) we used Wc =16.54 for the box distribution, Wc = 21.3
for the Gaussian distribution, and W = 4.26 for the
c
Note that the absolute scale of the localization length Lorentziandistributionwhensimulatingtheconductance
ξ cannot be determined with the method we have used distribution. An ensemble of samples of cubic shape was
0
2
generated, and for each sample, a Green’s function iter- for states with energiesin a band gap of the correspond-
ation technique was used to determine the transmission ing ordered (i.e. zero disorder W = 0) system21–23 . In
matrix t in a two probe measuring geometry15. The di- that case, single parameter scaling is only observed for
mensionlessconductanceg wascalculatedusingtheLan- sufficiently strong disorder. We suspect that this might
dauer formula alsoholdforthe higherLyapunovexponentsinthethree
dimensional systems we have studied here for energies
g =2trtt†, (12) outside the band.
wherethefactoroftwotakesaccountofspindegeneracy. To conclude, we have demonstrated that the higher
(We would like to emphasize that the t matrix that ap- Lyapunov exponents obey single parameter scaling at
pearsinthis formulaisanN×N matrix,notanL2×L2 the Anderson transition and presented evidence that
matrix, where N is the number of propagatingmodes at this scaling behavior is universaland independent of the
the Fermienergyinthe contactsattachedatthe left and choice of distribution of the random potential. Analysis
right of the sample13). of this scaling behavior yields estimates of the critical
In Fig. 5 and 6, the conductance distribution for en- exponent that are remarkably consistent between differ-
semblesof12×12×12cubicsamplesforthethreedistri- entLyapunovexponentsanddifferentdistributionsofthe
butionsofrandompotentialarepresented. Fixedbound- random potential. Finally, we presented evidence that
ary conditions were imposed on the wavefunction in the thecriticalconductancedistributionisalsouniversaland
direction transverse to current flow so that the system independent of the choice of the distribution of the ran-
size L = 12 is large enough that the distribution shown dom potential.
are a good approximation to the true critical distribu-
tions p (g) and p (lng) obtained in the asymptotic limit
c c
L → ∞16. (Larger system sizes are required if periodic
boundary conditions are considered and the asymptotic
form of the distribution is different10,16,17.) Note that
despite the appearance of Fig. 5, p (g) → 0 as g → 0.
c
1N.Goldenfeld,Lecturesonphasetransitionsandtherenor-
This can be deduced from the form of p (lng) in Fig. 6
c
or a more detailed plot of p (g) near g =016,17. malization group, (Addison-Wesley,Reading MA, 1992).
c 2P.A. Leeand T. V.Ramakrishnan, Revs.Mod. Phys. 57,
Looking at Figs. 5 and 6 we see that critical distri-
287 (1985).
butions obtained for the three different distributions of
3B.Kramer and A.MacKinnon,Rep.Prog. Phys. 56, 1469
the random potential are almost identical. This is con-
(1993).
sistent with the assertion that the critical conductance
4B. Bulka, M. Schreiber and B. Kramer, Z. Phys. B :Con-
distribution is universal. dens.Matter 66, 21 (1987).
5K.SlevinandT.Ohtsuki,Phys.Rev.Lett.82,382(1999).
6P. Markoˇs, J. Phys.A: Math. Gen. 33, L1 (2000).
IV. SUMMARY AND DISCUSSION
7B. Shapiro, Phys. Rev.Lett. 65, 1510 (1990).
8P.MarkoˇsandB.Kramer,Phil.Mag.B68,357(1993).P.
The critical exponent for the Anderson model with Markoˇs, Phys. Rev.Lett.83, 588 (1999). In thesetwo ref-
random hopping but no random potential has been erences the trace is not restricted only to the propagating
estimated18 to be ν = 1.61 ± .07. Since models with modes of the contacts when calculating trtt† . The reader
random hopping and random potentials are in the same may therefore notice small but statistically significant dif-
universalityclass,weshould,anddo,obtainestimatesfor ferences between the results in these references and our
the critical exponent that are consistent with Ref. 18. work.
9K.SlevinandT.Ohtsuki,Phys.Rev.Lett.78,4085(1997).
ThecomputertimerequiredtoestimatetheLyapunov
exponents with the method used here11 scales as L7 for 10K.SlevinandT.Ohtsuki,Phys.Rev.Lett.82,669(1999).
11A.MacKinnonandB.Kramer,Z.Phys.B:Condens.Mat-
a given accuracy. Thus, it is unlikely that this method
ter53, 1 (1983).
canbe extendedto muchlargersystemssizesinthe near
12K.Slevin and T. Nagao, Phys. Rev.Lett. 70, 635 (1993).
future. The critical exponent of the Anderson transition
13K.Slevin and T. Nagao, Phys. Rev.B 50, 2380 (1994).
has also been estimated by analyzing the scaling of en-
14W.H. Press, S.A. Teukolsky, W.A. Vetterling and B.P.
ergy level statistics (ELS)19. The most recent, and in
Flannery,Numericalrecipesinfortran,2ndedition,(Cam-
our opinion the most accurate, estimate of ν obtained
bridge University,Cambridge, England, 1992).
with ELS is ν = 1.45±0.220. This is consistent with 15T. Ando,Phys. Rev.B 44, 8017 (1991).
our results but the estimated uncertainty is an order of 16K. Slevin, T. Ohtsuki and T. Kawarabayashi, Phys. Rev.
magnitude worse than we have achieved here. Lett. 84, 3915 (2000).
In the simulations whose results we have presented 17C.M.Soukoulis, X. Wang, Q. Li and M.M. Sigalas, Phys.
here, the Fermi energy is always set at the band cen- Rev.Lett. 82, 668 (1999).
ter EF = 0. For one dimensional systems it has been 18P. Cain, R. A. R¨omer and M. Schreiber. Ann. Phys.
demonstratedthatsingleparameterscalingdoesnothold (Leipzig) 8 SI-33(1999).
3
19B. I. Shklovskii, B. Shapiro, B.R. Sears, P. Lambrianides 2 2821 (1990).
and H.B. Shore,Phys. Rev.B 47, 11487 (1993). 22L.I.Deych,D.ZaslavskyandA.A.Lisyansky,Phys.Rev.
20F.Milde,R.A.R¨omerandM.Schreiber.Phys.Rev.B61, Lett. 81, 5390 (1998).
6028 (2000). 23L. I. Deych, A. A. Lisyansky and B. L. Altshuler, Phys.
21K. M. Slevin and J. B. Pendry, J. Phys.:Condens. Matter Rev.Lett. 84, 2678 (2000).
2
n Nd Np Wc zc(n) ν
1 336 12 16.54(53,55) 1.737(733,742) 1.56(55,58)
2 336 12 16.55(54,55) 2.801(797,804) 1.57(56,58)
3 292 12 16.55(55,56) 3.596(591,602) 1.57(56,58)
4 250 10 16.56(55,57) 4.25(24,26) 1.58(57,59)
5 250 10 16.57(56,59) 4.83(82,84) 1.58(57,59)
10 162 10 16.59(51,71) 7.0(6.9,7.2) 1.55(52,59)
TABLEI. The results of the scaling analysis for the box distributed random potential. Here n indicates which Lyapunov
exponent is being considered, Nd the number of data available, Np the number of parameters used to fit the data, Wc is the
estimate of the critical disorder, zc(n) is defined in (11) and ν is the estimated critical exponent. The figures in brackets are
95% confidence intervalsfor thecorresponding critical parameter estimated using thebootstrap method (Ref. 14).
n Nd Np Wc zc(n) ν
1 200 10 21.30(28,31) 1.737(734,742) 1.58(56,61)
2 200 9 21.29(28,30) 2.796(794,799) 1.57(56,58)
3 175 9 21.31(30,32) 3.593(588,597) 1.58(57,59)
4 150 9 21.31(29,33) 4.244(236,252) 1.58(56,60)
5 125 9 21.29(27,33) 4.80(79,82) 1.57(55,60)
10 100 9 21.40(30,57) 7.05(6.98,7.18) 1.55(50,62)
TABLEII. The results of the scaling analysis for the Gaussian distributed random potential. The notation is explained in
thecaption for Table 1.
n Nd Np Wc zc(n) ν
1 256 10 4.25(24,26) 1.70(68,71) 1.57(52,61)
2 192 10 4.24(23,26) 2.71(64,75) 1.58(53,65)
3 127 10 4.29 3.62 1.58
TABLEIII. The results of the scaling analysis for the Lorentzian distributed random potential. The notation is explained
in thecaption for Table 1. (Wecould not executethe bootstrap method for n=3 so no confidenceintervals are available.)
4
3.4
3.2
3.0
22.8
z
2.6
2.4
2.2
2.0
15 16 17 18
W
FIG.1. z vs disorder W for thebox distributed random potential.
2
3.6
3.4
3.2
3.0
2
z2.8
2.6
2.4
2.2
0.001 0.01 0.1 1
L/ξ
FIG.2. The same data as in Figure 1 after corrections to scaling have been subtracted and plotted versus L/ξ where ξ is
thelocalization (correlation) length.
5
5.5
5.0
5
z
4.5
4.0
15 16 17 18
W
FIG.3. z vsdisorder W for theGaussian distributed random potential.
5
6.0
5.5
5.0
5
z
4.5
4.0
0.0001 0.001 0.01 0.1 1
L/ξ
FIG.4. The same data as in Figure 3 after corrections to scaling havebeen subtracted and plotted versusL/ξ.
6
2.0
1.5
)
g
1.0
(
p
0.5
0.0
0.0 0.5 1.0 1.5 2.0 2.5
2
g [e /h]
FIG.5. Thedistributionoftheconductanceforensemblesof100,00012×12×12cubesforbox(solidline),Gaussian(dotted
line) and Lorentzian (dashed line) distributed random potentials.
0.3
0.2
)
g
n
l
(
p
0.1
0.0
-6 -5 -4 -3 -2 -1 0 1
ln g
FIG.6. Thedistributionofthelogarithmoftheconductanceforensemblesof100,000 12×12×12cubesforbox(solidline),
Gaussian (dotted line) and Lorentzian (dashed line) distributed random potentials.
7
