Correlation-Strength Driven Anderson Metal-Insulator Transition

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by  Alexander Croy

Correlation-Strength Driven Anderson Metal-Insulator Transition
Alexander Croy∗
Institute of Physics, Chemnitz University of Technology, D-09107 Chemnitz, Germany and
Department of Applied Physics, Chalmers University of Technology, S-412 96 G¨oteborg, Sweden
Michael Schreiber
Institute of Physics, Chemnitz University of Technology, D-09107 Chemnitz, Germany
(Dated: January 17, 2012)
ThepossibilityofdrivinganAndersonmetal-insulatortransitioninthepresenceofscale-freedis-
orderbychangingthecorrelationexponentisnumericallyinvestigated. Wecalculatethelocalization
length for quasi-one-dimensional systems at fixed energy and fixed disorder strength using a stan-
2
dard transfer matrix method. From a finite-size scaling analysis we extract the critical correlation
1
exponent and the critical exponent characterizing the phase transition.
0
2
PACSnumbers: 71.30.+h,72.15.Rn,71.23.An

n
a
J The Anderson model of localization1 has been subject where |i(cid:105) denotes a localized state at lattice site i. The

6 ofintensestudyoverthepastdecades. Inparticular, the hopping matrix elements t are restricted to nearest
ij
1 occurrenceofametal-insulatorphasetransition(MIT)in neighbors. As usual, we set these elements to one and
three dimensions (3D) has attracted a lot of interest2,3. thereby fix the unit of energy. The on-site potentials εi
]
n Theoretical studies of the MIT have focused mainly on are taken as random numbers with a Gaussian probabil-
n situations with uncorrelated disorder3–6. Therefore, one ity distribution. Specifically, we use random potentials
- oftheopenquestionsinthefieldistheroleoflong-range withmean(cid:104)ε (cid:105)=0andacorrelationfunctionoftheform
s i
i correlated disorder in the Anderson MIT.
d C((cid:96))≡(cid:104)ε ε (cid:105)∝|(cid:96)|−α , (3)
Foruncorrelateddisorder,theAndersontransitioncan i i+(cid:96)
.
t be driven either by increasing the disorder strength or
a where α is the correlation exponent. In the con-
by changing the Fermi energy3. In the former case, for
m text of Anderson localization, this correlation function
sufficientlystrongdisorderstrengthW >W (E)allelec-
- c has been used to study localization in the presence
d tronic states are exponentially localized, where the value of long-range correlations for one-dimensional8–15, two-
n Wc(E) depends on the Fermi energy E. On the other dimensional16–20 and three-dimensional7,21 systems.
o hand,atfixeddisorderstrengthstateswith|E|<E (W)
c For the numerical calculations we generate the on-site
c are extended and otherwise localized.
[ potentials for systems of size M ×M ×L using a modi-
The presence of correlations provides an additional fied Fourier filtering method (FFM)22. Additionally, we
1 possibility to achieve the MIT. Depending on the na-
shift and scale the resulting random numbers to have
v ture of the correlations, a transition may, in principle,
vanishing mean and variance C(0) = W2/12. We fo-
4 be also driven by a change of the correlation strength
cus on quasi-one-dimensional systems with L = 400000
3
or correlation length. Such a scenario might be relevant
3 and M = 5,7,9,11 and 13. The localization length λ
in situations where the disorder is induced by a complex
3 is calculated using a standard transfer-matrix method
environment surrounding the system of interest.
. (TMM)3. Monitoring the variance of the change of the
1 In this Brief Report we study the possibility of a
0 Lyapunov exponent during the TMM iterations gives a
correlation-strength driven Anderson MIT in 3D. We
2 measure of the accuracy of the localization length23. We
consider the case of scale-free disorder, which is char-
1 use a new seed for each parameter combination (E, W,
: acterized by a power-law with correlation exponent α. α, M). Lastly, the critical exponent and the critical cor-
v
We find at fixed energy and fixed disorder strength that
i relation strength are obtained from a finite-size scaling
X the localization length behaves as (FSS) analysis24. We expand the one-parameter scaling
ar λ(α)∝|αc−α|−ν , (1) law for the reduced localization length Λ = λ/M into a
Taylor series
where the critical exponent ν depends on the values of
W and E. The obtained critical values αc are consistent Λ(M,τ)= (cid:88)nI φnM−nyF (χM1/ν), (4)
n
with results for disorder and energy driven MITs in the
n=0
presence of scale-free disorder7.
Tostudytheinfluenceofscale-freedisorderontheAn- where χ is a relevant scaling variable, φ is an irrelevant
derson MIT, we use the usual tight-binding Hamiltonian scaling variable, y > 0 is the irrelevant scaling expo-
in site representation1,3 nentandτ measuresthedistancefromthecriticalpoint.
However, instead of using energy or disorder strength to
(cid:88) (cid:88)
H= ε |i(cid:105)(cid:104)i|− t |i(cid:105)(cid:104)j|, (2) measure this distance, we utilize the correlation expo-
i ij
nent, i.e., τ = |α−α |/α . The functions F , χ and φ
i ij c c n
2
30
the band center.
Qualitatively, the correlation-strength driven transi-
25 0.9 tion can be understood by assuming an effective disor-
1.5 der strength, Weff(α), which depends on the correlation
20 2.5 exponent. An effective smoothening of the disorder po-
uncorrelated tential has, for example, been observed for 1D systems,
wherethelocalizationlengthinthebandcenterincreases
Wc 15
for smaller correlation exponents8,15. It is also in ac-
cordance with the shift of the phase boundary towards
10
higherenergiesandstrongerdisordershowninFig.1. Ac-
cordingly, thetransitionoccurswhenW (α )=W (E).
eff c c
5 Close to the transition, the localization length would di-
verge according to λ∝|Weff(α)−Wc|−ν0 ∝|α−αc|−ν0,
0 wherewehaveexpandedtheeffectivedisorderstrengthto
-6 0 6
firstorder,W (α)≈W +(α−α )∂W /∂α| . Bycon-
E eff c c eff αc
c structionthisprocedureyieldsthecorrectcriticalcorrela-
tionstrength,butitdoesnotexplainthedeviationofthe
FIG. 1. Schematic phase diagram based on the results re- observed critical exponents from the universal value ν .
0
portedinRef.7. Thephase-spacepointsdiscussedinthetext Providedtheone-parameterscalinglawholdsinthepres-
are indicated by symbols (∗ and ×).
ence of long-range correlations, this discrepancy might
also indicate that the FSS method in the normally used
arefurtherexpandeduptoordern ,m andm ,respec-
R R I
tively. Taking n > 0 allows us to consider corrections
I
to scaling due to the finite size of the sample, which is
0.7 5
reflected in a systematic shift of Λ with M in Eq. (4).
7
Using a least squares fit of the expansion of the reduced 9
localization length Λ to the numerical data allows us to 11
obtainthecriticalparameters. Althoughonedoesadhoc 0.6 13
not expect the FSS analysis to be valid in the present Λ
case, we find that it is working surprisingly well, as we
will show in the following.
0.5
We set E = const. and W = const. while varying α.
The chosen values of E and W are indicated in Fig. 1, (a)
whichshowsaschematicphasediagramfortheAnderson
0.4
MIT in the presence of scale-free disorder. The position
0.5 1 1.5 2 2.5
relative to the transition boundaries provides a first esti- α
mate of the expected critical correlation exponent.
In Fig. 2 the reduced localization length is shown in 0.58
the vicinity of the band center, E = 0, setting W = 26. 5
0.56 7
From the dependence on the system size M, a clear
9
transition can be seen. For small correlation exponents
0.54 11
(α < 1.5) the reduced localization length increases with 13
increasing size M, while for large exponents (α > 1.5) Λ0.52
it decreases. The former is characteristic for a metal-
lic phase and the latter for an insulating phase. The 0.5
FSS procedure yields for the critical correlation strength
0.48
α = 1.44±0.04, which agrees very well with the value
c (b)
expected from the phase diagram. The critical exponent
0.46
is ν = 0.98 ± 0.09 (y = 2.0 ± 1.3), which is different
2 2.4 2.8 3.2 3.6
fromthevalueν0 =1.58±0.03obtainedforuncorrelated α
disorder24 and from ν(α = 1.5) = 1.69±0.22 reported
for scale-free disorder7, both taken at E =0.
FIG.2. ReducedlocalizationlengthΛvscorrelationexponent
Also, for E = 6.0 and W = 16.5 we find a transition, α. Solid lines show FSS fit to numerical data. (a) Taking
asshowninFig.2. Inthiscasethecriticalvalueisfound corrections to scaling into account (nR = 2, nI = 1, mR =
to be αc = 2.85±0.03, again consistent with the phase 2, mI = 0) for E = 0.0,W = 26.0. (b) Without taking
diagram in Fig. 1. The critical exponent is ν = 0.62± corrections to scaling into account (nR =2, nI =0, mR =2,
m =0) for E =6.0,W =16.5.
0.03, which is even smaller than the exponent found at I
3
form is not suitable to extract the critical exponent in we found an increasing reduced localization length for
the present case. α < α and a decreasing reduced localization length for
c
α>α when increasing the system size. A FSS analysis
c
Insummary,wehavestudiedtheinfluenceofscale-free yielded critical exponents which depend on the values of
disorder on the Anderson MIT at fixed energy and fixed E and W and are smaller than the universal value ν
0
disorder strength. By varying the correlation exponent found previously for uncorrelated disorder24.
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