Table Of ContentProbability distribution of the conductance at the
9
9 mobility edge
9
1
n Peter Markoˇs
a Institute of Physics, Slovak Acad. Sci, Du´bravsk´a cesta 9, 842 28 Bratislava, Slovakia
J
6
February 1, 2008
2
]
l Abstract
l [9]. The main conclusion of these studies is that the
a
h symmetry ofthe systeminfluences the formofthe dis-
- Distribution of the conductance P(g) at the critical tribution at the critical point more strongly than in
s
e point of the metal-insulator transition is presented for the metallic or localized regime. Nevertheless, P(g) is
m three and four dimensional orthogonal systems. The invariant with respect to the choice of the microscopic
. form of the distribution is discussed. Dimension de- model within the same universality class [7].
at pendence ofP(g)is proven. The limiting casesg →∞ Studies of the statistics of the conductance have
m andg →0arediscussedindetailandrelationP(g)→0 their counterpart in the analysis of the level statistics
- in the limit g →0 is proven s = Ei+1 −Ei of the eigenvalues of Hamiltonian [11].
d
PACS numbers: 71.30.+h, 71.23.-k,72.15.Rn The critical distribution P(s) is also the subject of in-
n
tensive studies within last years [12]. In particular, its
o
c As the conductance g in disordered systems is not dependence on the symmetry [13], and dimension [14]
[ the self-averaged quantity, the knowledge of its prob- have been studied numerically.
1 ability distribution is extremely important for our un- Inthis Letterwepresentnew numericaldataforthe
v derstanding of transport. This problem is of special 3D and 4D Anderson model (orthogonal ensembles).
5 importance at the critical point of the metal-insulator Although data prove the dimension dependence of the
8 transition [1]. While the distribution of the conduc- distribution, their enables us to discuss the common
2
tancein the metallic phaseis knowntobe Gaussianin features of the critical distribution. In particular, we
1
0 agreement with the random-matrix theory [2] and the provethat P(g) decreasesmore quickly than exponen-
9 localizedregimeischaracterizedbythelog-normaldis- tially for large g. This assures that there are no huge
9 tribution of g [2], the form of the critical distribution fluctuations of the conductance, discussed in [3]. We
t/ remainsstillunknown. Amongtheproblemswhichare prove also that P(g)→0 in the limit of g →0.
a
not solved yet we mention e.g. the number of param- We calculated the conductance as
m
eters which characterize distribution, the existence of
d- huge fluctuations of the conductance, and the form of g =Tr t†t= cosh−2(zi/2) (1)
X
n P(g) for small values of g.
o Severalattemptshasbeenmadetocharacterizecon- wherequantities z determine eigenvaluesofthe trans-
i
c ductance distribution at the critical point. Using the mission matrix t†t. Details of the method have been
:
v Migdal - Kadanoff renormalization treatment, huge published elsewhere [5]. For a givensystem size L, the
i conductancefluctuationshasbeenpredictedin[3]. The probability distribution of g has been calculated from
X
same conclusion was found also in systems of dimen- an ensemble of N samples. The list of used ensem-
r stat
a sion d = 2+ε. In the limit ε << 1 the form of the bles together with mean and variances of g are given
distribution P(g) was found analytically [4]. However, in Table 1.
numericalstudiesofdisordered3Dsystem[5]indicated The last column of Table 1 presents parameter
thatitisnotpossibletogeneralizetheseanalyticalcon- hz i,whichcorrespondstotheparameterΛintroduced
1
clusions for realistic 3D systems (ε=1). in the finite size scaling theory by MacKinnon and
The form of P(g) for 2D symplectic models was Kramer [15] as hz i = 2Lt in the quasi-one dimen-
1 LΛ
found in [6, 7]. Recently, P(g) has been studied also sionallimit Ld−1×L , L >>L. When neglecting the
t t
for system in magnetic field, both in 3D [8] and in 2D smallestsystemsize,ourdataconfirmtheL-invariance
1
100 Table 1: Review of ensembles studied in the present
P(log g/<g>) work. L-sizeof the d-dimensionalcube, Nstat: number
P(g/<g>)
ofsamplesinagivenensemble,varg =hgi2−hg2i,hz i
1
10−1 is mean of the smallest of z’s. Data for 3D AM are in
good agreement with [9] (up to the spin degeneracy
factor 2).
10−2
L Nstat g √varg logg varlogg z1
h i h i h i
3D Anderson model: Wc 16.5
≈
10−3 6 20.000 0.375 0.324 -1.481 1.344 2.901
4D 8 (cid:1)◦ 20.000 0.400 0.333 -1.384 1.251 2.803
3D 2D symplectic 2D 3D 4D 10 (cid:3) 10.000 0.410 0.337 -1.347 1.229 2.770
10−4 −7 −5 −3 −1 1 3 5 7 9 1142 ♦ 25..500000 00..441261 00..333480 --11..330062 11..112929 22..772254
log(g/<g> g/<g> △
18 500 0.418 0.329 -1.279 1.083 2.717
▽
4D Anderson model: Wc 34.5 [12]
≈
4 22.000 0.190 0.247 -2.569 2.301 4.130
Figure 1: Probability distribution of logg/hgi (left)
5 30.000 0.229 0.270 -2.275 2.006 3.838
and g/hgi (right) for 4D (full symbols) and 3D (open 6 ▽2 15.000 0.225 0.269 -2.291 2.054 3.852
symbols) Anderson model. For comparison, we plot
7 7.000 0.239 0.275 -2.193 1.971 3.748
also data for 2D (symplectic) Ando model. The last 8 △ 200 0.227 0.274 -2.188 1.692 3.790
exhibit the best convergence for both small and large
valuesofg. Formeaningofsymbols,seeTable1. Solid
line is Poissondistribution P(g)=exp−g/hgi.
of the localizedstate than that ofthe metallic one (re-
maining critical). This is in agreement with studies of
the level statistics in 4D [12].
of hz i as well as of hgi and hloggi and their standard
1
Thesmall-g behaviorofP(g)canbeestimatedfrom
deviations. Owing to higher critical disorder, hz i is
1
Fig 1. Instead of P(g/hgi), we plot in the left side of
largerin4Dthanin3D.Thisguaranteesthatthefinite-
Fig. 1. the distribution P(γ) of γ = logg/hgi. Ev-
sizeeffectsdisappearmorequicklyin4D.Therefore,in
idently, logP(γ) = γ + logP(expγ). Therefore, an
spite of the fact that computer facilities limited the
assumption P(g = 0) = c 6= 0, implies P = γ +logc
system size to L ≤ 8 for d = 4, obtained data provide
for γ →−∞.
us with the relevant information about all parameters
Fig 1. shows clearly that logP(γ) decreases more
of interest.
quicklythanγforallensemblesweconsider. Thisguar-
We presented in Table 1 both mean values of g and
antees that P(g) → 0 as g → 0. Let us note that it is
logg to underline the common features of 3D and 4D
almostimpossibletoobtainlastresultformthestudies
distribution: thevarianceofloggisoforderofitsmean
of P(g) on the linear scale [10].
value. This relation is typical for localized state. On
Thesmall-gbehaviorofP(g)iseasytoestimatealso
the other hand, standard deviation of g is also ∼ hgi.
from the distribution P(z ) of the smallest parameter
Its value for 3D samples, 0.334, is smaller than the 1
z . Indeed, smallvalues of g requirelarge values ofz .
samequantity calculatedfor3D inthe metallic regime 1 1
Neglecting contributions of other channels, we have
[5].
Numerical data for P(g) are presented in Figure 1 2ε 1 ∞
P(g)dg = P(z )dz (2)
1. They confirm that the critical distribution of g is 2εZ 2εZ 1 1
0 z˜1
system-size independent, in agreement with previous
studies. Fig. 1. shows also that P(g) depends on the with ε = exp−z˜1. In the limit ε → 0 the integral on
dimension of the system within the same symmetry the LHSreads∼P(g),g =ε. RHS couldbe foundan-
class. Althoughthe distributionhas the same formfor alytically for special form of P(z1). In particular, for
3Dand4Densembles,itbecomesbroaderforhigherd: Wigner surmises P(z1) = π/2hz1i2 × z1exp(−π/4 ×
the probability to find g << hgi or g >> hgi growths [z1/hz1i]2) we obtain that P(g) ∼ g−1−const×logg/2
withdimension. Thisis duetohighercriticaldisorder, with const = 4hzπ1i2. Consequently, P(g = 0) = 0.
whichcausesthatelectronicstatepossesmorefeatures Fig. 2. assuresthatP(z )decreasesmorequicklythan
1
2
1.0 100
P(z/<z>)
1 1
0.8 4D
10−1 2
0.6
var z
10−2 i 3D
0.4 1
4D 3D 10−3
0.2
0.0 10−4 0
0.0 0.5 11..00 2.0 3.0 1 6 11
z/<z> <z>
1 1 i
Figure 2: Probability distribution of (normalized) z Figure3: Varz asafunctionofhz i(onlyforhzi<15)
1 i i
for 4D and 3D Anderson model. Solid line is Wigner for 3D (open symbols) and 4D (full symbols) orthog-
surmises P (z) = πzexp[−πz2]. For the mean value onal systems. Note the system-size invariance of pre-
W 2 4
hz i see Table 1. sented parameters (at least for i≤L).
1
than for 3D,. Then, the probability to find g ≈ n is
Wignersurmiseforlargez inorthogonalensemblefor
1 less than exp[−hz i/2varz ] ∼ exp[−const×nd/(d−1)]
both 3D and 4D systems. This assures that P(g)→0 n n
and
as g →0.
P(g)∼exp−const×gd/(d−1). (3)
Linear behavior of the distribution P(z ) for small
1
We conclude that presented numerical data for 3D
z (see left side of Fig. 2) guarantees nonzero prob-
1
and 4D Anderson model prove the system size invari-
ability that the first channel is fully open. Indeed,
ance of the conductance distribution at the critical
if P(z ) ∼ C ×z for z → 0, then the probability,
1 1 1
point. Although the distribution depends on the di-
thatthe firstchannelcontributionto the conductance,
g = 1/cosh2(z ), equals to 1, is C. This explains the mension and symmetry of the system, we found its
1 1
commonfeatures,namelyexponentialdecreaseofP(g)
origin of the characteristic bump in the distribution
forg >1,andadecreaseofP(g)tozeroforg =0. We
P(g) for g = 1. In Fig. 1, the bump is clearly visible
show that the form of P(g) can be analyzed on the
for both 3D and 4D systems.
basis of the statistics of parameters z introduced by
Fig. 1 (right) confirms that P(g) decreases more
relation (1). This analysis is more simple for higher
quickly than exponentially for large g. This is easy to
dimension, where the statistical correlations of zs are
understand on the basis of the analysis of the statis-
supposed to be less important.
tics of z’s presented in [5]. Fig 3. shows mean values
andvariancesofsome smallestz’sfor both 3Dand4D Acknowledgement This work has been supported by
system. Evidently, hz i ∼ O(1) and variances var z Slovak Grant Agency, Grant n. 2/4109/98.
i i
decreasesquickly with index i. Consequently, the con-
tribution to the conductance from the second (higher)
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