Although the initial formulation of the hypothesisof one�parameter scaling [1] implied its consistencyunder quite general conditions, including the presenceof an external magnetic field, discovery of the phe�nomenon of the integer quantum Hall effect showedthat the diagonal component of the conductivity ten�sor σxx cannot obey the renormalization�group equa�tion without including the tensor σxy [2–4] (fordetails, see [5]). This fact, which contradicts the phe�nomenological concepts of Abrahams et al. [1] and thequantitative results of Hikami [6], was assigned tosome “nonperturbative” corrections to σxx and σxy.Although the hypothesis of two�parameter scaling for�mulated in [3, 4] provides an elegant recipe for over�coming the aforementioned difficulty, the theoreticalmodel, on which the hypothesis is based, features anumber of shortcomings and the consequences of thismodel have not been verified thus far. It is, in particu�lar, the derivation of an additional term in the nonlin�ear σ�model (this term is proportional to σxy) thatrequires consideration of the edge currents in contrastto the bulk nature of the Kubo formula for conductiv�ity; due to this, the above derivation cannot beregarded as indisputable. In addition, at present, thereare no experimental evidence (or data of numericalexperiments) supporting the hypothesis of two�parameter scaling, as was discussed in [7].
In this publication, we show that violation of one�parameter scaling occurs in the first nonvanishingorder of the perturbation theory under conditions of ahigh quantizing magnetic field, ωcτ � 1, which is typ�ical of experiments with the quantum Hall effect. Theresults of calculations (carried out via the direct dia�grammatic method) of the quantum second�order
correction to σxx are in agreement with numericalresults in contrast to predictions of the unitary nonlin�ear σ�model (see [6]). Thus, consideration of Landauquantization is of fundamental importance for correctdescription of the integer quantum Hall effect; this is adrawback of the theory [4] based on the two�compo�nent nonlinear σ�model.
2. SELF�CONSISTENT BORN APPROXIMATION, DRUDE–ANDO CONDUCTIVITY, AND DIFFUSON
In order to calculate the diagonal component of theconductivity tensor σxx, we apply the standard dia�grammatic method for averaging over the disorder,which includes the Kubo formula in the form1
(1)
to the zero�spin two�dimensional electron gasdescribed by the Hamiltonian
(2)
Here, S is the area of the sample and GR, A are the exact(not averaged) Green’s functions (the horizontal dashindicates averaging over configurations of scatterers);we use the standard assumption that the shape of theweight functional for a random potential is Gaussian
with the point correlator = .
1 It is assumed that the system’s temperature makes it possible touse a stepwise approximation for the Fermi–Dirac distributionfunction.
σxxe2
�4πS��������Tr vx G
RG
A–( )vx G
RG
A–( ){ },–=
H p e/cA–( )2
2m���������������������� U r( ).+=
U r( )U r '( ) Wδ r r '–( )
SEMICONDUCTOR STRUCTURES, LOW�DIMENSIONAL SYSTEMS, AND QUANTUM PHENOMENA
Quantum Corrections to Conductivity Under Conditions of the Integer Quantum Hall Effect
A. A. GreshnovIoffe Physical–Technical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia
e�mail: [email protected] October 24, 2011; accepted for publication October 26, 2011
Abstract—Quantum corrections to the conductivity of a two�dimensional electron gas under conditions ofthe integer quantum Hall effect have been studied. It is shown that violation of the one�parameter scalingunder conditions of quantizing magnetic fields, ωcτ � 1, occurs at a level of the perturbation theory. Theresults of diagrammatic calculation of the quantum correction are in agreement with the numerical depen�dences of the peaks in the longitudinal conductivity on the effective size of the sample, in contrast to earliercalculations based on the unitary nonlinear σ�model. Due to this, consideration of Landau quantization rep�resents a criterion for correct description of the quantum Hall effect.
DOI: 10.1134/S1063782612060115
760
SEMICONDUCTORS Vol. 46 No. 6 2012
GRESHNOV
Since the integer quantum Hall effect does not requireconsideration of the Coulomb and spin–orbit interac�tions, we use this model as the minimal one, takinginto account the possibility of the degeneracy or split�ting of spin sublevels at each of the Landau levels, thegeneralization onto which is performed trivially on thebasis of the spin�free case.
In order to determine the Green functions at largenumbers of Landau levels, N � 1, we follow the previ�ous approach [8] and use the self�consistent Bornapproximation (SCBA) as shown in Fig. 1a. Withinthe limit of quantizing magnetic fields, ωcτ � 1, onlythe terms related to the Fermi Landau level on theright�hand side of the equation in Fig. 1a are impor�tant; therefore, we obtain the following expressionsillustrated in Fig. 1b:
(3)
(4)
(5)
(6)
GEF
R A, r r',( ) GnR A, x( )Kn r r',( ),
n
∑=
GNR A, x( ) 2
Γ��� x i 1 x2–+−( ),=
GN Δ+R A, x( ) 1
Δ�ωc
����������� i Γ 1 x2–
2Δ2�ωc( )2
���������������������,+−–=
Kn r r ',( ) Ψnk r( )Ψnk* r '( ),k
∑=
(7)
Here, N is the number of the Landau level correspond�ing to the Fermi level, Ψnk(r) are the eigenfunctions ofcyclotron motion [9], and –1 ≤ x ≤ 1 is the relativeposition of the Fermi level within the real Landaulevel. The width of the Landau levels Γ is given by theformula
(8)
where lB and τ are the magnetic length and the scatter�ing time in the zero magnetic field (reversely propor�tional to W), respectively. Drude conductivity underconditions of quantizing magnetic fields consists of
two components, i.e., σ0 = + . The generalexpressions for these components
(9)
with equations (4) and (5) and explicit expressions formatrix elements of the rate operator taken intoaccount, are transformed into responses
(10)
Thus, and provide equal contributions tothe final response for Drude conductivity in a highmagnetic field, as was first obtained by Ando [8] (seeFig. 1c).
Thus,
(11)
in contrast to the case of the zero magnetic field when
is parametrically smaller than . Moreover,summation of these contributions brings about can�cellation of the doubtful term, which is small withrespect to the parameter (2N + 1)–1 and is large withrespect to the parameter (ωcτ)–1. Qualitatively, theresponse (11) can be obtained from the conventionalexpression for Drude conductivity using the replace�
ments kF → and ltr → Rc = ;these replacements represent the physics of cyclotronmotion under conditions of quantizing magneticfields. Indeed, as a result, the large parameter kFl istransformed into 2N + 1 and the well�known expres�sion valid in the case of a zero magnetic field σ0 =
yields equation (11) up to a factor thatreflects the fine structure of the self�consistent broad�ening of Landau levels.
xEF �ωc N 1/2+( )–
��������������������������������������.=
Γ 4W
2πlB2
��������� 2π��
�2ωc
�������,= =
σ0RR σ0
RA
σ0RR RA, e2
�4πS��������Tr vxG
RvxG
R A,{ }+− c.c.+=
σ0RR RA, 2N 1+( ) 1 x2–( )
2������������������������������� x
π��
�ωc
Γ�������± e2
h���.=
σ0RR σ0
RA
σ02N 1+( ) 1 x2–( )
�������������������������������e2
h���,=
σ0RR σ0
RA
2N 1+ /lB 2N 1+ lB
1/2( )kFl e2/h( )
0.8
20 1 3 4 5 6 7 8E/�ωc
0.4
0
DO
S,
arb.
un
its
2
1
0
3
4σ0 =
(a)
(b)
(c)
x = 0
x = −1 x = 1
N � 1Γ � �ωc
σxx(0
) , e2 /h
Σ
Σ
+
=
= = +
� − �
� − �
�
�
Fig. 1. Basic equations and results of the self�consistentBorn approximation (SCBA) within the limit of ωcτ � 1.(a) SCBA equations for the Green functions averaged overthe disorder for the intrinsic�energy part and a diffuson;(b) the density of states (DoS) in the shape of a semicircle;and (c) the Drude–Ando conductivity according to for�mula (11).
=
SEMICONDUCTORS Vol. 46 No. 6 2012
QUANTUM CORRECTIONS TO CONDUCTIVITY 761
According to the above correlation between kFl and2N + 1, quantum corrections to conductivity underconditions of a quantizing magnetic field can be clas�sified with respect to smallness over the parameter(2N + 1)–1 in analogy with the case of a zero field.Since a magnetic field suppresses the divergence of thefirst�order quantum correction at zero temperature(which corresponds to L → ∞) due to its Coopernature, we pay our attention to contributions of thesecond order with respect to (2N + 1)–1, which areformed by diffusion sequences. In the context of theSCBA, the diffuson is represented by a conventionalgeometric progression, as shown in Fig. 1a; as a result,we have
(12)
(13)
where w = . At the limit ωcτ � 1, the main con�tribution to P is made by n1 = n2 = N; since N � 1, wecan replace (in these and similar expressions) theLaguerre polynomials with Bessel functions:
(14)
where t = . At the limit of the small wave
vector P ≈ 1 – ,and equation (12) is reduced tothe standard diffusion form:
(15)
in which the transport length l is replaced by the cyclo�tron radius Rc (here, ε ∝ L–2 is the cut�off parameter,which has the meaning of the reverse dephasing time.)Since integration of a separate diffuson D0 with respectto the wave vector q brings about ultraviolet diver�gence, it is useful to introduce the reduced diffusonsDm(q) in which case the summation starts from them + 1 impurity lines rather than from a single line, i.e.,
(16)
3. DIVERGING QUANTUM CORRECTIONS TO THE CONDUCTIVITY:
DIFFUSON DIAGRAMS
Since each diffuson brings a factor of smallness(2N + 1)–1, all logarithmically diverging contributionsto the quantum correction of the second order δσ2 arerepresented by the sum of one�, two�, and three�diffu�son sets of diagrams shown in Fig. 2. In order to calcu�late the coefficient at the temperature logarithm, wehave to, first of all, localize the contributions made by
D0 q( ) W1 P–����������,=
P Γ2
4���� Gn1
R Gn2
A e w– Ln1w( )Ln2
w( ),n1 n2,
∑=
q2lB2
/2
P PN≈ e w– LN w( )[ ]2 J0 t( )[ ]2,≈=
2 2N 1+( )w
Rc2q2
/2
D0 q( ) ≈ Γ2
4���� 1
2N 1+������������� 2π
q2/2 ε+
����������������,q → 0
Dm q( ) WPN[ ]m
1 PN–������������ W
J0 t( )[ ]2m
1 J0 t( )[ ]2–���������������������� .≈=
the regions where there are small values for each of thewave vectors, applying the rule
(17)
δσiq( ) d
2q
2π( )2����������� d
2reiqrD q( )F q r,( ) d2q
2π( )2�����������D q( )∫∫∫=
× d2rF 0 r,( )∫
Γ2
4���� 1/ε( )ln
2N 1+��������������� d
2rF 0 r,( )∫=
Fig. 2. A set of diagrams for the quantum correction to con�ductivity of the second order over the parameter (2N + 1)–1;these corrections diverge at zero temperature (this is equiv�alent to ε = 0). Solid lines denoted by � and � representthe retarded and advanced Green functions, the wavy lineimplies a valid diffuson, and the dotted line represents thelogarithmic contribution originating in the small wave vec�tor region of the relevant diffuson.
0 01 2
3
45
12
3
46
7
5
+ +δσ2 =
= +
+ + + +
+ + + +
+ + + +
B CE F
G H
LK M N
O
DA
I J
OP
=
+ +
+=
+=
+= +R S T
+ +U V
+== = = +
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
+
+
762
SEMICONDUCTORS Vol. 46 No. 6 2012
GRESHNOV
to all diffuson wave vectors involved in the diagramunder consideration (q is for the one�diffuson dia�grams; q and Q are for the two�diffuson diagrams; and
q1, Q, and q + Q are for the three�diffuson diagrams.)Then, it is necessary to exclude all possible integrationswith respect to intermediate points, using the rule
(18)
where the effective scattering time �(x) in the contextof the SCBA at ωcτ � 1 is given by the formula
(19)
Using expressions (4) and (5), it is not difficult toobtain the following formulas for the spectral Greenfunction GB, which appears on the right�hand sideof (18):
(20)
(21)
(22)
With the use of the above operations, all diagramsare radically simplified, as shown in Figs. 3–5. At thelimit ωcτ � 1, the residence of charge carriers at Lan�dau levels differing from the Fermi level with the num�ber N yields, as can be seen from expressions (5) and(22), a parametric smallness; therefore, all electronlines in each diagram in Figs. 3–5 refer to the index Nwith the exception of one line in each current apexcorresponding to the neighboring Landau levels withnumbers N ± 1. Technically, all diagrams in Figs. 3–5consist of paired loops, their geometric progressions(staircases RR/AA), diffusons, and two types of apexes,V3 and V4 (introduced in the second inset in Fig. 3).Since all these structural elements are closed, the tran�sition to the momentum representation becomes use�ful. Then, the mutual contraction of diagrams with asmall number of lines, assembled in Fig. 6, eliminatesthe problem of the ultraviolet divergence of separatecontributions shown in Figs. 3–5. As a result, diffu�sons in diagram (A) should be counted from threeimpurity lines (D2), while those in diagrams (B, C, E–H)should be counted from two impurity lines (D1). Inaddition, it is necessary to subtract the sum of dia�grams (a), (b), and (q) from the three�diffuson dia�gram with two “pure” (without additional lines)Hikami squares and subtract the sum of diagrams (m–p)and (q) from diagrams (R–V). Implementation of theabove scheme using expressions (A.1, A.2) for theapexes V3, 4 yields the following expression for thecoefficient C at the logarithm:
(23)
As a result, C(x) is represented by the following sumof integrals:
GR r1 r,( )GA r r2,( )d2r∫ i� x( ) GR GA–( ) GB,= =
� x( ) 1
Γ 1 x2–������������������ .=
GB r r',( ) GnB x( )Kn r r',( ),
n
∑=
GNB x( ) 4
Γ2����,=
GN Δ+B x( ) 1
Δ2�ωc( )2
������������������.=
δσ2C x( )2π
��������� 1/ε( )ln2N 1+���������������e2
h���.=
A = =+ +
q Q
B = =+ +
+= +=
V3 V4= =B
AC0
12 3
C
DA
B0
12
== + +
D = =+ +
E FG H+ + +
=
=
+ + +
+ +
+ + +
I
K L M N
J
= + + +
� �
�
�
� �
�� �
� �
����
��
� �
�
���
�
� � � �
�� �
�
� �
���
�
��
�� �
�
��
��� �
�
��
� �
��
�
�
�
��
�� �
�
��
��
����
�
� �
�
�
� �
�
� � � �
����
� �
����
� �
��
��
��
� �
��
�
��
��
�
�
�
�
��
��
� �
��
��
� �
� �
� �
�
��
�
�
��
�
� �
��
� �
��
��
Fig. 3. Simplification of two�diffuson diagrams as a resultof separating out the contributions of the small momentuaof each diffuson with formula (18) used. Solid linesdenoted as � represent the spectral Green function (20).The staircases RR/AA and the apexes V3, 4 are outlined inthe insets.
SEMICONDUCTORS Vol. 46 No. 6 2012
QUANTUM CORRECTIONS TO CONDUCTIVITY 763
(24)
(25)
(26)
(27)
(28)
(29)
(30)
where α = = 2x2 – 1 – origi�nates from the staircases RR/AA characteristic of theSCBA (see the bottom line in Fig. 2), while the func�tions
(31)
(32)
(33)
describe the quantum�mechanical dynamics of chargecarriers at separated Landau levels. It can be seen thatthe expressions under the integral sign in CABCD andCOP behave as ~ω–1 at small values of ω and give rise tothe behavior ~ln2ε for the two� and three�diffusoncontributions in the case of their independent calcula�tion; however, if the contributions are summed, diver�gence of such a kind disappears since
(34)
It is worth noting that the right�hand side ofexpression (34) can be represented as a function of thesingle argument σ0, which is indicative of the validity
of scaling.2 However, neither this fact, which is proba�bly related to the diffusion nature of these contribu�tions (in contrast to the ballistic essence of the sumexpressions for the coefficient C), nor elimination ofthe terms of the type ln2ε prove the concept of scalingeven in the context of the second�order perturbationtheory, in contrast to the case of a zero magnetic fieldconsidered by Gorkov et al. [10]. Such a difference isdirectly related to Landau quantization since theDrude–Ando conductivity defined by formula (11)depends not only on the number of the Fermi Landaulevel N (playing the role of the effective parameter kFl)but also on the parameter x, which defines the rela�tive position of the Fermi level within a given Lan�dau level. In order to really check the one�parame�ter scaling hypothesis, we consider the limit N � 1in what follows.
4. THE LIMIT OF HIGH LANDAU LEVELS
Although all expressions presented in the previoussection were obtained for an arbitrary Fermi Landaulevel N (under the single condition ωcτ � 1); their
������������ .2 The factor (e2/h)2 is omitted in the right�hand side of for�
mula (34) and in some other formulas from the consideration ofcompactness.
764
SEMICONDUCTORS Vol. 46 No. 6 2012
GRESHNOV
practical use begins to make sense only in the limitN � 1 taking into account the principle of diagramselection within the context of the method used. Dueto this, all Laguerre polynomials in formulas (24)–(33)should be substituted with Bessel functions, namely,F1(w) ≈ (2N + 1)P, F2, 3(w) ≈ (2N + 1)Q, P ≈ [J0(t)]2,and Q ≈ [J1(t)]2. Since dw = tdt/(2N + 1), the addi�tional factor 2N + 1 arising as a result of the above sub�stitutions disappears due to the transition to integra�tion with respect to the variable t. The final expressionsfor the coefficient C(x) can be written as
Although the above expressions appear to be cum�bersome, it is easy to analyze them using numericalintegration. The case of the Fermi level at the center ofone of the Landau levels (the point of transitionbetween two plateaus of the integer quantum Halleffect), which corresponds to x = 0 (see Fig. 1b), is themost interesting. At this point, all contributionsdescribed by formulas (24)–(30) are quantities on theorder of unity or exactly vanishing, specifically
However, the total coefficient, equal at x = 0 to
(40)
CABCD x = 0( ) COP x = 0( )+ 0.53526,–≈
CRST x = 0( ) 0.61274,≈ CUV x = 0( ) 0.07862,–≈
CE–J x = 0( ) CK–N x = 0( ) CQ x = 0( ) 0.= = =
C0P2tdt
1 P2–����������� P 3 2P–( ) Q 3 2P P2 2P3–+ +( )
1 P2–�����������������������������������������– ,
0
∞
∫=
Fig. 4. Simplification of the left�hand apexes in three�dif�fuson diagrams (“square Hikami boxes”) in the case ofoperations described for Fig. 3 (the right�hand apexes aretransformed similarly and, because of this, are not shown.)The contributions of the regions corresponding to smallwave vectors q, Q, and q + Q are shown sequentially.
R S =+ +
=
+ +=VU
T
� �
�� �
�
��
� �
��
�� �
�
�
��
�
��
� �
� �
�
�
�
�
�
�
�
�
Fig. 5. Similar to Fig. 3 but for one�diffuson diagrams.
a b c d
e f g h
i k l
m n o p
rq
+
j
+ +
+ + ++
+ + ++
+ + ++
+ + = 0
Fig. 6. Reduction of one�diffuson diagrams with all possi�ble combinations for two additional impurity lines elimi�nates the ultraviolet divergence of some diagrams in Fig. 2.
+ +==
+ +==
+ +=
+ +=
q + Q
Q
q ��
�
�
�
�
�
�
��
�
�
�
��
�
��
�
�
��
�
�
��
��
�
�
�
�
�
SEMICONDUCTORS Vol. 46 No. 6 2012
QUANTUM CORRECTIONS TO CONDUCTIVITY 765
is found to be smaller by more than two orders of mag�nitude, C0 ≈ –0.00115. The total dependence of thecoefficient C on the relative position of the Fermi level xis shown in Fig. 7, together with the density of statesand the Drude–Ando conductivity. The behavior ofthe coefficient under consideration near the edges ofthe given Landau level (|x| = 1) is represented by theasymptotic expression C(x) ∝ –1/[2(1 – x2)]; as aresult, the quantum correction to conductivity can beexpressed in terms of σ0 with a coefficient that coin�cides with that reported in [6, 11] and is related to thelimit of the nonquantizing magnetic field,
(41)
However, a comparison of Fig. 7 with the formula
(42)
shows that, in fact, scaling (at least, a one�parameterone) is disrupted at any position of the Fermi levelwithin a given Landau level (in addition to the specificpoints |x| = 1, at which the SCBA becomes essentially
δσ2 ≈ 1
4π 1 x2–( )�������������������� 1/ε( )ln
2N 1+���������������e2
h���– Lln
2π2σ0
������������ .–=|x| → 1
δσ2C x( )4π
��������� Lln2N 1+�������������e2
h��� 1 x2–( )C x( )
π2σ0
������������������������ Lln= =
inapplicable). Thus, it is shown that the fact of disrup�tion of the one�parameter scaling follows from consid�eration of the first nonvanishing order in the perturba�tion theory and does not require a nonpertubative con�sideration, in contrast to what was claimed in [4]. Inorder to assess the upper limit of the value of δσ2 in thecase of the Fermi level position being at the center ofthe Landau level under consideration, let us assumethat, in formula (42), x = 0 and σ0 ~ e2/h. As a result,we have
(43)
in the entire range of reasonable effective sizes of thesample, L � 105lB.3 This estimate shows that thedependence of the maxima in the longitudinal con�
ductivity on the effective size of the sample(described in the context of the perturbation theory byformula (42) with C(x = 0) ≈ –0.001) is so weak that it
3 In an actual experiment, the effective size of the sample is gov�erned by temperature. From scaling�related considerations,there is a power law L ∝ T –p/2 with the experimentally deter�mined exponent p ≈ 3. According to Koch et al. [12], the effec�tive size L = 64 μm ~ 104lB corresponds to the temperature T ≈40 mK.
δσ2 10 4– Le2
h��� � 10 3– e2
h���ln∼
σxxn( )
0−1.0−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1.0
Relative Fermi level position x
0.2
0.4
0.6
0.8
1.0
Dru
de
con
du
ctiv
ity,
arb
. un
its
DO
S,
arb.
un
its
1
2
(b)
σ0(x) ~ 1 − x2
−5
−4
−3
−2
−1
0T
ota
l co
effi
cien
t C
(x)
1
2
(a)
C0 = −0.001
C(x) = −1/2(1 − x2)
Fig. 7. Dependences of (a) the coefficient C in the regionsof (1) high and (2) low magnetic fields and (b) (1)theDrude–Ando conductivity and (2) the density of states onthe relative Fermi level position (x) within a given Landaulevel.
00
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
200 400 600 800 1000Sample size L/lB
(1) Numerical experiment(2) Theory of current paper(3) Hikami β�function
N = 0
N = 1
N = 2
Longitudinal conductivity peaks σxx, e2/h
Fig. 8. Dependences of the peak values of longitudinalconductivity for relevant Landau levels with the numbersN = 0, 1, 2 on the effective size of the sample under simu�lation: (1) numerical experiment, (2) the theory of thisstudy, and (3) the theory advanced by Hikami [6].
766
SEMICONDUCTORS Vol. 46 No. 6 2012
GRESHNOV
cannot be detected either in an actual or in a numeri�cal experiment, similar to that the results of which areshown in Fig. 8. Discarding the details of the numeri�cal study (described in [7]), we state that the flat
dependences of the peaks on the sample size(obtained in a numerical experiment) (deviation fromhorizontal is no larger than 0.02e2/h as L is varied from50lB to 1000lB) are in excellent agreement with theresults of calculations of the quantum corrections bythe method of diagrams, in contrast to the theory,which disregards Landau quantization [6]. In addi�tion, taking into account that Lagrangians of the effec�tive field theory used in previous studies [4, 6] differonly by a term proportional to σxy, we conclude thatthe study by Pruisken [4] also cannot account for thedata of a numerical experiment in contrast to the the�ory outlined in this publication.
5. CONCLUSIONS
Thus, we studied the quantum corrections (diverg�ing within the limit of zero temperature) to the longi�tudinal conductivity of a two�dimensional electron gasunder conditions of quantizing magnetic fields, ωcτ � 1;these conditions are typical for observation of the inte�ger quantum Hall effect. It is shown that disturbanceof the one�parameter scaling occurs in the first nonva�nishing order of the perturbation theory, in contrast tostatements by Pruisken [4] claiming the nonperturba�tive nature of the corrections. We assume that thecause of this divergence is the different approaches toconsideration of the effect of a high magnetic field inthis study and in the context of the nonlinear σ�model.The theoretical consideration in this study is sup�ported by numerical calculations of the dependencesof the peaks in the longitudinal conductivity on theeffective size of the sample. Flat dependencesobtained for three lower Landau levels are in excellentagreement with the estimate of the quantum correc�tion to conductivity, in contrast to the theory based onthe nonlinear σ�model [4–6]. Thus, it can be con�cluded that the standard diagram�based technique ofaveraging over disorder represents a reliable and pro�ductive tool for studying the transport properties ofquantum Hall systems.
APPENDIX
Expressions for Apexes and Calculation of Diagrams
In the APPENDIX, we report the necessary detailsfor calculating the contributions to the quantum cor�rection to conductivity δσ2. First of all, it is technicallyvery convenient to join the current apexes and someGreen functions into V3, 4 blocks on account of theircloseness. This makes it possible to eliminate thephase factor related to the presence of a magnetic fieldand facilitates the transition to the momentum repre�sentation. Explicit expressions for V3, 4 in the mainorder with respect to (ωcτ)–1 are written as
(A.1)
(A.2)
where vB = �/mlB, q is the wave vector associated withr12 for V3 and with r13 for V4 (see the illustration in thefirst inset in Fig. 3), and the factor eiϕ = (qx + iqy)/qrepresents the vector essence of V3 containing only onevector apex from a total of two. The designations A, B,C, and D in formulas (A.1) and (A.2) represent theretarded, advanced, or spectral Green’s functions, GR,G A, and GB; the positions of these functions are deter�mined by a specific diagram. The use of these formulasbrings about the following expressions for partial con�tributions to the total coefficient C(x):
where Ξ = �2Γ2/( ) is a factor common to the dia�grams under consideration, while the functions P, Dm,and F1, 2, 3 are described by formulas (14), (16) and(31)–(33), respectively. The apexes L1, 2, 3 appearing inexpressions (A.9) and (A10) (see Fig. 4) are defined bythe following expressions:
(A.14)
(A.15)
(A.16)
4lB2
L1 q( ) V3ABR{ } q( ) V3
ARR{ } q( )Γ2
/4( )BNRNPN q( )1 αPN q( )–
�������������������������������������+=
= vB/2
�ωc
�������������BNRN we w– LN w( )
1 αP–�������������������������������������–
× eiϕ LN1 w( ) PLN 1–
1 w( )+[ ] e iϕ– LN 1–1 w( ) PLN
1 w( )+[ ]+{ },
L2 q( ) V3BRA{ } q( ) V3
ARA{ } q( )Γ2
/4( )BNANPN q( )1 α*PN q( )–
�������������������������������������+=
= vB/2
�ωc
�������������BNANe w– wLN w( )
1 α*P–�������������������������������������–
× eiϕ LN1 w( ) PLN 1–
1 w( )+[ ] e iϕ– LN 1–1 w( ) PLN
1 w( )+[ ]+{ },
L3 q( ) V3ARB{ }
q( ) V3ARR{ }
q( )Γ2
/4( )RNBNPN q( )1 αPN q( )–
�������������������������������������+=
+ V3ARA{ } q( )
Γ2/4( )ANBNPN q( )
1 α*PN q( )–�������������������������������������
vB/2
�ωc
������������� 1 P2–
1 αP– 2�����������������–=
× BN we w– LN w( ) eiϕ ANLN1 w( ) RNLN 1–
1 w( )+[ ]{
+ e iϕ– ANLN 1–1 w( ) RNLN
1 w( )+[ ] }.
ACKNOWLEDGMENTS
This study was supported by the fund “Dinastiya”and the Russian Foundation for Basic Research (projectno. 11�02�00573). I thank I.V. Gornii, V.Yu. Kacho�rovskii, and A.P. Dmitriev for their invaluable partici�pation in discussions.
REFERENCES
1. E. Abrahams, P. W. Anderson, D. C. Licciardello, andT. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
2. B. Huckestein, Rev. Mod. Phys. 67, 357 (1995).
3. D. E. Khmel’nitskii, JETP Lett. 38, 552 (1983).
4. A. M. M. Pruisken, Phys. Rev. B 32, 2636 (1985).
5. H. Levine, S. B. Libby, and A. M. M. Pruisken, Nucl.Phys. B 240, FS12.30 (1984).
6. S. Hikami, Phys. Rev. B 24, 2671 (1981).
7. A. A. Greshnov, G. G. Zegrya, and E. N. Kolesnikova,J. Exp. Theor. Phys. 107, 491 (2008); A. A. Greshnovand G. G. Zegrya, Physica E 40, 1185 (2008).
8. T. Ando and Y. Uemura, J. Phys. Soc. Jpn. 36, 959(1974); T. Ando, J. Phys. Soc. Jpn. 37, 1233 (1974).
9. L. D. Landau and E. M. Lifshitz, Quantum Mechanics(Pergamon, Oxford, 1977), § 112.
10. L. P. Gor’kov, A. I. Larkin, and D. E. Khmel’nitskii,JETP Lett. 30, 228 (1979).
11. A. A. Golubentsev, JETP Lett. 41, 644 (1985).
12. S. Koch, R. J. Haug, K. v. Klitzing, and K. Ploog, Phys.Rev. Lett. 67, 883 (1991).
