Table Of ContentLevel correlations driven by weak localization in 2-d systems
1 2
VladimirE. Kravtsov and Igor V. Lerner
1
InternationalCentre for TheoreticalPhysics,34100Trieste,Italy
andInstituteof Spectroscopy,RussianAcademyof Sciences,142092Troitsk, Moscow r-n, Russia
2
Schoolof PhysicsandSpaceResearch,Universityof Birmingham,Edgbaston,BirminghamB15 2TT, UnitedKingdom
We consider the two-level correlation function in two-dimensional disordered systems. In the non-
ergodic di(cid:11)usive regime,atenergy (cid:15)>Ec (Ec is theThouless energy), itis showntobecompletely
determinedbytheweaklocalization e(cid:11)ects,thusbeingextremelysensitive totime-reversal andspin
symmetrybreaking: itdecreasesdrastically inthepresenceofmagnetic(cid:12)eldormagneticimpurities
and changes its sign in the presence of a spin-orbit interaction. In contrast to this, the variance of
the levels number (cid:13)uctuations is shown to be almost una(cid:11)ected by the weak localization e(cid:11)ects.
(Submittedto Phys.Rev.Lett. 19 Septmeber1994)
AremarkableobservationofWignerandDyson[1]was Ec <(cid:24)"<(cid:24)(cid:22)h=g(cid:28) (1)
the existence of universal statistics that govern energy
level spectra in a wide variety of quantum systems [2]. This is another of a few known examples (as an anoma-
TheWigner{Dysonstatisticsofeigenvaluesintheinvari- lousmagnetoresistanceorthe Aharonov-Bohme(cid:11)ect [7])
ant ensembles of random matrices are applicable also to where the weak-localization corrections determine the
2
disorderedelectronsystemsinametallicphase[3{6]. The main e(cid:11)ect. In Eq. (1), Ec = (cid:22)hD0=L0 is the Thouless
level repulsion, characteristic of these statistics, results energy,L0 isthe samplesize, D0 =`vF=2isthedi(cid:11)usion
5 fromthe overlaping of one-electron states corresponding coe(cid:14)cient, ` = vF(cid:28) is the elastic scattering length, and
9 to di(cid:11)erent energies. With the disorder increasing such g is the conductance in units of e2=(cid:25)h.
n systemsundergotheAnderson metal-insulatortransition We consider the two-level correlation function:
a [7]. On an insulator side of the transition, uncorrelated (cid:10) (cid:11)
J (cid:26)(E)(cid:26)(E +")
energy levels are described by the Poisson statistics. RE(")= (cid:10) (cid:11)(cid:10) (cid:11) (cid:0)1: (2)
7 Recently, the existence of the third universal spec- (cid:26)(E) (cid:26)(E+")
1
tral statistics, applicable to the disordered systems in a
critical regime near the Anderson transition point, has Here h:::i denote averagingover allthe realizations,and
3 been predicted [8{11]. These statistics emerge[10]inthe X
7 crossoverregionnearthetransition,whereelectrondi(cid:11)u- (cid:26)(E) =pL(cid:0)d (cid:14)(E(cid:0)"n) (3)
0
sionisanomalous,andturn outtobe completelynew[9] n
1
and di(cid:11)erent from the Poisson and Wigner-Dyson ones,
0 isanexactdensityofstates foraparticularrealizationof
5 ratherthanauniversalhybridofboth,ashasbeenearlier
disorder,where "n areexactenergylevels,andpisaspin
9 conjectured [8]. They are still characterized by the level
degeneracy. ForE,E+"inthebulkofconductionband,
/ repulsion [11], albeit it is weaker than in the metallic
t h(cid:26)(E)i (cid:17) (cid:26) is energy-independent, and RE(") (cid:17) R(") is
a region. It is essential that nontrivial level correlations
m a function of (") only.
[9,10] turn out to be driven by small scale-dependent
The averaginginEq.(2)isperformedwithinthe impu-
- corrections to the conductance which by itself is almost
d scale-independent near the Anderson transition. rity diagradmmR atic t+echnique [13] with (cid:26)((cid:6)E) represented
n by (ip=(cid:25)L ) ImG (r;r;E)dr (here G are exact re-
The conductance has only a weak scale-dependence
o tarded or advanced one-particle Green's functions). It
c also in the case of a two-dimensional disordered system
is convenient to use a representation in which slow dif-
in a weak-localization (WL) regime. There is no metal-
fusion modes are explicitly separated from fast \ballis-
insulator transition at d=2, and, with the size increas-
tic" ones [14,15]. In Fig. 1, there are shown the lowest
ing, such a system crosses over to a strong-localization
order (one-loop) diagrams containing one or two di(cid:11)u-
regime [12]. So one can expect some analogies between
sionorcooperonpropagators(shownbywavylineswhich
the level correlations in this case and those in the criti-
correspond to the ladder series in the conventional tech-
cal regime in the higher dimensionality. However, while
nique[5]). Triangularandsquare\Hikamiboxes"repre-
the one-parameter scaling hypothesis [12] was taken for
sent the motionat the ballistic scale as all the averaged
grantedtomakeconsiderationsinthecriticalregimepos- (cid:6) 0
Green's functions G (r(cid:0)r ;") (represented by edges of
sible[9,10],theWLregimeatd=2maybetreatedwithin 0
(cid:0)jr(cid:0)rj2`
a rigorous and conjecture-free perturbative approach. the boxes) decay as e . At the di(cid:11)usion scale,
In this Letter we consider level correlations in a 2d these boxes are reduced to certain constants (see Ref.
disordered electron system and show that they are to- [16] for review). Taking into account the total number
tallygovernedbytheWLcorrectionstotheconductance, of cooperon and di(cid:11)uson contributions that depends on
when the distance between the levels, ", is in the region the universality class, one obtains for the one- and two-
di(cid:11)uson diagrams:
1
2
1b (arisen when the corrections to the boxes /(q`) are
taken into account). It gives:
(cid:20) (cid:21)
2
(cid:28)(cid:1) ("(cid:28)=(cid:22)h)
Rbal(")= 2(cid:25)(cid:12)g(cid:22)h ln 1+("(cid:28)=(cid:22)h)2 : (7)
This result corresponds to that obtained by Altlandand
Gefen [17]whohave thoroughlyconsidered the level cor-
(a) (b)
relations in the ballistic regime (in their technique both
the diagramsofFig.1 were described by asingle expres-
sion so that no explicit cancellation has been seen). In
FIG.1. One-loop diagrams contributed to R(").
the di(cid:11)usive regime, it gives only a logarithmic depen-
2
dence on " witha smallamplitude,(cid:28)(cid:1)=g(cid:22)h(cid:24)(1=kFL0) .
A surprising situation at d= 2 is that in a wide re-
gion,Eq. (1), the contribution of the one-loop diagrams,
2(cid:1)2(cid:28) X 1
Eq. (7), is small compared to that of the two-loop dia-
R(1)(")= (cid:25)2(cid:12)(cid:22)hRe (h(cid:22)D0q2(cid:0)i"), (4a) gram containing four propagators (Fig. 2a). The shaded
q
2
(cid:1)2 X 1 Hikami box is / q [15], so that this diagram describes
R(2)(")= (cid:25)2(cid:12)Re q (h(cid:22)D0q2(cid:0)i")2 . (4b) t(h4)e.WItLcocnotrariencstiaoncotooptehroend,it(cid:11)huussiobneicnoge(cid:14)abcsieennttiDn0thien(cid:12)E=qs2.
case where the next-order (three-loop) diagrams should
where q=(2(cid:25)=L0)(nx;ny), and nx;y run over allthe in- beconsidered. Therefore, to(cid:12)ndthe WLcontributionto
tegers, (cid:12)=1; 2;or4isfortheDysonorthogonal,unitary, the correlation function (2), we calculate the correction
d
and symplectic ensembles, respectively, (cid:1)=p=(cid:26)L is the to D0, given for (cid:12) =1; 4 by
mean level spacing.
(cid:1) X 1
The two-di(cid:11)uson diagram of Fig. 1b is well-known to
(cid:14)D(")= ; (8)
dominate at d> 2 in the interval (cid:1)<(cid:24) " <(cid:24) (cid:22)h=(cid:28). Indeed, (cid:25)(cid:22)h((cid:12)(cid:0)2) q q2(cid:0)iL(cid:0)"2
in the ergodic region, " (cid:28) Ec, where all the terms with
q 6=0 maybe neglected in Eq. (4b), it gives and substitute D =D0+(cid:14)D intothe integralinEq.(6).
Naturally,its real part would not change the zero result
1 "
atd=2. SoitistheimaginarypartoftheWLcorrection
R(")'(cid:0) s(cid:17) (5)
(cid:25)2(cid:12)s2, (cid:1), thatgovernsthelevelcorrelationsintheregion(1). With
the same accuracy as in Eq. (6),
which is the envelope of the Wigner{Dyson correlation
function. The contribution of the one-di(cid:11)uson diagram,
d (cid:0)2(d(cid:0)1)
Eq. (4a), is of order (l=L0) (lkF) in this region,
i.e. totally negligible. In the di(cid:11)usive region, Ec (cid:28) " (cid:28)
(cid:22)h=(cid:28), the summation in Eqs. (4) may be changed by in-
tegration, and the two-di(cid:11)uson diagram,Eq. (4b), gives
the well-known result [5]:
Z
2 d d
(cid:1) L d q Cd
R(")= (cid:25)2(cid:12)(2(cid:25))d Re (h(cid:22)D0q2(cid:0)i")2 = (cid:12)gd=2jsj2(cid:0)d2 . (6)
Here the numericalcoe(cid:14)cient Cd depends only on d. (a)
For d > 2, Eq. (6) makes the main contribution into
R("), as that of the diagram of Fig. 1a turns out to be
4(cid:0)d 2
smallerby(l=L") , where L"=(cid:22)hD0=". It becomes rel-
evantforthecorrelationfunction(2)onlyattheballistic
energy scale " (cid:24) (cid:22)h=(cid:28), where the appropriate di(cid:11)usion
length L" is of order ` and the di(cid:11)usion approximation
is no longer valid.
For d=2 such an estimation fails as C2=0 in Eq.(6).
A similar cancellation (of a contrubution that would be (b) (c)
main if not a vanishing coe(cid:14)cient) happens at the mo-
bility edge for d > 2, although much more subtle con- FIG. 2. Two-loop diagrams contributed to R("). The dia-
siderations were required [9]. On the face of it, what is grams (b)and (c),that describe renormalization ofthe e(cid:11)ec-
left at d=2 is the contribution of the diagram of Fig. tivevertex,canceleachother. Thediagram(a)yieldstheWL
1a and the next order correction for the diagramof Fig. correction to the di(cid:11)usion coe(cid:14)cient.
2
(cid:26) p (cid:2) (cid:3)
i(cid:17)(g;(cid:12)) sign(") (cid:25)=(cid:11) 1+2exp((cid:0)(cid:25)2=(cid:11)) ; (cid:11)(cid:28)1;
Im(cid:14)D(")= : (9) '((cid:11))= (12)
4(cid:25)(cid:22)h(cid:26) 1+2e(cid:0)(cid:11); (cid:11)(cid:29)1:
(cid:0)1
Here (cid:17)(g;(cid:12)) = ((cid:12)(cid:0)2) for (cid:12) = 1; 4, and =(cid:0)1=2g for Fors(cid:28)g,Eq.(11a)isthusreduced totheergodicresult
p
(cid:12)=2 where the above result is obtained with due regard
(5), while for s (cid:29) g, Ref(s) / exp[(cid:0)(cid:25) 2s=g]. There-
to the third-loop corrections, and thus contained an ad-
fore, for large s using the accurate summation instead
ditional factor of 1=g. Calculating the integral (6) with
of integration gives only the exponentially small correc-
allowance for the correction (9), we obtain
tion to the WL result of Eq. (10). (Note that with the
accuracy up to the same exponent, Imf(s) = (cid:25)=s, so
(cid:17)(g;(cid:12)) 1 1
RWL(")= (cid:12) g2jsj: (10) that usingthe integrationforcalculatingthe WL correc-
tion is justi(cid:12)ed). In the transient region of the width
2
There exist two more two-loop diagrams, Fig. 2b,c, but "0 = 2Ecln g, the contribution of Ref(s) is important
they exactly cancel each other. It has been expected, (and dominant for j" (cid:0) Ecj (cid:28) "0), and it describes a
smooth and universal crossover from the Wigner-Dyson
as these diagrams would yield a singular correction to a
to the WL regimein R(s).
renormalizedtriangularvertex ofthe diagramofFig.1b.
Now we can combine the results in the ergodic re-
The vertex has zero scaling dimensionality, and should
gion, Eq. (5), the WL region, and the transient region
remain unrenormalized, as its renormalizationwould vi-
described above to represent them as
olate the particle conservation law. Thus all the vertex
corrections must cancel each other, as has been noted in (cid:20) 2 (cid:21)
1 (cid:25) (cid:17)
Ref. [9] for the diagramsat the mobilityedge. The can- R(s)= Ref(s)+ (13)
(cid:25)2(cid:12)g2 jsj
cellation of the diagrams in Fig. 2b,c provides a pertur-
bativeproofoftheabsence ofthevertexrenormalization.
Inthe region(1)thiscontributiondominatesoverthat Onlyfors>(cid:24)(h(cid:22)=g(cid:28)(cid:1)),thiscontributionbecomessmaller
2 than the quasi-ballisticone, Eq. (7), found in Ref. [17].
of Eq. (7), provided that g < (cid:22)h=Ec(cid:28) (cid:24) (L0=`) . (For
(cid:12)=2,itdominatesforEc"<(cid:24)(cid:22)h=g2(cid:28),such regionexisting The two-level correlation function R(2s) studied above
for g <(cid:24)L0=`). This conditionmaybe violatedfor a very determines the variance (cid:6)2 = h(N(cid:0)N) i of the number
oflevelsN inanenergystripofthegivenwidthE =N(cid:1)
smallor a very pure samplewhich is almostballistic. In
(here hNi(cid:17)N). Its derivative is given by
general, however, it iseasily satis(cid:12)ed. The twocontribu-
tionsarematchingfor"(cid:24)(cid:22)h(cid:17)=g(cid:28),andthelatterbecomes Z N
dominatingforthehigherenergies, inthe`quasi-ballistic' d(cid:6)2
= R(s)ds: (14)
region h(cid:22)(cid:17)=g(cid:28) <(cid:24) " <(cid:24) (cid:22)h=(cid:28). There is, however, a paramet- dN (cid:0)N
ric mismatching of the results of Eqs. (5) and (10) at
the boundary between the ergodic and di(cid:11)usive regime, In order to calculate the r.h.sofEq.(14),it is convenient
"(cid:24)Ec, where the ergodic expression, Eq.(5),isoforder touse therepresentation ofR(s)inthe formofasum,as
2 3 in Eq.(11b), and integrate (cid:12)rst over s. Then, neglecting
(cid:0)1=g while di(cid:11)usive one is of order (cid:0)1=g for (cid:12) = 1,
4 3 the WL corrections, one obtains for N (cid:29)g:
(cid:0)1=g for (cid:12) =2, and +1=g for (cid:12) = 4. The reason is
that one can justify neglecting all the q 6= 0 terms in
the sum (4b) only for " (cid:28) Ec, and changing summa- d(cid:6)2 = 2N X 1 ' 1 ; (15)
tion by integration only for " (cid:29) Ec. There exists some dN (cid:25)2(cid:12)g2 n [(N=g)2+(n2)2] (cid:12)g
transient region near " (cid:24) Ec where R(") crosses over
from the Wigner{Dyson behavior, Eq. (5), to the WL so that (cid:6)2 (cid:24) N=g [5]. Allowing for the WL term, one
behavior, Eq. (10). In this region one should evaluate (cid:12)nds with the logarithmicaccuracy:
accurately the sum in Eq. (4b). Changing there to the
(cid:20) (cid:18) (cid:19)(cid:21)
dimensionless variables n and s, and using the transfor- N 2(cid:17)(g;(cid:12)) N
(cid:0)2 R1 (cid:0)(cid:11)b (cid:6)2(E)= 1+ ln : (16)
mationb = 0 (cid:11)e d(cid:11),oneperformsthesummation (cid:12)g g g
over nx and ny in Eq. (4b) to obtain the result in terms
(cid:0)1
of the elliptic theta-function [18](cid:18)3: As in the WL regime g ln(L0=l) << 1, the variance
(cid:6)2 is almost linear in N in the whole region Ec < E =
1
N(cid:1)<(cid:22)h=(cid:28). Indeed, the WLlogarithmictermin Eq.(16)
R(2)(s) =X(cid:25)2(cid:12)g2 R1ef(s); Z 1 i(cid:11)s=g 2 (11a) cgoivuelsdebge<(cid:24)im(cid:22)hp=oErtca(cid:28)nt(cid:24)on(lLy0f=olr)2Nw>(cid:24)hilgeegth.eFoWrLEa<(cid:24)pp(cid:22)hr=o(cid:28)a,chthiiss
f(s) =nx;ny (n2(cid:0)is=g)2 = 0 (cid:11)e ' ((cid:11))d(cid:11); (11b) applicable in the opposite limit,g(cid:0)1ln(L0=l)<(cid:24)1.
Note that theN-independent result ofEq.(15)isonly
+1
X 2 valid in the di(cid:11)usive region (cid:22)h=((cid:28)(cid:1)) (cid:29) N (cid:29) g. For
(cid:0)(cid:11)n (cid:0)(cid:11)
'((cid:11))= e =(cid:18)3(0;e ): (11c) N !1theintegralinther.h.s.ofEq.(14)must vanish.
(cid:0)1
Thereasonisthatthereexistsasumrulewhichfollows
from the conservation of the total number of states N.
The asymptoticalvalues of ' are given by
3
Multiplying Eq. (2) by h(cid:26)(E +")i and integrating over thankful to the INT at the University of Washington,
R
d
all"withtheidentityL (cid:26)(E+")d"=N,oneobtains: Seattle, for kind hospitality extended to them at di(cid:11)er-
ent stages of this work. The partial support of the EEC
Z Z +1
grant CT90 0020 (V.E.K. and I.V.L.) and the NATO
RE(")h(cid:26)(E +")id"= R(s)ds=0; (17)
grant CRG.921399(I.V.L.) is gratefully acknowledged.
(cid:0)1
where the second identity inEq.(17)results fromh(cid:26)i be-
ing a constant in the relevant interval, "(cid:28)"F.
By the derivation, the sum rule Eq.(17) holds exactly
as long as the total number of states N is (cid:12)nite. How-
ever,itmaybeviolatedifthethermodynamic(TD)limit,
N !1, were taken before the integration in Eq.(17). [1] E.P.Wigner,Proc.CambridgePhilos.Soc.47,790(1951
To illustrate this point let us consider an ensemble of F. J.Dyson, J.Math. Phys. 3, 140 (1962).
diagonal random N (cid:2) N-matrices. Suppose that ma- [2] See for reviews M. L. Mehta, Random matrices (Aca-
trixelements,xn,areindependentrandomvariableseach demic Press, Boston, 1991; F. Haake, Quantum signa-
having the same probability distribution: P(x) = N(cid:0)1 tures of chaos(Springer-Verlag, Berlin, 1991).
for jxj<N=2 and P(x)=0 otherwise. Using the de(cid:12)ni- [3] L. P. Gor'kov and G. M. Eliashberg, Sov. Phys. JETP
tions (2, 3), one immediately(cid:12)nds the correlation func- 21, 940 (1965).
(cid:0)1 [4] K. B.Efetov, Adv.Phys. 32, 53 (1983).
tion,R0(s) =(cid:14)(s)(cid:0)N (cid:18)(N=2(cid:0)jsj),where(cid:18)(x)isastep
[5] B. L. Altshuler and B. I. Shklovskii, Sov. Phys. JETP
function. For (cid:12)nite N, R0(s) obeys the sum rule (17),
64, 127 (1986).
sinceapositivecontributionofthe(cid:14)-functioniscancelled
[6] N. Argaman, Y. Imry, and U. Smilansky, Phys. Rev. B
by a negative one given by the (cid:13)at small\tail"in R0(s).
47, 4440 (1993).
The universal Poisson distribution, R0(s) = (cid:14)(s), arises
[7] See forreviewsP.A.Lee andT.V.Ramakrishnan, Rev.
only in the TD limit and obviously does not obey the
Mod. Phys. 57,287 (1985).
sum rule.
[8] B.I.Shklovskii,B.Shapiro,B.R.Sears,P.Lambrianides,
The above toy model illustrates what happens in the
and H.B. Shore, Phys. Rev. B 47,11487 (1993).
present problem. The integral in Eq. (14) is really van-
[9] V. E.Kravtsov, I.V. Lerner, B. L.Altshuler, and A.G.
ishing in the limitN !1, as required by the sum rule,
Aronov, Phys. Rev. Lett.72, 888-891 (1994).
(cid:0)1
due to the negative contribution, ((cid:0)(cid:12)g) , of the long [10] A. G. Aronov, V. E. Kravtsov, and I. V. Lerner, Phys.
and small \ballistic" tail in R(s) given by Eq. (7) for Rev. Lett.,in print.
"(cid:28)=(cid:22)h(cid:24)1. Ifthe TDlimit,L0 !1,istakeninR(s)this [11] A. G. Aronov, V. E. Kravtsov, and I. V. Lerner, JETP
ballistic tail vanishes, and the sum rule breaks down. Letters 59, 39-44 (1994).
A situation similar to the present 2d case takes place [12] E. Abrahams, P. W. Anderson, D. C. Licciardello, and
at d>2 at the mobilityedge where the sum rule for the T.V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
limitingfunction R(s) breaks down for the same reason. [13] A.A.Abrikosov, L.P.Gor'kov,and I.E.Dzyaloshinskii,
Thus, in contrast to the statement made in Ref. [9], it Methodsof QuantumField Theory in StatisticalPhysics
could not be applied to calculating the variance. This (Pergamon Press, NewYork, 1965).
leadstoalinearinN terminthe variance[19],missedin [14] L. P. Gor'kov, A. I. Larkin, and D. E. Khmel'nitskii,
Ref.[9],whichissimilartothatinEq.(16). Adi(cid:11)erence JETP Letters 30,228 (1979).
is that g (cid:24) 1 at the mobilityedge so that the coe(cid:14)cient [15] S. Hikami, Phys. Rev. B 24, 2671 (1981).
[16] B. L. Altshuler, V. E. Kravtsov, and I. V. Lerner, in
of proportionality is just a certain number. The long-
Mesoscopic Phenomena in Solids, edited by B. L. Alt-
range levels correlations lead to the contribution to the
(cid:13) shuler, P.A. Lee, and R. A.Webb, North-Holland, Am-
variance / N ( (cid:13) is a universal critical exponent), so
(cid:13) sterdam, 449 (1991).
that(cid:6)2 =AN+BN where AandB arecontributedby
[17] A. Altland and Y. Gefen, Phys. Rev. Lett. 71, 3339
allthe diagramsandthus cannot be exactly determined.
(1993).
Weconclude that in the di(cid:11)usive region of Eq.(1) the
[18] I. S. Gradstein and I.M. Ryzhik, Table of Integrals,Se-
level correlations, Eq. (10), are totally governed by the
ries, andProducts(Academic Press, New York,1980).
weaklocalizatione(cid:11)ects whilethelevelnumbervariance,
[19] B. L. Altshuler, I. K. Zharekeshev, S. A. Kotochigova,
Eq. (16), is governed by the total conductance with the
and B. I.Shklovskii, Sov. Phys. JETP67, 625 (1988).
WL e(cid:11)ects resulting only in smallcorrections.
ACKNOWLEDGMENTS
We are grateful to A. Aronov, D. Khmelnitskii,
G. Montambaux, B. Shapiro, and B. Shklovskii for use-
ful discussions on this and related subjects. V.E.K. is
thankful to the University of Birmingham,and I.V.L. is
4
