Spectral statistics in disordered metals: a trajectories approach

9 Pages·0.21 MB·English

by  Robert A. Smith

Spectral statistics in disordered metals: a trajectories approach
R. A. Smith1, I. V. Lerner1, and B. L. Altshuler2,3
1School of Physics and Space Research, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom
2NEC Research Institute, 4 Independence Way, Princeton, NJ 08540, USA
3 Physics Department, Princeton University, Princeton, NJ 08544, USA
8
(January 22, 1998)
9
9
1
Weshowthattheperturbativeexpansionofthetwo-levelcorrelationfunction,R(ω),indisordered

n conductorscanbeunderstoodsemiclassically intermsofself-intersectingparticletrajectories. This
a requires the extension of the standard diagonal approximation to include pairs of paths which are
J non-identical but have almost identical action. The number of diagrams thus produced is much

2 smaller than in a standard field-theoretical approach. We show that such a simplification occurs
2 becauseR(ω)hasanaturalrepresentationasthesecondderivativeoffreeenergyF(ω). Wecalculate
R(ω) to 3-loop order, and verify a one-parameter scaling hypothesis for it in 2d. We discuss the
]
possibility of applying our “weak diagonal approximation” to generic chaotic systems.
l
l
a
PACS numbers: 73.23.-b, 73.20.Fz, 05.45+b
h
- The relationship between the quantum properties of In addition the leading-order perturbative term5 in the
s
e disordered and classically chaotic systems has been the non-ergodic regime gives a result equivalent to that ob-
m focus of much recent research1–6. The main reason for tainedinthediagonalapproximation12totheGutzwiller
. this is that both types of system show the same un- trace formula11. The same approximation when applied
t
a derlying behavior – their energy spectra in appropriate to the disordered systems1 also gives the leading order
m regions have statistics given by random matrix theory perturbative term first derived in Ref. 13. One therefore
- (RMT)7,8. The spectral properties in these universal expectsthathigherorderperturbativetermsinquantum
d
regions are well understood. The challenge now is to chaotic systems would be analogous to those occurring
n
understand whether generic features also emerge in the in disordered systems (where they are known as weak-
o
deviations from universality. Answering this question is localization corrections). The ballistic sigma model is,
c
[ made difficult because two very different languages are however, ill-defined in the ultraviolet limit, making per-

used to characterize the two types of system. In quan- turbative analysis currently ambiguous.
1
v tumdisorderedsystemsoneaveragesoverallrealizations On the other hand, various authors14–16,1 have pro-
9 of disorder to obtain an effective field theory: the non- ceededintheoppositedirection,anddevelopedsemiclas-
2 linearsigmamodel8–10. Inquantumchaoticsystems,(i.e. sical methods for the quantum disordered system. The
2 quantumsystemswhicharechaoticintheclassicallimit), goalistogainamoreintuitivepictureofphenomenasuch
1 one considers a particular system using the Gutzwiller as weak localization and universal conductance fluctua-
0 trace formula11, which involves a sum over classical pe- tions.
8
riodic orbits. Each language has its own strengths and
9 Inthispaperweshowhowthediagrammaticsforspec-
weaknesses. The field theory technique is rather formal
/ tral correlations in disordered conductors can be rewrit-
t and somewhat opaque physically, but has a well-defined
a ten in terms of particle trajectories which self-intersect
m perturbation expansion. The trace formula appears to
in real space. Each diagram for the two-level corre-
be more physically transparent in that one is summing
- lation function is represented as consisting of two tra-
d over classical trajectories. However, this sum is difficult
jectories that are identical everywhere except at self-
n toperform,itsconvergencepropertiesarenotwellunder-
intersection regions, where they are rejoined in different
o stood,andnocontrolledexpansioniscurrentlyavailable.
c ways. As the two trajectories are identical for most of
Itisthereforenaturalthatoneshouldattempttousethe
: their length, they are phase coherent, and interfere con-
v strengths of one language to compensate the weaknesses
structively. The perturbation parameter, 1/g, where g
i of the other.
X is the the dimensionlessconductance, canbe understood
r Onerecentattemptwastotrytocarryoverthepower- as the probability for a self-intersection to occur. The
a
ful calculational techniques developed for the disordered greatadvantageofthis approachis the drastic reduction
systemstothechaoticsystems4,5. Inthemostconsistent in the number of diagrams which occur in a given or-
form,oneaveragesoveracertainenergyintervaltogener- der of perturbation theory This reduction arises because
ate a field-theoreticalfunctional in the Wigner represen- our approach does not distinguish the starting points of
tation (the “ballistic” sigma-model). From this one can the (closed) trajectories,whereas the standard approach
derive, e.g., the two-level correlation function for quan- does. Weshowthatwhentranslatedintofieldtheorylan-
tumchaoticsystems. Intheuniversalergodicregime,the guage,thismeansthatthetwo-levelcorrelationfunction,
non-perturbative derivation is equivalent to one which R(ω), is the second derivative with respect to energy, ω,
Efetov8 developed for disordered systems, as expected. of the free energy, F(ω). As a particular problem we
1
calculate the two-level correlationfunction to three-loop The question that we address here is how to get reg-
orderinperturbationtheory,whichyieldstheleadingor- ular corrections to the diagonal approximation which
der level statistics for the unitary system in 2d. Finally are due to contributions of relatively long trajectories.
weshowthatthethree-loopresultsforR(ω)inbothuni- Such corrections are of fundamental importance, espe-
tary and orthogonal systems can be derived from a one- cially for two-dimensional disordered systems: the lead-
parameter scaling hypothesis in which the renormalized ing order term corresponding to the diagonal approxi-
conductance g(ω) is substituted into the one-loopresult. mation vanishes17 for d = 2. However, the possibility
Notethatinourapproachweareusingthetrajectorypic- of getting the regular corrections could be interesting in
turetoclassifythefieldtheorydiagrammaticsincontrast a much wider context. To evaluate the corrections one
to the work discussed in the previous paragraph where has to consider pairs of paths with almost equal actions,
field theory is being used to classify periodic orbits. S ≈ S , like in the diagonal approximation, but allow
p q
We consider the relation of the picture of trajectories p6=q.
to the perturbation expansion of R(ω) which is defined
To construct the weak diagonal approximation which
by
allows for such pairs of trajectories we start with a path
1 that self-intersects in real space at one or more points.
R(ω)= ν(E+ω)ν(E) −1. (1)
ν2 Atapointofintersectionwebreakupthepathandmake
D E differenttrajectoriesbyjoiningthepiecestogetherindif-
Here ν(E) is the density of states per unit volume which ferent ways. We see at once that this gives rise to a per-
can be written, given the spectrum {En} of the system, turbation expansion, with the perturbative order given
as by the number of loops in real space created by self-
1 intersections, and the perturbation parameter given by
ν(E)= Ld δ(E−En), (2) the probability of having a self-intersection. In the case
n of a disordered electronic system, we will show that this
X
expansion is none other than the usual field-theoretical
and ν = hν(E)i = 1/(∆Ld), where ∆ is the mean level
loop expansion. Before we do this, let us speculate on
spacing. Here h...i denotes the averaging,either over all
thenatureofthisexpansionforagenericchaoticsystem.
realizations of disorder for a disordered system or over a
certainenergywindowforachaoticsystem. Suchenergy Inthiscasetheabovepictureofself-intersectingtrajec-
windows are always narrow enough so that ν has no en- toriesrequiresclarification. Firstofall,thepictureoftra-
ergy dependence. The semiclassical approachallows one jectories is formulatedin phase space rather than in real
to write ν(E) as a sum over classical paths by means of space. Ofcourse,classicaltrajectorieswhichareidentical
the Gutzwiller trace formula11, along part of their length in phase space cannot diverge
inthewaydiscussedabove–thisprocessisquantumme-
ν(E)= Ap(E)exp(iSp(E)/¯h) (3) chanicalinnature. Theuncertaintyprinciplemeansthat
p phasespaceiscoarse-grainedintoboxeswithsizeoforder
X
¯hd. Two trajectories which were originally nearly iden-
where A (E) and S (E) arethe amplitude andactionof
p p tical (i.e. passing through the same phase space boxes)
thep-thperiodicorbitatenergyE. Onecanthensubsti-
start to deviate significantly at the Ehrenfest time, t ,
tute this semiclassical formula into the definition of the Ehr
due to quantum processes such as diffraction. Then we
two-levelcorrelationfunction,R(ω),andtaketheFourier
can make pairs of piecewise identical trajectories as de-
transform to obtain the spectral form factor, K(t),
scribed above. For this picture to make sense we need
to work at time-scales greater than t . This has been
dω Ehr
K(t)= exp(−iωt/¯h)R(ω). (4) noted in Ref. 18 which recently considered the weak lo-
2π¯h
Z calization correction in a disordered system where the
Since all factors vary slowly with E except the action Ehrenfest time is determined by the diffraction at ran-
S (E), one expands S (E+ω) to first order to obtain domlydistributedscatteringspheres. Intheusualmodel
p p
of a disordered conductor with point-like scatterers t
Ehr
1
K(t)= A A∗ei(Sp−Sq)/h¯δ[t− (T +T )] (5) coincideswiththeelasticscatteringtimeτ sincethepar-
p q 2 p q ticle’s direction is totally randomized after each scatter-
p,q
X ing. In general, we expect that the classification of tra-
where T (E)=dS (E)/dE is the period of the orbit p. jectories in the weak diagonal approximation might be
p p
ToproceedfurtherBerry12introducedthediagonalap- valid for chaotic system with t ≪ t where t is
Ehr erg erg
proximation assuming that only terms with p = q con- the ergodic time scale at which the trajectory samples
tribute to the sum in Eq. (5). The reasoning is that the entire phase space. This condition is necessary in
termswithp6=q haverandomlyvaryingphasesandcan- order to treat the self-intersection region as a perturba-
cel each other. In the disordered metal such a diagonal tion. It simply means that the length ofalmostidentical
approximation was shown1 to reproduce the leading or- regions of the two trajectories is much larger than the
der perturbative contribution13 to R(ω). length of the self-intersection region.
2
At time scales t ≫ t , one expects completely uni-
erg
versalbehaviorwhichisdescribedbythe randommatrix
theory.19 This has been conjectured by Bohigas et al7
and partially proved by Andreev et al.5,6 This univer-
sal behavior is completely non-perturbative and cannot
FIG. 2. The 2-loop diagram for the free energy F(ω) cor-
be described by the diagonal approximation. Moreover,
respondingtopathswith oneself-intersection. First weshow
evenfort≪t ,wheret ≡¯h/∆istheHeisenbergtime,
H H the two different ways of linking up the paths at the self-
the diagonal approximation does not reproduce the de-
intersection point. Then we rewrite this in the field theory
viation fromthe linear behavior of K(t) knownfrom the
language so that regions where the two paths are identical
random matrix theory. We will show that we can ex-
become wavy lines, whilst the region of self-intersection be-
actly allow for this deviation within our weak diagonal
comes a closed box. Finally we twist the latter diagram into
approximation.
its usual form.
At time scales t ≪ t ≪ t , one expects non-
Ehr erg
universal corrections to the diagonal approximation. It
The pair of identical or time-reversed trajectories be-
is not clearwhether this is true for a genericchaoticsys-
comes a diffusion or Cooperon propagator, respectively,
tem. Attempts20 togeneratethesecorrectionswithinthe
ballistic nonlinearσ model4–6 haveled to results vanish- drawn as a wavy line. This yields the one-loop diagram
shown in Fig. 1a.
ing to all orders in perturbation theory. These results
are dubious, however, because they depend strongly on The two-loop case is shown in Fig. 2 and involves one
the short-time regularization procedure. In particular, self-intersection region. At this point the paths can be
there exist regularizations where the corrections do not linked up in two different ways, and the diagram con-
vanish and are totally governed by short-time processes sists of these two trajectories. Again we rewrite in field
attime scalest<∼tEhr. Inaddition,the ballisticσ model theory language with the identical or time-reversed por-
itself might not take into account all relevant quantum tions of path becoming diffusion or Cooperon propaga-
processes. tors. The regionaroundthe self-intersectionbecomes an
In the disordered conductor these corrections can be effectiveinteractionbetweenthesepropagatorsknownas
calculated both for t ≫ t and for t ≪ t . Here a Hikami box21. If we put arrows on the trajectories to
erg erg
t =L2/Disthetimetakenforanelectrontodiffuseto show their direction of traversal we see that they go in
erg
theboundaries. We willshowthatthetrajectorypicture the same directionoverpartofthe diagram. This means
is naturally related to a standard diagrammatic expan- that time-reversalinvariance is requiredfor their actions
sionand,moreover,allowsconsiderablesimplificationsin to be identical. Hence there is no two-loop term in the
this expansion. unitary case, where time-reversal invariance is broken;
We will now draw all closed trajectories with up to in fact, all even loop contributions vanish for the same
two self-intersections regions and show their relation to reason.
thediagramsuptothree-looporder. Letusfirstrewritea The three-loop case is shown in Fig. 3, and involves
pairofidentical(uptodiscretesymmetriesoftheensem- twoself-intersections. Thesituationhereismorecompli-
ble) trajectories, which correspond to Berry’s diagonal cated since there are three distinct topological ways for
approximation,12,1 asadiagraminfieldtheorylanguage. two self-intersections to occur. They can occur either at
the same point, or at two different points A and B. In
the latter case there are two distinct orderings in which
the trajectory can be traversed – AABB and ABAB.
Finallythereismorethanonewayoflinkingupthepar-
tial paths to form trajectories. There are five diagrams
in total: F has AABB form; F and F have inter-
3a 3b 3c
sectionatonlyonepoint;F andF haveABAB form.
3d 3e
Putting arrows on the trajectories we find that only F
3b
and F contribute to the unitary case.
(a) (b) 3d
Weseethattheaboveprocedurecanobviouslybegen-
eralized to any order of perturbation theory, and is a
FIG. 1. (a) The 1-loop diagram for the free energy F(ω),
powerfulwayofensuringthatallcontributionshavebeen
which consistsof pairsof trajectories that areidenticalupto
considered.
time reversal symmetry. In the field theory language it con-
sists of a single closed wavy line. (b) The same diagram in Let us now relate these pictures of semiclassical tra-
the disordered metal. Since the closed wavy line in (a) con- jectories to the standard diagrammatic approach13 for
sists of asum overimpurityladders, andthediagram with n evaluatingR(ω). Thestartingpointis the expressionfor
impurity ladders has symmetry factor 1/n, the ladder gives thedensityofstates,Eq.(2),intermsofelectronGreen’s
logarithmic contribution −ln(Dq2−iω). functions,
3
(a)
(b)
(c)
(d)
(e)
FIG.3. Thefive3-loop diagrams for thefree energy F(ω)corresponding topathswith twoself-intersections. Wefollow the
formatofFig.2,showingfirstthepairsofdistincttrajectories; thenrewritingthesewithwavylinesrepresentingregionswhere
the two paths are identical, and closed boxes at the self-intersection points; then finally twisting the latter diagrams in their
usual form.
4
i
ν(E)= ddr GR(r,r;E)−GA(r,r;E) . (6)
2πLd
Z
(cid:2) (cid:3)
It follows from this expression and the definition (1) that
∆2
R(ω)= ℜe ddrddr′ GR(r,r;E )GA(r′,r′;E ) , ω ≡E −E . (7)
2π2 1 2 c 1 2
Z
(cid:10) (cid:11)
Here only the connected average of GRGA remains: paths, r and r′, are fixed in the beginning, even though
the sum of all unconnected averages is absorbed by −1 we finally integrate over all r and r′ in Eq. (7). Con-
in the definition (1), while the connected average of trarily, there are no starting points on the semiclassical
GRGR+GAGA vanishes. diagrams of Fig. 1. Obviously, one can obtain the dia-
Equation (7) directly corresponds to the semiclassical gramofFig.4 fromthe semiclassicalpicture of Fig.1by
expressions (4) and (5), since GR(r,r;E) is the quan- insertingpointsrandr′ intodifferentelectronlinesinall
tum mechanical propagatorfor a particle of energy E to possibleways. Insertinganexternalpointisequivalentto
start and finish at the same point r. We can then rep- takingaderivativewithrespecttoenergy: insertingrre-
resent GR(r,r;E) as the sum of all closed paths while placesGR(E )byGR(E )GR(E ),whichcanbeachieved
1 1 1
GA(r,r;E) will be the sum of all closed paths traversed bytheactionof−∂/∂E . Similarly,insertingr′ givesthe
1
intheoppositedirection. Averagingoverdisorderresults same result as −∂/∂E . After averaging R(ω) depends
2
in the vanishing of all contributions apart from those only upon ω =E −E . Therefore,one may representit
1 2
corresponding to pairs of coherent paths. This coher- as the second derivative of a certain function F(ω),
ence arises when the pair of paths involves scattering
off the same impurities although not necessarily in the ∂2
R(ω)=−∆2 F(ω). (9)
same order. Pictorially, scattering off the same impuri- ∂ω2
ties is shown by impurity lines between GR and GA. In
the lowest perturbative order, the closed paths for GR We will show below that this F(ω) corresponds to the
and GA are identical, apart from their different starting ‘free energy’ of the appropriate field-theoretical func-
points r and r′, while the impurity lines form a ladder tional. NotethatsucharepresentationofR(ω)hasprevi-
ously been used in the evaluation of first-order diagrams
which corresponds to a diffusion propagator
in the ballistic regime22. Here we show that Eq. (9) is
1 1 valid to all orders of perturbation theory.
P(q,ω)= , qℓ≪1, ωτ ≪1 (8)
2πντ2Dq2−iω Higher order perturbative contributions to R(ω) arise
from two paths where scattering occurs from the same
or its time-reversed counterpart, a Cooperon. impurities but in a different order. The parts of the di-
The key feature of the diagram for R(ω) generated in agrams where the sequence of impurity scatterings coin-
such a way, Fig. 4, is that the starting points of the two cides (or is time reversed) for the two paths are again
represented by diffuson (or Cooperon) ladders. These
ladders are connected by Hikami boxes which represent
the changeinthescatteringsequence. Intermsofpaths,
this corresponds to the regions of self-intersection de-
scribed above. The only difference between these higher
r r r r order contributions to R(ω) and semiclassical diagrams
in Figs. 2 and 3 is the necessity to distinguish starting
points r and r′. Therefore these semiclassical diagrams
describe higher order contributions to F(ω), and R(ω)
can be obtained with the help of Eq. (9).
Finallyletusfurtherjustifytheabovepictorialdiscus-
sion with a rigorous derivation using the standard field-
FIG.4. Thefield theory diagram for thelowest order con- theory machinery9,10,21,23. We perform the average over
tribution to the two-level correlation function R(ω) in the disorder by the replica trick. This can be done with ei-
standard approach. This is identical to Fig. 1 except that ther bosonic (commuting) or fermionic (anticommuting)
the starting points of the two trajectories, r and r′ are dis- variables,andbothmustyieldthesameresultsforphysi-
tinguished. In general inserting points in such a way yields calquantities. Inthispaperwewillusebosonicvariables.
many more diagrams for R(ω)than for F(ω). The two-level correlation function is then given by
∆2 2
R(ω)=− lim DQ ddrTr[ΛQ(r)] exp(−F[Q;ω])−1, (10)
N→016π2N2
Z (cid:18)Z (cid:19)
5
where F[Q;ω] is the non-linear sigma model functional To generate perturbation theory we must introduce a
parameterizationofQthatsatisfiesthesaddle-pointcon-
πν
F[Q;ω]= ddrTr D(∇Q)2−2iωΛQ (11) straints. There are several parameterizations available,
8 andthey will givedifferent contributions for a givendia-
Z
(cid:2) (cid:3) gram,butthesumofallcontributionsatagivenorderof
ThematrixQhasrank2N: wemadeN replicaseachfor
perturbationtheorywillalwaysbethesame. Thepartic-
retardedandadvancedGreen’s functions. It satisfiesthe
ular one used depends upon the application. The choice
standard saddle-point conditions:
Q2 =I, Tr(Q)=0. 0 2iV
Q=Λ(W + 1+W2) W = (13)
2iV+ 0
The matrix Λ is diagonal with elements +1 for retarded (cid:18) (cid:19)
p
indicesand−1foradvancedindices. Thenitfollowsthat
reproduces the results of the original impurity diagram-
Tr(ΛQ(r))=N(GR−GA)=−2πiNν(r), matics in the diffusive regime21, and is thus useful for
direct comparison between the results of impurity dia-
so that the prefactor in Eq. (10) is the product of two grammatics and the non-linear σ-model. Here V is an
densities of states. If we now introduce the free energy unconstrained N ×N matrix with elements of the form
F(ω) as appropriate for the given Dyson ensemble. The func-
tional integration is then carried out over the indepen-
1
F(ω)= lim DQexp(−F[Q;ω]), (12) dent variables V. Another parameterization,23
N→0N2
Z
Q=Λ(1+W/2)(1−W/2)−1 (14)
we see from Eqs. (10) and (11) that R(ω) can indeed be
written in terms of F(ω) as in Eq. (9).
We can now apply the usual perturbative methods to where W has the same propertiesas before,is more con-
expand F(ω) in powersof the coupling constant (inverse venient for calculation because many terms then vanish,
dimensionlessconductance)1/g ≡1/4π2νDandthenap- and the Jacobianof the transition from Q to W is unity
plythe relation(9)toobtainR(ω). Thegreatadvantage inthereplicalimit. SincetheJacobiancontributesterms
of this method is the much smaller number of diagrams necessary to remove ultraviolet divergences, this means
weneedtoevaluate. Forexample,inthree-loopperturba- that the sum of all terms at any order in perturbation
tion theory for the orthogonal case there are 5 diagrams theorywill haveno ultravioletdivergenceandsothere is
for F(ω) compared to 41 for R(ω). noneedfor regularization. Considerationsinthe ergodic
Inpreviouswork23 a sourcefield forν(E) has beenin- regimearealsomuchsimpler,sinceJacobiantermswould
troduced with a complex index structure similar to that contribute there. In what follows we will, therefore, use
necessary for calculating the conductance moments. A these parameterization.
considerable simplification here is that ω itself suffices When we substitute a parameterization into the free
as a source field for calculating ν(E). We note that this energy functional F[Q] we obtain a sum of vertices,
methodcannotbeextendedtothesupersymmetricsigma
model8 where the free energy is always unity owing to ∞
the fact that the supersymmetry is preserved in the ef- F[Q]= F2n[W] (15)
fectivefunctional. Tobreakthesupersymmetryonemust n=1
X
introduce the k-matrix in the expression for the prefac-
tors,STr(kΛQ),orequivalentlyintroduceanappropriate where the F2n[W] vertex contains 2n powers of W. The
sourceterm in the effective functional2. Although this is first term F2 is the quadratic part, and leads to the lad-
straightforward, all ‘savings’ in terms of the number of der propagators. The perturbation expansion up to the
diagrams would be lost. Thus, for perturbative calcula- third loop order is then obtained by expanding out the
tions the replicated σ model turns out to be more eco- exponential in the other F2n,
nomical than the supersymmetric one. Finally we note
thatthe freeenergyhaspreviouslybeencalculatedupto exp(−F4−F6−F8−...)=1−F4− F6−F42/2
four-looporder24 in order to find β(g). Our result above
(cid:0) (cid:1)
givesa directphysicalmeaning to the field-theoretic free and averaging with respect to exp(−F ). Upon substi-
2
energy. tuting the parameterizationwe get
F =πν ddrTr −D∇V∇V+−iωVV+
2
Z
(cid:2) (cid:3)
F =πν ddrTr −2D∇V∇V+VV+−iωVV+VV+ (16)
4
Z
(cid:2) (cid:3)
F =πν ddrTr D(2∇V∇V+VV++∇VV+V∇V+VV+)−iωVV+VV+VV+
6
Z
(cid:2) (cid:3)
6
If we then Fourier transform into momentum space we withintheparameterization(14)arevalidfortoboththe
recover the expected diffusion propagator of Eq. (8) as ergodic and diffusive regimes.
the average of VV+ with respect to exp(−F ), Let us first check that these formulae reproduce the
2
known results for the ergodic regime to this order. The
1 1
hV (q)V+(−q)i= δ δ (17) ergodicregimecorrespondstothezero-modeapproxima-
iα βj 2πνDq2−iω ij αβ tion in which we set all q equal to zero. Doing this and
i
differentiating twice with respect to ω yields
The higher-order terms yield the effective four- and six-
point vertices 1 1 3
Rort(x)=ℜe + + (22a)
(ix)2 (ix)3 2(ix)4
V (q )=2πν −D∆1−2iω (18) (cid:20) (cid:21)
4 i 4 1
V6(qi)=−2π(cid:2)ν −D(∆16+∆(cid:3)36)−3iω (19) Runi(x)=ℜe 2(ix)2 (22b)
(cid:20) (cid:21)
where ∆m is the sum(cid:2)of all distinct scalar(cid:3)products of where x = πω/∆. These are the correct expansions of
n
pairs qi and qi+m of the n incoming momenta which are the exact results.8 Indeed, for K(t), which is the Fourier
separated by m vertices, transform of R(x), one has8 in the region 0≤t≪1:
1
∆mn = qi.qi+m (20) Kort(t)=t[2−ln(t+1)]≈2t−t2+ t3 (23a)
2
Note that in the derivXation of Eq. (18) we have sym- Kuni(t)=tθ(1−t)=t (23b)
metrized over all incoming momentum variables.
Here t is measured in units of t = h¯/∆. (To compare
Ifweusetheaboveformulastocalculatetheexpansion H
the results of Eqs. (22b) and (23a), one uses the result
offreeenergyF(ω)tothree-looporder,thediagramsob-
thattheinverseFouriertransformoftnisin+1n!/2xn+1.)
tained are simply those we derived earlier using the tra-
Recall the famous result that there are no corrections to
jectorylanguage. Thefieldtheoryenablesustoassociate
theuniversalRMTresultintheunitarycasetoanyorder
analgebraicexpressionto eachcomponentina diagram,
in perturbation theory. This happens due to the cancel-
and we obtain the results
lationbetweenratherthantheabsenceofappropriatedi-
agrammaticcontributions,andisthereforeausefulcheck
F =− lnP
1 1 onourcalculation. Notealsothatthecompleteperturba-
Xq1 tive expansion for R(ω) would give, upon Fourier trans-
1 iω
F = formation, the exact K(t) only for t < 1. The pertur-
2
(2πν) P P
1 2 bation theory cannot give the discontinuity point, t=1,
qX1,q2 becausethisiscontrolledbythenon-trivialsaddlepoint2.
2 (iω)2
F = We will now derive the leading-order contributions to
3a (2πν)2 P2P P
q1X,q2,q3 1 2 3 R(ω) in 2d. This was previously done17 for the orthog-
1 P onal case, and the form in the unitary case conjectured
1
F = − (21)
3b (2πν)2 P P P fromaone-parameterscalinghypothesis. Wereplacethe
1 2 3
q1X,q2,q3 sums over q in Eq. by integrals and use dimensional reg-
1 P1−2iω ularizationtoevaluatetheseintegralsind=2+ǫdimen-
F = −
3c (2πν)2 P P P sions. (Note that all the relevantintegrals may be found
1 2 3
q1X,q2,q3 inRef.24). Thenwecarefullytakethelimitǫ→0,keep-
1 (P −iω)2
12 ing only the terms divergent in this limit. The two-loop
F =
3d 4(2πν)2 P1P2P3P123 orthogonalresult is
q1X,q2,q3
F3e = 2(2π1ν)2 (P12P−1Piω2P)(3PP1132−3 iω) F2ort(ω)= ∆π22Lβd(2πν)(12πD)2(−iωǫ2)ǫ+1
q1X,q2,q3
Taking the secondderivative with respectto ω, and not-
where we use the notation
ing that dimensionless conductance g = G/(e2/πh) =
P =D(q +...q )2−iω. 4π2νD, gives
1...n 1 n
2∆ (−iω)ǫ−1
Of the above terms only F , F and F contribute in Rort(ω)= ℜe (24)
1 3b 3d 2 πg2 ǫ
the unitary case. All diagrams contribute in the orthog- (cid:20) (cid:21)
onalcase,andthereisanextrafactoroftwoarisingfrom Finally we let ǫ → 0 and note that (−iω)ǫ/ǫ becomes
the two possible relative directions of the electron lines.
ln(−iωτ). Hence
SymbolicexpressionsforF similartothoseinEqs.(21)
havebeenpreviouslyderivedinRef.25wheretheparam- 2∆ ln(−iωτ) 1 ω
Rort(ω)= ℜe =− , s≡
eterization (13) has been used. Equations (21) derived 2 πg2 iω g2s ∆
(cid:20) (cid:21)
7
The symplectic result would simply follow upon multi- Runi(ω)= 3∆ ℜe ln(−iωτ) =− 3 . (26)
plying by −1/2. 3 4πg3 iω 8g3s
(cid:20) (cid:21)
Next we consider the three-loop unitary case. Careful
evaluation of the integrals yields Finally we calculate the three-loop orthogonalresult:
∆2Ld 1 (−iω)3ǫ/2+1 ∆2Ld 1 (4−3ǫ)(−iω)3ǫ/2+1
Funi(ω)= . Fort(ω)= ,
3 π2β (2πν)2(2πD)3 3ǫ2 3 π2β (2πν)2(2πD)3 3ǫ3
Thus so that
∆ (−iω)3ǫ/2−1 2∆ (4+3ǫ)(−iω)3ǫ/2−1
Runi(ω)= ℜe (25) Rort(ω)= , (27)
3 πg3 2ǫ 3 πg3 2ǫ2
(cid:20) (cid:21)
which in the limit ǫ→0 becomes which in the limit ǫ→0 gives
9∆ ln2(−iωτ)+ln(−iωτ) 9
Rort(ω)= ℜe = 1+2ln(s∆τ) .
3 2πg3 iω 4g3s
(cid:20) (cid:21) h i
We now show that the same results can be obtained using a one-parameter scaling hypothesis in which the
renormalized conductance, g(ω), is substituted into the lowest order R(ω) diagram. The latter has the form
R(ω)=(∆2/π2β)ℜe P−1, and substituting here D =D +δD(ω) gives upon the expansion in δD
q 1 0
P
∆2 1 δD D q2 δD 2 (D q2)2
0 0
R(ω)= −2 +3 +... (28)
π2β (D q2−iω)2 D (D q2−iω)3 D (D q2−iω)4
( q 0 (cid:18) 0(cid:19) q 0 (cid:18) 0(cid:19) q 0 )
X X X
We obtain the perturbative correctionto the diffusion arising from stronger mesoscopic divergences. To pick
constant,δD(ω),fromthe β(g)functionwhichisdefined upalllogarithmiccorrectionsonemustworkind=2+ǫ
by and take carefully the limit ǫ→0, as we did here.
dlng(L) In conclusion, we have shown that the language of
β(g)= semiclassical trajectories suggests the most economical
dlnL
wayofdrawingdiagrammaticcorrectionstospectralcor-
To two-looporderthe orthogonalandunitary beta func- relations in disordered electronic systems up to a high
tions areβort(g)=2/g andβuni(g)=2/g2 fromwhichit order of perturbation theory. We have introduced the
follows that weak diagonal approximation which includes the con-
tributions of pairs of trajectories which are made from
δD δg 2 2 (−iω)ǫ/2
ort ort
= =− ln(L/ℓ)→ identicalpieces joined together in different waysat some
D g g g ǫ
0 0 0 0 self-intersection (in real space) points. This gives physi-
δDuni δguni 2 1 (−iω)ǫ calmeaningtotheloopexpansionofthefield-theoretical
= =− ln(L/ℓ)→
D g g2 g2 ǫ ‘free energy’: the second derivative of this free energy
0 0 0 0
with respect to frequency is the two-level correlation
We can then substitute these results into Eq. (28), and
function. Wehaveshownthis directlybyusingthe repli-
use the following results for the q-integrals,
cated nonlinear σ model. Note that such a derivation
D q2 2+ǫ doesnotworkforthesupersymmetricnonlinearσ model
0 =− (−iω)ǫ/2−1 as the free energy equals zero unless the supersymmetry
(D q2−iω)3 8πD
q 0 is broken. Naturally, all the perturbative results may be
X
(D q2)2 4+3ǫ reproduced within the supersymmetric model but in a
0 =− (−iω)ǫ/2−1 much less economical way. Using the method described
(D q2−iω)4 16πD
0
q above, we have calculated the leading order contribu-
X
tions to the two-level correlation function in 2d in the
We find that this exactly reproduces the results of Eqs.
non-ergodicregime wherethe standarddiagonalapprox-
(24), (27) and (25), proving the validity of the scaling
imation gives vanishing results.
hypothesis to three-loop order. Note that the conjecture
of Ref. 17 for Runi(ω) does not produce the correct nu- In the ergodic regime we have found that the loop ex-
3
mericalfactorinEq.(26). Thereasonisthatthecalcula- pansionreproduces the deviation from linear in time be-
tioninRef.17hasbeenperformedinexactlytwodimen- havior of the spectral form factor in the orthogonalcase
sionswhereitwaspossibletotakeintoaccountallweak- known from random matrix theory. In the unitary case,
localization logarithms but not logarithmic corrections wehavedemonstratedthatallsuchcorrectionsaremutu-
8
ally cancelled. The random matrix theory is believed to 6A. V. Andreev, B. D. Simons, O. Agam, and B. L. Alt-
describecorrectlylevelstatisticsofclassicallychaoticsys- shuler, Nucl.Phys. B 482, 536 (1996).
temsintheuniversalregime.5−7Itisthereforereasonable 7O.Bohigas, M.Giannoni,andC.Schmit,Phys.Rev.Lett.
to conjecture that the weak diagonal approximation in- 52, 1 (1984).
troducedhereshouldbevalidforgenericchaoticsystems. 8K.B. Efetov, Adv.Phys.32, 53 (1983).
Although the universal regime is well understood within 9F. Wegner, Z. Phys.B 35, 207 (1979).
thesupersymmetricapproach,theweakdiagonalapprox- 10K. B. Efetov, A. I. Larkin, and D. E. Khmel’nitskii, Zh.
imation might be extended to the non-ergodic regime in Eksp. Teor. Fiz. 79, 1120 (1980) [Sov. Phys. JETP 52,
568 (1980)].
chaotic systems.
11M. C. Gutzwiller, J. Math. Phys. 12, 343 (1971).
12M. V. Berry,Proc. R. Soc. Ser. A 400, 229 (1985).
ACKNOWLEDGMENTS 13B. L. Altshuler and B. I. Shklovskii, Zh. Eksp. Teor. Fiz.
91, 220 (1986) [Sov.Phys. JETP 64, 127 (1986)].
14A. I. Larkin and D. E. Khmelnitskii, Sov. Phys. Usp. 25,
Work in Birmingham was supported by EPSRC grant
185 (1982).
GR/J35238. I.V.L. and B.L.A. thank the ITP in Santa 15D.E. Khmelnitskii, Physica B&C 126, 235 (1984).
Barbara for kind hospitality and partial support under 16S. Chakravarty and A. Schmid, Phys. Repts. 140, 193
NSF Grant No. PHY94-0719. R.A.S. and I.V.L. thank
(1986).
the Newton Institute at Cambridge for kind hospitality 17V.E.KravtsovandI.V.Lerner,Phys.Rev.Lett.74,2563
at the final stage of this work. (1995).
18I. L. Aleiner and A. I. Larkin, Phys. Rev. B 54, 14423
(1996).
19M. L. Mehta, Random matrices (Academic Press, Boston,
1991).
20O.Agam, I. V.Lerner, B. D.Simons, unpublished(1997).
21S.Hikami, Phys. Rev. B 24, 2671 (1981).
1N.Argaman,Y.Imry,andU.Smilansky,Phys.Rev.B47, 22A.Altland and Y. Gefen, Phys. Rev. B 51, 10671 (1995).
4440 (1993). 23B.L.Altshuler,V.E.Kravtsov,andI.V.Lerner,inMeso-
2A. V. Andreev and B. L. Altshuler, Phys. Rev. Lett. 75, scopic Phenomena in Solids, edited by B. L. Altshuler,
902 (1995). P. A. Lee, and R. A. Webb, North-Holland, Amsterdam,
3O. Agam, A. V. Andreev,and B. L. Altshuler, Phys. Rev. 449 (1991).
Lett. 75, 4389 (1995). 24F. Wegner, Nucl.Phys. B 280, 193 (1987).
4B.A.MuzykantskiiandD.E.Khmelnitskii,JETPLetters 25J. T. Chalker, I. V. Lerner, and R. A. Smith, J. Math.
62, 76 (1995). Phys.37, 5061 (1996).
5A. V. Andreev, O. Agam, B. D. Simons, and B. L. Alt-
shuler, Phys.Rev. Lett. 76, 3947 (1996).
9