Mean-field theories for disordered electrons: Diffusion pole and Anderson localization

0.23 MB·English

by  V. Janis

Mean-field theories for disordered electrons: Diffusion pole and Anderson localization
V. Janiˇs and J. Kolorenˇc
Institute of Physics, Academy of Sciences of the Czech Republic,
Na Slovance 2, CZ-18221 Praha 8, Czech Republic∗
(Dated: February 2, 2008)
5
0 We discuss conditions to be put on mean-field-like theories to be able to describe fundamental
0 physicalphenomenaindisorderedelectronsystems. Inparticular,weinvestigateoptionsforaconsis-
2
tentmean-fieldtheoryof electron localization and forareliable description oftransport properties.

n We argue that a mean-field theory for the Anderson localization transition must be electron-hole
a symmetric and self-consistent at the two-particle (vertex) level. We show that such a theory with
J local equationscan bederivedfrom theasymptoticlimit tohigh spatial dimensions. Theweight of

5 thediffusionpole,i.e.,thenumberofdiffusivestatesattheFermienergy,inthismean-fieldtheory
2 decreases with the increasing disorder strength and vanishes in the localized phase. Consequences
of thedisclosed behavior for our understandingof vanishing of electron diffusion are discussed.

]
n
PACSnumbers: 72.10.Bg,72.15.Eb,72.15.Qm
n
-
s
i I. INTRODUCTION tureandthermodynamicpropertiesofrandomalloysnot
d
only at the model level but also in realistic settings.1
.
t Later on, the CPA was shown to fit the modern defini-
a Mean-field theories play an important role in the de-
m scription of thermodynamic systems. They are intended tion of the mean-field theory as an exact solution of the
model system in infinite spatial dimensions.2,3
- and used as a first approximation offering a qualitative
d picture of the physics of the studied phenomena. The The coherent potential approximation is nowadays
n
mean-field concept has developed from its initial intu- considered as a paragon for mean-field theories of quan-
o
itive ideas of van der Waals and Weiss through the Lan- tum disordered and interacting systems. Its generalized
c
[ dau theory of critical phenomena to its present sophisti- form4 offers one possible interpretation of equations of
cation and systematics provided by the limit to infinite- motion in the Dynamical Mean-Field Theory (DMFT).5
1
dimensional lattice models. At present, a modern mean- In spite of the proved reliability of the CPA to produce
v
field theory is no longer a weak-coupling approximate anaccurateequilibriumelectronicstructureofdisordered
6
8 treatment neglecting spatial fluctuations. It represents systems, it fails in encountering for inter-site quantum
5 a comprehensive theory providing a phase diagram in coherence and backscattering effects. The CPA is es-
1 the whole range of the input parameters and simulating sentially unable to go beyond the semi-classical descrip-
0 the exact behavior in specific limiting situations. With- tionoftransportpropertiesqualitativelycapturedbythe
5 out a mean-field theory we are mostly unable to identify Boltzmann equation. This inability is due to the fact
0
the relevant fluctuations the mean values of which are that the CPA does not include vertex corrections to the
/
t reflected by thermodynamic (order) parameters. Mean- electrical conductivity independently of how strong the
a
m fieldtheoryisparticularlyimportantforcriticalphenom- disorder may be.6 It is hence unsuitable for the descrip-
ena with divergent correlation functions, where it allows tion of one of the most prominent features of disordered
-
d ustohandlesingularitiesinaconsistentandmanageable systems: Anderson localization.
n way and to select the proper low-temperature phase, at
Anderson localization in disordered or amorphous
o least above the lower critical dimension.
solids takes place when there are available electronic
c
Mean-field theories were primarily developed for col- states at the Fermi surface but no diffusion or charge
:
v lectivephenomenaininteractingsystems. Nontrivialand transportatlongdistancesisobserved. Possibilityofthe
Xi sometimesnoteasilyunderstandableeffectsare,however, absenceofdiffusioninimpuremetalsandalloyswaspro-
alsoinducedbyrandomness. Randomness,inconnection posedbyP.W.Andersononasimpletight-bindingmodel
r
a withinteractionorwithquantuminterference,cancause of disordered noninteracting electrons.7 Since then, van-
significantandsometimesevenunexpectedchangesinthe ishing of diffusion, now called Anderson localization,
behavior of the system. Since mostly no exact solutions has attracted much attention of both theorists and ex-
areavailablefordisorderedsystems,amean-fieldapprox- perimentalists.8,9 In spite of a considerable portion of
imation has become one of the most powerful tools to amassedexperimentaldata,disclosedvariousspecificand
handlefluctuationsinthechemicalcompositionofsolids. general aspects of the Anderson metal-insulator transi-
Milestonesofamean-fieldtheoryfordisordered(non- tion, anda number oftheoreticaland computationalap-
interacting) electron systems were laid at the end of the proaches so far developed we have not yet reached com-
sixties andthe beginning of the seventiesof the lastcen- plete understanding of Anderson localization. Although
tury. The so-called Coherent Potential Approximation many features of the critical behavior at the Anderson
(CPA)developedatthattimeisaself-consistentapprox- localization transition have been disclosed, the position
imationdescribingratheraccuratelytheelectronicstruc- of this disorder-driven metal-insulator transition within
2
the standard classification scheme of phase transitions sary ingredients for a theory being able to describe the
withcontrolandorderparametershasremainedunclear. Anderson localization transition.
It has been mainly due to the nonexistence of an appro- The layout of the paper is as follows. In Section II
priate mean-field-like theory for this phenomenon. Only we summarize basic properties of the CPA defined from
very recently the present authors demonstrated that a thelimittoinfinitespatialdimensions. Weshowhowthe
mean-field-likesolutionforthe Andersonmetal-insulator averaged grand potential is derived from the local one-
transition can be derived from the asymptotic limit to particlepropagatorandtheself-energy. Thehigher-order
high (but finite) spatial dimensions.10 This solution is Green functions are then determined via local external
very close to the CPA in the resulting electronic struc- perturbations. Themean-fieldtheorywithatwo-particle
ture, it reduces to the CPA in infinite dimensions. It, self-consistence is constructed in Section III. First, in-
however, differs from the CPA significantly in transport ability of the CPA to reproduce the proper infinite-
properties derived from two-particle functions. In addi- dimensional limit for two-particle Green functions is
tion to the one-electron self-consistence of the CPA, the demonstrated. Then, using the parquet scheme and the
newmean-fieldtheoryisendowedwithatwo-particleself- electron-hole symmetry we derive a self-consistent (non-
consistence. Thatis,thetwo-particleirreduciblevertices linear) equation for the irreducible two-particle vertex.
are determined from self-consistent nonlinear equations. Thisequationisthensolvedatthe mean-fieldlevel,i. e.,
in the leading nontrivial order of the high-dimensional
Themean-field-liketheoryforthedisorderdrivenvan-
limit. The explicitformofthe diffusionpoleinhighspa-
ishingofdiffusion ofRef.10showssome unexpected fea-
tialdimensionswithitsweightdependentonthedisorder
tures. Itcontradictsthewidelyextendeddogmathatthe
strengthisfinallyobtained. Consequencesofourfindings
weight of the diffusion pole, i. e., the number of diffus-
forunderstandingofthedisorder-drivenvanishingofdif-
ing particles, does not depend on the disorder strength.
fusionandAndersonlocalizationarediscussedinthelast
The weight of the diffusion pole is conserved if all the
Section IV.
states near the Fermi energy are finite combinations of
Bloch waves for any configuration of the random poten-
tial. Or equivalently if a Ward identity between self-
II. THERMODYNAMIC MEAN-FIELD
energy and the irreducible electron-hole vertex holds for
THEORY: ONE-PARTICLE SELF-CONSISTENCE
all transfer energies.11,12 We demonstrated in Ref. 13
that once the electron-hole irreducible vertex contains
A. One-particle functions and generating
theso-calledCooperpole,thenumberofextendedstates
functional
at the Fermi level decreases with increasing the disorder
strength. Hence,thereisnochanceforatheorywiththe
Cooper pole to fully satisfy the Ward identity between To construct a comprehensive mean-field theory for
the averagedone- and two-electron Green functions and thermodynamic properties ofa randomsystemmeans to
tokeepthenumberofdiffusive statesindependentofthe findanapproximaterepresentationinclosedformforthe
disorder strength. grand potential averagedover random configurations
The aim of this paper is to clarify the ambiguities 1
connected with the mean-field concept applied to two- Ω(µ)= lnTrexp βH +βµN (1)
−β − av
particlefunctionsindisorderedelectronsystems. Wede- D n oE
limit the content and the range of validity of the two b b
where µ is the chemical potential and N is the particle
existing mean-field theories for the Anderson model of
number operator. We will consider in this paper only a
noninteractingelectrons—the CPAandthatofRef.10.
noninteracting lattice electron gas scattbered on random
Thecommongroundforboththeoriesisthelimittohigh
impurities and described by the Anderson tight-binding
spatial dimensions. We show that the ambiguity in the
Hamiltonian
identification of the mean-field theory for two-particle
functions results from two different ways how it can be H = t c†c + V c†c , (2)
ij i j i i i
derived: either via the generating local functional and
<ij> i
Ward identities, or via a direct diagrammatic construc- X X
b bb bb
tion in high dimensions. The former construction, CPA, where V is a local, site-independent random potential.
i
is suitable only for one-electron spectral and thermody- It has become evident since the introduction of the
namic properties. Since the CPA lacks the electron-hole conceptofthelimittoinfinitespatialdimensionsinquan-
symmetry and does not correctly reproduce the high- tum itinerant systems that a controllable comprehensive
dimensional limit of two-particle (vertex) functions, it mean-field theory of itinerant models should be defined
becomes unreliable when applied to the electrical con- via this formal limit.14 In high spatial dimensions the
ductivity and other transport properties. We demon- diagonal (local) and off-diagonal (nonlocal) elements of
strate that the latter approach, when properly formu- the one-particle propagator separate from each other.
lated,canleadtoatheorybeingself-consistentandman- The former are of order O(d0), while the latter vanish
ifestly electron-holesymmetricatthe one-andaswellat as d−1/2, where d is the spatial dimension. The full one-
the two-particlelevel. The lasttwo conditionsare neces- particlepropagatorandtheself-energyhavethefollowing
3
high-dimensional asymptotics stationarityequationsarediagonalintheMatsubarafre-
quencies and can be solved for each frequency indepen-
G=Gdiag[d0]+Goff[d−1/2] , (3a) dently. Inserting the solution for all Matsubara frequen-
Σ=Σdiag[d0]+Σoff[d−3/2] . (3b) cies to Eqs. (4) we obtain the equilibrium thermody-
namicpotentialfornoninteractingelectronsscatteredon
We can classify contributions to the many-body per- random impurities. The equilibrium thermodynamics of
turbationexpansionfortheself-energyaccordingtotheir the systems is then determined only by the local irre-
high-dimensional asymptotic contribution and obtain in ducible part of the averagedone-particle resolvent. This
the leading order a local approximation for the irre- irreducible part is self-consistently determined from the
ducible part of the one-electron propagator. The inter- Soven equation (5).
acting part of the thermodynamic potential in infinite
spatialdimensions is then a functional of onlyGdiag and
Σdiag.3 B. External sources and two-particle functions
In disordered systems the inter-particle interaction is
replaced by correlations between scatterings on impuri- Thermodynamicsofdisorderedsystemsisnotofmuch
ties. The self-energy is here a coherent potential of an interestunlessinter-particleinteractionsarepresent. But
effective homogeneous (nonrandom) medium represent- even then the averaged thermodynamic potentials de-
ing the effect of impurity scatterings on the motion of pend on only one-electron functions. One-electron func-
electrons. Since thescatteringsarestatic,wecanfindan tions, however, do not contain the complete information
explicitrepresentationoftheaveragedgrandpotentialin about the behavior of statistical ensembles, in particu-
infinite spatial dimensions. We can write lar of disordered systems. The equilibrium thermody-
namic potentials do not contain sufficient information
1 from which we could derive transport properties of the
Ω G,Σ =F Gdiag −1+Σdiag TrlnGdiag
µ − β systemanditsresponsetoweakexternalelectromagnetic
h i n 1 o perturbations. Toincludetheelectricalconductivityinto
b b b Trln Gb(0)−1 Σdiag+µb (4a) themean-fielddescription,thethermodynamicconstruc-
− β −
h i tion from the preceding subsection must be extended to
where we denoted b b include averagedtwo-particle propagators.
Averaged two-particle propagators in disordered sys-
1
F X = Trln X V (4b) tems containatleasttwo energyarguments (two innon-
−β − av interacting and three in interacting systems). The best
n o D h iE
way to guarantee that one- and two-particle functions
thelocal”interabctingpart”ofthetbhermbodynamicpoten-
are approximated consistently within a single approxi-
tial, in this case the effect of multiple scatterings.4 The
mate scheme is touse the Baym-Kadanoffconceptofex-
trace operator Tr extends over the lattice space as well
ternal sources added to the equilibrium thermodynamic
as over the Matsubara frequencies. The only nonlocal
potential.15 To introduce higher-order Green functions
contributionto the generatingfunctionalΩ comesfrom
µ with severalenergies (chemical potentials) into the ther-
the barepropagatorG(0). The site-diagonal(local)com-
modynamic description we have to replicate the original
plex vectorsGdiag(µ) andΣdiag(µ) infermionic Matsub-
n n system so as for each energy we have an independent
ara frequencies (2n+b1)π/β are variational parameters, replica of the original system, that is, of creation and
the physical values of which are attained at stationarity
annihilation operators.
points of the generating functional (4).
We replicate the creation and annihilation operators
The defining equationfor the localelementof the av-
andintroduceexternalperturbationsintothe thermody-
eragedone-particlepropagatorisobtainedfromanequa-
namic description via a generalized grand potential of a
tionδΩµ/δΣdniag(µ)=0andtheself-energyisdetermined ν-times replicated system Ων(E ,E ,...,E ;U) with ν
fromδΩ /δGdiag(µ)=0. After straightforwardmanipu- 1 2 ν
µ n chemical potentials E ,...,E . An external perturba-
1 ν
lations the two equations reduce to
tion U is used to couple different replicas and to break
the initial replica independence. We then can write
1
1= (5a)
1+[Σ (µ) V ]G (µ)
(cid:28) n − i n (cid:29)av Ων(E1,E2,...,Eν;U)
and
ν
1
= lnTrexp β H(i) δ
Gn(µ)= N1 G(k,iωn)= iω +µdǫρ(Σǫ)(µ) ǫ . −β * − iX,j=1(cid:16) AD ij
k Z n − n − b
X (5b) EiN(i)δij +∆H(ij) (6)
− av
We dropped the superscript diag in the local functions (cid:17)oE
andintroducedthe electronic density ofstates ρ(ǫ). Due whereweassignedtoeachrepblicacharactberizedbyenergy
to the static character of the impurity scatterings the (chemical potential) E a separate Hilbert space and de-
i
4
noted ∆H(ij) = U(ij)c(i)†c(j) an external perturba- whereγisthelocalpartofthetwo-particlevertexΓ. The
kl kl k l
tiontobe setzeroattheend. Thermodynamicpotential irreducible vertex λ in equilibrium (U = 0) determined
Ων(E1,Eb2,...,EνP;U)isabgenebratingfunctionalforaver- via Eq. (10) obeys an equation
agedproductsofGreenfunctionsuptotheνthorder. In
practice,wewilluselinear-responsetheorywithone-and δΣ (z ,z ) 1
U 1 2
idtswueocxe-ppoaanrntldyiceltdewGuoprreeteponliUcfua2ns..cTtihonerse,fio.ree.,itΩiνs(Esu1ffi,Eci2e,n.t..t,oEinν;trUo)- λ(z1,z2)=1δGU(z1,z2)(cid:12)(cid:12)(cid:12)U=0 = G1(z1)G(z2)(cid:20)1−−1
(cid:12) .
In fact we are interested only in corrections to the (cid:28)1+[Σ(z1)−Vi]G(z1)1+[Σ(z2)−Vi]G(z2)(cid:29)av#
products of the averaged one-particle propagators ex- (11)
pressed via vertex functions. The two-particle vertex Γ
isdefinedfromthetwo-particleresolventG(2) inmomen- Wecaneasilyverifythatthisequationcoincideswiththe
tum representation as CPA solution for the irreducible vertex λ(z ,z ).6
1 2
There is no ambiguity in the mean-field construction
G(2)(z ,z ;q)=G(k,z )G(k+q,z )[δ(k k′) of local one- and two-particle functions. But a mean-
kk′ 1 2 1 2 − field treatment has a physical relevance only if it is able
+Γkk′(z1,z2;q)G(k′,z1)G(k′+q,z2)] . (7) toproducenonlocalcorrelationfunctions,thelong-range
fluctuationsofwhichmaysignificantlyinfluencethether-
The external disturbance U mixes different replicas
modynamic and dynamical behavior. There is not, how-
and propagators in the replicated space are matrices in
ever, a unique way how to generate the two-particle ver-
the replica indexes. Since we are interested only in the
tex with non-local contributions within the local mean-
averagedtwo-particlefunctions, wecanresortto twoen-
field approach. The simplest and most straightforward
ergies and to a two-by-two matrix propagator
way is to use the Bethe-Salpeter equation with the CPA
irreducible vertex λ, Eq. (10), and to replace the local
G−1(k1,z1,k2,z2;U)=G(0)−1+U Σ propagatorswith the full nonlocalone-electronpropaga-
−
z ǫ(k ) Σ (U) U Σ (U) tors G(k,z). Such a Bethe-Salpeter equation remains
b= 1−U 1Σ−(U1)1 b z ǫ(−bk )1b2Σ (U) (8) algebraic in momentum representation and results in a
21 2 2 22
(cid:18) − − − (cid:19) two-particle vertex with only one transfer momentum.
where ǫ(k) is the lattice dispersion relation and the self- We obtain
energy elements Σ generally depend on both energies
ij λ(z ,z )
Γ±(z ,z ;q±)= 1 2 (12)
z1,z2. The matrix G represents the averaged resolvent 1 2 1 λ(z ,z )χ±(z ,z ;q±)
that is to be used in the grand potential Ω2(E ,E ;U) − 1 2 1 2
1 2
fromEq.(6). Itisnobwastraightforwardtasktoderivea where we denoted the two-particle bubble
matrix Soven equation generalizing Eq. (5a). We obtain
1
χ±(z ,z ;q)= G(k,z )G(q k,z ) . (13)
1 2 1 2
N ±
G(z1,z2;U) Xk
−1
b = G−1(z1,z2;U)+Σ(z1,z2;U) Vi (9) The ambiguity in this definition of the full mean-field
− vertex is in the type of nonlocal multiple scatterings we
(cid:28)h i (cid:29)av
include into the Bethe-Salpeter equation. They are here
b b b
where G(z ,z ;U) = N−2 G(k ,z ,k ,z ;U) is denotedbythesuperscript . Theplussigncorresponds
1 2 k1k2 1 1 2 2 ±
the local element of the matrix one-particle propaga- to multiple scatterings of electron-hole pairs, while the
P
tor. Invbersions in Eq. (9) have mabtrix character in the minus sign to electron-electron pairs. In case of elastic
replicaspace. Thediagonalelementsofthematrixequa- scatterings the electron-hole and electron-electron bub-
tion (9) determine the one-particle propagatorsfor ener- bles produce numerically the same number. However,
gies z and z . The off-diagonal elements, proportional the difference between the two types of pair scatterings
1 2
to the perturbation U, determine the local two-particle lies in the respective transfer momentum q±. Using the
resolvent that is defined as the coefficient at the linear notation for momenta in the two-particle resolvent from
term in the expansion of the local matrix propagator Eq. (7) we have q+ =q and q− =q+k+k′.
G(z ,z ;U) in the external perturbation U. The local Thisambiguityinthedefinitionofthemean-fieldtwo-
1 2
two-particle Green function can be represented with the particle vertex is not usually acknowledged in the lit-
abid of the irreducible vertex (two-particle self-energy) λ erature, since the electron-hole scattering channel, rele-
viaaBethe-Salpeterequation. WefindfromEq.(9)that vant for the electrical conductivity, is preferred and di-
the Bethe-Salpeter equation in the mean-field approxi- rectlyderivedfromtheBaym-Kadanoffapproach.6 How-
mation reduces to an algebraic one ever, when the mean-field theory is viewed upon as the
limit to infinite spatial dimensions, both electron-hole
λ(z1,z2) and electron-electron multiple scatterings possess the
γ(z ,z )= , (10)
1 2 1 λ(z ,z )G(z )G(z ) same high-dimensional asymptotics.16 A priori, neither
1 2 1 2
−
5
electron-hole nor electron-electron multiple scatterings Gloc contains the local self-energyΣloc(z)from Eq.(5a),
should be discarded. The appropriate form of the ver- whilethefulloneaself-energyΣ(k,z)thatistobedeter-
texis thenselectedbythe physicalquantitiesinwhichit minedlaterfromaDysonequation. Weapplytheexpan-
appears,suchasisthecaseoftheelectricalconductivity. sionaroundtheCPAtotwo-particlefunctions,wherethe
Incapability of the thermodynamic mean-field the- one-particlepropagatorswillbetreatedasexternalfunc-
ory to determine uniquely the nonlocal part of the two- tions. It means, that we first disregard the consistence
particle vertex results from the degeneracy of the local between the one- and two-particle functions. This con-
theory with elastic scatterings only (noninteracting sys- sistence will be restored later via Ward identities once a
tems). Multiplesingle-sitescatterings,theonlyonesrele- suitable approximationforthe vertexfunctions hasbeen
vantinthemean-fieldapproach,areunabletodistinguish fixed.
betweenelectronsandholes. Onlyifweincludeexplicitly We can classify nonlocal contributions to the two-
scatterings on distinct lattice sites we are able to distin- particle vertex by the type of the correlatedtwo-particle
guish between electrons and holes. Hence, the standard propagation. We either simultaneously propagate an
thermodynamic mean-field theory of quantum itinerant electron and a hole or two electrons (holes). Diagram-
systems uniquely defines only the local two-particle ver- maticallyitmeansthatweconnectspatially distincttwo-
tex, while it remains ambiguous in the determination of particle scattering events with antiparallel or parallel
the full nonlocal two-particle vertex. pairsofone-particlepropagators. Multiple scatteringsof
pairs of the same type define a channel of a two-particle
irreducibility. We call a diagramtwo-particle irreducible
III. MEAN-FIELD THEORY FOR VERTEX if it cannot be split into separateparts by cutting simul-
FUNCTIONS: TWO-PARTICLE taneously either an electron-hole or an electron-electron
SELF-CONSISTENCE pair of propagators. The two definitions of the two-
particle irreducibility lead to topologically nonequiva-
A. Nonlocal contributions to the vertex function lentirreduciblefunctions andto differentBethe-Salpeter
equations for the full vertex. In each Bethe-Salpeter
equation the two-particle functions are interconnected
Aratherinaccuratewaytothemomentum-dependent
via one-particle propagators in a different manner. We
two-particle vertex is not the only imperfection of the
can generically represent the channel-dependent Bethe-
thermodynamic mean-field theory. This theory com-
Salpeter equations as
pletely fails to account for backscattering effects, ver-
tex corrections to the electrical conductivity, and the
Anderson metal-insulator transition. All these effects Γkk′(z+,z−;q)=Λ¯αkk′(z+,z−;q)
are induced by strong nonlocal quantum coherence and + Λ¯αG¯G¯ Γ (z ,z ;q) . (14)
⊙ kk′ + −
spatial correlations reflected in the momentum behav-
ior of the two-particle (vertex) functions. To capture We used the symbol (cid:2) for the ch(cid:3)annel-dependent mul-
⊙
these phenomena we have to go beyond a perturbative tiplication of the two-particle functions represented by
description and to employ a self-consistent scheme for specific momentum convolutions. Here Λ¯α is the irre-
the (irreducible) vertex functions. The best local ap- ducible vertex in the α-channel.
proximation for the irreducible vertex, CPA, is non-self- We will specify the momentum convolutions in the
consistentatthe two-particlelevel. We hence haveto go generic Bethe-Salpeter equation (14) for electron-hole
beyond the CPA and include nonlocal (long-range) con- and electron-electron multiple scatterings only. There is
tributions to the vertex function in a non-perturbative also a third two-particle channel, the so-called vertical
manner. Thereby a question arises whether we are able channel with one-particle self-correlating scatterings.17
to reach a reasonably simple approximation with a two- Thesescatteringsare,however,unimportantforthephe-
particleself-consistencethatcouldbecalledamean-field nomenonof Andersonlocalization,since the correspond-
theory. Itisclearthatsuchatheorymustbemomentum ingtwo-particlepropagatordoesnotcontainthediffusion
dependent,butthemomentumdependenceshouldbere- pole.
duced to a necessary minimum. We will demonstrate in Using the notation from Eq. (7) for the momentum
the next subsections that the desired momentum depen- dependence of the two-particle functions we can write
dence can be very effectively reduced by the asymptotic explicitly the Bethe-Salpeter equation in the electron-
limit to high spatial dimensions. hole channel with barred functions as
The best way to construct a mean-field-like approx-
imation for momentum-dependent functions is to build Γkk′(z+,z−;q)=Λ¯ekhk′(z+,z−;q)
up the theory within a self-consistent expansion around
1
the CPA.If we denote the localCPAone-particlepropa- + N Λ¯ekhk′′(z+,z−;q)G¯+(k′′)G¯−(k′′+q)
gator Gloc(z)=N−1 Gloc(k,z), the small parameter k′′
k X
controllingtheexpansionaroundtheCPAisaperturbed Γk′′k′(z+,z−;q) . (15a)
propagatorG¯(k,z)=PG(k,z) Gloc(z), where G(k,z) is ×
−
the full one-electron propagator. The CPA propagator The Bethe-Salpeter equation with the electron-electron
6
multiple scatterings then analogously reads The standard nonlocal CPA vertex can be recovered
from the above expression if we neglect the contribution
Γkk′(z+,z−;q)=Λ¯ekek′(z+,z−;q) from the electron-electron multiple scatterings, the sec-
1 ond term in the brackets on the right-hand side (r.h.s.)
+ N Λ¯ekek′′(z+,z−;q+k′−k′′)G¯+(k′′) ofEq.(18). Thereis,however,noreasonforthisneglect,
Xk′′ since both terms within the brackets on the r.h.s. of
G¯−(Q k′′)Γk′′k′(z+,z−;q+k k′′) , (15b) Eq.(18)producethesameasymptoticbehaviorinpowers
× − −
of the inverse dimension, O(d−1). Their only difference
wherewedenotedQ=q+k+k′ thetransfermomentum
is in the momentum dependence.
betweenthetwoelectronsofthescatteredcorrelatedpair.
In these equations we abbreviated G¯(k,z ) G¯ (k). The limit to infinite spatial dimensions reduces the
± ±
→ momentum dependence of two-particle functions but
Equations (15) constitute the fundamental building
does not lead to a mean-field-like approximationfor ver-
blocksforthe constructionofsystematicapproximations
tices. There is no self-consistence in the two-particle pa-
for the two-particle vertex. They are analogues of the
rametersandthelocalirreducibleverticesaredetermined
Dysonequationandenableustodeterminethefullvertex
from the CPA. We hence cannot expect that this high-
from irreducible vertices. We hence can apply the per-
dimensional two-particle non-self-consistent asymptotics
turbation (diagrammatic) expansion to the two-particle
would lead to major deviations from the CPA. The only
irreducible vertices, i. e., two-particle self-energies. As a
significant change in the vertex function, Eq. (18), with
first step toward a mean-field-like theory for these ver-
respect to the CPA vertex from Eq. (12) are the vertex
tices we have to maximally simplify the momentum de-
corrections to the electrical conductivity in the form of
pendenceofthevertexfunctions,butstillstayingbeyond
weaklocalization properlydescribedbymultipleelectron-
the local CPA. This will be achieved by the asymptotic
electron scatterings (maximally crossed diagrams).
limit to high (finite) spatial dimensions.
C. Parquet equations and electron-hole symmetry
B. Asymptotic limit to high lattice dimensions
To go significantly beyond the CPA predictions for
Bethe-Salpeter equations (15) use only off-diagonal
transport properties and response functions we have to
one-particle propagators and hence are suitable for per-
introduceaself-consistencethatwouldextendalsotothe
forming the limit to high spatial dimensions. We use
two-particleirreduciblevertices. Thatis,thetwo-particle
the hypercubic lattice that has a straightforward high-
dimensional limit. The one-electron propagator G¯ has irreducible vertices Λ¯eh and Λ¯ee should be determined
fromnonlinearequations. This effectcanbe achievedby
the following asymptotics
introducing the so-calledparquet equations. The concept
. t d dǫρ(ǫ) of parquet equations is based on the observation that
G¯(k,z)= √d cos(kν) (z Σ(z) ǫ)2 , (16) two-particle reducible diagrams in one scattering chan-
ν=1 Z − − nel are irreducible in the other distinguishable scattering
X
where Σ(z) is the CPA (d= ) self-energy.18 channels. Parquet equations were introduced in many-
The irreducible two-part∞icle vertices must collapse bodytheories19,20,21 butrecentlytheywereadjustedalso
to local quantities in the limit to infinite dimensions. to disordered systems.17 The reducible diagrams from
Since the Bethe-Salpeter equations (15) use only the off- onechannelcanbecomeirreducibleintheotherchannels
diagonalpropagatorsvanishing in the limit d= , both only if different channels are indeed distinguishable or
vertices Λ¯eh and Λ¯ee must coincide with the f∞ull local nonequivalent. The idea of parquet equations cannot be
two-particle CPA vertex in d= . We hence have applied to local propagators of noninteracting particles
∞ with indistinguishable electrons and holes, hence within
Λ¯ekhk′(z+,z−;q)=Λ¯ekek′(z+,z−;q)=γ(z+,z−) . (17) the CPA. It, however, works very efficiently for nonlo-
calvertexfunctions inthe Andersonmodelofdisordered
We further denote χ¯(z ,z ;q) = χ(z ,z ;q) G G
+ − + − + −
with χ(z ,z ;q) = χ+(z ,z ;q), G = Gloc(−z ), and electrons.
+ − + − + +
G =Gloc(z ). Ifwetakeintoaccountonlytheelectron- If we again take into account only the electron-hole
− −
and the electron-electron scattering channels, we can
hole and electron-electron multiple scatterings we can
write the basic parquet equation for the full two-particle
representthe leading asymptoticsofthe full two-particle
vertex in high dimensions
vertex in high dimensions as follows
Γkk′(z+,z−;q)=. γ(z+,z−)+λ(z+,z−) Γkk′(z+,z−;q)=Λ¯ekhk′(z+,z−;q)
γ(z+,z−)χ¯(z+,z−;q) +Λ¯ekek′(z+,z−;q)−γ(z+,z−) . (19)
× 1 λ(z ,z )χ(z ,z ;q)
(cid:20) − + − + − Equation (19) tells us that the full vertex is decomposed
γ(z ,z )χ¯(z ,z ;Q)
+ + − + − . (18) intoirreducibleandreduciblediagramsintheeitherscat-
1−λ(z+,z−)χ(z+,z−;Q)(cid:21) tering channel and that the reducible diagrams consist
7
of only the irreducible diagrams from the other chan- The time-reversal (electron-hole) symmetry reduces
nel from which the completely irreducible vertex, i. e., the number of parquet equations to just one nonlinear
the vertex irreducible in both channels, was subtracted. integral equation for a single vertex that we define as
The limit to highlattice dimensions then determines the Λ¯kk′(z+,z−;q) ≡ Λ¯ekek′(z+,z−;q) = Λ¯ekhk′(z+,z−;−Q).
completely irreducible vertex to be the full local CPA The resulting equation for this vertex reads
(d= ) vertex γ.
Pa∞rquetequation(19) can now be used in the Bethe- Λ¯kk′(q)=γ
Salpeter equations (15) to exclude the full vertex Γ from 1
them. Therebywereachaclosedsetofnonlinearintegral + Λ¯kk′′( q k k′′)G¯+(k′′)G¯−(q+k′′)
N − − −
equations for the irreducible vertices Λ¯eh and Λ¯ee. This k′′
X
set of equations is generally not solvable without further Λ¯k′′k′( q k′ k′′)+Λ¯k′′k′(q) γ . (22)
× − − − −
approximations. To approximate the resulting parquet
equations in a systematic and controlled way we again Equati(cid:2)on (22) can now be simplified in h(cid:3)igh spa-
utilize the mean-field idea — limit to high dimensions. tial dimensions by using the Gaussian summation rules,
Fortwo-particlefunctionsandparquetequationswehave Eqs. (20). It is clear from these rules that the fermionic
to use this limit only in the asymptotic sense so that momenta from different two-particle functions must be
nonlocal fluctuations do not go lost completely. summed independently in the leading asymptotic order.
One can make an important observation in high spa- Anycorrelatedmomentumsummationinvolvingtwodif-
tial dimensions. The off-diagonal one-particle propa- ferent two-particle functions costs a factor 1/d. Then
gators G¯ behave in the leading asymptotic order as only the conserved bosonic transfer momenta survive as
Gaussian random variables with respect to momentum in the case ofthe high-dimensionalvertexfromEq.(18).
summations.10 Using representation (16) we can easily We hence have to sum both sides of Eq. (22) over in-
prove the following relations coming and outgoing fermionic momenta k,k′ so as to
extract the high-dimensionallimit of the irreducible ver-
1 χ¯(q′+q)G¯ (q′+k)=. Z G¯ (q k) , (20a) texΛ¯. Toreachanequationfortherelevantvariableswe
± ±
N 4d − introduce
q′
X
1 . Z 1
N χ¯(q+q1)χ¯(q+q2)= 4dχ¯(q1−q2) , (20b) Λ¯(q)= N2 Λ¯kk′(q) . (23a)
q kk′
X X
where we used abbreviations Z = t2 G2 G2 with Further on we have in the leading order
G2 =N−1 G (k)2. Theequalitieshin+Eiqh.(−20i)hold
h ±i k ± 1 1
ofunnlyctiwointshiGn¯ tPhaenldeaχ¯difnogrmasyamcplotsoetdicsoetrdoefrGda→uss∞ian. rTahne- N2 Λ¯kk′(q+k+k′)= N Λ¯(q)=Λ¯0 . (23b)
± kk′ q
X X
dom variables with respect to momentum convolutions.
WehencecanuseEqs.(20)tosimplifytheparquetequa- Since the fermionic momenta from different two-particle
tions for the irreducible vertices Λ¯eh and Λ¯ee. functions are summed independently in high spatial di-
Before we attempt to resolve the parquet equations mensions, the parquet equation (22) reduces with the
in high dimensions, we utilize the time-reversal invari- above definitions to
anceofthe system. Itisanimportantfeatureofelectron Λ¯ χ¯(q)
systemswithoutspin-andorbital-dependentscatterings. Λ¯(q)=γ+Λ¯ 0 . (24a)
01 Λ¯ χ¯(q)
According to this invariance the physical (measurable) 0
−
results should not depend on the orientation of propa-
We see that the high-dimensional irreducible vertex is
gators. We hence can write for one- and two-particle
completely determined from a single local (mean-field)
propagators parameter Λ¯ and the two-particle bubble χ¯(q). Sum-
0
G¯(k,z)=G¯( k,z) , (21a) ming bothsidesofEq.(24a)overmomentaweobtainan
− equation for the local two-particle irreducible vertex
Γkk′(z+,z−;q)=Γkk′(z+,z−; Q)
−
=Γ−k′−k(z+,z−;Q) . (21b) Λ¯ =γ+Λ¯2 1 χ¯(q) . (24b)
0 0N 1 Λ¯ χ¯(q)
In case of the two-particle vertex, the time reversal q − 0
X
was applied only to one fermion propagator. The
Knowingthelocalpartofthetwo-particleirreduciblever-
time reversal leaves the full two-particle vertex invari-
tex Λ¯ we can reconstruct the full two-particle vertex in
ant but it transforms the Bethe-Salpeter equation (15a) 0
high spatial dimensions. We have
toEq.(15b)andviceversa. Itmeansthattheirreducible
vertices transform as follows
Γkk′(q)=γ
Λ¯ekek′(z+,z−;q)=Λ¯ekhk′(z+,z−;−Q) Λ¯0χ¯(q) Λ¯0χ¯(k+k′+q)
+Λ + (25)
=Λ¯e−hk′−k(z+,z−;Q) . (21c) 0(cid:20)1−Λ0χ(q) 1−Λ0χ(k+k′+q)(cid:21)
8
where, in analogy to the non-self-consistent high dimen- the irreducible vertexdoes notleadto bifurcationpoints
sional vertex, Eq. (18), Λ = Λ¯ /(1 + Λ¯ G G ) and and multiple solutions, i. e., to a phase transition, un-
0 0 0 + −
χ¯(q)=χ(q) G G . less we find them in the self-energy itself. It is normally
+ −
−
Equations(23)–(25)formanapproximationfortwo- very difficult to determine bifurcation points in the self-
particle functions with a single, local parameter deter- energy that is a bounded function. It is more convenient
mined self-consistently. Such an approximation is a self- to search for possible bifurcation points in two-particle
consistent extension of the high-dimensional limit of the functions that may display divergences.
two-particle vertex, Eq. (18). We hence can call it a It is clear that we need self-consistent equations for
mean-field approximation for two-particle functions of functions thatcouldhavemultiplesolutions. Suchaself-
noninteracting disordered electron systems. The self- consistent approximation for the irreducible vertex was
consistentlydeterminedmean-fieldparameterΛ¯ exactly achieved in the preceding subsection. A self-consistent
0
reproduces the leading 1/d correction to the CPA irre- equation for the irreducible vertex cannot be derived
ducible vertex λ. The fundamental self-consistent equa- from a self-energy directly. To achieve consistence be-
tionofthisapproximation,Eq.(24b),hasatypicalmean- tweenthe one- andtwo-particlefunctions inthis casewe
fieldcharacter. Thatis, itcanbe usedinanydimension. have to determine the self-energy as a functional of the
The lattice structure enters the equation only through irreducible vertex. This must be done in concord with
the momentum summation running over the first Bril- the Ward identity (26).
louin zone. Notice, however, that we cannot reduce The self-energy as a functional of the irreducible ver-
the momentum summation in two-particle functions to tex is overdetermined from identity (26). We can nev-
an integral over the density of states. In high but fi- ertheless use the Vollhardt-W¨olfle identity to determine
nite spatialdimensions we haveto sum overmomenta in the self-energy from the vertex function as suggested in
Eq. (24b) by using the Gaussian rules only asymptoti- Ref. 22 and used in the parquet approach from Ref. 17.
cally for d .10 We use only a specific element of the Vollhardt-W¨olfle
→∞
identity to determine the imaginary part of the self-
energy. Inthemean-fieldcase,wheretheirreduciblever-
D. Self-energy and diffusion pole in high tex is local, we can write a generalized CPA equation
dimensions
The mean-field theory constructed in the preceding ℑΣR(E)=ΛR0A(E,E)ℑGR(E) . (27a)
subsectionisaself-consistentapproximationforthetwo-
Consistence, or negative definiteness of the self-energy,
particle vertex where one-particle propagators are as-
demands that the local element of the irreducible vertex
sumed to be external functions. Such a situation cannot
bethefinalstageofthetheory,sinceduetoconservation ΛR0A(E,E) = N−2 kk′Λkk′(E +i0+,E −i0+;0), de-
terminedfromthe mean-fieldequation(24b),is positive.
laws and thermodynamic consistence the one- and two-
P
Wecannotfindtherealpartoftheself-energydirectly
particle functions are correlated. Actually, the electron-
from the irreducible vertex. Instead, we use causality
hole irreducible vertex is connected with the self-energy
of the self-energy and the Kramers-Kronig relation ex-
fornoninteractingdisorderedelectronsviatheVollhardt-
W¨olfle Ward identity11 pressingtherealpartofananalyticfunctionasaHilbert
transform of its imaginary part. We have
1
ΣR(k,E+ω)−ΣA(k,E)= N ΛRkkA′(E+ω,E) Σ(E)=Σ +P ∞ dE′ ℑΣR(E′) . (27b)
GR(k′,E+ω)Xk′ GA(k′,E) . (26) ℜ ∞ Z−∞ π E′−E
× − Equations (27) now determine the self-energy from the
HerewedenotedΣR(cid:2),ΣA the retardedandadvan(cid:3)cedself- mean-field (local) irreducible vertex Λ . Notice, how-
0
energyandΛRA(E+ω,E) Λeh (E+ω+i0+,E i0+;0). ever, that to determine the self-energy at one frequency
kk′ ≡ kk′ −
Note that the irreducible vertex Λ in Eq. (26) does not we have to know the irreducible vertex in the whole fre-
have bar, i. e., it is defined via the Bethe-Salpeter equa- quency range. Equations (27) complete the mean-field
tion with the full one-electronpropagatorsGR,GA. The theory for vertex functions, Eqs. (23) and Eqs.(24), and
Ward identity says that if we modify the electron-hole make it a consistent approximationwith proper analytic
irreduciblevertexwehavetochangeadequatelythe one- properties of one- and two-particle Green functions.
electron self-energy and vice versa. We hence cannot The Vollhardt-W¨olfle identity, its specific form from
approximate independently the irreducible vertex with- Eq. (27a), not only serves as a means for a consistent
outchangingappropriatelytheself-energy. Ifwehavean determination of a causal self-energy from a given irre-
analytic expression for the self-energy as a functional of ducible vertex, but it is also indispensable for the exis-
the one-electronresolvent,we can use a differential form tence of the diffusion pole in the electron-hole correla-
of the Ward identity (11) to determine the irreducible tion function. We now show in what form the diffusion
vertex. Then the vertex function is determined directly polesurvivesinthemean-fieldtheoryforvertexfunctions
fromtheone-electronpropagator. Suchaconstructionof with the self-energy determined by Eqs. (27).
9
The electron-holecorrelationfunction is definedfrom density of states calculated from the one-electron Green
the averagedtwo-particle Green function as function, n = GR(E)/π .
E
−ℑ
The dependence of the weight of the diffusion pole
1
ΦRA(q,ω)= G(2)(E+ω+i0+,E i0+;q). (28) on the disorder strength found in the mean-field theory
E N2 kk′ kk′ − for averaged two-particle functions could be an artifact
X
of this specific approximation. We could still hope that
Thetwo-particleGreenfunctionisevaluatedwiththefull the full exact solution recovers the invariant weight of
two-particle vertex via Eq. (7). the diffusion pole expected from the unrestricted con-
FromEq.(27a)andfromχRA(0)= GR/ ΣRweob- servation laws for averaged Green functions. Based
ℑ ℑ
tainthatboththe denominatorsin the high-dimensional onlyonapproximation-freeargumentswefoundthatthe
limitofthe two-particlevertexΓRA vanishatzerotrans- Vollhardt-W¨olfle identity (26) for finite frequencies can-
fer momenta. Hence, the mean-field approximation notbefulfilledinanyfinitedimensioniftheelectron-hole
for vertex functions contains the diffusion pole in the irreducible vertexΛeh containsthe Cooperpole. Enforc-
electron-hole channel (first term in the brackets on the ing the full form of the Vollhardt-W¨olfle identity with
r.h.s. of Eq. (25)) and the Cooper pole in the electron- the Cooper pole in the electron-hole irreducible vertex
electron channel (second term). Only the diffusion pole inevitablyleadstoaself-energybeinganonanalyticfunc-
survives as a singularity in the electron-hole correlation tion of its energy argumentfor almost all Fermi energies
function, Eq. (28). within the energy bands.23,24 It hence seems that the
Toderivethelow-energybehavioroftheelectron-hole high-dimensionalbehaviorofthe two-particlevertexand
correlationfunction we denote thedisorder-dependentweightofdiffusionpole,disclosed
∂ΛRA(E+ω,E) by the asymptotic mean-field solution, are generic fea-
A =1+2i GR(E) 0 (29a) tures of the Anderson model of disordered electrons. At
E
ℑ ∂ω
(cid:12)ω=0 leastfortheoriesthatcanbe analyticallycontinuedfrom
(cid:12)
and (cid:12) the limit to high spatial dimensions.
(cid:12)
∂χRA(q)
D0(ω)=2 ΣR(E)ΛRA(E+ω,E) . (29b)
E ℑ 0 ∂(q2) IV. DISCUSSION AND CONCLUSIONS
(cid:12)q=0
(cid:12)
(cid:12)
With these two definitions we find the high-(cid:12)dimensional
We presented in this paper two ways how to reach
asymptotics of the low-energy limit of the electron-hole
mean-field-like approximations for noninteracting disor-
correlationfunction at zero temperature to be
deredelectronsystems. Thefirstone,beingthestandard
2πn thermodynamic mean-field theory known from many-
ΦRA(q,ω) E (30)
E ≈ iA ω+D0(ω)q2 body systems, uses the limit to infinite spatial dimen-
− E E sions applied to the generating, configurationally aver-
where n is the density of states at the Fermi energy E. agedthermodynamicpotential. Thelimittoinfinitespa-
E
The low-energy asymptotics of the electron-hole cor- tial dimensions enables one to separate the diagonaland
relation function serves as an important tool for testing off-diagonal elements of the one-particle propagator and
consistence of approximations. We find from gauge in- its self-energy and to find an explicit representation for
variance and the (unrestricted) Ward identity (26) that the generating functional. The local one-particle func-
the electron-hole correlation function should display the tions from the generating functional serve as generalized
diffusion pole in form of Eq. (30) with A = 1 for ar- variational parameters, the physical values of which are
E
bitrary disorder strength. It then means that the low- determined from stationarity equations for the generat-
energy behavior of the electron-hole correlation func- ing functional. The higher-order Green and vertex func-
tion is controlled by a single parameter, the bare dy- tionsaredeterminedfromresponsesofthesystemtolocal
namical diffusion constant D0(ω).11,12 However, we al- external perturbations. In this way the construction of
E
ready found in Ref. 10 that the constant A increases a mean-field approximation is consistent and unambigu-
E
with the disorder strength and becomes infinite at the ousinthedeterminationofone-particlefunctionsasthey
Anderson localization transition. The disorder depen- aretheonlyonesenteringthegeneratingfunctional. The
dent weight of the diffusion pole n /A is the cen- higher-orderGreenfunctionsaredefineduniquelyonlyin
E E
tral unexpected feature of the mean-field theory for ver- their local parts.
tex functions. This mean-field approximation obeys the Thenonlocalpartsoftwo-particleGreenfunctionsare
Vollhardt-W¨olfle identity only in the limit to zero fre- nolongerdeterminedfromthelocalthermodynamicthe-
quency, Eq. (27a), and not for finite energy differences. ory uniquely. We can either use the standard construc-
We found a consistent explanation for the decrease of tion of Baym and Kadanoff or we can directly apply the
the weight of the diffusion pole with increasing disorder asymptotic limit to infinite spatial dimensions to two-
strength.13 The weight of the diffusion pole, n /A , ex- particle functions. The outcome of these two construc-
E E
presses a portion of extended (diffusive) states from all tions is not identical. In the former way we miss some
available states at the Fermi energy determined by the of the leading-order (maximally crossed) diagrams and
10
losetheelectron-holesymmetryatthetwo-particlelevel. disorder limits and to cover the split-band limit as well
Thesedeficienciesseverelydiscreditreliabilityofthether- as vanishing of diffusion. The other existing approaches
modynamic mean-field theory in the calculation of spa- have concentrated on only one of the two phenomena.
tialcoherenceandtransportpropertiesofdisorderedsys- They either miss the two-particle self-consistence or do
tems. Toovercomethesedrawbacksweproposedanother not consistently match the one- and two-particle func-
route toward a mean-field-like approximation for vertex tions.
functions based ona direct analysis of two-particlefunc- The consistence between the one-electron self-energy
tions in high spatial dimensions. and the electron-hole irreducible vertex is essential for
The incapability of the thermodynamic mean-field credibility of approximate treatments of the Anderson
theoryto correctlydescribe nonlocalcorrelationsintwo- metal-insulator transition. Only with this relation cor-
particle functions is caused by the degeneracy of local rectly taken into account we obtain the proper form of
theories with elastic scatterings. Static local approxi- thediffusionpoleandelectrondiffusiononlongdistances.
mations are unable to distinguish between electrons and Inthis respectthe mean-fieldtheoryfor vertexfunctions
holes. Only quantum dynamics or multiple scatterings leads, surprisingly against the common expectations, to
onspatiallydistinctimpurities candiscernthe motionof a nonconserving weight of the diffusion pole and its de-
an electronfrom the motionof a hole. The distinguisha- pendence on the disorder strength. The thermodynamic
bilityofelectronsandholesisofprincipalimportancefor mean-fieldtheoryandallotherapproachestotheAnder-
encounteringbackscatteringeffectsandforatwo-particle sonlocalizationtransitionassumeorusetheunrestricted
self-consistence used in the construction of a mean-field form of the Ward identity (26) being a consequence of
theory for vertex functions. conservation of the norm of the wave function for all
A mean-field approximation for vertex functions in configurations of the random potential. We argued al-
disordered electron systems was constructed from the ready earlier that the Hilbert space of Bloch waves is
asymptoticlimittohighspatialdimensions,where,unlike incomplete in the sense that it does not encompass the
the thermodynamic mean field, the lattice dimension is eigenstates of all configurationsof the randompotential.
high but finite. Alike the strict limit d= , the asymp- Ata givenenergyweobserveeveninthe metallic regime
∞
toticbehaviorinhighdimensionsleadstosignificantsim- macroscopically relevant numbers of configurations with
plifications in momentum convolutionsthat enable us to localized states.13
reduce the approximation to a mean-field-like one with VanishingofthediffusionpoleattheAndersonmetal-
alocalgeneratordeterminedfromaself-consistentequa- insulator transition and in the localized phase modifies
tion. We applied the asymptotic limit directly within a the existing picture of the criticalbehavior for vanishing
diagrammaticexpansionaroundtheCPAfortwo-particle ofdiffusion. Inthestandardapproaches,suchasthenon-
functions. The basic ingredients for the construction linearsigmamodelortheVollhardt-W¨olfleself-consistent
were parquet equations for the two-particle irreducible approximation, the dynamical diffusion constant is the
vertices from the electron-hole and the electron-electron only parameter controlling the low-energy behavior of
scattering channels. Using the electron-hole symmetry the electron-hole correlation function. In our mean-field
atthetwo-particlelevelandtheasymptoticlimittohigh theory we have apart from the diffusion constant also
dimensions we succeeded in producing a self-consistent the weight of the diffusion pole that significantly influ-
approximation for the local part of the electron-hole ir- ences the description of the long-range correlations and
reducible vertex. It is a self-consistent 1/d extension of diffusion. In the critical region, however, only the con-
the CPA irreducible vertex. The mean-field theory for stant A from Eq.(29a) is relevant and allcritical
E
→∞
vertexfunctionsthendeterminesinauniquewaythefull scales can be derived from it. For instance the renor-
two-particlevertexthatcorrectlyreproducesthelimitto malizeddiffusionconstantD =D0/A vanishesatthe
E E E
infinite dimensions with the electron-hole symmetry at localization transitionwith the diverging parameterA ,
E
both one- and two-particle levels. etc. It means that the Anderson localization transition
The mean-field theory for vertex functions is an ap- is compatible with a one-parametric scaling theory.
proximationfor two-particlefunctions. The one-electron Althoughtheone-parameterscalingholdsfortheAn-
functions,usedasaninputforthetwo-particleequations, derson localization transition, two relevant parameters
are then calculated from the vertex function via a spe- in the critical region, n /A and D0/A , nevertheless
E E E E
cific form of the Vollhardt-W¨olfle Ward identity and the lead to a modification of the critical behavior deduced
Kramers-Kronigrelation. Withthisextensionofthethe- from the field-theoretic approaches. The two parame-
orytoone-particlefunctionsweaccomplishedanapprox- ters stand for two quantities influencing the electrical
imate descriptionof disorderedsystems fulfilling all con- conductivity. The former expresses the number of ex-
sistency conditions on one- and two-particle functions. tended states at the Fermi energy E and the latter the
The most important achievement of this mean-field averaged velocity of the diffusive particles. Both quan-
theoryis its abilitytodescribethe disorder-inducedvan- tities go simultaneously to zero at the Anderson metal-
ishing of diffusion, that is, the Anderson localization insulator transition. It is straightforward to verify that
transition. This theory succeeded for the first time to the mean-field theory for vertex functions predicts that
bridge qualitatively correctly the weak and the strong A λ λ−1/2, where λ is the bare disorderstrength
E c
∼| − |