Integrability of the diffusion pole in the diagrammatic description of noninteracting electrons in a random potential

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by  V. Janis

Integrability of the diffusion pole in the diagrammatic description of noninteracting
electrons in a random potential
V. Janiˇs
Institute of Physics, Academy of Sciences of the Czech Republic,
Na Slovance 2, CZ-18221 Praha, Czech Republic ∗
(Dated: January 20, 2009)
We discuss restrictions on the existence of the diffusion pole in the translationally invariant di-
9 agrammatic treatment of disordered electron systems. We use the Bethe-Salpeter equations for
0 thetwo-particlevertexintheelectron-holeandtheelectron-electron scatteringchannelsandderive
0 for systems with time reversal symmetry a nonlinear integral equation the two-particle irreducible
2
vertices from both channels must obey. We use this equation to test the existence of the diffusion

n poleinthetwo-particlevertex. Wefindthatasingularity ofthediffusionpolecanexistonlyifit is
a integrable, that is only in themetallic phase in dimensions d>2.
J
0 PACSnumbers: 72.10.Bg,72.15.Eb,72.15.Qm
2

] Introduction. Scattering of free charge carriers on rived and to which restrictions they are subject. One of
n impurities and lattice imperfections can lead at low- themostimportantfeaturesusedinthedescriptionofthe
n
temperaturestoametal-semiconductortransition. There critical behavior of the Anderson localization transition
-
s are two qualitatively different scenarioshow a metal can is a singular low-energy behavior of the density-density
i turninsulatingduetoexcessivescatteringsonimpurities. correlation function of disordered systems. This singu-
d
. In the first case the metal-insulator transition material- larity has form of a resolventof a diffusion equation and
t
a izes in substitutional alloys when charge carriers are ex- iscalledthediffusionpole. Theexistenceofthediffusion
m pelledfromtheFermisurfaceandanenergygapdevelops. pole andaconnectionofthe diffusionconstantwithcon-
- Thistransition,calledsplitband,isqualitativelywellun- ductivity are consequences of conservation laws in ran-
d derstoodandquantitativelywellmodeledbyamean-field dom systems.10 Conservation laws should be a firm part
n solution.1,2 The second type of a metal-insulator transi- ofanyreliabletheory. We,however,showedrecentlythat
o
tion is much more complicated and up to now not com- an asymptotic solution of the Anderson model of nonin-
c
[ pletely understood. Electrons in a metal with random teracting electrons in high spatial dimensions does not
impuritiescanlosetheirabilitytodiffuseonmacroscopic fully obey conservation of probability.11 We suggested a
1 scales. Such scenario was first suggested by Anderson3 qualitative explanation for such an unexpected behavior
v
and is now called Anderson localization transition. but moreimportantly, weamassedargumentsthatunre-
7
5 Oneofprincipleobstaclesoffullunderstandingofthe stricted compliance with the conservation law is in ran-
0 Anderson localization transition is inability to describe dom systems in conflict with analyticity of the spectral
3 vanishing ofdiffusion ofelectrons analyticallyevenin its function.12,13Sincediscussionabouttheformofthediffu-
1. simplest model version and reconcile results from ana- sionpoleisstillongoing,14,15 wetracedowninthispaper
0 lytic and numerical approaches. Analytic, mostly dia- the origin and set exact restrictions on the form of the
9 grammaticandfield-theoretic approachesinthe thermo- diffusionpolederivedwithinthetranslationallyinvariant
0 dynamic limit indicate that the critical behavior at the description of disordered systems in the thermodynamic
v: Anderson localization transition fits the one-parameter limit. We first thoroughly analyze the assumptions used
i scaling scheme with a single correlation length control- to derive the diffusion pole and then prove an assertion
X
ling the long-range fluctuations.4,5 On the other hand, about the acceptable formof this singularitywithout re-
r an increasing number of numerical studies of the Ander- ferring or resorting to any specific approximation. We
a
son localization transition in finite volumes suggest that find thatinsystems invariantwithrespecttotime inver-
insteadofhomogeneous,translationallyinvariantparam- sion the diffusion pole must be integrable in momentum
eters one has to take into consideration distributions of space.
conductances or local particle densities.6,7 The two dif- Definitions and assumptions. We model the sys-
ferent methodological approaches, analytic and numeri- temofnon-interactingelectronsbyalatticegasdescribed
cal, disagree not only on the number of relevant control- by an Anderson Hamiltonian3
ling parameters needed to understand Anderson local-
H = |kiǫ(k)hk|+ |iiV hi| (1)
ization but also on the critical behavior and the values i
of the critical exponents.8,9 Neither of these approaches Xk Xi
is, however, absolutely conclusive in delivering ultimate used to captbure the impact of randomness on the elec-
answers. tronic structure of metallic alloys as well as to under-
Incaseofdisagreementofresultsfromtworatherwell stand vanishing of diffusion in the limit of strong ran-
establishedandotherwisereliablemethodsonehastore- domness. The first, homogeneous, part of this Hamilto-
visittheassumptionsunderwhicheitherresultswerede- nian is kinetic energy and is diagonalized in momentum
2
space (Bloch waves). The second sum runs over lattice particle resolvents G(k,z) and G(2)(z ,z ;q), where
kk′ + −
sitesanddescribesasite-diagonalrandompotential. Val- z =E+ω+iηandz =E−iηarecomplexenergieswith
+ −
uesV atdifferentpositionsareuncorrelatedandfollowa E standingfortheFermienergy,ω forthebosonictrans-
i
probabilitydistributionP(V ). Thistermisdiagonalized fer frequency (energy), and η is a (infinitesimally) small
i
in the direct space by local Wannier states. The two op- damping (convergence) factor. We adopt the electron-
erators do not commute, quantum fluctuations become hole representation for the two-particle Green function
important and the full Anderson Hamiltonian cannot be with k and k′ for incoming and outgoing electron mo-
easily diagonalized. The only way to keep analytic con- menta. The bosonic momentum q measures the differ-
trol of the behavior of equilibrium states of the Ander- ence between the incoming momenta of the electron and
son model is to go directly to the thermodynamic limit. the hole. Energies of the electron and the hole z ,z
+ −
Standardlyitis approachedby applying the ergodicthe- in systems with noninteracting particles are externalpa-
orem, that is, summation over lattice sites equals the rameters.
configurational averaging. This means that we assume
Theaveragedone-electronresolventindisorderedsys-
self-averagingpropertyforallquantitiesofinterest. This
tems canberepresentedasinmany-bodytheoriesviaan
need not be, however, always fulfilled as we know from
irreduciblevertex–theself-energyΣ(k,z). Wecanwrite
studies of Anderson localization. Presently we disregard
a Dyson equation for it
this option from consideration as well as the problem of
the existence of the thermodynamic limit.
1 δ(k−k′)
Ergodicity itself, however, does not simplify the pro- k k′ = . (2)
cess of averaging over randomness. Another assumption z1−H z−ǫ(k)−Σ(k,z)
(cid:28)(cid:28) (cid:12) (cid:12) (cid:29)(cid:29)av
must be adopted to master this problem. We assume (cid:12) (cid:12)
(cid:12) (cid:12)
that the thermodynamic limit can be performed inde- Theself-en(cid:12)ebrgyΣb((cid:12)k,z)standsfortheimpactofthescat-
pendently term by term in the expansion in powers of teringsofthe electrononrandomimpurities. Knowledge
the random potential. It means that we expect that the of the self-energy is then sufficient to determine the en-
configurationallyaveragedperturbationexpansioninthe ergyspectrum,spectraldensityandingeneralallaspects
random potential converges for all quantities of interest. of propagationof single particles in disordered media.
Thermodynamic limit has an important simplifying
The two-particle resolvent G(2) can then be repre-
consequence for macroscopic (averaged) quantities. The
sented via a two-particle vertex Γ defined from an equa-
spectrum of a random Hamiltonian in the thermody-
tion
namic limit is invariant with respect to lattice transla-
tions. It means that operators H and T HT†, where
R R
TR is the operatoroftranslationwith a lattice vectorR, G(k2k)′(z+,z−;q)=
have identical spectrum of eigenvbalues wbithbtrbanslation-
1 1
abllyshiftedeigenvectors. Alatticetranslationbyavector q+k,k ⊗ k′,q+k′
Rn applied to the Anderson Hamiltonian from Eq. (1) ** (cid:12)(cid:12)z+−H z−−H(cid:12)(cid:12) ++av
generatesa new one, |kiǫ(k)hk|+ |i+niV hi+n| (cid:12) (cid:12)
k i i 1(cid:12) (cid:12)1
having the same distribution of random energies. Un- ≡ k (cid:12) k′ b q+k′ b(cid:12) q+k
less we break translatPional symmetry iPn thermodynamic ** (cid:12)z+−H(cid:12) +* (cid:12)z−−H(cid:12) ++
(cid:12) (cid:12) (cid:12) (cid:12) av
states,weareunabletodistinguishtranslationallyshifted (cid:12)(cid:12)=G(k,z(cid:12)(cid:12)+)G(q+k,z−(cid:12)(cid:12))[δ(k−(cid:12)(cid:12)k′)
Hamiltonians. We cannot, however, break translational (cid:12) b(cid:12) (cid:12) b(cid:12)
invarianceof the thermodynamic states arbitrarily,since + Γkk′(z+,z−;q)G(k′,z+)G(q+k′,z−)] (3)
their symmetry should be in concord with the spatial
distributionofthe eigenstatesofthe Hamiltonianforthe where ⊗ denotes the direct product of operators. The
givenconfigurationoftherandompotential. Sincewedo two-particle vertex introduces a disorder-induced corre-
notknowthisspectrum,wemusttreatalllatticetransla- lation into the two-particle propagation. Analogously to
tionsoftheHamiltonianasequivalentandinsteadofone the self-energy it measures the net impact of scatterings
Hamiltonianwe areable to describe only the whole class on impurities on the motion of particles in the presence
ofequivalentHamiltoniansT HT†. In this waywe can- of other particles.
R R
not distinguish directly between extended and localized The two-particle vertex Γ can further be simplified
eigenstates of the random pbotebnbtial, since the localized by introducing an irreducible vertex Λ playing the role
states are represented by a class of vectors differing by ofa two-particleself-energy. The irreducible andthe full
lattice translations. vertex are connected by a Bethe-Salpeter equation. Un-
The natural basis for translationally invariant quan- like the one-particle irreducibility, the two-particle irre-
tities is formed by Bloch waves labeled by quasimo- ducibility is ambiguous.17 There are two types of two-
menta. Wegenericallydenotek,qfermionicandbosonic particle irreducibility in systems with elastic scatterings
(transferred) momenta respectively. The fundamental only, electron-hole and electron-electron. They are char-
building blocks of the translationally invariant descrip- acterized by different Bethe-Salpeter equations. The
tion of disordered electrons are averaged one- and two- Bethe-Salpeter equation in the electron-hole scattering
3
channel then reads average this identity and the averaging procedure need
not conserve all its aspects when projected onto transla-
Γkk′(q)=Λekhk′(q) tionallyinvariantstates.12Whenusingtheaboveidentity
1 intheevaluationofthehomogeneouspartoftheelectron-
+ N Λekhk′′(q)G+(k′′)G−(q+k′′)Γk′′k′(q) . (4a) hole correlationfunction, that is q =0, we obtain
k′′
X
We suppressed the frequency variables in Eq. (4a), since ΦRA(0,ω)=. 2πnF . (7)
theyarenotdynamicalones. Theycanbeeasilydeduced E −iω
from the one-electron propagators G (k) = GR,A(k) ≡
±
No spatial fluctuations (q 6= 0) of the correlation func-
G(k,z ) used there.
±
tion in the low-frequency limit can be deduced from the
We can introduce another nonequivalent representa-
Velicky´-Ward identity. To derive the spatial behavior
tion of the two-particlevertex. If we sum explicitly mul-
of the diffusion pole in Eq. (5) one has to resort to an-
tiplescatteringsoftwoelectrons(holes)wecanconstruct
other relation introduced by Vollhardt and W¨olfle.18 It
an alternative Bethe-Salpeter equation17
utilizes the DysonandBethe-Salpeter equations,Eq.(2)
1 and Eq. (4a), and relates the one- and two-particle irre-
Γkk′(q)=Λekek′(q)+ N Λekek′′(q+k′−k′′) ducible functions Σ and Λeh, respectively. It reads
k′′
X
×G+(k”)G−(q+k+k′−k′′)Γk′′k′(q+k−k′′) . 1
(4b) ΣR(q+k,E+ω)−ΣA(k,E)= N ΛRkkA′(E+ω,E;q)
k′
X
We introduced an irreducible vertex in the electron-
× GR(q+k′,E+ω)−GA(k′,E) . (8)
electron scattering channel Λee. Irreducible vertices Λeh
and Λee do not include isolated pair electron-hole and and was proved(cid:2)diagrammatically (perturbativel(cid:3)y). Us-
electron-electronscatterings, respectively.
ingtheBethe-Salpeterequationonecanshowthatinthe
Diffusion pole and electron-hole symmetry.
homogeneous limit q = 0 this identity reflects the con-
Noninteracting particles scattered on impurities are
tinuity equation and hence is equivalent to the Velicky´-
marked by a diffusion pole. The low-energy limit
Ward identity. Equation (8) together with the Bethe-
of a special matrix element of the two-particle resol-
Salpeter equation are then used to show that the long-
vent,electron-holecorrelationfunction,hasthefollowing
distance fluctuations of the low-energy limit of the cor-
asymptotics for q →0 and ω/q →0
relation function are controlled by a diffusion constant.
1 Introducing a dynamicaldiffusion constantD(ω) we can
ΦREA(q,ω)= N2 GRkkA′(E+ω,E;q) represent the full leading low-energy asymptotics of the
Xkk′ electron-hole correlation function as in Eq. (5).10 Note
=. 2πnF +O(q0,ω0) (5) that identity (7) holds for both pure and random sys-
−iω+D(ω)q2 tems, the actual diffusion pole, however, is only the sin-
gularityfromEq.(5)withthe momentumdependence of
wheren isthedensityofone-particlestatesattheFermi
F
level.10 We used an abbreviation for the energy argu- the low-energy behavior. To prove such a spatially dif-
mentsGRA(E+ω,E;q)≡G(2)(E+ω+i0+,E−i0+;q). fusive behavior the Bethe-Salpeter equation becomes an
kk′ kk′ indispensable tool.
The low-energy electron-hole correlation function be-
Anotherimportantfeatureofnoninteractingelectrons
comes a propagator of a diffusion equation.
onabipartitelatticewithoutexternalmagneticfieldand
Suchalow-energybehaviorisnotevidentandtoprove
spin-orbit coupling is the time reversal symmetry. Time
it one has to use Ward identities connecting one- and
inversion is equivalent to reversing the direction of the
two-particle averaged functions. Ward identities reflect
particle propagation, that is k → −k. The electron
conservation laws. In disordered noninteracting systems
and the hole interchange their roles. The time-reversal
we have probability (mass or charge) conservation. It is
invariance for the one-particle propagator then means
mathematicallyequivalenttocompletenessoftheHilbert
G(k,z) = G(−k,z). Time inversion leads to nontriv-
space of Bloch waves. First Ward identity due to charge
ial symmetries when applied onto one of the fermion
conservation was derived for disordered systems within
propagators in two-particle functions. The electron-
the mean-field approximation by Velicky´16 and later ex-
hole transformation can be represented either by revers-
tended beyond this approximation in Ref. 17. It is a
ing the electron line leading to a transformation k →
consequence of an operator identity
−k′,k′ → −k,q → Q or by reversing the hole propa-
1 1 1 1 1 gator k → k,k′ → k′,q → −Q for the electron-hole
= − (6) function. Here we denoted Q = q+k+k′. We then
z+−H z−−H z−−z+ "z+−H z−−H# obtain two symmetry relations for the full two-particle
where the multiplication is the standard operator (ma- vertex
b b b b
trix)one. Thisidentityholdsforanyone-particleHamil-
tonian. In the thermodynamic limit we must, however, Γkk′(q)=Γ−k′−k(Q)=Γkk′(−Q). (9a)
4
The two-particle irreducible vertices are not invariant to the electron-hole transformation can alternatively be
with respect to time inversion, since the electron-hole decomposed by means of the so-called parquet equation
vertex is transformed onto the electron-electron one and that can be represented in various equivalent ways17
viceversa. Wethenhavethefollowingelectron-holesym-
metry relations Γkk′(q)=Λekek′(q)+Kkeek′(q)=Λekhk′(q)+Kkehk′(q)
Λekek′(q)=Λe−hk′−k(Q)=Λekhk′(−Q). (9b) ==KΛekkeehkk′′((qq))++ΛKekkehkek′′((qq))−+IIkkkk′′((qq)) (13)
This relation says that Bethe-Salpeter equation (4a)
transforms upon time inversion in one particle line onto where Kkehk′(q) and Kkeek′(q) are two-particle reducible
Bethe-Salpeter equation(4b). When the invariance with vertices in the electron-hole and electron-electron chan-
respect to the electron-hole transformation is applied to nels, respectively. We denoted I = Λeh ∩ Λee a two-
the correlationfunction we obtain particle fully irreducible vertex, that is, a vertex irre-
duciblesimultaneouslyforboththeelectron-holeandthe
1
ΦREA(q,ω)= N2 GRkkA′(E+ω,E;−q−k−k′). (10) elecTtrhoenp-ealreqctureotneqpuaiartipornosphagoaldtifoonr(tmheusltyisptleemscsawttheerrinegtsh)e.
kk′
X
electron-hole and the electron-electron multiple scatter-
This representation together with the Ward identity, ings are nonequivalent, that is, the corresponding two-
Eq. (8), tell us that the same low-energy singularity for particle irreducibilities are unambiguous and excluding
ω,q2 →0mustemergewiththesameweightintheaver- definitions of diagrammatic contributions. The con-
agedtwo-particle resolventGRkkA′(E+ω,E;q) also in the cept of the parquet theory based on nonequivalence of
limit ω,(k+k′+q)2 →0. two-particle irreducibility can at best be understood
The uncorrelated propagation of electrons in a ran- in terms of sets of diagrams where addition of func-
dom potential does not contain the diffusion pole, and tions is represented by union of sets of diagrams the
hence it must emerge in the vertex function Γ. Taking functions stand for. Nonequivalence of the electron-
into account the time-reversal invariance we can single holeandtheelectron-electronmultiplescatteringsmeans
outthesingularpartsoftheelectron-holesymmetrictwo- Kee ∩ Keh = ∅. We trivially have in each α-channel
particle vertex and obtain Λα ∩Kα = ∅. Further on, we have Λeh = Λeh ∩Γ =
(Λeh∩Λee)∪(Λeh∩Kee)⊂I∪Kee. On the other hand,
ΓRkkA′(q,ω)=γkRkA′(q,ω)+ −iω+ϕRkDkA′(ω)q2 KHeeenc=eKKeeee∩⊂ΓΛ=eh(K. eCeo∩mΛbeihn)in∪g(Ktheee∩abKoevhe)t=woKereel∩atΛioenhs.
ϕRA we obtain Λeh = I ∪Kee from which we reach the par-
+ kk′ . (11) quet representations via irreducible or reducible vertices
−iω+D(ω)(q+k+k′)2
in Eq. (13), Λeh∪Λee\I =I∪Kee∪Keh =Γ.
The reduced vertex γRA has a marginal and thermody- One must be careful when using the parquet decom-
namically irrelevant singularity for ω → 0 at k = k′ = position for noninteracting electrons with elastic scat-
terings only. In this case multiple scatterings on a sin-
q = 0. It can, nevertheless, display another singular
behavior in fermionic variables k,k′ that is not deriv- gle site are identical for both channels. Hence, the two
Bethe-Salpeterequations(4)areidentical,whentheone-
able from the diffusion pole. Such a singularity must
electron propagators are purely local. We then obtain
not, however, affect the form of the diffusion pole in the
Λeh = Λee = I. It means that irreducible and reducible
electron-hole correlation function for q →0. The second
local diagrams coincide and the concept of two-particle
termon the right-handside ofEq.(11) dominates in the
irreducibility becomes ambiguous. To amend this prob-
leading order of the limit q → 0, ω → 0 while the third
one in the limit q+k+k′ →0, ω →0. We used the dy- lem we introduce a stronger full two-particle irreducibil-
ity includingalsolocalscatteringswheretheelectronand
namical form of the diffusion constant D(ω) so that the
the hole are indistinguishable. We denote this vertex J.
localization phase would fit. Equation (11) is the most
TheirreducibleverticesI,Λeh andΛeefornoninteracting
general form of the two-particle vertex reproducing the
electronsarethentransformedinparquetequations(13)
diffusion pole in the correlation function Φ. The singu-
to
larity for q → 0 is the diffusion pole while the other for
q+k+k′ → 0 is the Cooper pole caused by multiple J0G G
electron-electron scatterings. To conform this represen- Ikk′(q)=Jkk′(q)+ 1−J0+G −G J0 , (14a)
+ −
tation with Eq. (7) we have to satisfy a normalization
condition that in the metallic phase (D(0)>0) reads Λαkk′(q)=Λαkk′(q)+ 1−J0JG0+GG−G J0 (14b)
+ −
1
N2 |G+(k)|2ϕRkkA′|G+(k′)|2 =2πnF . (12) where J0 = N−3 kk′qJkk′(q) and G± =
Xkk′ N−1 kG±(k) are the Pappropriate local (momentum-
α
Parquet equations with time-reversal symme- indepPendent) parts. Vertex Λkk′(q) is irreducible in
try. The fulltwo-particlevertexsymmetricwithrespect channel α but does not contain multiple scatterings on
5
the same site. It is important that the fully irreducible theBethe-Salpeterequationsintheelectron-holeandthe
vertex Jkk′(q) contains only cumulant averaged powers electron-electron channels, Eqs. (4) and the two equa-
of the random potential on the same lattice site so that tions are not identical, that is Λeh (q) 6= Λee (q). Non-
kk′ kk′
double counting is avoided. linearity of the fundamental equation for the irreducible
We now use the symmetries from Eq. (9) to replace vertex poses restrictions on the admissible form of the
the two irreducible vertices by a single function. We de- singularbehaviorinitssolutions. Singularitiesinthefull
fine vertex Γ emerge only via singularities in the irreducible
vertex Λ.
Λkk′(q)≡Λekek′(q)=Λekhk′(−q−k−k′) . (15) Assertion. Two-particle vertex Γ of noninteracting
electrons in a random potential can be decomposed into
We usethis definitioninparquetequation(13)wherewe
irreducible vertices as
representthefullvertexbyBethe-Salpeterequation(4a).
We then obtain a fundamental equation for the irre- Γkk′(q)=Λkk′(q)+Λkk′(−q−k−k′)−Ikk′(q) ,
ducible vertex
(18)
Λkk′(q)=Ikk′(q) ifelectronsandholesaredistinguishable (non-equivalent)
1 quasiparticles and the system is invariant with respect to
+ Λ (−q−k−k”)G (k”)G (q+k”)
N kk” + − time inversion (electron-hole symmetric). We denoted I
Xk” the two-particle fully irreducible vertex. Irreducible ver-
×[Λk”k′(q)+Λk”k′(−q−k”−k′)−Ik”k′(q)] . (16) tex Λ obeys Eq. (16). The diffusion pole in the full two-
particle vertex Γ may materialize only if it appears in
This is a nonlinear integral equation for vertex Λ from
the irreducible vertex Λ. Consequently, the diffusion and
an input I that may have multiple solutions. We choose
Cooper poles from Eq. (11) can exist in Γ only in the
the physical one by matching it to a perturbative solu-
metallic phase in spatial dimensions d>2.
tion reached by an iterative procedure with an auxiliary
Proof. Equation (18) is a direct consequence of
coupling constant λ and a starting condition Λ(0) = λI.
parquet equation (13) where the electron-hole symme-
The iteration procedure for a fixed coupling constant λ
try, Eq. (9), is used. The parquet equation holds if the
is determined by a recursion formula
electron and the hole are distinguishable quasiparticles
via their multiple mutual scatterings. That is, electron-
1
N δk”,k′ −Λk(nk−”1)(−q−k−k”)G+(k”) etilceactlrroensualntsd.electron-holescatteringsdonotleadtoiden-
k” (cid:20)
X
Weneednotfindthemostgeneralformoflow-energy
×G−(q+k”) Λ(kn”)k′(q)−λIk”k′(q) (ω →0)singularities compliantwith Eq.(16) but rather
(cid:21)(cid:16) (cid:17) check whether and when singularities from representa-
1
= Λ(n−1)(−q−k−k”)G (k”) tion (11) can emerge in solutions of Eq. (16).
N k” kk” + Vertex Λkk′(q) contains the diffusion pole of the full
X vertex Γ, Λeh (q) the Cooper pole and the fully irre-
×G (q+k”)Λ(n−1)(−q−k”−k′) . (17) kk′
− k”k′ ducible vertexIkk′(q)isfreeofthesepoles. Thisconclu-
sion follows from an alternative form of Eq. (16)
In this way vertex Λ = Λ(∞) is completely determined
from the input, the fully irreducible vertex λI. A phys-
ical solution for λ = 1 is reached only if the iteration Λkk′(q)=Ikk′(q)
procedure converges for 0 < λ ≪ 1 and the result can 1
+ Γkk”(q)G+(k”)G−(q+k”)Γk”k′(q)
analytically be continued to λ=1. This construction of N
k”
the physical solution corresponds to the linked-cluster X
1
expansion from many-particle physics.19 The iteration − [Λ (q)−I (q)]
kk” kk”
N
scheme from Eq. (17) is the only available way to reach k”
X
a physical solution and hence its convergence and ana- ×G+(k”)G−(q+k”)Γk”k′(q) (19)
lyticityareofprincipalimportanceforthe diagrammatic
descriptionofdisorderedsystems. UsingEqs.(14)wecan whereweusedthe fundamentalparquetequation(13)to
rewrite the above equation to another one for the irre- represent the integral kernel Λ. The electron-hole sym-
duciblevertexΛdeterminedfromJ. Thelattervertexis metry leadsin the limitq →0andω →0to anequation
thegenuineindependent input. Notice thatinsingle-site for the complex conjugate of the full two-particle vertex
theories with local one-electron propagatorswe obtain a
solution Λ=J0 to Eq.(17). ΓRkkA′(q,ω)∗ =ΓRk′A+q,k+q(−q,−ω) (20)
Equation(16)(alternativelyEq.(17))is afundamen-
tal equation of motion for the two-particle irreducible that we use to evaluate the convolution of the diffusion
vertex being electron-hole symmetric. The correspond- poles fromthe fullvertexΓ inthe firstsum onthe right-
ing full two-particle vertex obeys simultaneously both handsideofEq.(19). Weobtainfork=k′intheleading
6
order of q →0 and ω →0 frequency limit as (−iω)−1 for any value of the exter-
nal momenta q and k. Due to the normalization condi-
1 tion, Eq. (12), we find to each vector k a set (of mea-
ΓRA(q,ω)G (k”)G (q+k”)ΓRA(q,ω)|
N kk” + − k”k sure one) of momenta q so that ϕRA 6= 0. If the
k,−q−k
Xk” homogeneous case, q = 0, falls into this set then from
1 |ϕRAG (k”)|2
−−−−−−→ k”k + . Eq. (22a) we obtain integrability of the diffusion pole.
q→0,ω→0 N ω2+D(ω)2q4 If not, then for ϕRA 6= 0 we obtain SL (q,ω) ∝
k” k,−q−k kk
X (−iω)d/2−1/(−iω +Dq2) and SR (q,ω) ∝ (−iω)d/2−2.
kk
This squareddiffusion pole must be compensated by the For low dimensions d ≤ 2, both functions SL (q,ω) and
kk
second sum on the right-hand side of Eq. (19). It means SR (q,ω) have a stronger divergence than (−iω)−1 (for
kk
that the diffusion pole must be completely contained in q = 0) and Eq. (17) cannot be satisfied by any func-
functionΛkk′(q)−Ikk′(q)=Kkehk′(q). Fromtheelectron- tion Λkk(q,ω). The diffusion pole can hence exist in the
holesymmetrywethenobtainthattheCooperpolemust metallic phase only in dimensions d>2.
completely be contained in function Kkeek′(q) and conse- In the localized phase we expect the following low-
quently the sum of the diffusion and the Cooper poles energy asymptotics (q → 0,ω → 0) of the dynamical
fromthefullvertexΓkk′(q)inEq.(11)isalreadypartof diffusion constant20
function Γkk′(q)−Ikk′(q). The fully irreducible vertex
D(ω)
Ikk′(q) is hence free of the diffusion and Cooper poles. ξ2 = >0 (23)
We discuss first the behavior of the diffusion pole in −iω
the metallic phase with D(0) = D > 0. When inserting where ξ is a localization length. Using this asymptotics
the singular part of the two-particle vertex due to the we can representthe singularpart of the irreducible ver-
diffusion pole we obtain the leading singularity on the tex Λ as follows
left-hand side of Eq. (17)
Λsing(q,ω)=. ϕkk′ 1 . (24)
kk′ −iω 1+ξ2q2
1
SkLk′(q,ω)=− −iω+Dq2 We utilize the electron-hole symmetry to evaluate the
1 ϕRAϕRA G (k”)G (q+k”) complexconjugateoftheirreduciblevertexΛinthelow-
× kk” k”k′ + − (21a) frequency ω →0 and momentum q →0 limit
N −iω+D(q+k+k”)2
k”
X ΛRkkA′(q,ω)∗ =ΛR−Ak−q,−k′−q(q,−ω) (25)
and on its right-hand side
anduseittoderiveaconditionforvanishingofquadratic
1 ϕRAϕRA singularity of order ω−2 on the right-hand side of
SkRk′(q,ω)= N −iω+Dk(kq” +k”kk′+k”)2 Eq. (16). After substituting the representation of ver-
k” tex Λ from Eq. (24) and setting k′ = k and q = 0 in
X
G (k”)G (q+k”) Eq. (16) we obtain
+ −
× . (21b)
−iω+D(q+k′+k”)2
1 ϕRAG (k′′) 2
kk′′ + 2+ξ2(k+k′′)2 =0. (26)
SincethesingulartermfromEq.(21a)containsthecom- N 1+ξ2(k+k′′)2
plete form of the diffusion pole, the sum over momenta Xk′′ (cid:12)(cid:12) (cid:12)(cid:12) (cid:2) (cid:3)
must not bring any new singular contribution in small This cond(cid:12)(cid:12)itioncanbe fulfi(cid:12)(cid:12)lled only ifthe irreduciblever-
frequencies and is of order O(ω0). To assess the low- tex Λkk′′(q,ω) is free of the singularity due to the dif-
frequency behavior (ω → 0) of the sum over momenta fusion pole for q → 0 and ω → 0, that is, ϕRA = 0
kk′
we equal external fermionic momenta k′ =k and use an point-wise. The diffusion pole hence cannot exist in the
asymptotic representation for the contribution from the localized phase.
singular part of the integrands Discussion and conclusions. The most severe
consequence of the Assertion is nonexistence of the dif-
SL (q,ω)=. ϕRk,A−q−kϕR−Aq−k,kGR(q+k)GA(k) fusion pole in the localized phase in any dimension. It
kk −iω+Dq2 meansthatwhenapproachingthelow-energylimitq →0
κ and ω/q →0 in low dimensions (d ≤2) we cannot meet
1 1
× , (22a) the diffusion-pole singularity. The localized phase must
N −iω+D(q+k+k”)2
k” bereachedinanon-criticaloralesscriticalmannerthan
. X
SR (q,ω)=ϕRA ϕRA GR(q+k)GA(k) that of the diffusion pole. Theories, such as the self-
kk k,−q−k −q−k,k
consistent theory of Anderson localization of Vollhardt
κ
1 1 and W¨olfle,20 leading to solutions with a nonintegrable
× , (22b)
N [−iω+D(q+k+k”)2]2 diffusion pole are in conflict with the Bethe-Salpeter
k”
X equation either in the electron-hole or in the electron-
where κ is an appropriate momentum cut-off. The electron channel or with the electron-hole symmetry at
two expressions cannot be more divergent in the low- the two-particle level.
7
The Assertion poses no restriction on the expected metallic phase of the most interesting spatial dimen-
form of the diffusion pole in the metallic phase in di- sions 2 < d < 4 contains apart from the diffusion and
mensions d > 2, since the singularity is integrable. The the Cooper pole also another low-energy singularity for
.
localized phase in d > 2 is, however, different. There ω → 0 and |k−k′| → 0. We found that SR (0,ω) =
kk
the widely accepted behavior of the diffusion pole, due (−iω)d/2−2 in d < 4 and hence a new singularity in
to vanishing of the diffusion constant (D = 0), becomes vertex ΛRA(q,ω) emerges for k−k′ → 0. Due to the
kk′
momentum independent and hence nonintegrable. The normalization condition, Eq. (12), it must be integrable,
fundamental equation (16) for the irreducible vertex Λ which is the case for d>2. This new singularity is com-
cannotleadtoatwo-particlevertexwithsuchasingular- patible withthedecompositionfromEq.(11)ofthe two-
ity. If the diffusion pole in d>2 survives in the metallic particlevertexΓintosingularitiescausedbythediffusion
phase unchanged till the Anderson metal-insulator tran- pole. Theexistenceofanewsingularitymakeseitherthe
sition, there must be a jump at the transition point at weight of the diffusion pole ϕRA or γRA(q,ω) or both
kk′ kk′
which the diffusion pole abruptly ceases to exist. singular with an integrable singularity. A new singular-
Numerical simulations nevertheless seem to confirm ity in the two-particlevertexindicates that the averaged
theexistenceofthediffusionpoleinthelocalizedphase.15 two-particle functions in spatial dimensions 2 < d < 4
There are two possible conclusions we can draw from behave qualitatively differently and have a richer ana-
these incommensurable results. One can speculate that lytic structure from those in higher dimensions. How far
some of the assumptions on which the diagrammatic this singularity influences the macroscopic behavior and
translationallyinvariantdescriptionofrandomsystemsis transport properties of disordered systems and in par-
baseddonotholdneartotheAndersonlocalizationtran- ticular criticality of the Anderson localization transition
sition. Either ergodicity may be broken or one cannot remains to be investigated.
average the expansion in the random potential term by
Toconclude,weprovedinthispaperthatthediffusion
term,orthe conceptofdistinguishabilityofthe electron-
pole in the two-particlevertex canexist in the models of
electron and electron-hole scatterings is invalid. If this
noninteractingelectronsinarandompotentialwithtime
was true, then one had either to revisit the derivation of
reversal symmetry only in the metallic phase in dimen-
the diffusion pole, being presently heavily based on the
sions d > 2. An equation of motion for the two-particle
Bethe-Salpeter representationof the two-particle vertex,
irreducibleverticespreventstheexistenceofthediffusion
or to question the concept of electrons and holes as dis-
pole in the localized phase. The existing translationally
tinguishable quasiparticles in the localized phase.
invariant descriptions of electrons in a random potential
On the other hand, numerical simulations are per-
predicting the existence of a pole in the localized phase
formable only on rather small lattices where one can-
should hence be revisited. In view of our result, it seems
not effectively reach the diffusive regime q → 0 with
very difficult, if not impossible, to build up a consistent
ω/q → 0. Ref. 15 investigates the opposite limit ω → 0
analytic theory of Anderson localization with the diffu-
withq/ω →0. Asweknow,10 thetwolimitsdonotcom-
sion pole in the localized phase.
mute and the latter has no relevance for the existence of
the diffusionpole. The numerically observed1/ω behav- Acknowledgments. Research on this problem
iorreflectsonly the Velicky´ identity (7) validforrandom was carried out within project AVOZ10100520 of the
as well as pure systems. A conflict between the form of Academy of Sciences of the Czech Republic and sup-
the diffusionpoleandthe Bethe-Salpeterequationarises ported in part by Grant No. 202/07/0644 of the Grant
onlyinthecriticalregionofthelatter. Wardidentity(7) Agency of the Czech Republic. I acknowledge a fruitful
cannotbe extendedto inhomogeneouslong-rangefluctu- collaborationand extensive discussions with J. Kolorenˇc
ations and the homogeneous low-frequency limit ω → 0 on the problem of Anderson localization. I profited
with q = 0 may be a singular point having no macro- a lot particularly from his critical remarks. I thank
scopic relevance for nonzero spatial fluctuations q > 0 the Isaac Newton Institute for Mathematical Sciences
in the thermodynamic limit. in Cambridge (UK) for hospitality extended to me dur-
Last but not least, we obtain as a consequence of ingmyparticipationintheProgrammeMathematics and
Eq. (16) that the two-particle vertex Γkk′(q) in the Physics of Anderson Localization.
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