Density and current response functions in strongly disordered electron systems: Diffusion, electrical conductivity and Einstein relation

16 Pages·0.33 MB·English

by  V. Janis

Density and current response functions in strongly disordered electron systems:
Diffusion, electrical conductivity and Einstein relation
V. Janiˇs, J. Kolorenˇc, and V. Sˇpiˇcka
Institute of Physics, Academy of Sciences of the Czech Republic,
Na Slovance 2, CZ-18221 Praha 8, Czech Republic∗
3
(Dated: February 1, 2008)
0
0 Westudyconsequencesofgaugeinvarianceandchargeconservationofanelectrongasinastrong
2 random potential perturbedbyaweak electromagnetic field. Weusequantumequationsofmotion
and Ward identities for one- and two-particle averaged Green functions toestablish exact relations
n
betweendensityandcurrentresponsefunctions. Inparticularwefindpreciseconditionsunderwhich
a
J wecanextractthecurrent-currentcorrelationfunctionfromthedensity-densitycorrelationfunction
andviceversa. Weusetheseresultsintwodifferentwaystoextendvalidityofaformularelatingthe
3
densityresponsefunctionandtheelectricalconductivityfromsemiclassical equilibriumtoquantum

nonequilibrium systems. Finally we introduce quantum diffusion via a response function relating
]
n the averaged current with the negative gradient of the averaged charge density. With the aid of
n thisresponsefunctionwederiveaquantumversion oftheEinstein relation andprovetheexistence
- of the diffusion pole in the zero-temperature electron-hole correlation function with the long-range
s
i spatial fluctuations controlled by thestatic diffusion constant.
d
t. PACSnumbers: 72.10Bg,72.15Eb,72.15Qm
a
m
- I. INTRODUCTION librium functions. Unlike the response functions, the
d equilibrium Green functions obey equations of motion
n
and are suitable for developing systematic approxima-
o Low-energy physics of equilibrium systems with
tions within many-body diagrammatic techniques.
c weakly interacting electrons is well understood both
[ qualitatively and quantitatively. The relevant informa- Kuboformalism,however,providesdifferentrepresen-

3 tion about the equilibrium system is contained in one- tations for different response functions that we have to
v and two-particle Green functions, in particular in their approximate separately. The problem that emerges with
4 behavior around the Fermi energy. A number of re- independent approximations for different response func-
1 liable approximate methods have been developed for tions via different Kubo formulas is that we are usually
4 the calculation of these functions. Among them, sys- unable to keep exact relations between response func-
3 tematic renormalized perturbation expansions based on tions that may hold due to special symmetries of the
0
the many-body Feynman diagrammatic technique have system under investigation. Most pronounced case with
2
0 provedmosteffective. Whenimpuritiesorquenchedcon- “hidden” symmetries is the response to a weak electro-
/ figurational randomness are added we still are able to magnetic perturbation where we have to satisfy gauge
at describe equilibrium properties of such a system quite invarianceandchargeandcurrentconservations. Instan-
m reliably within diagrammatic schemes and a mean-field- taneous field-dependentdeviationsofchargeandcurrent
- type coherent-potential approximation(CPA).1 densities from their equilibrium values are described by
d density and current response functions. Within linear-
Thesituationgetslessstraightforwardifwecomeout
n response theory these functions are obtained from inde-
o of equilibrium. This is the case when the system is dis-
pendentKuboformulaswithdensity-densityandcurrent-
c turbed by a time-dependent external force but does not
current correlation functions, respectively. Although
: managetoreachanewequilibriumwithinthe relaxation
v bothcorrelationfunctionsarederivedfromthesametwo-
times of the experimental setup. We then have to de-
i
X termine the response of the system to the external per- particleGreenfunction,theyarenormallyapproximated
independently according to the purpose they serve to.
r turbation to obtain experimentally relevant data. How-
a ever, we do not have at our disposal many established When quantum coherence is negligible we can calcu-
methods to calculate nonequilibrium response functions. late the transportpropertiesfromthe coherent-potential
When the perturbation is weak, which is the most com- approximation.1 There is no contribution to the homo-
mon situation in practice, we can use linear response geneous current-current correlation function beyond the
theory to calculate effectively the response functions.2 A single electron-hole bubble in the single-band coherent-
fundamentaltoolforthecalculationofresponsefunctions potential approximation.4 Hence most theories beyond
withinthelinear-responsetheoryistheKuboformalism.3 the CPA,eitheronmodelorrealisticlevel,usethe Kubo
Itdetermineshowtheresponsefunctionsofthedisturbed formula for the electrical conductivity with the current-
nonequilibriumsystemcanbe expressedinterms oftwo- currentcorrelationfunctiontodeterminetransportprop-
particleGreen(correlation)functionsthatarecharacter- erties of disordered solids.5,6
istics of an equilibrium state. Kubo formulas are means On the other hand, when quantum coherence ef-
for the description of nonequilibrium systems with equi- fects are substantial and we expect the Anderson metal-
2
insulatortransition,electricalconductivityisusuallycal- obey Heisenbergequationsofmotionwiththe perturbed
culated from the electron-hole correlation function with Hamiltonian. The perturbation is decoupled from the
theaidofthediffusionconstantcontrollingitslow-energy equilibriumHamiltonianandis treatedonly tolinearor-
behavior near the diffusion pole.7,8 The diffusion pole der. Second, strongly disordered electron systems can
is crucial and enables application of scaling arguments be desribed only by lattice models with a nonquadratic
and the renormalization group approach to Anderson dispersion relation. Continuity equation (2) is then to
localization.9 be modified even in equilibrium. Hence the above, in
Thereisanumberofmoreorlessheuristicarguments the literature broadly disseminated reasoning leading to
intheliteraturethatrelatethedensityresponsewiththe Eq. (4) cannot be fully trusted in quantitative studies of
conductivity.10 They are essentially based on gauge in- strongly disordered systems out of equilibrium.
variance and charge conservationof a semiclassicalequi- Presentlythereis noreliabletheoryofstronglydisor-
libriumdescriptionofanelectrongasexposedtoanelec- deredelectronsbeyondthe mean-fieldCPA.Sincemean-
tromagneticfield. Inweaklydisorderedquantumsystems field approximationsdo notinclude vertexcorrectionsto
(described by continuum models) a relation between the the electrical conductivity they are not suitable for the
density response and conductivity can formally be de- investigation of localization effects in three spatial di-
rived as follows.10 Gauge invariance is used to relate the mensions. Unlike low dimensions (d ≤ 2), Anderson lo-
externalscalarpotentialwiththeelectricfieldE=−∇ϕ. calizationind=3mayoccuronlyinstronglydisordered
The current density generated by the external field then systems. Anderson localization in bulk systems has not
is yetbeenunderstoodorquantitativelydescribedinasat-
isfactory manner. Its quantitative description demands
j(q,ω)=σ(q,ω)·E(q,ω)=−iσ(q,ω)·q ϕ(q,ω) (1)
bridging the gap between the mean-field (CPA) trans-
where σ(q,ω) denotes tensor of the electrical conduc- port theory and theories for weakly and strongly local-
izedelectronstates. Itisclearthatsuchaninterpolating
tivity. Charge conservation is expressed by a continuity
scheme shouldbe basedonadvanced approximationsfor
equation. In equilibrium we can use its operator form
two-particle functions.
thatfollowsfromthe Heisenbergequationsof motionfor
Recently a diagrammatic method for summations of
thecurrentanddensityoperators. ForHamiltonianswith
classes of two-particle diagrams has been proposed.11
quadratic dispersion relation we have
This theory has a potential to interpolate between the
e∂ n(x,t)+∇·j(x,t)=0 . (2) mean-field and localization theories provided exact rela-
t
tionsbetweenthedensityandcurrentresponsefunctions
Energy-momentum representation of the continuity have been proved for this scheme in the whole range of
b b
equation in the ground-state solution is the disorder strength. Criteria for validity of relations
between density-density and current-current correlation
−iωeδn(q,ω)+iq·j(q,ω)=0 . (3) functions in approximate treatments of strongly disor-
dered systems have not yet been established. When we
We have to use a density variation of the equilibrium
wanttogobeyondtheCPAwestillhavetoestablishsuch
density, i. e., the externally induced density δn(q,ω) =
relations in the metallic regime of tight-binding models
n(q,ω) − n in the continuity equation with averaged
0 with extended electron states. Simultaneously we have
values of operators. From the above equations and for
toformulateconditionsunderwhichthederivedrelations
linear response δn(q,ω) = −eχ(q,ω)ϕ(q,ω) we obtain
hold or may be broken.
in the isotropic case
The aim of this paper is threefold. First, we de-
−ie2ω rive various relations between current-current, current-
σ(q,ω)= χ(q,ω) . (4) density, and density-density correlation functions for
q2
stronglydisorderedelectronsoutofequilibrium,i.e.,be-
The derived equality formally holds for complex func- yond the reach of the existing derivations. Second, we
tions without restrictions on momenta or frequencies. articulate conditions under which these relations hold or
Frequencies can, in principle, be even complex. Rela- towhichextenttheymaybebrokeninquantitativetreat-
tion (4) is often taken as granted for the whole range mentsandshowwhentheconductivitycanbecalculated
of the disorder strength and used for the definition of fromthedensityresponsefunctionandviceversa. Third,
the zero-temperature conductivity when describing An- we introduce a generalizationof diffusion via a quantum
derson localization transition.7,8 response function, test validity of the Einstein relation
Althoughtheabovederivationmayseemverygeneral for this quantum diffusion, and relate quantum diffusion
it suffers from a few flaws. First, the operator continu- with its classicalcounterpartand with the diffusion pole
ity equation (2) cannot be directly used out of equilib- in the electron-hole correlation function.
rium. A new term due to causality of the response func- To reach this goal we use Kubo formalism and av-
tions enters the continuity equation when the equilib- eraged many-body Green functions. In this approach a
rium is disturbed. Moreover the nonequilibrium density weakerformofthecontinuityequationexpressedinterms
andcurrentoperatorsinlinear-responsetheorynolonger ofone-andtwo-particleGreenfunctions replacestheop-
3
erator identity. It is derived from equations of motion a function depending on the random potential V is av-
i
for Green functions and Ward identities between them. eraged as
WeusetwoWardidentitiesthatarenotfully equivalent.
One expresses conservation of probability and the other ∞
hX(V )i = dV ρ(V )X(V ). (6)
reflects charge conservation. Each Ward identity is used i av i i i
Z−∞
in a different manner to relate the conductivity with the
density response. The averaged two-particle propagator (resolvent) is de-
The layout of the paper is as follows. In Sec. II we finedastheaveragedproductofone-particlepropagators
summarize the definitions and useful representations of
the density and current response functions. In Sec. III
G(2) (z ,z )=
we show how the current-current correlation function ij,kl 1 2
emerges from a momentum expansion of the density −1 −1
z 1−t−V z 1−t−V . (7)
response function. Then in Sec. IV we derive continuity 1 2
equations expressed in Green functions and use them to (cid:28)h iij h ikl (cid:29)av
show how the density response function can be revealed b b b b b b
Fouriertransformtomomentumspaceisnotuniquelyde-
from the conductivity. Based on the exactly derived re-
fined, sinceduetomomentumconservationwehaveonly
lationbetweenthedensityandcurrentresponsefunction
threeindependentmomenta. Forourpurposeswechoose
we introduce in Sec. V quantum diffusion, derive the
thefollowingnotationanddefinitionoftheFouriertrans-
diffusion pole in the hydrodynamic limit of the density
form
responsefunctionandrelatethediffusionconstanttothe
caonnedxuaccttivlyitys.olIvnabSleec.limViItwoef iilnlufisntirtaetespgaetniaerladlirmeseunlstisonosn. G(k2k)′(z1,z2;q)= N1 e−i(k+q/2)Riei(k′+q/2)Rj
InAppendixwediscussassumptionsforandtherangeof ijkl
X
validityoftheVollhardt-W¨olfle-Wardidentityusedinthe ×e−i(k′−q/2)Rkei(k−q/2)RlG(2) (z ,z ). (8)
derivationofthecontinuityequationforGreenfunctions. ij,kl 1 2
The density response function is determined from
thefollowingthermodynamicrepresentationoftheKubo
formula13
II. DENSITY AND CURRENT RESPONSE
FUNCTIONS IN DISORDERED SYSTEMS
χ(q,iν )=
m
A simplest description of the electron motion in im- 1 ∞
pure, weakly correlated metals is provided by a tight- − N2 kBT G(k2k)′(iωn,iωn+iνm;q) . (9)
bindingAndersonmodel. Itassumesnoninteractingspin- kk′ n=−∞
X X
lesselectronsmovinginarandom,site-diagonalpotential
Vi. Its Hamiltonian reads Here ωn = (2n+1)πT are fermionic and νm = 2mπT
bosonic Matsubara frequencies at temperature T. Due
H = t c†c + V c†c . (5) totheanalyticpropertiesofthetwo-particleGreenfunc-
AD ij i j i i i
tion we can perform analytic continuation from imagi-
<ij> i
X X
nary Matsubara to real frequencies using contour inte-
b
The values of the random potential V are site- grals. Only the contribution along the real axis remains
i
independentandobeyadisorderdistributionρ(V). I.e., and we obtain
1 ∞ dE
χ(q,ω+i0+)= N2 2πi [f(E+ω)−f(E)]GAkkR′(E,E+ω;q)
kk′ Z−∞
X (cid:8)
+ f(E)GRkkR′(E,E+ω;q)−f(E+ω)GAkkA′(E,E+ω;q) . (10)
(cid:9)
We used the following definition energy half-plane, respectively. We denoted the Fermi
functionf(E)=[1+exp{β(E−µ)}]−1 withthechemical
GAkkR′(E,E+ω;q)=G(k2k)′(E−i0+,E+ω+i0+;q) potential µ.
and analogously for the functions GRR and GAA, where The two-particle Green function is generally deter-
both energies are from the upper and lower complex minedfromaBethe-Salpeterequationinwhichtheinput
4
is a two-particle irreducible vertex. The latter is known with n as the density of states at the Fermi level and
F
onlyapproximately,exceptforspeciallimits. However,a GR(ω) = N−1 GR(q,ω). Note that Eqs. (13) hold
q
few specific elements of the full two-particle Green func- only in the limit ω/q →0. In the inverse case, q/ω →0,
P
tion can be evaluated without knowing the two-particle Eq. (12) applies. Noncommutativity of the limits ω →
irreducible vertex. It is sufficient to know only the one- 0 and q → 0 indicates that the point q = 0,ω = 0 is
particle Green function to determine them. Determina- not analytic in the same manner as in the Fermi liquid
tion of two-particle functions from one-particle ones is theory.14
enabled by Ward identities. The Ward identity relating
theaveragedone-andtwo-particleGreenfunctionsreads Thecurrent-currentcorrelationfunctionΠ (q,t)re-
αβ
lates the current of the system j(q,t) perturbed by an
1 G(2)(z ,z ;0)= 1 [G(k,z ) external vector potential A(q,t). Tensor of the elec-
N kk′ 1 2 z −z 1 trical conductivity σ (q,t) determines the current re-
k′ 2 1 αβ
X sponse to an external electric field. It can be obtained
− G(k,z )] . (11)
2 from the current response function by using a relation
E(x,t) = A˙(x,t). In the Fourier representation we
and was proved for the first time within the coherent-
obtain15
potential approximation by Velicky´.4 It is a nonpertur-
bative identity valid quite generally beyond the CPA.11
As a consequence of this identity we obtain vanishing of
the density response function for a homogeneous global i
perturbation,i.e.,forq=0. Using(11)in(10)weeasily σαβ(q,ω+)= [Παβ(q,ω+)−Παβ(q,0)] (14)
ω
find withω =ω+i0+. ThecompensationtermΠ (q,0)on
+ αβ
theright-handsideofEq.(14)isrealandwasintroduced
χ(0,ω+i0+) to warrant finiteness of the complex conductivity. This
1 ∞ dE additive term is not generated from the Kubo formula
= f(E+ω) GA(E+ω) with a commutator of current operators but originates
ω 2πi
Z−∞ from gauge invariance of the electron gas in an external
− GR(E+ω) −f(E)(cid:8)GA(E)−(cid:2)GR(E) =0 . (12) electromagneticfield.16 AccordingtoEq.(14)thereisno
current in the system with a static vector potential and
Another situ(cid:3)ation wh(cid:2)ere we do not nee(cid:3)d(cid:9)to know the thestatic(complex)conductivitycanbedefinedonlyvia
two-particle irreducible vertex is the static limit, ω = a dynamical one from the limit ω →0.
0. First, the static density response function is real and
reads Unlike the density response function the current re-
1 ∞ dE sponse function cannot be simplified in any limit and
χ(q,0)= N2 π f(E)ℑGRkkR′(E,E;q) (13a) to determine the conductivity from Eq. (14) we have to
kk′ Z−∞ knowthetwo-particleirreduciblevertex. Thecurrentre-
X
sponse function alike the density response function is a
that in the limit q→0 goes over to
complex quantity with the same analytic properties. Of
∞ dω ∂f(ω) interest for us are the parallel components that for real
χ(0,0)= ℑGR(ω) −−−→n (13b) frequencies read
F
Z−∞ π ∂ω T→0
e2 1
σ (q,ω)=− [v (k +q/2)−v (−k+q/2)] v (k′+q/2)−v (−k′+q/2)
αα 4 N2 α α α α
k,k′
X (cid:2) (cid:3)
∞ dE
× 2πω [f(E+ω)−f(E)]GAkkR′(E,E+ω;q)+f(E)GRkkR′(E,E+ω;q))
Z−∞
(cid:8) −f(E+ω)GAkkA′(E,E+ω;q)−f(E)[GRkkR′(E,E;q)−GAkkA′(E,E;q)] . (15)
(cid:9)
We denoted the group velocity v (k) = m−1∂/∂k ǫ(k), III. DYNAMICAL CONDUCTIVITY
α α
where ǫ(k) is the dispersion relation of the underlying CALCULATED FROM THE DENSITY
lattice. RESPONSE
As first we show how the dynamicalconductivity can
be obtained from the long-range behavior of the density
5
response function (hydrodynamic limit of small transfer δ(k′ −k) (q·∇ )2ǫ(k)/8. We will be interested in the
k
momentum). To this purpose we use the Velicky´-Ward following function of two energies
identity,Eq.(11). Itenablesustoevaluateaveragedma-
trixelementsofthetwo-particleGreenfunctionwithzero 1
I(z ,z ;q)= Tr G (z )G (z ) (17)
transfer momentum in terms of the one-particle propa- + − N + + − −
gator. Zero transfer momentum is a severe restriction h i
ontheapplicabilityandutilizationofthis Wardidentity. thatcanbe,afteraveragingovebrtheconbfigurationsofthe
Physically we are interested in the hydrodynamic limit, random potential, expressed via the two-particle Green
i. e., the asymptotics q → 0. We are unable to extend function
identity (11) beyond the homogeneous case. However,
if we assume analyticity of the asymptotics q → 0, we 1
hI(z ,z ;q)i = G(2)(z ,z ;q) . (18)
can use momentum q as an expansion parameter and + − av N2 kk′ + −
investigate the hydrodynamic limit perturbatively. The Xkk′
hydrodynamic limit is analytic if frequency ω 6= 0, i. e.,
we are in the regime q/ω ≪1. We then can expand the We assume that the expansionin the operators ∆ t
1,2
density response function in powers of momentum q. commutes with the configurational averaging. We first
We define two configuration-dependent resolvent op- expand the quantity I and then average the series termb
erators byterm. Itissufficientforourpurposestoexpandonlyto
secondorderinmomentum q. This precisiondetermines
−1
G (z)= z 1−t∓∆ t−∆ t−V (16) the leading small-momentum behavior.
± ± 1 2
Expanding quantity I in ∆ t we have to keep the
h i 1,2
where ∆1bt ±∆2t is abdiffberencebin thbe dibspersion rela- order of operators in the products, since ∆1,2t and the
tion of the two resolvents and is defined via its matrix resolventG(z)donotcommute. Tbheexpansiontosecond
elements hbk|∆1t|bk′i=δ(k′−k)v(k)·q/2, hk|∆2t|k′i= order reads b
b
b b
I (z ,z q)=Tr G(z )G(z )+G(z ) ∆ t+∆ t G(z )G(z )
2 + − + − + 1 2 + −
n − G(z )G(z ) ∆(cid:2)t−∆ t G((cid:3)z )−G(z )∆ tG(z )G(z )∆ tG(z )
b b + −b 1 b 2 b −b b + 1 + − 1 −
(cid:2) + G(cid:3)(z ) ∆ tG(z )∆ tG(z )+G(z )∆ tG(z )∆ t G(z ) . (19)
b b b b b+ 1 b + 1bb + b − bb1 − 1 −
h i o
Each direct product of the resolvent operators G(bz±)G(z∓)bcban be sibmbplified ubsing idenbtibty (11d). Dobing this conse-
quently we end up with a sum of products of two resolvents. We have three different terms to analyze:
b b
Tr G(z ) ∆ t+∆ t G(z )−G(z ) ∆ t−∆ t G(z ) =
+ 1 2 − + 1 2 −
nb h(cid:0) b b(cid:1)1b Trb G′(z(cid:0)+)+b G′(z−b(cid:1))i b∆1t +o Tr G′(z+)−G′(z−) ∆2t
z −z
+ −
n h(cid:16) (cid:17) i h(cid:16) (cid:17) i
2
b b b − b Tbr ∆ t G(zb )−G(z ) , (20a)
1 + −
z −z
+ − h (cid:16) (cid:17)i(cid:27)
b b b
1
Tr G(z )∆ tG(z )G(z )∆ tG(z ) = Tr G(z )∆ tG(z )∆ t
+ 1 + − 1 − (z −z )2 + 1 + 1
+ −
n o n h i
b bb b bb + Tr G(z b)∆ tG(zbb)∆ t −b2Tr G(z )∆ tG(z )∆ t , (20b)
− 1 − 1 + 1 − 1
h i h io
b d b b b bb b
Tr G(z ) ∆ tG(z )∆ tG(z )+G(z )∆ tG(z )∆ t G(z ) =
+ 1 + 1 + − 1 − 1 −
n h 1 i o
b bb bb Tbr G′(z+b)b∆1tG(z+b)∆b1t −Tr G′(z−)∆1tG(z−)∆1t
z −z
+ −
n h i h i
1
b − bb Tr bG(z )−bG(z ) b∆bt G(zb)−G(z ) ∆ t . (20c)
+ − 1 + − 1
z −z
+ − h(cid:16) (cid:17) (cid:16) (cid:17) i(cid:27)
b b b b b b
We insert Eqs. (20) in expansion (19) and average over the configurations of the random potential V . To
i
6
recover the density response function we have to limit wherea,bstandforA,Rdependingonwhethertheimag-
the complex energies to the real axis from above or be- inary part of the corresponding frequency argument is
low with which we distinguish the causality. We define positive or negative. We then can represent the leading
correlation functions that can be represented as a trace asymptoticsinsmallmomentaqoftheaveragedelectron-
of the averagedtwo-particle Green function hole correlationfunction
1
ΦaEb(q,ω)= N2 Gakbk′(E,E+ω;q) . (21)
kk′
X
1
ΦAR(q,ω)−ΦAR(0,ω)=− ∂ GR(E+ω)+GA(E) q·v
E E 2ω E
1 (cid:2)(cid:10) (cid:0) (cid:1) 1(cid:11)(cid:3)
+ ∂ GR(E+ω)−GA(E) (q·∇)2ǫ + GR(E+ω)−GA(E) q·v
8ω E ω2
1 (cid:2)(cid:10) (cid:0) (cid:1) (cid:11)(cid:3) (cid:2)(cid:10)(cid:0) (cid:1) (cid:11)(cid:3)
− 4ωN2 q·v(k)q·v(k′) ∂ε GAkkA′(E+ε,E;0)−GRkkR′(E+ω,E+ω+ε;0) ε=0
kk′
X (cid:8) (cid:2) (cid:3)
2
+ ω GAkkA′(E,E;0)+GRkkR′(E+ω,E+ω;0)−2GAkkR′(E,E+ω;0) (22a)
(cid:27)
(cid:2) (cid:3)
and analogously of the electron-electron (ΦRR) and hole-hole (ΦAA) ones that can be written generically as
1
Φaa(q,ω)−Φaa(0,ω)=− [h∂ (Ga(E+ω)+Ga(E))q·vi]
E E 2ω E
1 1
+ ∂ (Ga(E+ω)−Ga(E))(q·∇)2ǫ + [h(Ga(E+ω)−Ga(E))q·vi]
8ω E ω2
1 (cid:2)(cid:10) (cid:11)(cid:3)
− 4ωN2 q·v(k)q·v(k′) {∂ε[Gakak′(E+ε,E;0)−Gakak′(E+ω,E+ω+ε;0)]ε=0
kk′
X
2
+ ω [Gakak′(E,E;0)+Gakak′(E+ω,E+ω;0)−2Gakak′(E,E+ω;0)] . (22b)
(cid:27)
In the above equations we used angular brackets to de- electron-hole with electron-electron and hole-hole corre-
notesummationoverfermionicmomentainthefirstBril- lationfunctions appropriatelytobuildupthedensity re-
louin zone of one-electron functions, that is sponse function.
Forisotropicsituationswedefineinthehydrodynamic
1 limit
hG(ω) f i= G(k,ω) f (k) . (23)
q q
N ω
Xk g(ω)=−ilim χ(q,ω) . (24a)
q→0q2
Note thatonly the termwith ω−2 fromthe electron-hole
This quantity can be generalizedto anisotropic cases as
correlation function diverges in the limit ω → 0 while
in the electron-electron (hole-hole) functions it remains ∂2
finite even in the static limit. This is a manifestation gαβ(ω)=−iω ∂q ∂q χ(q,ω) q=0 . (24b)
α β
of the diffusion pole missing the latter two correlation
(cid:12)
functions. Function g(ω) measures leading long(cid:12)-range correlations
In Eq. (22) different powers of momentum q and fre- ofthedensityresponsefunction. Itsrealpartcanbeiden-
quency ω appear. There is no restriction on validity of tified with the diffusive conductivity or mobility of the
Eqs.(22) in frequency but it holds only for small mo- system. It is now easy to find an explicit representation
menta, more precisely only perturbatively up to second ofthis functionusingEqs.(22). Allcontributionsexcept
order. Equations (22) establish relations between the forthelasttermsontheright-handsidesofEqs.(22)can-
density-densityand the current-currentcorrelationfunc- cel and we obtain equality σ (0,ω) = e2g (ω) being
αα αα
tions in the asymptotic limit q = 0. Generally, however, just Eq. (4) in the limit q = 0. Note that in the course
these two functions are not directly proportional, since of the derivation we had to sum contributions from the
theright-handsidesofEqs.(22)containone-particlecon- electron-hole,electron-electron,andhole-holecorrelation
tributions. They cancel each other if we combine the functions and integrate over energies.
7
Ifweresorttothe low-frequencylimitω →0,the dif- Note that identity (25) for q = 0,z = E +i0+,z =
+ F −
fusionpoleintheelectron-holecorrelationfunctiondom- E −i0+ was already used in Ref. 18. Identity (25) is
F
inatesinthedensityresponseandwerecovertheconduc- relatedto Eq.(11), however,the twoWardidentities are
tivity directly fromΦAR. Itis interesting to note that in identicalneitherinthederivationnorintheapplicability
this static limit we derive the mobility (dc-conductivity) andvaliditydomains. Thelatterholdsfornonzerotrans-
with contributions from GAR and GRR solely from the fer momentum q, i. e., for an inhomogeneous perturba-
electron-hole correlationfunction ΦAR. tion while the formeronly for q =0. Onthe other hand,
we show in the appendix that Eq. (25), unlike Eq. (11),
holds only perturbatively within an expansion in powers
IV. DENSITY RESPONSE FUNCTION of the random potential. I. e., we can prove this iden-
CALCULATED FROM THE CONDUCTIVITY tity for nonzero transfer momentum q only by assuming
that perturbation expansions for the self-energy Σ and
We showed in the preceding section how the conduc- simultaneously for the two-particle irreducible vertex Λ
tivity can be revealedfrom the small-momentum behav- converge. It was shown earlier that in the homogeneous
ior of the density response function. In this section we case, q = 0, the Vollhardt-W¨olfle-Ward identity follows
express the current-currentcorrelationfunction in terms from the Velicky´-Ward one.11
of the density-density correlation function. The aid we
use for this task is a continuity equation and another
To derive a continuity equation for Green functions
Ward identity
we start with an equation of motion for the two-particle
Green function. It is a Bethe-Salpeter equation where
1
Σ(k+,z+)−Σ(k−,z−)= Λkk′(z+,z−;q) the input is a two-particle irreducible vertex. We have
N
k′ threepossibilities(topologicallydistinctscatteringchan-
X
× G(k′ ,z )−G(k′ ,z ) . (25) nels) how to construct a Bethe-Salpeter equation.11 For
+ + − −
our purposes the electron-hole channel is the relevant or
provedforretardedand(cid:2)advancedfunctionsbyV(cid:3)ollhardt most suitable one. There the Bethe-Salpeter equation in
and W¨olfle in Ref. 17. We denoted k = k ± q/2. momentum representation reads
±
1
G(k2k)′(z+,z−;q)=G(k+,z+)G(k−,z−) δ(k−k′)+ N Λkk′′(z+,z−;q)G(k2′′)k′(z+,z−;q) (26)
" k′′ #
X
where Λkk′(z+,z−;q) is the two-particle irreducible ver-
tex from the electron-hole channel. One-electron propa-
gatorsG(k ,z )=[z −Σ(k±q/2,z )−ǫ(k±q/2)]−1
± ± ± ±
are the averagedresolvents.
∆ G(k;z ,z )=G(k ,z )−G(k ,z ) . (28b)
Theproductoftheone-electronpropagatorsfromthe q + − + + − −
right-hand side of Eq. (26) can be decomposed into
G(k+,z+)G(k−,z−)= and analogously the difference ∆qΣ(k;z+,z−). We mul-
∆ G(k;z ,z ) tiply both sides of the Bethe-Salpeter equation (26) by
q + −
− (27)
thedenominatorfromtheright-handsideofEq.(27)and
∆z−∆ Σ(k;z ,z )−∆ ǫ(k)
q + − q
assume validity of Eq. (25) relating the two-particle ir-
where we denoted reducible vertex Λ with the one-particle irreducible self-
energy Σ. We then obtain a ”difference” equation of
∆ ǫ(k)=ǫ(k )−ǫ(k ) , (28a) motion
q + −
1
[∆qǫ(k)−∆z]G(k2k)′(z+,z−;q)=∆qG(k;z+,z−)δ(k−k′)+ N Λkk′′(z+,z−;q)
k′′
X
× ∆ G(k;z ,z )G(2) (z ,z ;q)−G(2)(z ,z ;q)∆ G(k′′;z ,z ) . (29)
q + − k′′k′ + − kk′ + − q + −
h i
8
We now define correlation functions generalized to com- is not the momentum or velocity that is summed with
plex frequencies describing density-density and density- the two-particle Green function. We used an energy
current correlations by summing over the fermionic mo- difference ∆ ǫ(k) from Eq. (28a). Only in the case of
q
menta in Eq. (29) quadratic dispersion relation we have ∆ ǫ(k) = q·k/m
q
and the energy difference is proportional to the group
1
Φ(z ,z ;q)= G(2)(z ,z ;q) , (30a) velocity. Otherwise Φǫǫ equals the current-currentcorre-
1 2 N2 kk′ 1 2 lationfunctiononlyinthesmall-momentumlimit,q →0.
kk′
X We use
1
Φ (z ,z ;q)= ∆ ǫ(k)G(2)(z ,z ;q) , (30b)
ǫ 1 2 N2 q kk′ 1 2
Xkk′ ∆qǫ(k)≈q·v(k) (32)
1 to convert the energy difference in the hydrodynamic
Φ¯ (z ,z ;q)= G(2)(z ,z ;q)∆ ǫ(k′) . (30c)
ǫ 1 2 N2 kk′ 1 2 q limit to a multiple of the group velocity v(k) and de-
Xkk′ note
The contributions from the two-particle irreducible ver-
tex Λ in Eq. (29) cancel each other provided the two- Φ (z ,z ;q)
αβ + −
Λpakr′kti.clIef isror,edwueceibnlde uvperwteixthisascyomntmineutirtiyc,eqi.uae.t,ioΛnkrke′la=t- = N12 vα(k)vβ(k′)G(k2k)′(z+,z−;q) . (33)
ing the generalized density-density and density-current kk′
X
correlationfunctions
With the aid of Eq. (33) we rewrite continuity equation
(31c) to
Φ (z ,z ;q)−∆zΦ(z ,z ;q)
ǫ + − + −
1
= N ∆qG(k;z+,z−) . (31a) qαqβΦαβ(z+,z−;q)− qαhvα∆qG(z+,z−)i
Xk Xαβ Xα
Another continuityequationcanbe derivedby multiply- =∆z[∆zΦ(z+,z−;q)+h∆qG(z+,z−)i] (34)
ingEq.(29)withtheenergydifferenceandsummingover
the fermionic momenta. We obtain an equation relating wheretheangularbracketsstandforthesummationover
the generalized current-current and density-current cor- the fermionic momenta from the first Brillouin zone as
relation functions defined in Eq. (23). In the isotropic case we obtain
Φ (z ,z ;q)−∆zΦ¯ (z ,z ;q) 1
ǫǫ + − ǫ + − Φ (z ,z ;q)= q·hv∆ G(z ,z )i
αα + − q2 q + −
1
= ∆qG(k;z+,z−)∆qǫ(k) . (31b) ∆z
N + [∆zΦ(z ,z ;q)+h∆ G(z ,z )i] . (35)
Xk q2 + − q + −
Combiningtheabovetwocontinuityequationsweobtain
It is the most general relation between the current-
the resulting relation between the generalized current-
current and density-density correlation functions for ar-
current and density-density correlation functions replac-
bitrary complex frequencies and non-zero momenta. It
ing the operator continuity equation (3)
holds for finite momenta as far as the dispersion law re-
mains quadratic. Forgeneraldispersionrelationsthe va-
Φ (z ,z ;q)−(∆z)2Φ(z ,z ;q)
ǫǫ + − + − lidity of Eq. (35) remains in the asymptotic regime of
1 small momenta.
= ∆ G(k;z ,z )[∆z+∆ ǫ(k)] . (31c)
q + − q
N To returnto measurable quantities we limit the com-
k
X plexfrequenciestotherealaxis. Toderiveexactrelations
Actually, correlation function Φ is strictly speak- we first define a generalization of the conductivity with
ǫǫ
ing not the current-current correlation function, since it the correlation functions Φab:
ǫǫ
∞ dE
σ (q,ω)=−e2 [f(E+ω)−f(E)]ΦAR(E,E+ω;q)+f(E)ΦRR(E,E+ω;q)
ǫǫ 2πω ǫǫ ǫǫ
Z−∞
(cid:8) −f(E+ω)ΦAA(E,E+ω;q)−f(E) ΦRR(E,E;q)−ΦAA(E,E;q) . (36)
ǫǫ ǫǫ ǫǫ
(cid:2) (cid:3)(cid:9)
9
We use continuity equation (31c) to relate σ and χ and find
ǫǫ
e2 ∞ dE
σ (q,ω)+ie2ωχ(q,ω)= f(E) ΦRR(E,E;q)−ΦAA(E,E;q)
ǫǫ ω 2π ǫǫ ǫǫ
Z−∞
ie(cid:2)2 ∞ dE 1 (cid:3)
− f(E) ∆ ǫ(k) ℑGR(k ,E)−ℑGR(k ,E) =0 . (37)
q + −
ω π N
Z−∞ k
X (cid:2) (cid:3)
Vanishingoftheright-handsideinEq.(37)canbeexplic- via the semiclassical limit where the electron-hole corre-
itlymanifestedwhencontinuityequation(31c)isapplied lation function becomes a Greenfunction ofthe classical
to the two-particle correlation functions ΦRR and ΦAA. diffusion equation.10
ǫǫ ǫǫ
The two terms on the right-hand side of Eq. (37) add to Here we propose a nonperturbative way to define dif-
zero. fusionfromfirstprinciples. Weintroduceaquantumgen-
We can extract the electrical conductivity from eralizationofFick’slawwithadiffusionresponsefunction
σ (q,ω) in the limit q → 0 or for quadratic dispersion relating the averaged current density with the negative
ǫǫ
law by replacing Φ = q2Φ leading to σ (q,ω)= gradientofthechargedensityofaperturbedsystem. We
ǫǫ α α αα ǫǫ
q2σ (q,ω). In the isotropic case we then reveal hence use
α α αα P
Eq. (4) from Eq. (37). It is important to state that this
P ∞
proofofvalidityofEq.(4)isstronglybasedontheWard hj(x,t)i =− e dt′ ddx′D(x−x′,t−t′)
identity (25) for nonzero momenta. The latter can be av
Z−∞ Z
proved only perturbatively, see Appendix, and hence it ×∇hδn(x′,t′)i (38a)
is not clear whether Eq.(4) holds beyond the perturba- av
tive regime near and beyond the Anderson localization as a definition of the diffusion response function D(x,t)
e
transition. being an extension of the diffusion constant to which it
shouldreduceinthesemiclassicallimit. Intheabovedef-
inition we used a perturbed charge density δn(x,t) that
V. DIFFUSION, DIFFUSION POLE, AND is not the total change of the charge density δn(x,t) =
EINSTEIN RELATION IN QUANTUM SYSTEMS n(x,t)−n0. It is a “dynamical” density chanege obeying
a condition
IntheprecedingsectionsweprovedvalidityofEq.(4) ∞
forarbitraryfrequenciesandsmallmomentafromthere- dthδn(x,t)iav =0 (38b)
gion where quadratic dispersion relation is accurate in Z−∞
strongly disordered lattice systems. It is a useful rela- sayingthatthetotalchangeeatanyplaceduringtheentire
tioninparticularinapproximatetheories. Itenablesone
time evolutionis zero. This restrictionis dictatedbythe
to approximate only either the density response func-
same condition obeyed by the averaged current density
tion or the conductivity and to determine the other one
defined from the conductivity (14).
from Eq. (4). However, both quantities in Eq. (4) are
Definition of diffusion (38) is nonperturbative, has
response functions that are not well suited for system-
quantum character and does not demand any assump-
atic approximations. We would prefer exact relations or
tion about the behavior in the semiclassical limit or the
identities for equilibrium correlationor Green functions.
existence of the diffusion pole in the electron-hole corre-
This can be achieved when we introduce diffusion. Dif-
lation function. Moreover, it offers a microscopic deter-
fusion in classical systems is defined by Fick’s law relat-
mination of the diffusion function from first principles,
ing the current with the negative gradient of the charge
i. e., quantum response functions of the perturbed sys-
density.3,13Thediffusionconstantintroducedinthisway,
tem and via the Kubo formalism from correlation and
i. e. via Fick’s law, has in classical physics an intu-
Green functions of an equilibrium solution. It is hence
itivephenomenologicalcharacterwithoutapropermicro-
directly related to measurable quantities.
scopic backing that should come from quantum physics.
In Fourier representation we obtain for the averaged
The study of diffusion in quantum disordered systems
current
waslaunchedbytheseminalworkofAndersonanalyzing
destructive effects of quantum coherence on diffusion of hj(q,ωi =−ieqD(q,ω)hδn(q,ω)i . (39)
disordered electrons.20 Since then diffusion has become av av
part of transport studies of disordered and interacting The dynamicalchargedensity δn isinduced byanexter-
e
quantum itinerant systems.8,12,21 However even in these nal dynamical scalar potential, which can be expressed
quantum approaches diffusion is introduced either via in Fourier representation as
e
(self-consistent) perturbation expansions leading to the
diffusion pole in the electron-holecorrelationfunction or hδn(q,ω)i =− e[χ(q,ω)−χ(q,0)]ϕ(q,ω) . (40)
av
e
10
It is easy to verify that with this definition Eq. (38b) is known from other treatments of diffusion.19 Eq. (44)
fulfilled. WeuseEq.(1)togetherwithEqs.(39)and(40) holds for arbitrary frequencies and momenta within the
to find an identity between the conductivity, diffusion range of quadratic dispersion relation. This represen-
and density response tation directly indicates a singularity in the density re-
sponse function in the limit ω → 0,q → 0. Since the
σ(q,ω)=−e2D(q,ω)[χ(q,ω)−χ(q,0)] . (41)
order of the limits is relevant for the result, we have to
specify it explicitly. To single out the singular contribu-
We have derived a general relation between quantum
tion in the density response function we have to choose
diffusion and the electrical conductivity. It is now to
ω/q ≪1, that is, the inverse situation to Sec. III and to
show that quantum diffusion response function D(q,ω)
Eqs. (43).
behaves in the semiclassical limit as expected from the
Thesingularityinthedensityresponsefunctionleads
diffusion constant in the diffusion equation. Namely,
toapoleinthetwo-particleGreenfunction. Itisdemon-
we have to show that the homogeneous static diffusion
strated in the low-energy and small-momentum asymp-
D(0,0)obeysEinstein’s relationandentersthe diffusion
totics of the density response function at zero tempera-
equation in the semiclassical limit. To come to it we
ture. We find from Eq. (10)
use Eq. (4) to exclude the density response χ(q,ω) from
Eq. (41) and Eq. (13a) to determine χ(q,0). Doing so
iω
we obtain in the isotropic case χ(q,ω)=χ(q,0)+ ΦAR(q,0)+O(q0)
2π EF
iq2 (cid:0) +O((cid:1)ω2) . (45)
1+ D(q,ω) σ(q,ω)
ω
(cid:20) (cid:21) Using this asymptotics in Eq. (44) we obtain an ex-
∞ dE
=e2D(q,ω) f(E)ℑΦRR(q,0) . (42) plicit manifestation of the diffusion pole in the zero-
π E
Z−∞ temperature electron-hole correlationfunction
If we define the homogeneous dynamical conductivity
2πn
σ(ω)=σ(0,ω) (in the same way also the dynamical dif- ΦAR(q,ω)≈ F (46)
EF −iω+Dq2
fusion) and use the Velicky´-Ward identity (11) in the
limiting case z −z → 0 we obtain a dynamical gener-
1 2 representing the singular part of the electron-hole corre-
alization of the well known Einstein relation
lation function. We denoted a diffusion constant D =
∞ dE limq→0D(q,0). This representationwas derived for zero
σ(ω)=e2D(ω) f′(E)ℑGR(E) temperature in the asymptotic limit q → 0,ω → 0 with
π
Z−∞ the restriction ω/q ≪ 1. To extend it to the opposite
∂n limit, q/ω ≪1, we have to show that the diffusion func-
=e2D(ω) (43a)
∂µ tion D(q,ω) is analytic in the limit q → 0,ω → 0. It is
(cid:18) (cid:19)T
not a priori clear that the diffusion constant in Eq. (46)
that at zero temperature reduces to
equals the diffusion constant obtained from the inverse
σ(ω)=e2n D(ω) . (43b) order of limits D = limω→0limq→0D(q,ω) used in the
F
Einstein relation, Eq. (43b). However, the diffusion re-
A classical version of this formula was proved by Ein- sponse function D(q,ω) was introduced in such a way
stein for the Brownian motion of a particle in a random that the limits ω → 0 and q → 0 commute and we have
medium22 and later on it was re-derived in the frame- only one definition for the static diffusion constant from
work of nonequilibrium statistical mechanics by Kubo.3 the density response function
HerewederivedEinstein’srelationforquantumresponse
functions and showed that it is a consequence of gauge i limω→0∇2qχ(q,ω)|q=0
D =
invariance of the system, namely Eq. (4). 2 lim ∂ χ(q,ω)|
q→0 ω ω=0
It is not yet clear from the above reasoning how the
2πn
diffusion function D(q,ω) is related to the classical dif- = F . (47)
lim q2ΦAR(q,0)
fusion equation and to the diffusion pole. The existence q→0
of the diffusion pole in quantum systems is usually de-
The left equality in Eq. (47) holds quite generally while
ducedfromthesemiclassicallimitleadingtothediffusion
the right one only at zero temperature. Note that repre-
equation. Wecanprovetheexistenceofthediffusionpole
sentation (46) holds only at zero temperature for small
entirelyfromfirstquantumprinciplesandfromthequan-
momenta and in the low-frequency limit.
tumgeneralizationofFick’slaw,Eq.(38). WeuseEq.(4)
We can see from Eq. (46) that the diffusion func-
to exclude now conductivity σ(q,ω) from Eq. (41). It is
tion introduced in the quantum version of Fick’s law
easy to come to a representation of the density response
reduces to the diffusion constant determining the long-
function
range fluctuations of the electron-hole correlation func-
D(q,ω)χ(q,0)q2 tion and hence in the semiclassical limit also the diffu-
χ(q,ω)= (44)
−iω+D(q,ω)q2 sive behavior in the diffusion equation. However, only