Single-site Anderson Model. I Diagrammatic theory

0.19 MB·English

by  V. A. Moskalenko

Single-site Anderson Model. I Diagrammatic theory
V. A. Moskalenko1,2,∗ P. Entel3, D. F. Digor1, L. A. Dohotaru4, and R. Citro5
1Institute of Applied Physics, Moldova Academy of Sciences, Chisinau 2028, Moldova
2BLTP, Joint Institute for Nuclear Research, 141980 Dubna, Russia
3University of Duisburg-Essen, 47048 Duisburg, Germany
4Technical University, Chisinau 2004, Moldova and
4Dipartimento di Fisica E. R. Caianiello, Universit´a degli Studi di Salerno and CNISM,
7
Unit´a di ricerca di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy
0
(Dated: February 6, 2008)
0
2 The diagrammatic theory is proposed for thestrongly correlated impurityAnderson model. The
stronglycorrelatedimpurityelectronsarehybridizedwithfreeconductionelectrons. Forthissystem
n
a the new diagrammatic approach is formulated. The linked cluster theorem for vacuum diagrams
J is proved and the Dyson type equations for electron propagators of both electron subsystems are
established, together with such equations for mixed propagators. The approximations based on
3
the summing the infinite series of diagrams are proposed, which close the system of equations and
1
permit the investigation of thesystem’s properties.

]
l PACSnumbers: 78.30.Am,74.72Dn,75.30.Gw,75.50.Ee
e
-
r
st I. INTRODUCTION H0 = H0c+H0f,
.
at H0c = ǫ(k) Ck+σCkσ.
m The study of strongly-correlated electron systems be- Xkσ
come in the last decade one of the most active fields
d- of condensed matter physics. The properties of these H0f = ǫfXfσ+fσ+Unf↑nf↓, (1)
n systems can not be described by Fermi liquid theory. σ
1
[co OtArnnondeseorfissotnhth[e1e]imsinipnogtrlhete-asni1tt9em6o1ordaiemnldspuodfriisstctyurosmnseogddlyeilncoitnertnrresoliavdteuelcydedeinlebcya- Hint = √N Xkσ (cid:0)Vkσfσ+Ckσ+Vk∗σCk+σfσ(cid:1),
1 lot of papers[2−15]. It is a model for a system of free nfσ = fσ+fσ,
v conduction electrons that interact with the system of lo-
6 cal spin, treated as just another electrons of d- or f- whereCkσ(Ck+σ)andfσ(fσ+)-annihilation(creation)op-
9 shells of an impurity atom. The impurity electrons are eratorsofconductionandimpurity electrons withspin σ
2 correspondingly. ǫ(k) is the kinetic energy of the con-
strongly correlatedbecause of strong Coulomb repulsion
1 duction band state (k,σ) , ǫ is the local energy of f -
and they undergo the exchange and hybridization with f
0 electrons,U -isthe on-siteCoulombrepulsionoftheim-
7 conduction electrons. This model has some properties purityelectronsandN isthenumberoflatticesites. H
0 similar to those of Kondo modelhaving more interesting int
is the hybridization interaction between conduction and
/ physics[16−18]. It has the application for heavy fermion
t localizedelectrons. Summationoverkwillbechangedto
a systemswherethelocalimpurityorbitalisf -orbital. In-
m vestigationsofimpurityAndersonmodelhaveusedinten- anintegraloverthe energyǫ(k) withthe density ofstate
ρ (ǫ)ofconductionelectronsandthematrixelementswill
- sivelythemethodsandresultsobtainedforKondomodel 0
d by Nagaoka[18] andotherauthors[19,20]. All the citedpa- be consideredas the function of energyV(ǫ). Becauseof
n the hybridization term of the Hamiltonian down and up
pers are based on the method of equation of motions for
o spinsofconductionelectronscomeandgointhelocalor-
retarded and advanced quantum Green’s functions pro-
c bitalandthereisnoappearanceofspinflipprocess. Thus
: posed by Bogoliubov and Tiablikov[21] and developed in
v theimportantparametersoftheAndersonmodelarethe
papers[22−24].
i bandwidthW,theconductiondensityofstatesρ(ǫ),the
X The first attempt to develop the diagrammatic theory
local site energy ǫ and the on-site Coulomb interaction
r for this problem was realized in the paper[25]. These au- f
a thors used the expansion by cumulants for averages of U. The electron energies are counted of chemical poten-
tial µ of the system: ǫ(k) = ξ(k) µ, ǫ = ǫ µ .
products of Hubbard transfer operators and their alge- − f f −
There is also an energy parameter Γ(ǫ) associated with
bra.
the hybridization term
With introduction of Dynamical Mean Field Theory
the interest for Anderson impurity model increases be- π
cause infinite dimensional lattice models can be mapped Γ(ǫ)= Vk2δ(ǫ ǫ(k))=πV2(ǫ)ρ0(ǫ). (2)
N −
onto effective impurity models together with a self- Xk
consistency condition[26,27].
Thisfunctionisassumedtobeaconstant,independentof
The Hamiltonian of the model is written as
energy. The term in the Hamiltonian involving U comes
H = H +H , from on-site Coulomb interaction between two impurity
0 int
2
electrons. U it is far to large to be treated by pertur- tion have the form:
bation theory. It must be included in Ho which is non
interacting Hamiltonian. The existence of this term in-
validates Wick’s theorem for local electrons. Therefore,
first of all, we formulate the generalized Wick’s theorem G(k,σ,τ |k′,σ,′τ′) = − TCkσ(τ)Ck′σ′(τ′)U(β) c0,
(tGheWorTem) ffoorrlcoocnadluecletciotrnoenlse,cptrroensesr.vOinugrtGheWoTrdrienaalrlyyiWstihcke g(σ,τ |σ′,τ′) = − (cid:10)Tfσ(τ)fσ′(τ′)U(β) c0.(cid:11) (3)
(cid:10) (cid:11)
identity which determines the irreducible Green’s func-
tions or Kubo cumulants. Such definitions have already
beenusedbyusfordiscussingthepropertiesofone-band
Hubbard model[28−30] and the formulation of the new Besides them there are also anomalous ones:
diagram technique for it[31−34].
In SectionII, we startby introducing the temperature
Green’s functions for the conduction and impurity elec-
trons in interaction representation, formulate the gen- F(k,σ,τ |−k,−σ′,τ′) = − hTCkσ(τ)C−k′−σ′(τ′)U(β)ic0,
edfuriaallgliprzareomdpWacgaialcctkuorltasht.eioTonrheemforreastnuhdletrspmraoorvdeiydanenaaemlxyipzcleaidclitpinoetxSeaenmcttipiaollneasInoIdIf F(−k,−fσ(,στ,τ|k|−′,σσ′′,,ττ′′)) == −− (cid:10)hTTfCσ−(τk)−fσ−(στ′)(Cτ′k)′Uσ′((βτ)′)iUc0,(β)(cid:11)c0,
and comparedto the other data in Section IV. Some ap- f(−σ,τ |σ′,τ′) = − Tf−σ(τ)fσ′(τ′)U(β) c0,
proximations are discussed in Section V and in Section (cid:10) (cid:11)
(4)
VI there are the conclusions.
II. DIAGRAMMATICAL THEORY if the system is in superconducting state. Here τ and τ′
stand for imaginary time with 0 < τ < β, β - inverse
TheMatsubararenormalizedGreen’sfunctionsofcon- temperature and T is the chronological ordering opera-
ductionandimpurityelectronsininteractionrepresenta- tor. The evolution operator U(β) is given by
β β β
∞ ( 1)n
U(β) = T exp( H (τ)dτ)= − dτ ... dτ T(H (τ )...H (τ )). (5)
−Z int n! Z 1 Z n int 1 int n
X
n=0
0 0 0
Thestatisticalaveragingiscarriedoutin(3)and(4)with and the energy E =U +2ǫ . By using the relation
2 f
respecttothezero-orderdensitymatrixoftheconduction
and impurity electrons. fσ =χ0σ+σχσ2, (7)
we obtain the diagonalized form of the impurity Hamil-
e−βH0 e−βH0c e−βH0f tonian
= . (6)
Tre−βH0 Tre−βH0c × Tre−βH0f 2 2
Hf = E χnn, χnn =1. (8)
0 n
X X
n=−1 n=−1
The thermodynamic perturbation theory for H re-
int
Inzeroorderapproximation,whenweneglectthepro-
quires the generalization adequate for calculation of the
cessofhybridizationoftheconductionandimpurityelec-
statistical averages of the T - products of localized f -
trons,thecorrespondingGreen’sfunctionshavetheform
electron operators. This necessity appears for the rea-
(ω ω =(2n+1)π/β)
son that cannot be diagonalized with free f - electron n
≡
operators. This Hamiltonian can be diagonalized by 1
using the algebra of Hubbard[28−30] transfer operators G0σσ′(k,k′ |iω) = δσσ′δkk′iω ǫ(k),
χmn = m><n when the m> with m = 1,0,1,2 −
ethneumemerpat|tyesorfovuarcus|utamtesstaotfetwhei|thimepnuerrigtyyEato=m:0,−|0th>e 1- >is gσ0σ′ =δσσ′gσ0(iω) = λ1σ−(inωσ) + λσn(σiω), (9)
0
|
and 1> or >and > arethe states withone parti-
|− |↑ |↓ where (σ = σ)
cle with energy E = ǫ and spin σ = 1 and the state −
σ f
±
2>= >containstwof -electronswithoppositespins λ (iω) = iω+E E
σ 0 σ
| |↑↓ −
3
λ (iω) = iω+E E , In the case of f - electrons we formulate the identity
σ σ 2
−
Z0 = e−βE0 +e−βEσ +e−βEσ +e−βE2, which is just our GWT in this simple case:
e−βEσ +e−βE2
n = ,
σ
Z
0
e−βE0 +e−βEσ
1 n = .
σ
− Z
0
ir
Tf f f f = Tf f Tf f Tf f Tf f + Tf f f f , (10)
1 2 3 4 0 1 4 0 2 3 0− 1 3 0 2 4 0 1 2 3 4 0
(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
or
g0(1,23,4) = g0(14)g0(23) g0(13)g0(24)+g(0)ir(1,23,4), (11)
2 | | | − | | 2 |
where n stands for (σn,τn). The generalizationfor more terms of the form g0(1|5)g3(0)ir(234|678) and finally one
complicate averages of type formg(0)ir(12345678). Thetotalnumberoftermsis131.
g0(1,...,n n+1,...,2n) = ( 1)n Tf ...f f ...f 4 |
n | − 1 n n+1 2n 0 The signs of all these contributions can be easily deter-
is straightforward, namely the righ(cid:10)t - hand part of th(cid:11)is mined. Thus the definition of the irreducible Green’s
quantity will contain n! term of ordinary Wick type
functionsorKubocumulantsisjustourGWT.Intheab-
(chain diagrams) and also the different products of ir- sence of Coulomb repulsion U all these irreducible func-
reducible functions with the same total number of oper-
tions are equal to zero. When U = 0 they contain all
ators. The full irreducible functions in the spin, charge and pairing fluctu6 ations produced by
g0(1,...,nn + 1,...,2n) also appears. For example
n | the strong correlations. These definitions are the sim-
g30(123|456) contains the contribution of 3! = 6 terms plification of ones for Hubbard and other lattice models.
of ordinary Wick kind, then appear 9 terms of the form
The calculation of the simplest irreducible functions for
g.0(1T|h4)eg2(t0o)tira(l23n|u5m6)bearndofthteerlmasstitser1m6.isIgn3(0)tihre(12ca3s|4e56o)f feoxrawmaprdle.gI2t(0i)sirn(e1c2e|3ss4a)riystroatfihnedrtchuemvbaelurseosmofecbhurtonstorlaoiggihcat-l
g0(12345678) there are 4! = 24 terms of ordinary Wick
4 | averages for 4! = 24 different orders of τ1,τ2,τ3 and τ4
kind,the72termsofthetypeg0(15)g0(26)g(0)ir(3478), times and then to determine its Fourier representation
| | 2 |
then 18 terms of type g(0)ir(1256)g(0)ir(3478), then 16
2 | 2 |
1
g(0)ir[σ ,τ ;σ ,τ σ ,τ ;σ ,τ ] = exp( iω τ iω τ +iω τ +iω τ )
2 1 1 2 2| 3 3 4 4 β4 − 1 1− 2 2 3 3 4 4 ×
X
ω1ω2ω3ω4
(12)
g(0)ir[σ ,iω ;σ ,iω σ ,iω ;σ ,iω ].
× 2 1 1 2 2| 3 3 4 4
The Fourier representation conserves the frequencies
g(0)ir[σ ,iω ;σ ,iω σ ,iω ;σ ,iω ] = βδ(ω +ω ω ω )
2 1 1 2 2| 3 3 4 4 1 2− 3− 4 ×
(13)
g(0)ir[σ ,iω ;σ ,iω σ ,iω ;σ ,iω +iω iω ].
× 2 1 1 2 2| 3 3 4 1 2− 3
e
There is also the spin conservation σ +σ = σ +σ . III. THERMODYNAMIC POTENTIAL
1 2 3 4
Thus we have the rules to deal with chronological aver-
ages of thermodynamic perturbation theory. First of all we can determine the thermodynamic po-
tential F of the system
1
F = F ln U(β) ,
0− β h i0
4
(14) depicted by the thin (solid and dashed) lines with two
1 2 opposite directions at the end of them.
F = lnZ ln[1+exp( βǫ(k))],
0 0
−β − β −
Xk
whereZoreferstothefreeimpurityatom. Thediagrams IV. RENORMALIZED PROPAGATORS
which determine the thermodynamic potential have not
the external lines and are named vacuum.
Now we shall consider the diagrammatical analysis of
InFig.1areshownthe simplest vacuumconnecteddi-
the perturbation series for renormalized propagators (3)
agrams of the normal state. In the diagrams we shall
and (4). The simplest contributions to such series are
depict the process of hybridizationof C and f electrons.
represented on the Figures 3 6. All such diagrams
The zero order propagators of conduction and impu- −
contain two external points with attached arrows deter-
rity electrons are represented by their solid and dashed
mined by the arguments of Green’s functions and their
lines correspondingly. These lines connect the crosses
kind.On the inner points of diagrams is supposed sum-
whichdepicttheimpuritystates. Tocrossesareattached mation on σ ,k , and integration on τ .
n n n
two arrows, one of which is ingoing and other outgoing.
In the same second order approximation of perturba-
They depict the annihilation and creation electrons cor-
tion theory the diagrams for impurity electron propaga-
respondingly. The index n means (σ ,τ ) for impurity
n n torscontainnewdiagrammaticalelementsnamelytheir-
and (k ,σ ,τ ) for conduction electrons. The rectan-
n n n reducibletwoparticleGreen’sfunctions. Thesefunctions
gles with 2n crosses depict the irreducible g(0)ir Green’s
n also can be normal or anomalous. The process of their
functions.
renormalization will be not considered by us, supposing
Besidesthevacuumdiagramsoffourthordershownon
thenecessityofrenormalizationonlyforthepropagators.
the Fig.1 b) and c) there is also one disconnected dia-
In Fig.5 the diagrams for impurity electron normal
gramcomposedfrom two diagramsof the type Fig.1 a)
propagator are shown.
and containing additional factor 1/2!. Such situation is
ThelasttwoirreducibleGreen’sfunctions ofFig.5are
repeatedinhighorderofperturbationtheoryandpermit
anomalous ones because they contain non equal number
us to formulate linked cluster theorem. It has the form
of annihilation and creation f-operators enumerated in
c the left and right parts about the vertical bare corre-
U(β) =exp U(β) , (15)
h i0 h i0 spondingly. Thanks the summation of infinite series dia-
c gramsthe renormalizednormaland anomalouspropaga-
where U(β) contains only connected diagrams and is
h i0 tors appear and now it is necessary to put equal to zero
equal to zero when hybridization is absent. If we ad-
the source of electron pairs and simultaneously the bare
mittheexistenceofthepairingmechanismofconduction
electrons, thanks the hybridization, the paring mecha- f0 and f0 together with anomalous irreducible Green’s
nism appear also for impurity electrons. This mecha- functions. Thecorrespondingcontributiontotheanoma-
nismresultsinappearanceoftheanomalouspropagators lous impurity electron function fσσ′(τ τ′) is depicted
−
of both kind of electrons. on the Fig.6
Fig.2 shows some of the simplest connected anoma- The final equations for renormalized functions it is
lous vacuum diagrams. The anomalous propagators are more convenient to write down in Fourier representation
G(k,σ,τ k′,σ′,τ′) = 1 Gσσ′(k,k′ iω) exp iω(τ τ′) ,
| β X | h− − i
ω
F(k,σ,τ|−k′,−σ′,τ′) = β1 XFσσ′(k,−k′|iω)exph−iω(τ −τ′)i.
ω
The complete equations for the conduction electrons propagatorshave the form:
Gσσ′(k,k′|iω) = δkk′δσσ′G0σ(k|iω)+ Vk∗NVk′(G0σ(k|iω)gσσ′(iω)G0σ′(k′|iω)−
− G0σ(k|iω)fσσ′(iω)Fσ0′σ′(−k′|iω))−Fσ0σ(k|iω)gσ′σ(−iω)F0σ′σ′(−k|iω)−
− Fσ0σ(k|iω)fσσ′(iω)G0σ′(k′|iω)), (16)
Fσσ′(k,−k′|iω) = Fσ0σ(k|iω)δkk′δσσ′ + Vk∗NVk′(G0σ(k|iω)gσσ′(iω)Fσ0′σ′(k′|iω)+
+ G0σ(k|iω)fσσ′(iω)Gσ0′(−k′|−iω)+Fσ0σ(k|iω)gσ′σ(−iω)Gσ0′(−k′|−iω)−
− Fσ0σ(k|iω)fσσ′(iω)Fσ0′σ′(k′|iω)). (17)
5
∗
V1 G0 V2 ∗ G0
G0 1 2 V2 V3
V1 1 2 V∗2 −12 g0 g0 +21 2g2(0)ir(12|34)3
1 4
g0 ∗ 4 G0 3 V∗1 G0 V4
(a) V4 V3
(c)
(b)
FIG. 1: The simplest connected vacuum diagrams in normal state. The diagram a) is of second and b), c) of fourth order of
thetheory. Here Vn =Vn/√N.
∗
V1 G0(1|2) V2
−12V11 F0(1|2) 2V2 −12V∗11 F0(1|2)2V∗2 + 12 f10(4|1) 2 f0(3|2)
f0(1|2) f0(1|2) 4 3 ∗
(a) (b) V4 V3
G0(4 3)
|
c
( )
V1 F0(1 2) V2
| 0
1 2 V∗1 F (1|2) V∗2
+ 21 g0 g0 + 14 1 2
4 3
4 3
∗ ∗ V4 V3
V V F0(3|4)
4 0 3
F (3 4)
(d|) (e)
FIG.2: Thesimplest vacuumanomalous diagrams. Thediagrams a)and b) areof second andc), d) ande) of fourth orderof
perturbation theory.
These renormalized propagators are expressed through the correlation functions Λσσ′ = gσ0σ′ +Zσσ′, Yσσ′ and
the full propagators g,f and f of impurity electrons. Y : The secondequationweshalldepictforanomalous
σσ
Now it is necessary to obtain the corresponding equa- propagator f of the impurity electrons (see Fig.8). In
tions for the full impurity electronpropagators. Because both these equations the bare conduction electron prop-
the subsystem of f-electrons is strongly correlated we agators G0(kiω), F0 (kiω) and
σ | σσ |
havetointroducethecorrelationfunctionsZσσ′,Yσσ′ and F0 ( kiω) play the role of mass operators for the f-
Yσσ′ whicharerepresentedbystrongconnecteddiagrams eleσcσtr−on|propagators. It is easy to see that these func-
with irreducible Green’s functions[31−35]. The process of
tions participate in above equations being averaged on
renormalizationoff-electronpropagatorsisshownonthe
the Brillouin cell with matrix elements of hybridization.
Figures 7 and 8, where the double dashed lines depict
Therefore we define the new quantities
the full f-electronfunctions andthe rectanglesrepresent
1 1
N Vk∗2Vk1G0(k1,σ1,τ1|k2,σ2,τ2) = N |Vk1|2G0σ1σ2(k|τ1−τ2)≡δσ1σ2G0σ1(τ1−τ2),
kXk Xk
1 2 1
N1 Vk∗1Vk∗2F0(k1,σ1,τ1|k2,σ2,τ2) = N1 |Vk1|2F0σ1σ2(−k1|τ1−τ2)≡δσ1σ2F0σ1σ1(τ1−τ2), (18)
kXk Xk
1 2 1
6
∗
NG(2)(k,σ,τ k′,σ′,τ′)= G0 V1 g0 V2 G0
|
k,σ,τ 1 2 k′,σ′,τ′
G0 V1∗ f0 V2∗ F0 F0 V1 g0 V2∗ F0
− k,σ,τ 1 2 k′,σ′,τ′ − k,σ,τ 1 2 k′,σ′,τ′
F0 V1 f0 V2 G0
− k,σ,τ 1 2 k′,σ′,τ′
FIG. 3: Second order perturbation theory contribution for conduction electron normal propagator.
NF(2)(k,σ,τ k′, σ′,τ′) = G0 V1∗ g0 V2 F0
|− − k,σ,τ 1 2 k′, σ′,τ′
− −
+ G0 V1∗ f0 V2∗ G0 + F0 V1 g0 V2∗ G0
k,σ,τ 1 2 k′, σ′,τ′ k,σ,τ 1 2 k′, σ′,τ′
− − − −
F0 V1 f0 V2 F0
− k,σ,τ 1 2 k′, σ′,τ′
− −
FIG. 4: Second order perturbation theory contribution for conduction electron anomalous propagator.
1 1
N Vk1Vk2F0(k1,σ1,τ1|k2,σ2,τ2) = N |Vk1|2Fσ01σ2(k1|τ1−τ2)≡δσ1σ2Fσ01σ1(τ1−τ2).
kXk Xk
1 2 1
These definitions gives us the possibility to simplify the structure of equations for the f-electron propagators. By
using these average bare propagators G0, F0 and F0 and Fourier representation for τ-variables we obtain
σ σσ σσ
Λ (iω) G0( iω)[Λ (iω)Λ ( iω) + Y (iω)Y (iω)]
g (iω) = σ − σ − σ σ − σσ σσ , (19)
σ
d (iω)
σ
0
Y (iω) + F (iω)(Λ (iω)Λ ( iω) + Y (iω)Y (iω))
f (iω) = σσ σσ σ σ − σσ σσ , (20)
σσ d (iω)
σ
Y (iω) + F0 (iω)[(Λ (iω)Λ ( iω) + Y (iω)Y (iω)]
f (iω) = { σσ σσ σ σ − σσ σσ }, (21)
σσ
d (iω)
σ
d (iω) = (1 Λ (iω)G0(iω))(1 Λ ( iω)G0( iω))+Y (iω)F0 (iω)+
σ − σ σ − σ − σ − σσ σσ
+ Y (iω)F0 (iω)+F0 (iω)F0 (iω)[Y (iω)Y (iω)+Λ (iω)Λ ( iω)]+
σσ σσ σσ σσ σσ σσ σ σ −
+ G0( iω)G0(iω)Y (iω)Y (iω). (22)
σ − σ σσ σσ
In the previous part of the paper we supposed the existence of pairing potential of conduction electrons with order
parameter and with the bare propagators:
G0(kiω) = iω + ǫ(k) ; F0 (kiω)=F0 ( kiω)= ∆ ; E(k)= ǫ2(k)+∆2. (23)
σ | (iω)2 E2(k) σσ | σσ − | (iω)2 E2(k)
p
− −
Now we shall discuss the case when the pairing potential is absent and the superconducting state appears simultane-
ously with bothsubsystems asaconsequenceofthe brokensymmetry andphasetransition. In this moresimple case
the renormalized conduction electron propagators have the form
Gσσ′(k,k|iω) = δkkδσσ′G0σ(k|iω)+ VkN∗VkG0σ(k|iω)gσσ′(iω)G0σ′(k|iω), (24)
7
Ng(2)(σ,τ σ′,τ′) = g0 V1 G0 V2∗ g0
| σ,τ 1 2 σ′,τ′
g0 V1 F0 V2 f0 f0 V1∗ G0 V2 f0
− ′ ′ − ′ ′
σ,τ 1 2 σ,τ σ,τ 1 2 σ,τ
f0 V1∗ F0 V2∗ g0
− ′ ′
σ,τ 1 2 σ,τ
0
F (1 2)
V1∗ G0 V2 V1 F0(1|2) V2 V1∗ | V2∗
1 2
1 2 1 2
− στ σ′τ′ − στ σ′τ′ − στ σ′τ′
g2(0)ir(στ;σ1τ1|σ2τ2;σ′τ′) g2(0)ir(στ |σ1τ1;σ2τ2;σ′τ′) g2(0)ir(στ;σ1τ1;σ2τ2 |σ′τ′)
FIG. 5: The second order perturbation contribution for theimpurity electron normal propagator.
Nfσ(2,σ)′(τ−τ′) = g0 V1 F0 V2 g0
σ,τ 1 2 σ′,τ′
−
+ g0 V1 G0 V2∗ f0 + f0 V1∗ G0 V2 g0
′ ′ ′ ′
σ,τ 1 2 σ,τ σ,τ 1 2 σ,τ
− −
f0 V1∗ F0 V2∗ f0
− ′ ′
σ,τ 1 2 σ,τ
−
0 F0(1 2)
G0 F (1 2) |
|
1 2 1 2 1 1 2
−V1∗V2 −σ′τ′ στ −12V1∗V2∗ στ −σ′τ′ −2V1V2 −σ′τ′ στ
g2(0)ir(στ,−σ′τ′,σ1τ1 |σ2τ2) g2(0)ir(στ,σ1τ1,σ2τ2,−σ′τ′|) g2(0)ir(στ,−σ′τ′|σ1τ1,σ2τ2)
FIG. 6: Anomalous impurity electron Green’s function in the second orderperturbation theory.
Fσσ′(k,−k|iω) = VkN∗VkG0σ(k|iω)fσσ′(iω)G0σ′(−k|−iω), (25)
G0(kiω) = (iω ǫ(k))−1. (26)
σ | −
The renormalized propagatorsof impurity electron in this case are:
Λ (iω) G0( iω)[Λ (iω)Λ ( iω) + Y (iω)Y (iω)]
g (iω) = σ − σ − σ σ − σσ σσ , (27)
σ
d (iω)
σ
Y (iω) Y (iω)
σσ σσ
f (iω) = ; f (iω)= , (28)
σσ d (iω) σσ d (iω)
σ σ
d (iω) = (1 Λ (iω)G0(iω))(1 Λ ( iω)G0( iω))+
σ − σ σ − σ − σ −
+ G ( iω)G0(iω)Y (iω)Y (iω). (29)
σ − σ σσ σσ
The equation (24) has been established many years ago in the paper of Anderson[1] by using the equation
8
g = Λ + Λ V1 G0 V∗2 g
στ σ′τ′ στ σ′τ′ στ 1 2 σ′τ′
V1 F0 V2 f Y V∗1 G0 V2 f
Λ
− −
στ 1 2 σ′τ′ στ 1 2 σ′τ′
Y V∗1 F0 V∗2 g
− στ 1 2 σ′τ′
FIG. 7: Dyson type equation for the normal propagator of impurity electrons. Double dashed lines depict full electron
propagators. The arrows on them distinguish the normal and anomalous ones. The squares with attached arrows depict the
correlated functions. On doublerepeated indices 1 and 2 is supposed summation by σn and kn and integration by τn .
f = Y + Λ V∗1 G0 V2 f
στ σ′τ′ στ σ′τ′ στ 1 2 σ′τ′
+ Λ V∗1 F0 V∗2 g + Y V1 G0 V∗2 g
στ 1 2 σ′τ′ στ 1 2 σ′τ′
Y V1 F0 V2 f
−
στ 1 2 σ′τ′
FIG. 8: Dyson typeequation for one of anomalous Green’s functions of f -electrons.
of motion of conduction electron operators. In this their average values. After that truncation the Green’s
equation the propagator g (iω) has the role of t-matrix functions of low order remain. This approximation has
σ
for non-spin-flip scattering. By setting k = k′ in been proposed by Bogoliubov, Tiablikov, Zubarev and
G (k,k′ iω) Tserkovnikov[21−24] and used by other authors[2−14,18].
σ
|
The hybridization of conduction and impurity electrons
G (k,k′ iω)= 1 + |Vk|2gσ(iω) (30) causes the appearance of mixed Green’s functions:
σ | iω − ǫ(k) N(iω − ǫ(k))2 Gm(k,σ,τ|σ′,τ′) = − TCkσ(τ)fσ′(τ′)U(β) c0,
aisndpocsosnibsliedetroincgonthceluLdeehtmhaatnnthsepedcitsrcaolnrteinpureitsyenotfatgion(Ei)t Fm(k,σ,τ|σ′,τ′) = −(cid:10)hTCkσ(τ)fσ′(τ′)U(β)i(cid:11)c0, (32)
across the real axis is pure imaginary[8] σ Fm(−k,σ,τ|σ′,τ′) = − TC−kσ(τ)fσ′(τ′)U(β) c0,
(cid:10) (cid:11)
g (E+iδ)=[g (E iδ)]∗. (31) and also
σ σ
−
apTprhoexGimreaetne’fsofrumncatsioanrgeσsu(ilωt)ofhsapsebceiaenldkencoowunpltinilglnmowechin- Gm(σ,τ|k,σ′,τ′) = − Tfσ(τ)Ckσ′(τ′)U(β) c0,
anism used for equation of motion of quantum Green’s Fm(σ,τ|−k,σ′,τ′) = −(cid:10)hTfσ(τ)C−kσ′(τ′)U(β(cid:11))ic0(3,3)
ftuionnctsioomnse. cAomsbisinkantioownns oifnospuecrhatdoercsoiusptlainkgenaopffprtohxeimava-- Fm(σ,τ|k,σ′,τ′) = −(cid:10)Tfσ(τ)Ckσ′(τ′)U(β)(cid:11)c0.
erage value of product of operators and are replaced by Let Gmσσ′(kiω), Fmσσ′(kiω) and Fmσσ′(kiω) be the
| | |
9
G(k2σ2τ2|k1σ1τ1)
∗
V1 V2
1 2
Zσσ′(τ τ′) = 1 g2(0)ir(στ;σ1τ1|σ2τ2;σ′τ′)
− − στ σ′τ′
F(k1σ1τ1|−k2σ2τ2)
V1 V2
1 2
Yσσ′(τ −τ′) = −−21 σ′τ′ στ g2(0)ir(στ;σ′τ′|σ1τ1;σ2τ2)
F(−k1σ1τ1|k2σ2τ2)
∗ ∗
V V
1 2
1 2
Yσσ′(τ −τ′) = −−21 σ′τ′ στ g2(0)ir(σ1τ1;σ2τ2|στ;σ′τ′)
FIG. 9: Schematic representation of the main approximations for the correlated functions The solid double lines with arrows
depict the renormalized one-particle Green’s functions of conduction electrons. The rectangles depict the irreducible Green’s
functions of impurityelectrons.
G0
G0
g(0)ir
2
G0
g(0)ir g(0)ir
G0 G0 2 2
G0
στ σ′τ′
g(0)ir (b)
2
στ a σ′τ′
( )
FIG. 10: Examples of thesimplest ladder diagram for g-function.
Fourier representationof the first groupof Green’s func- Inthepresenceofsuperconductingpairingofconduction
tionsandGmσσ′(k|iω),Fσmσ′(−k|iω)andFσmσ′(k|iω)ofthe electrons we obtain the following results:
second group.
10
V∗
Gmσσ′(k|iω) = √kN G0σ(k|iω)gσσ′(iω)−Fσ0σ(k|iω)fσσ′(iω) ,
(cid:2) (cid:3)
V∗
Fmσσ′(k|iω) = √kN G0σ(k|iω)fσσ′(iω)+Fσ0σ′(k|iω)gσ′σ(−iω) , (34)
(cid:2) (cid:3)
Fmσσ′(−k|iω) = √VkN∗ hGσ0(−k|−iω)fσσ′(iω)+F0σσ(−k|iω)gσσ′(iω)i.
For the second group of mixed propagatorswe have:
Gmσσ′(k|iω) = √VNk hgσσ′(iω)G0σ′(k|iω)−fσσ′(iω)Fσ0′σ′(−k|iω)i,
Fσmσ′(−k|iω) = √VNk gσσ′(iω)Fσ0′σ′(k|iω)+fσσ′(iω)Gσ0′(−k|iω) , (35)
(cid:2) (cid:3)
Fmσσ′(k|iω) = √VNk hfσσ′(iω)G0σ′(k|iω)+gσσ′(−iω)Fσ0′σ′(−k|iω)i.
Now we multiply the system of operators (33) by V∗/√N and sum after k, use the definitions (18) and suppose
k
the paramagnetic phase of the system. Then we obtain:
Gmσ(iω) = √1N Xk Vk∗Gmσ(k|iω)=gσ(iω)G0σ(iω)−fσσ′(iω)F0σσ(iω),
1
Fm(iω) = V∗Fm( kiω)=g (iω)F0 (iω)+f (iω)G0( iω), (36)
σσ √N Xk k σσ − | σ σσ σσ σ −
Fm (iω) = 1 V∗Fm (kiω)=g ( iω)F0 (iω)+f (iω)G0(iω).
σσ √N Xk k σσ | σ − σσ σσ σ
When the superconducting state is established in the type equations for these correlated functions Z, Y and
bothsubsystemssimultaneouslyandthebareanomalous Y don’t exist. Therefore to close the system of equa-
Green’s functions of conduction electrons are equal to tions and to determine the order parameters of the sys-
zero the above equations become more simple: tem state it is necessary to make some approximations.
Our main approximations are determined by the dia-
Gmσ (iω) = gσ(iω)G0σ(iω), grams shown on the Fig.9.
Fm(iω) = f (iω)G0( iω), (37) Our approximations correspond to the summation of
σσ σσ σ −
Fm (iω) = f (iω)G0(iω). ladderdiagramsinverticaldirectionshownontheFig.10
σσ σσ σ a).. Weneglectthesummationofladderdiagramsinthe
Forthesecondgroupofmixedfunctionsinthesamecon- horizontal direction (see Fig.10 b))
ditions we obtain:
G (iω) = G0(iω)g (iω),
mσ σ σ VI. CONCLUSIONS
F (iω) = G0(iω)f (iω), (38)
mσσ σ σσ
Fmσσ(iω) = Gσ0(−iω)fσσ(iω). The diagrammatic theory has been developed for one-
site Anderson model in which strong correlations of im-
purity electrons and their hybridization with conduction
V. APPROXIMATIONS electrons is taken into account.
The definition of irreducible Green’s functions or
In previous part of the paper we have formulated the Kubo cumulants is used as a generalized Wick theorem
Dyson type equations for the propagators of the system for strongly correlated subsystem of localized electrons.
in general case of superconducting phase. These equa- These irreducible functions contain all spin, charge and
tions contain the correlation functions which take into pairing fluctuations. On this base the linked cluster the-
accountcharge,spinandpairingfluctuationsandarede- orem has been proved to determine the thermodynamic
termined by infinite sums of strong connected diagrams potential of the system and Dyson type equations were
composedfromirreducibleGreen’sfunctions. TheDyson established for one-particle propagators of the electrons