Diagrammatic theory for Periodic Anderson Model: Stationary property of the thermodynamic potential

0.23 MB·English

by  V. A. Moskalenko

Diagrammatic theory for Periodic Anderson Model.
Stationary property of the thermodynamic potential
V. A. Moskalenko1,2,∗ L. A. Dohotaru3, and R. Citro4
1Institute of Applied Physics, Moldova Academy of Sciences, Chisinau 2028, Moldova
2BLTP, Joint Institute for Nuclear Research, 141980 Dubna, Russia
3Technical University, Chisinau 2004, Moldova and
4Dipartimento di Fisica E. R. Caianiello, Universita´ degli Studi di Salerno and CNISM,
Unit´a di ricerca di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy
(Dated: January 12, 2010)
0 Diagrammatic theory for Periodic Anderson Model has been developed, supposing the Coulomb
1 repulsion of f− localized electrons as a main parameter of the theory. f− electrons are strongly
0 correlated and c− conduction electrons are uncorrelated. Correlation function for f− and mass
2 operatorfor c− electronsare determined. TheDyson equationfor c−and Dyson-typeequation for
f−electronsareformulatedfortheirpropagators. Theskeletondiagramsaredefinedforcorrelation
n
function and thermodynamic functional. The stationary property of renormalized thermodynamic
a
J potential about the variation of the mass operator is established. The result is appropriate as for
normal and as for superconducting state of the system.
1
1
PACSnumbers: 71.27.+a,71.10.Fd

]
l
e
I. INTRODUCTION themostessentialstagesinthedevelopmentofthismodel
-
r because exists a number of consistent reviews [2−7] and
t
s books[8,9] onthis field andweshalluse the referencesto
The study of the systems with strongly correlated
at. electrons has become in the last time one of the cen- previous our papers.
m tral problem of condensed matter physics. One of the
most important models of strongly correlated electrons
- II. MODEL HAMILTONIAN
d is periodic Anderson model (PAM)[1]. This model is
n used to describe the physics of mixed valence systems,
o We considerthe simplestformofPAMwith aspinde-
heavy fermion compounds, high-temperature supercon-
c generationoftheleveloflocalizedf electrons,asimple
ductivityaswellasotherphenomenainwhichthestrong
[ −
Coulomb repulsion of the localized electrons is present. energy band of conducting c electrons, Coulomb one-
−
1 Thismodeldescribestheintermetalliccompoundswhich site repulsionU ofcorrelatedf electronswith opposite
−
v spins and one-site hybridization between both group of
contain magnetic moments of rare earth or actinide ions
4 electrons of this system. The hamiltonian of the system
included in the host metal. This ions have a partially
4
reads:
filled f shell and can be considered as scattering cen-
6
−
1 ters for conduction electrons of the host metal. Because H = H0+H0+H ,
. ofthestrongCoulombrepulsionoftheelectronswithop- c f int
1
0 posite spins located at the same site of lattice the mag- Hc0 = ǫ(k) Ck+σCkσ,
0 netic ion electrons are strongly correlated. There is also Xkσ
1 thehybridizationofstatesbetweentheuncorrelatedcon-
H0 = ǫ n +U n n , (1)
: ductionelectronsandlocalizedcorrelatedoneswhenboth f f iσ i↑ i↓
v of them are present on the same lattice site. Magnetic Xiσ Xi
i
X propertiesoftheimpuritiesionsaffectinadifferentman- Hint = V Ci+σfiσ+fi+σCiσ ,
r ner the properties of the host matrix and of the system Xiσ (cid:0) (cid:1)
a
as a whole. For different regime of physical parameters,
where
determined by Coulomb local interaction, hybridization
ofthe wavefunctions andexchangeinteraction,itispos-
ǫ(k) = ǫ(k) µ,ǫ =ǫ µ,
sible to obtain different classes of the system phases. − f f −
n = f+f , (2)
Therearealreadyanenormousnumberofapproximate iσ iσ iσ
1
methods and approaches devoted to PAM, as perturba- Ciσ = exp( ikRi)Ckσ.
tion expansions, static and dynamic mean field theories, √N −
Xk
variational and numerical approaches, large N expan-
sion, slavebosonmethods, non crossingapproximations Here V is the hybridization amplitude assumed con-
(NCA), Bogoliubov inequality method and others. Also stant. We have indicated with C+(f+) the creation op-
iσ iσ
some exact results are known, obtained in special with eratorforanuncorrelated(correlated)electronwithspin
the Bethe ansatz, renormalization group methods and σ and i lattice site, n is the number operator for f
iσ
Bogoliubovinequalitymethod. Wewillnotenlargeupon electrons, ǫ(k) is the band energy with momentum k o−f
2
conductivity electrons spread on the entire width W of chronologicalproductofelectronoperators(AB...). Such
the band. ǫ is the energy of localized electrons. Both averagesarecalculatedindependentlyforc andf op-
f
− −
these energiesareevaluatedwithrespectto the chemical eratorswith using for c electronsthe Wick Theoremof
−
potential µ. weakquantumfieldtheoryandbyusingforf electrons
−
The approach proposed in this paper generalizes the the Generalized Wick Theorem (GWT) proposed by us
diagrammatic theory of normal and superconducting inpapers[10−13] forstronglycorrelatedelectronsystems.
phases of strongly correlated systems proposed in pre- In the superconducting state, unlike the normal one,
vious papers [10−19]. nontrivial statistical averages of operator products with
The strong on-site repulsion U between f electrons even total number but inequal number of creation and
−
of opposite spins is the main term in the Hamiltonian. annihilation electron operators are possible. They re-
Astheconductionelectronscanbelongnotonlytothe alize the Bogoliubov quasi-averages [22] or Gor’kov [23]
s but also to the d atomic shell, their Coulomb re- anomalous Green’s functions. To unify the calculation
− −
pulsion can also be important. In this case the extended of statistical averages for normal and superconducting
PAM must be used [20]. For simplicity the correlations phases it is useful to assignan additionalquantumnum-
of c electrons are not considered and one subsystem ber α, called by us charge number [15], with the values
−
is of c uncorrelated and the second of f correlated 1,whichcanbeaddtoelectronoperatorsaccordingthe
− − ±
electrons. Because of strong localization of the f elec- rule (a=c,f):
−
trons they cannot hope from one lattice site to another
and their delocalization is due only to the hybridization a(x), α=1;
of the f and c states with matrix element V. It is aα(x)= (6)
− − (cid:26)a+(x), α= 1.
obvious that at V =0 in the given model with two sub- −
6
systems superconductivity arises simultaneously in both InthisrepresentationtheinteractionoperatorH be-
int
subsystems. comes:
In the present paper we develop the thermodynamic
perturbation theory for the system in the superconduct- Hint = V αfi−σαCiασ. (7)
ingstatewithHamiltonian(1)undertheassumptionthat Xiσα
thetermresponsibleforhybridizationofc andf elec- Obviously,introducinga new quantumchargenumber
− −
trons is a perturbation. leads to additional summation over it values in all dia-
The Hamiltonian Hc0 of the uncorrelated c− electrons gram lines and to an additional factor α in the vertices
is diagonal in band representation, where as the Hamil- of diagrams.
tonianHf0 is diagonalizedby using Hubbardtransferop- Now, after such introducing, it is irrelevant whether
erators [21]. Therefore in the zeroth-order of the pertur- one deals with creation or annihilation operators. First
bation theory the statistical operator of grand canonical of all we shall enumerate the main results of diagram-
ensemble of the system is factorized in the momentum matic theory obtained in the previous paper [15] neces-
representation for c and in local representation for f sarytoourprovingofstationarytheorem. Suchtheorem
− −
electrons: for uncorrelated many-electron systems in normal state
has been proved by Luttinger and Word [24].
exp[ β(H0+H0)] = exp( βH0 (k))
− c f − cσ ×
Ykσ
exp( βH0(i)), III. PERTURBATIVE TREATMENT [15]
× − f
Yi
Hc0σ(k) = ǫ(k) Ck+σCkσ, (3) We use the definition of the one-particle Matsubara
Green’s functions for c and f electrons
H0(i) = ǫ n +Un n , − −
f fXσ iσ i↑ i↓ Gcαα′(x|x′) = − Tcα(x)c−α′(x′)U(β) c0,
We use the series expansion for evolution operator: (cid:10) (cid:11) (8)
Gf (xx′) = Tfα(x)f−α′(x′)U(β) c,
β αα′ | − 0
(cid:10) (cid:11)
U(β) = T exp( Hint(τ)dτ). (4) where index c for ... c means the connected of the di-
−Z h i0
agrams which are taken into account in the right-hand
0
part of definition (8).
in the interaction representation for electron operators
The following condition is fulfilled
(a=c,f):
a(x)=eτH0a(x)e−τH0,a(x)=eτH0a+(x)e−τH0. (5) Gaαα′(x|x′) = −Ga−α′,−α(x′|x),a=(c,f). (9)
Between this new definition and traditional one [23]
Here by x means (x,σ,τ).
there is a relation
We shall denote by TAB... the thermodynamic av-
erage with zeroth-ordher statisit0ical operator (3) of the Ga (xx′) = Ga(xx′),
1,1 | |
3
Ga (xx′) = Fa(xx′), diagrams for f electron belong to the correlationfunc-
Ga−1,1−,11(x||x′) = Fa(x||x′), qtiuoannwtithyicΛhαisα′d(−xenxo′t)edisbdyefiunsedasbΛyαtαh′e(xe|qxu′)atfuionnction. The
Ga (xx′) = Ga(x′ x). (10) |
−1,−1 | − |
Inthepresenceofstrongcorrelationsoff electronsthe Λαα′(x|x′)=Gfα(α0′)(x|x′)+Zαα′(x|x′), (13)
−
(GWT)containsadditionaltermsnamelytheirreducible where the function Zαα′(xx′) contains the contribution
one-site many-particle Green’s functions or Kubo cumu- of strongly connected diag|ram based on the irreducible
lants of the form (x=x,σ,τ): many-particle Green’s functions.
The strong connected part of the c electron propa-
G(n0)ir[α1,x1;...;α2n,x2n]= Tfxα11...fxα22nn i0r = (11) gatorwithout the externallines is dete−rmined by us as a
=δx1x2...δx1x2n Tfσα11(cid:10)(τ1)...fσα22nn(τ2(cid:11)n i0r, misadsesnoopteedraatsorΣfαoαr′(uxncxo′r).related electrons. This quantity
(cid:10) (cid:11) |
A simple relation exists between these two functions:
whereinsimplesttwo-particlecasewehavethefollowing
definition of the irreducible function
Σαα′(xx′)=V2αα′Λαα′(xx′) (14)
| |
Tfα1(τ )fα2(τ )fα3(τ )fα4(τ ) ir = (12)
σ1 1 σ2 2 σ3 3 σ4 4 0 Theanalysisofthe propagatordiagramspermits usto
=(cid:10)Tfα1(τ )fα2(τ )fα3(τ )fα4(τ )(cid:11) formulate the following Dyson equation for uncorrelated
σ1 1 σ2 2 σ3 3 σ4 4 0 −
[ Tfα(cid:10)1(τ )fα2(τ ) Tfα3(τ )fα4(τ )(cid:11) + electron propagator
− σ1 1 σ2 2 0 σ3 3 σ4 4 0
+(cid:10)Tfα1(τ )fα4(τ )(cid:11) (cid:10)Tfα2(τ )fα3(τ )(cid:11)
For n−(cid:10)≥T(cid:10)f3σα1tσ1h1(τe1)i1rfrσαe3σ3d4(uτc3i)4b(cid:11)l0(cid:11)e0(cid:10)GT(cid:10)rfeσαe2σ2n2(’τs2)f2fuσnα4σc43(tτio4n)3(cid:11)0c(cid:11)]o0.nt−ains in Gcαα′(x|x′) += GαX1cαα(α20′)X(1x2|xG′)cα(α+01)(x|1)Σα1α2(1|2)Gcα2α′(2|(x1′5)).
its right-hand part besides the products of one-particle
propagatorsalsotheirproductswithirreduciblefunctions AtthesametimewecanformulatetheDyson-typeequa-
ofsmallernumberofparticles. Therearepresentalsothe tion for correlatedelectron propagator Gf:
product of irreducible functions G(0)ir and G(0)ir with
condition n1+n2 =n and so on [10n−113]. n2 Gfαα′(x|x′)=Λαα′(x|x′)+(16)
Asaresultofapplyingthesetheoremweobtainforthe +V2 α α Λ (x1)Gc(0) (12)Gf (2x′).
renormalized conduction electron propagator the contri- αX1α2X12 1 2 αα1 | α1α2 | α2α′ |
butions depicted on the Fig. 1 The contributions of the
diagrams Fig. 1 b) and e) are the following In equation (15) and (16) as in the previous equations
which contain repeated indices (1,2) is supposed sum-
V2αX1α2X12 α1α2Gcα(α01)(x|1)Gfα(10α)2(1|2)Gcα(20α)′(2|x′), mintaetgiornatiboynsbiytetsiminedvicaersiab(1le,s2()τ,1s,pτi2n)iinndthiceeisnt(eσr1v,aσl2()0,aβn)d.
V4 Unfortunate the Dyson-type equations far correlation
2 α1α2α3α4Gcα(α01)(x|1)Gcα(20α)3(2|3)× functionΛαα′ andmassoperatorΣαα′ don’texist. There-
α1X...α41X...4 forethecalculationofthec andf renormalizedpropa-
− −
Gc(0) (4x′) Tfα1f−α2fα3f−α4 ir gatorsneedstheapproximationsbasedonthesummation
× α4α′ | 1 2 3 4 0 of special classes of diagrams.
(cid:10) (cid:11)
correspondingly. It demonstrates the dependence of the On the Fig. 3 the skeleton diagrams for the correla-
diagrams from the charge quantum number α. tionfunctionΛαα′(xx′)aredepicted. Theydemonstrate
|
The contributions of perturbation theory for f elec- impossibility to formulate Dyson-type equation for this
tron propagator are depicted on the Fig. 2 The−contri- function. The number of skeleton diagrams depicted on
butions of diagrams Fig. 2 b) and c) are equal to the Fig. 3 for the correlationfunction is infinite.
The contribution of the diagrams Fig. 3 b) and c) is
V2 α α Gf(0)(x1)Gc(0) (12)Gf(0)(2x′), the following:
1 2 αα1 | α1α2 | α2α′ |
V22 αXα1Xα1α22X1X212α1α2Gcα(10α)2(1|2)DTfxαf1−α1f2α2fx−′α′Ei0r, 21αX11α2X12 α1α2Gcα1α2(1|2)DTfxαf1−α1f2α2fx−′α′Ei0r,
α α α α Gc (12)Gc (34)
correspondingly. −8α1X...α41X...4 1 2 3 4 α1α2 | α3α4 | ×
torBtehtwereeeanrtehsetrdoianggraamndswfoeratkhecoonnnee-pctaerdticolneefs.−Tphreopwaegaak- ×DTfxαf1−α1f2α2f3−α3f4α4fx−′α′Ei0r,
connecteddiagramscanbeseparatedintwopartsbycut-
tingonepropagatorline. Thesumofallstrongconnected correspondingly.
4
Gcαα′(x|x′)= +V2 +V4 +
αx −α′x′ αx 1 2 −α′x′ αx 1 2 3 4 −α′x′
a) b) c)
2 3
V4
+V6 + +
2
αx 1 2 3 4 5 6 −α′x′ αx 1 4 −α′x′
d) e)
3 4
+ V6 2 5 + V6 4 5 +
2 αx 1 6 −α′x′ 2 αx 1 2 3 6 −α′x′
f) g)
3 4
2 3
+ V6 − V6 2 5 + ...
2 αx 1 4 5 6 −α′x′ 8 αx 1 6 −α′x′
h) i)
FIG.1: Thefirstsix ordersofperturbationtheoryforconduction electron propagator. Thesolid anddashedthinlinesdepict
zero order propagators for c− and f− electrons correspondingly. The rectangles depict theirreducible Green’s functions. The
points of diagram are the vertices with α and V contributions.
1 2
Gf (x|x′)= + V2 +V2 +
αα′ αx a−)α′x′ αx 1 b) 2 −α′x′ 2 αx c) −α′x′
1 2
V4
+ V4 + +
αx 1 2 3 4 −α′x′ 2 αx 3 4 −α′x′
d) e)
2 3
3 4
+ V4 − V4 1 4 + ...
2 8
αx 1 2 −α′x′
f) αx −α′x′
g)
FIG. 2: The contributions of the first four orders of perturbation theory for the f− electron propagator.
Ifwetakeintoaccountonlythe firsttermoftheright- the condition of equality to zero of the sums of all α
−
hand part of Fig. 3 we obtain the simplest Hubbard I indices of every dynamical quantity. For example such
approximationwith considerationonly of the chain-type transitionof the diagramFig. 3 b) is conditioned by the
diagrams. equalities α α = 0 and α α +α α′ = 0 with
1 2 1 2
solution α =−α and α=α′. − −
The diagram Fig. 3 b) is the simplest contribution 1 2
tocorrelationfunctionwhichtakesintoaccounttheelec-
Thesummationbyα givesustwoequalcontributions
1
troniccorrelations. ThediagramsFig. 3b),c)andd)are
and the coefficient before diagram increases twofold and
localized and their Fourier representations in real space becomes 1 instead originally 1.
areindependentofmomentum. Therearealsootherdia- 2
gramsofthiskindwithirreduciblefunctionsG(0)ir,G(0)ir In normal state the correlation function Λ(xx′) has a
5 6 |
andsoon. Thecoefficientsbeforethese diagramsarede- form depicted on the Fig. 4.
termined by the number 1 , where 2n is the perturba-
2nn! The new coefficient before the diagrams take into ac-
tion theory order of diagram. The last diagram of Fig.
count the existence of different possibilities of transition
3 is not local and its Fourier representation depends of
from superconducting to normal state.
themomentum. Todynamicalmeanfieldtheoryonlythe
first group of local diagrams of Fig. 3 correspond.
After discussionofthe propagatorspropertieswe shall
The transition of the diagram contribution from su- proceed to the main part of our paper and investigate
perconducting version to the normal one is realized by the properties of evolution operator average.
5
3 4
1 2
Λ (x|x′)= + V2 − V4 1 2 +
αα′ αx −α′x′ 2 αx −α′x′ 8
a) b) αx −α′x′
c)
5 6
3 4
1 2
V6 3 4 − V6 + ...
+ 6
48 1 2 αx 5 6 −α′x′
e)
αx −α′x′
d)
FIG. 3: The skeleton diagrams for correlation function Λαα′(x|x′). The thin dashed line is zero-order f− electron Green’s
function. Therectanglesdepictthemany-particlesirreducibleGreen’sfunction. Thedoublesolidlinesdepicttherenormalized
conduction electron Green’s function Gcαα′(x|x′).
Gc
Gc
3 Gc 4
1 2
Gf(0) V4
Λ(x|x′)= + (−1)V2 − 1 2 −
2
x x′
x x′
x x′
Gc Gc
5 Gc 6
3 Gc 4
Gc 1 2
V6 3 4 − V6 Gc + ...
− 6
6 1 2 x 5 6 x′
x x′
FIG. 4: Correlation function Λ(x|x′) in normal state. All thelines correspond to normal propagators and havea direction of
propagators and of thearrays in thevertices.
IV. VACUUM DIAGRAMS In normal state the diagrams of Fig. 5 are changed.
The direction of propagator lines and of the arrays at
By using the perturbation theory we have obtained vertices points appear together with new coefficients be-
for the connected part of evolution operator average the forethediagrams. Thischangingisdemonstratedonthe
contributions depicted in the Fig. 5. Fig. 6, where some ofthe diagramsof Fig. 5 are demon-
The contributions of the first, third and fourth dia- strated. Vacuum diagrams in superconducting and nor-
grams of Fig. 5 are enumerated below: mal states contain the factor n1, where n is the order
of perturbation theory in which given diagram appears.
V2
α α Gc(0) (12)Gf(0) (21), Thisfactormakesdifficulttheinvestigationofthiscontri-
− 2 1 2 α1α2 | α2α1 | butions. Inordertoremovethiscoefficientitisnecessary
αX1α2X12
to use the trick of integration by constant of interaction
V4
α α α α Gc(0) (12)Gc(0) (34) V. The result of such integration is depicted on the
8 α1X...α41X...4 1 2 3 4 α1α2 | α3α4 | × Fig. 7.
Tf−α1fα2f−α3fα4 ir, Onthebaseofseriesexpansionsforrenormalizedprop-
× 1 2 3 4 0 agators of the conduction c electrons (see Fig. 1), lo-
V6 (cid:10) (cid:11) calized f electrons (see F−ig. 2) and definition of the
−48 α1X...α61X...6α1α2α3α4α5α6Gcα(10α)2(1|2)Gcα(30α)4(3|4)× ccoornrterliabtuiot−inonfuonfctthioeninΛteαgαr′a(nxt|xo′)f Fwieg.c7aninpervoevreytohradterthoef
Gc(0) (56) Tf−α1fα2f−α3fα4f−α5fα6 ir. perturbation theory can be presented itself as the prod-
× α5α6 | 1 2 3 4 5 6 0
(cid:10) (cid:11)
6
1 2 3 4
2 3 2 5
hU(β)ic0= −V22 1 2 −−VV4444 +V84 −V486 −
1 4 1 6
4 3
1 2
1 2
3
4 8 3
2 +
−V6 6 3 +V6 −V8
6 4 8
7 4
1
5
6
5 4 6 5
4
3 4 4 5 3
5 6 3 6
2 5
+V8 +V8 +V8 −
48 7 8 48 4
1 2 2 1 8 7 1 8 6
7
5 6
5 6 4 5
3 4 7
−V8 +V8 4 7 −V8 23 76 +...
16 16 16
3 8 1 8
2 1 8
2 1
FIG. 5: Vacuum diagrams of first eight orders of perturbation theory in superconductingstate.
1 2
2 3 3 4
hU(β)ic0normal= −V2 1 2 −V24 +V24 +V66 2 5 −
1 4 1 6
4 3
1 2
1 2
4 5
8 3
+
−V6 6 3 +V8 3 6 −V8
3 24 2 7 4 7 4
1 8
5 4 6 5
3 4 2 3 4 5
5 6
+V8 −V8 +...
8 7 8 2
1 2 1 8 7 6
FIG. 6: Some of thevacuum diagrams in normal state of the system.
uct of some contribution from Gc and some one from Λ. isfy this condition and all of them must be taken into
If the contribution of Gc is of n order of perturbation account.
1
theory and of Λ of n order when the order of U(β) c
2 h i0
is equal to n with the condition n +n +2 = n which For example there are three possibilities to compose
1 2
mustbe satisfied. There aredifferent possibilities to sat- from Gc and Λ the sixth diagrams of Fig. 7. These pos-
sibilities are enumerated below on the Fig. 8. Other ex-
7
1 2 3 4
2 3 2 5
hU(β)ic0=V0R dλλ{−λ2 1 2 −λ4 +λ24 −λ86 −
1 4 1 6
4 3
1 2
1 2
3
4 8 3
2 +
−λ6 6 3 +3λ6 −λ8
2
7 4
1
5
6
5 4 6 5
4
3 4 4 5 3
5 6 3 6
2 5
+λ8 +λ8 +2λ8 −
6 7 8
1 2 2 1 8 7 1 8 6
7
5 6
5 6 4 5
3 4 7
−λ8 +λ28 4 7 −λ488 23 76 +...}
3 8 1 8
2 1 8
2 1
FIG. 7: Vacuumdiagrams of thesystem in superconductingstate after integrating by interaction constant.
ampleofvacuumdiagramofeighthorderofperturbation demonstrate the generalstatement that the integrandof
theoryispresentedontheFig. 9. Onlyallthethreepos- the evolution operator averagecan be presented itself as
sibilitiesgiveusthecorrectcoefficient 1 1 1 = 1 of a product of λ2GcΛ of the form
−8−4−8 −2
diagram. Otherpossibilitiesdon’texist. Theseexamples
V
dλ
hU(β)ic0 =−Z λ αα′λ2Gcαα′(x|x′|λ)Λα′α(x′|x|λ)= (17)
Xαα′Xxx′
0
V V
dλ dλ
=−Z λ Gcαα′(x|x′|λ)Σα′α(x′|x|λ)=−Z λ Tr(Gˆc(λ)Σˆc(λ)).
Xαα′Xxx′
0 0
where the operators Gˆc(λ) and Σˆc(λ) have the matrix This expression for renormalized thermodynamic poten-
elementsGc (xx′ λ)andΣαα′(xx′ λ)correspondingly. tialofthe stronglycorrelatedsystemcontainsadditional
αα′ | | | |
Index λ underline that these quantities depend of the integration over the integration strength λ and is awk-
auxiliary constant of integration λ. ward because it. Equation (18) generalizes the result of
Therefore the thermodynamic potential of our system LuttingerandWard[24] provedfornon-correlatedmany-
F is equal to electron system in normal state.
Our generalization has been obtained for the case of
V strongcorrelationsofspecialkindwhichcontainsoneun-
1 dλ
F =F + Tr[Gˆc(λ)Σˆc(λ)]. (18) correlatedsubsystemandonestronglycorrelatedandwe
0 β Z λ
admit also the existence of superconductivity in both of
0
8
Vacuum diagram Gc Λ
λ4
2
λ2 λ2
3λ6
2
2
λ4
2
FIG. 8: Three possibilities to organize thevacuum diagram of sixth order of perturbation theory. The correct coefficient 3 is
2
obtained bysumming all thepossibilities.
them. tions. We shall use the skeleton diagrams of strongly
Luttinger and Ward have proved the possibility to correlated system which differ essential from Luttinger
transformthisexpressionintomuchmoreconvenientfor- and Ward [24] case and transform equation (18) to more
mula without such integration. For that they used a convenient form.
specialfunctionalconstrictedfromskeletondiagramsthe Inourstrongcorrelatedcaseweintroducethefollowing
linesofwhicharetherenormalizedelectronGreen’sfunc- functional
1
Y = Tr ln(Gˆc(0)Σˆ 1)+GˆcΣˆ +Y′, (19)
−2β { − }
which is the generalization of the Luttinger-Ward [24] dence of full Green’s function Gc(V). The contributions
equation just for the strongly correlated systems. Here of the diagrams a), b) and c) are following
operation Tr use the summation by α,σ,i and integra-
tion by τ.
V2
The quantity Y′ contains all peculiarities of the α α Gf(0) (1,2V)Gc (2,1V),
strongly correlated systems and is presented itself as a 2 αX1α2X12 1 2 α1α2 | α2α1 |
sumofclosedlinkedskeletondiagrams,constructedfrom
V4
irreducible Green’s functions of correlated electrons and α α α α Gc (1,2V)Gc (3,4V)
− 8 1 2 3 4 α1α2 | α3α4 | ×
full Green’s functions of uncorrelated electrons. α1X...α41X...4
On the Fig. 10 are depicted some of simplest skeleton Tf−α1fα2f−α3fα4 ir,
diagrams for functional Y′. These diagrams depend of × 1 2 3 4 0
V6 (cid:10) (cid:11)
the interaction strength V not only through the factors α α α α α α Gc (1,2V)Gc (3,4V)
in the frontofeachdiagrambut alsothroughthe depen- 48 α1X...α61X...6 1 2 3 4 5 6 α1α2 | α3α4 | ×
9
Vacuum diagram Gc Λ
−λ6
4
−λ8 λ2 −λ4
2
8
−λ6
8
FIG. 9: Three possibilities to obtain one of thevacuum diagram of eight order of perturbation theory.
3 4
2 3 2 5
V2 −V4 −V6 −
βY′ = 1 2 8 48
2
1 4 1 6
a) b) c)
4 5
3 4
5 6
−V8 −V8 23 76 +...
48 7 8 384
1 2 1 8
d) e)
FIG. 10: Skeleton diagrams for functional Y′.
Gc (5,6V) Tf−α1fα2f−α3fα4f−α5fα6 ir. from V we shall obtain the other property:
× α5α6 | 1 2 3 4 5 6 0
(cid:10) (cid:11)
From Fig. 3 and Fig. 10 it is possible to demonstrate dY′
the following equation: Vβ dV |Gc= Gcα1α2(2,1|V)Σα1α2(1,2|V)=Tr[ΣˆGˆc].
αX1α2X12
δβY′ α α 1 (21)
= 1 2V2Λ (2,1)= Σ (2,1). Becauseofthenecessitytohavethefunctionalderivatives
δG (1,2) 2 α2α1 2 α2α1
α1α2 over mass operator Σ we shall use the Dyson equation
(20) (15) rewritten in the form [x=(α,x,σ,τ)]:
Ifwetakeintoaccountonlytheexplicitdependenceofthe
functional Y′ of interaction constant V without consid-
eringthedependence ofthe fullGreen’sfunctions Gc(V) Gc(0)−1(x,x′)=Gc−1(x,x′)+Σ(x,x′) (22)
10
We obtain V. HEAT CAPACITY [25]
δGc(x,x′)
=Gc(x,y)Gc(y′,x′), (23) As a illustration of our results we shall consider the
δΣ(y,y′)
problem of finding the heat capacity of our strongly cor-
and related system in normal state and at the low tempera-
tures.
δ
Trln(1 Gˆc(0)Σˆ)= Gc(y′,y), The heat capacity at constant volume V is equal to
δΣ(y,y′) − −
δ ∂S
δΣ(y,y′)Tr(GˆcΣˆ)=Gc(y′,y)+(GˆcΣˆGˆc)y′y (24) CV =T(∂T)V, (34)
where the entropy S is given by
By summing these equations we obtain:
∂F
δ 1
( )Tr ln(Gˆc(0)Σˆ 1)+GˆcΣˆ = S = T( )µ,V. (35)
δΣ(y,y′) −2β { − } − ∂T
=−21β(GˆcΣˆGˆc)y′y. (25) TTh.e quantity µ is the chemical potential at temperature
At low temperature we may expand F(µ,V,T) and µ
OnthebaseofdefinitionofthefunctionalY′ Fig. 10and
in even powers of the temperature:
equation (20) we have
1
δΣδ(yY,′y′) =Xxx′ δGcδ(Yx,′x′)δδGΣc((yx,,yx′′)) = 21β(GˆcΣˆGˆc)y′y. µ(TF) == Fµ0(µ+0,µV′T,02)+−..2.,γ(µ0,V)T2+...,
(26)
As a result we obtain the stationarity property of the whereµ0 is the value ofchemicalpotentialforcorrelated
functional Y: system at T =0.
As a result we have
δY
=0. (27)
δΣ(y,y′) C =γ(µ ,V)T +... (36)
V 0
Now we shall discuss the derivative over interaction
the linear dependence of the heat capacity of the tem-
constant V of functional Y. We shall taken into account
perature at low its values. Therefore to evaluate the co-
the stationarity Y about Σˆ and Gˆc and equation (21).
efficient γ(µ ,V) it is necessary to obtain the expansion
0
We obtain:
of F in powers of T by using the expression (31), (19)
dY ∂Y δY ∂Σ ∂Y′ Tr(ΣˆGˆc) and Fig.10 for Y′ obtained by us in previous part of the
V =V +V =V = (.28) paper.
dV ∂V |Σ δΣ∂V ∂V |Σ β
We know that the expression (19) for Y and (31) for
From equation (18) we have: thermodynamic potential is stationary with respect to
changes in the proper self-energy Σk(iω0). Therefore be-
dF Tr(GˆcΣˆ) cause we are interested in the first corrections to F, we
V = , (29) canneglectthe explicit temperature dependence ofmass
dV β
operatorΣ(kiω) and Gc(kiω) andreplace them by val-
and as a consequence we establish uesΣ(kiω )| andGc(k|iω ) calculatedatT =0.
0 T=0 0 T=0
| | | |
Thus the first correction to the T = 0 value of F (31)
dF dY
V =V , (30) comes only from the difference between the ωn sums in
dV dV expression(31)andwhatwewouldgetifwereplacethem
with the solution by integrals according to the equation
F =Y +const. (31) 1 1 ∞
= dω. (37)
β 2π Z
This constant is F0. Therefore Xωn −∞
F =F0+Y, (32) Now it is necessary to consider the functional Y′ on
Fig.10. Since each line of a skeleton diagram of func-
with the stationary property
tional Y′ contains an ω sum, the total first correction
n
δF to Y′ is obtained by correcting the computation in each
=0, (33)
δΣ(x,x′) diagram for a single line and use equation (37) for the
other ω sums of the diagram, finally summing over ev-
n
as in superconducting and in normal states. ery line.