Diagram technique for the Hubbard model. Ladder diagram summation

A new diagram technique based on the generalized Wick theorem has been elaborated for the systems with strong electronic correlations. Coulomb repulsion of the electrons of the Hubbard model is considered as the main part of Hamiltonian and is taken into account in a zero order approximation. Th...

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Datum:	1998
Hauptverfasser:	Moskalenko, V.A., Kon, L.Z.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 1998
Schriftenreihe:	Condensed Matter Physics
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spelling	irk-123456789-1186282017-05-31T03:08:19Z Diagram technique for the Hubbard model. Ladder diagram summation Moskalenko, V.A. Kon, L.Z. A new diagram technique based on the generalized Wick theorem has been elaborated for the systems with strong electronic correlations. Coulomb repulsion of the electrons of the Hubbard model is considered as the main part of Hamiltonian and is taken into account in a zero order approximation. The hopping matrix elements are considered as a perturbation. One-particle Matsubara-Green function of the model has been investigated and Dyson equation has been obtained. New elements of the theory which are characteristic of this new approach are the local many-particle irreducible Green functions, or Kubo cumulants. They become zero when Coulomb interaction is zero. The main task of this paper is the summing of ladder diagrams which take into account the most essential charge and spin fluctuations of the system. The integral equations, which sum such diagrams, have been established for two different channels. The coherent potential approximation has been used to simplify and solve these equations. On this basis a metal-dielectric phase transition has been investigated. На основі узагальненої теореми Віка розроблено нову діаграмну техніку для систем з сильними електронними кореляціями. Кулонівське відштовхування електронів в моделі Хаббарда розглядається як головна частина гамільтоніана і враховується в нульовому наближенні. Матричний елемент міжвузлового перескоку електронів враховується як збурення. Досліджено одночастинкову мацубарівську функцію Гріна й отримано рівняння Дайсона. Новими елементами теорії, що характерні для даного наближення, є багаточастинкові незвідні функції Кубо. Вони обертаються в нуль, коли кулонівська взаємодія рівна нулю. Головна мета даної роботи – це сумування драбинкових діаграм, що враховують найбільш суттєві спінові і зарядові флуктуації системи. Отримані інтегральні рівняння для двох каналів розсіяння. Метод когерентного потенціалу використовується для спрощення і розв’язування цих рівнянь. На цій основі встановлена умова переходу метал-діелектрик. 1998 Article Diagram technique for the Hubbard model. Ladder diagram summation / V.A. Moskalenko, L.Z. Kon // Condensed Matter Physics. — 1998. — Т. 1, № 1(13). — С. 23-29. — Бібліогр.: 11 назв. — англ. 1607-324X PACS: 71.28.+d, 71.27.+a DOI:10.5488/CMP.1.1.23 http://dspace.nbuv.gov.ua/handle/123456789/118628 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	A new diagram technique based on the generalized Wick theorem has been elaborated for the systems with strong electronic correlations. Coulomb repulsion of the electrons of the Hubbard model is considered as the main part of Hamiltonian and is taken into account in a zero order approximation. The hopping matrix elements are considered as a perturbation. One-particle Matsubara-Green function of the model has been investigated and Dyson equation has been obtained. New elements of the theory which are characteristic of this new approach are the local many-particle irreducible Green functions, or Kubo cumulants. They become zero when Coulomb interaction is zero. The main task of this paper is the summing of ladder diagrams which take into account the most essential charge and spin fluctuations of the system. The integral equations, which sum such diagrams, have been established for two different channels. The coherent potential approximation has been used to simplify and solve these equations. On this basis a metal-dielectric phase transition has been investigated.
format	Article
author	Moskalenko, V.A. Kon, L.Z.
spellingShingle	Moskalenko, V.A. Kon, L.Z. Diagram technique for the Hubbard model. Ladder diagram summation Condensed Matter Physics
author_facet	Moskalenko, V.A. Kon, L.Z.
author_sort	Moskalenko, V.A.
title	Diagram technique for the Hubbard model. Ladder diagram summation
title_short	Diagram technique for the Hubbard model. Ladder diagram summation
title_full	Diagram technique for the Hubbard model. Ladder diagram summation
title_fullStr	Diagram technique for the Hubbard model. Ladder diagram summation
title_full_unstemmed	Diagram technique for the Hubbard model. Ladder diagram summation
title_sort	diagram technique for the hubbard model. ladder diagram summation
publisher	Інститут фізики конденсованих систем НАН України
publishDate	1998
url	http://dspace.nbuv.gov.ua/handle/123456789/118628
citation_txt	Diagram technique for the Hubbard model. Ladder diagram summation / V.A. Moskalenko, L.Z. Kon // Condensed Matter Physics. — 1998. — Т. 1, № 1(13). — С. 23-29. — Бібліогр.: 11 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT moskalenkova diagramtechniqueforthehubbardmodelladderdiagramsummation AT konlz diagramtechniqueforthehubbardmodelladderdiagramsummation
first_indexed	2025-07-08T14:20:53Z
last_indexed	2025-07-08T14:20:53Z
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fulltext	Condensed Matter Physics, 1998, Vol. 1, No 1(13), p. 23–39 Diagram technique for the Hubbard model. Ladder diagram summation V.A.Moskalenko 1,2 , L.Z.Kon 2 1 Joint Institute for Nuclear Research, 141980 Dubna, Russia 2 Institute of Applied Physics, 5 Grosul St., 277028 Kishinev, Moldova Received March 20, 1998 A new diagram technique based on the generalized Wick theorem has been elaborated for the systems with strong electronic correlations. Co- ulomb repulsion of the electrons of the Hubbard model is considered as the main part of Hamiltonian and is taken into account in a zero order ap- proximation. The hopping matrix elements are considered as a perturba- tion. One-particle Matsubara-Green function of the model has been investi- gated and Dyson equation has been obtained. New elements of the theory which are characteristic of this new approach are the local many-particle irreducible Green functions, or Kubo cumulants. They become zero when Coulomb interaction is zero. The main task of this paper is the summing of ladder diagrams which take into account the most essential charge and spin fluctuations of the system. The integral equations, which sum such di- agrams, have been established for two different channels. The coherent po- tential approximation has been used to simplify and solve these equations. On this basis a metal-dielectric phase transition has been investigated. Key words: generalized Wick theorem, ladder diagrams, Hubbard model PACS: 71.28.+d, 71.27.+a 1. Introduction In the previous papers [1-5] a new diagram technique for the systems with strong electron correlations was elaborated. Such systems contain as the main part of their Hamiltonian on-site Coulomb repulsion of electrons. This interaction has to be taken into account in the zero order Hamiltonian of such systems. The other terms of the Hamiltonian, such as the hopping matrix elements of the Hubbard model [6] or hybridization of electron states of the periodic Anderson model [7] are treated as perturbations. Because the zero order Hamiltonian which contains the above mentioned Cou- lomb interaction can be diagonalized by using the Hubbard and not a free electron operators, the ordinary Wick theorem proposed in the weak coupling field theory c© V.A.Moskalenko, L.Z.Kon 23 V.A.Moskalenko, L.Z.Kon for disentangling chronological averages of an electron operator is not valid. Instead of a weak coupling, the Generalized Wick Theorem (GWT) has been proposed for the first time for a one-band Hubbard model with the Hamiltonian H = H0 +Hint, H0 = ∑ i H0 i , H0 i = −µ ∑ σ niσ + Uni↑ni↓ , Hint = ∑ ijσ t (j − i) c+jσciσ, niσ = c+iσciσ, (1) where c+jσ(cjσ) are an electron creation (annihilation) operator with spin σ and site index j. Here U is Coulomb repulsion of an electron, µ - a chemical potential of the system and t(j − i) - a matrix element of hopping. As the zero order density matrix is factorized on the site indices, all the calcu- lations of the thermodynamic perturbation theory contributions are made in the local presentation. The sum of such contributions gives us the effect of delocaliza- tion and renormalization of the dynamical functions. The new elements of the GWT are one-site many-particle irreducible Green functions G (0)ir n or Kubo cumulants. These quantities contain all spin and charge fluctuations of a strong correlated system. They are identically equal to zero when Coulomb interaction is zero. The next simple example can be useful to elucidate this question. We apply the GWT to the following chronological average: 〈T c(x1)c(x2)c(x3)c(x4)〉0 = 〈T c(x1)c(x4)〉0〈T c(x2)c(x3)〉0− −〈T c(x1)c(x3)〉0〈T c(x2)c(x4)〉0 +G (0)ir 2 [x1x2\|x3x4] . (2) Here x stands for x, σ, τ. All the quantities of the right-hand side of this equation are local. The zero order or the local approximation of the Matsubara one-particle Green function is equal to G (0) 1 (x1x2) = −〈T c(x1)c(x2)〉0 = δx1,x2 G(0) σ1σ2 (τ1 − τ2) G(0) σ1σ2 (τ1 − τ2) = −〈T cσ1 (τ1)cσ2 (τ2)〉0, where c(τ ) and c(τ ) are interaction presentations of electron operators and τ – imaginary time of thermodynamic Green functions. Symbol 〈...〉0 stands for the statistical average with a zero order density matrix. A two-particle irreducible Green function G (0)ir 2 is a local quantity: G (0)ir 2 [x1, x2\|x3x4] = δx1x2 δx1x3 δx1x4 G (0)ir 2 [σ1, τ1;σ2τ2\|σ3, τ3;σ4, τ4] (3) with the structure of Kubo cumulant: G (0)ir 2 [σ1, τ1;σ2, τ2\|σ3, τ3;σ4, τ4] = 〈T cσ1 (τ1)cσ2 (τ2)cσ3 (τ3)cσ4 (τ4)〉0− −〈T cσ1 (τ1)cσ4 (τ4)〉0〈T cσ2 (τ2)cσ3 (τ3)〉0 + 〈T cσ1 (τ1)cσ3 (τ3)〉0〈T cσ2 (τ2)cσ4 (τ4)〉0. (4) 24 Diagram technique for the Hubbard model This quantity is just a new element of the GWT because the first two terms of the right-hand side of equation (2) are the ordinary Wick theorem contributions. The signs of these terms, and in general of all of them, are determined by the number of permutation P which is necessary to obtain the final from the initial order of electron operators. The Hubbard operators appear only at the last stage of such an investigation – when the local quantities are calculated. In the Generalized Wick theorem for Hubbard operators Xnm i , used in papers [8-10], the electrons operators are expressed with their aid at the initial stage of the investigation. Because the Hubbard operators have a more complicated algebraical structure, the diagram technique for them is more complicated. Now we shall consider our approach. In the higher orders of the perturbation theory there appear more complicated irreducible structures. For example, irre- ducible functions G (0)ir n with the number of particles n = 3, 4... will appear. Then, different products of irreducible functions G (0)ir m1 G (0)ir m2 G (0)ir m3 will be obtained, where the sum m1 +m2 +m3 = n is equal to the number of particles that participate in the process of delocalization. We can formulate the GWT in the following form: G(0) n (x1...xn\|x′ 1...x ′ n) = (−1)n〈T c(x1)...c(xn)c(x ′ 1)...c(x ′ n)〉0 = ∑ {P} (−1)PG(0)(x1\|x ′ 1)....G (0)(xn\|x′ n) + ∑ {P} (−1)P ∑ m1>1,m2>1, m1+m2+...=n G(0)ir m1 [x1...xm1 \|x′ 1...x ′ m1 ]× ×G(0)ir m2 [xm1+1...xm1+m2 \|x′ m1+1...x ′ m1+m2 ] +G(0)ir n [x1...xn\|x′ 1...x ′ n]. (5) Figure 1. Diagrams of the first two or- ders of the perturbation theory for one- particle Green-function. Diagram a) is of the zero order, b) of the first and c) and d) of the second order contributions. a), b) and c) diagrams are of a chain type and d) of a new kind, containing the first Kubo cumulant G (0)ir 2 . In (5) we mean that G (0)ir 1 (xi\|x′ i) = G(0)(xi\|x′ i). The first term of the right-hand side of (5) is of a usual Wick kind, but all the next ones are characteristic of strongly correlated systems. The diagrammatic rules for writing down the contributions of the pertur- bation theory series have been formu- lated in [1-2]. One-particle Green func- tion G(0)(x1\|x′ 1) is represented by a thin solid line directed from x′ 1 to x1 and the local quantity G (0)ir n [x1...xn\|x′ 1...x ′ n] – by a rectan- gle which surrounds n arrows directed to n points x1...xn and n arrows origi- nated from x′ 1...x ′ n. In these diagrams the hopping ma- trix elements are represented by thin dashed lines, the delocalized electron Green function – by a full solid line. 25 V.A.Moskalenko, L.Z.Kon In figure 1 the one-particle Green function’s diagrams of the first two pertur- bation theory orders is represented: The sign of the diagram 1d) corresponds to the enumeration of rectangle ar- guments G (0)ir 2 [x1\|2′x′]. We can see from the first three diagrams of figure 1 that the chain iterative process leads to the ordinary Dyson equation with hopping matrix element t(1′−1) as a self-energy. But new diagrams containing charge and spin fluctuations give rise to an ad- ditional renormalization process resulting in the necessity to introduce a new ir- reducible strongly linked function Z(x\|x′) in the theory. This function is the sum of all irreducible contributions which cannot be broken into two parts by cutting a single hopping line. In the second order of the perturbation theory the diagram of figure 1 d) is a contribution to this function Z(2)(x\|x′ ). In paper [1] it was proved that by introducing function Λ(x\|x′) (x = x, σ, τ ) Λ(x\|x′) = G(0)(x\|x′) + Z(x\|x′) (6) we get the Dyson equation for the renormalized one-particle Green functionG(x\|x′ ) of the Hubbard model: G(x\|x′) = Λ(x\|x′) + ∑ x1,x2 Λ(x\|x1)t(x1 − x2)G(x2\|x′), (7) where t(x1 − x2) = t(x1 − x2)δσ1σ2 δ(τ1 − τ2) and summations stand for summing by discrete indices and integration by imaginary time. In the Fourier representation we have Gσ(k\|iω) = Λσ(k\|iω) 1− ξ(k)Λσ(k\|iω) . (8) Here k is the momentum of an electron, ωn = (2n + 1)π/β – Matsubara odd fre- quency and ξ(k)− Fourier component of the hopping matrix element and electron band energy. Equation (8), in distinction from (7), supposes diagonality by spin indices of all the quantities. If we sum only chain diagrams, without taking into account correlation contributions, we obtain the so-called Hubbard I approximation: GI σ(k\|iω) = G (0) σ (iω) 1− ξ(k)G (0) σ (iω) , (9) where G (0) σ (iω) is the local Hubbard-Green function of an electron. As it is well known, the energy spectrum in this approximation consists of two lower and upper Hubbard subbands. The complicacy of irreducible contributions to one-particle Green function makes impossible the obtaining of the exact result of a Dyson-type for Z(x\|x′) function. There are some possibilities to sum a class of irreducible diagrams which contain the most important information about the role of spin and charge fluctu- ations. 26 Diagram technique for the Hubbard model 2. Ladder diagrams The task of this paper is to demonstrate the possibility to sum a special class of irreducible diagrams which take into account two-particle correlation effects. In figure 2 we demonstrate some diagrams for Z(x\|x′ ) function which are es- sentially for strong correlated systems and will be summed. Figure 2. Ladder diagrams for irreducible Z(x\|x′ ) function. They repeat the infinite number of times the irreducible two-particle Green functions. The dashed lines represent renormalized hopping matrix elements. The rectangles in these diagrams stand for two-particle irreducible Green functions G (0)ir 2 . The renormalized hopping matrix element t̃(x−x ′ ) is the result of the summing of some elements of diagrams and are determined by the equation: t̃(x ′ − x) = t(x ′ − x) + ∑ x1,x2 t(x ′ − x1)G(x1\|x2)t(x2 − x), (10) Fourier representation ξσ(k\|ω) of quantity t̃(x ′ − x) is equal to ξσ(k\|iωn) = ξ(k) + ξ2(k)Gσ(k\|iωn) = ξ(k) 1− ξ(k)Λσ(k\|iωn) , (11) where ξ(k) is a tight-binding dispersion law of non-interacting electrons. For some of the diagrams in figure 2, especially for the diagram in figure 2a, the renormalization process is very important because this diagram with the un- renormalized hopping line is equal to zero, noting that t(x ′ = x) = 0. The other diagrams in figure 2 are delocalized (x 6= x ′ ) and their contributions are not zero when single dashed lines are used. 27 V.A.Moskalenko, L.Z.Kon In figure 2 we have two different kinds of ladder diagrams. One of them rep- resented by diagrams a), b),c) and the next diagrams of the higher order of the perturbation theory are repetitions of the process of scattering two electrons with the conservation of their sum of moments and frequencies. The second ladder shown by the diagram in figure 2d and the next diagrams represent the process of repeated electron-hole scattering with the conservation of their differences of momenta and frequencies. In the weak coupling theory of the solid state such a summation is known very well and is the realization of the Random Phase Approximation. The difference between the classical RPA and our ladder generalized RPA consists in the instant in the first case and the retarded in the second case character of electron scattering, which in our case is determined by four imaginary times of irreducible two-particles Green functions represented in the diagrams by the rectangles. To sum the first class of ladder diagrams of figure 2 for Z(x\|x′ ) function we shall investigate a more simple process of delocalization of two-particle irreducible Green functions represented in figure 3. This renormalized function is determined as Gir 2 . The graphical integral equation (figure 3b) for the delocalized irreducible Green function Gir 2 [x1,x2\|x′ 2,x ′ 1] has the form: Gir 2 [x1, x2\|x′ 2, x ′ 1] = G (0)ir 2 [x1, x2\|x′ 2, x ′ 1] + 1 2 ∑ 11 ′ 22 ′ G (0)ir 2 [x1,x2\|1 ′ , 2 ′ ] ×t̃(1 ′ − 1)t̃(2 ′ − 2)Gir 2 [2, 1\|x ′ 2, x ′ 1). (12) Here xi stands, as usual, for x1, σ1, τ1; 1− for 1, α1, θ1. The sum ∑ 1 stands for the summing by lattice sites 1, spin α1 and integration by imaginary time θ1 in the (0, β) interval. The site dependence of the renormalized Gir 2 function is partially diagonal, as it can be seen from figure 3a: Gir 2 [x1, x2\|x′ 2, x ′ 1] = δx1,x2 δx′ 1,x ′ 2 Gir 2 [x1 − x′ 1;σ1τ1;σ2τ2\|σ ′ 2τ ′ 2;σ ′ 1τ ′ 1]. (13) The delocalized irreductible Green function, which depends on relative lattice sites, obeys the equation: Gir 2 [x1 − x′ 1;σ1τ1;σ2τ2\|σ′ 2τ ′ 2;σ ′ 1τ ′ 1] = δ x1x ′ 1 G (0)ir 2 [σ1τ1;σ2τ2\|σ′ 2τ ′ 2;σ ′ 1τ ′ 1]+ + 1 2 β∫ 0 ... β∫ 0 dθ1dθ2dθ ′ 1dθ ′ 2 ∑ α1α2 ∑ 1 G (0)ir 2 [σ1τ1;σ2τ2\|α1θ ′ 1, α2θ ′ 2]× ×t̃α1 (x1 − 1\|θ′1 − θ1)tα2 (x1 − 1\|θ′2 − θ2)G ir 2 [1− x′ 1;α2, θ2;α1θ1\|σ′ 2, τ ′ 2;σ ′ 1, τ ′ 1]. (14) To advance the solution of this equation we introduce the momentum and fre- quency representation of our functions: G2[σ1τ1;σ2τ2\|σ′ 2τ ′ 2;σ ′ 1τ ′ 1] = 28 Diagram technique for the Hubbard model = 1 β4 ∑ ω1ω2ω 2 ′ ω 1 ′ G2[σ1, iω1;σ2, iω2\|σ′ 2, iω ′ 2;σ, iω ′ 1]e −iω1τ1−iω2τ2+iω′ 2τ ′ 2+iω′ 1τ ′ 1, Z (x\|x′) = 1 βN ∑ k ∑ ω Zσσ1(k\|iω)e−ik(x−x′)−iω(τ−τ ′)). (15) That permits us to rewrite equation (14) in the form: Figure 3. Ladder diagrams (figure 3a) for the delocalized two-particle irreducible Green function Gir 2 and the graphical integral equation (figure 3b) for their sum- ming. The rectangles with full lines stand for delocalized and with thin lines – for localized irreducible functions. Gir 2,q [σ1, iω1;σ2, iω2\|σ′ 2, iω ′ 2;σ ′ 1, iω ′ 1] = G (0)ir 2 [σ1, iω1;σ2, iω2\|σ′ 2, iω ′ 2;σ ′ 1, iω ′ 1]+ + 1 2β2 ∑ α1α2 ∑ ω′′ 1 ,ω ′′ 2 G (0)ir 2 [σ1, iω1;σ2, iω2\|α1, iω ′′ 1 ;α2, iω ′′ 2 ]× 29 V.A.Moskalenko, L.Z.Kon ×ξ2,q[α1, iω ′′ 1 ;α2, iω ′′ 2 ]G ir 2,q[α2, iω ′′ 2 ;α1, iω ′′ 1 \|σ′ 2, iω ′ 2;σ ′ 1, iω ′ 1], (16) where ξ2,q[α1, iω ′′ 1 , α2, iω ′′ 2 ] = 1 N ∑ q1 ξα1 (q1\|iω′′ 1) ξα2 (q − q1\|iω′′ 2) . (17) All the irreducible functions demonstrate spin and frequency conservation accord- ing to the fact that the total spin and frequency of the annihilated electrons are equal to those of the created electrons: G (0)ir 2 [σ1,iω1;σ2,iω2\|σ3,iω3;σ4,iω4] = βδ(ω1 + ω2 − ω3 − ω4)× G̃ (0)ir 2 [σ1, iω1;σ2, iω2\|σ3, iω3;σ4, ω4] = δσ1σ3 δσ2σ4 [K1[σ1, iω1;σ2, iω2\|σ1, iω3;σ2, iω4]+ β2δ(ω1 + ω2 − ω3 − ω4)δ(ω1 − ω3)G̃ (0) σ1 (iω1)G̃ (0) σ2 (iω2)] + δσ1σ4 δσ2σ3 × [K2[σ1iω1;σ2iω2\|σ2iω3;σ1iω4]− β2δ(ω1 + ω2 − ω3 − ω4)δ(ω1 − ω4)× G̃(0) σ1 (iω1)G̃ (0) σ2 (iω2)]− σ1σ3δσ2,−σ1 δσ4,−σ3 K3 [σ1, iω1;−σ1, iω2\|σ3, iω3;−σ3, iω4] . (18) The functions Ki, i = 1, 2, 3, are the contributions of different spin channels from the two-particle on-site Green function, while the bilinear on G̃ (0) σ (iω) terms are the substracted from Kubo cumulant members. All the Ki functions are pro- portional to the factor βδ(ω1 + ω2 − ω3 − ω4) which expresses the frequencies conservation law. Some values of these Ki functions may be found in [1]. These laws of conservation are also valid for the renormalized Gir 2 quantity. To take into account the frequency conservation law we go from Gir to G̃ir functions and define new frequencies using the even quantity Ω = 2mπ/β G̃ir 2,q [σ, i(ω + Ω);σ1, i(ω1 −Ω1)\|σ2, iω1;σ ′, iω] = = G̃ (0)ir 2 [σ, i(ω +Ω);σ1, i(ω1 − Ω)\|σ2, iω1;σ ′, iω] + + 1 2β ∑ Ω1α1α2 G̃(0)ir [σ, i(ω + Ω);σ1, i(ω1 − Ω)\|α1, i(ω1 − Ω1);α2, i(ω + Ω1)] ×ξ2,q [α1, i(ω1 − Ω1);α2, i(ω + Ω1)] ×G̃ir 2,q [α2, i(ω + Ω1);α1, (ω1 − Ω1)\|σ2, iω1;σ ′, iω] . (19) Using the definition of renormalized Gir 2 function we can write down the contribu- tion of the first class of the ladder diagrams in figure 2 to the function Z(x\|x′) in the form: ZI(x\|x′) = − ∑ x1x ′ 1 Gir 2 [x, x1\|x′ 1, x ′] t̃(x′ 1 − x1) On the basis of (13) and (18) this equation may be rewritten as ZI σσ ′ (x− x′\|τ − τ ′) = − ∑ σ1,σ ′ 1 ∫ ∫ dτ1dτ ′ 1G ir 2 [x− x′;στ, σ1τ1\|σ′ 1, τ ′ 1;σ ′τ ′] ×t̃σ′ 1 σ1 (x′ − x, τ ′ 1 − τ1) = − 1 β3 ∑ σ1,σ ′ 1 ,ω,ω′,ω1 t̃σ′ 1 σ1 (x′ − x; iω1) ×Gir 2 [x− x′, σ, iω;σ1, iω1\|σ′ 1, iω1;σ ′, iω] e−iωτ+iω′τ ′ (20) 30 Diagram technique for the Hubbard model or ZI σ(q\|iω) = − 1 N 1 β ∑ k1ω1σ1 G̃ir 2,k1 [σ, iω;σ1, iω1\|σ1, iω1, σ, iω] ∗ ξσ1 (k1 − q\|iω1). (21) According to equation (21), to find ZI we need a more simple modification of renormalized Gir 2 function with equality in pairs of frequencies and spins. But it is necessary, at the beginning, to find this function for a more general case and then to put a specific condition Ω = 0 in it. Now we take into account that the spin of the propagating electron is conserved σ = σ′ (no magnetic fields and magnetic structures), and so is the conservation law of the spin in all the intermediate scattering processes. Then we have σ2 = σ1 and σ+ σ1 = α1 +α2. Because σ1 can be equal to ±σ, we shall consider these two possibilities separately. In the first case σ1 = σ we have α1 = α2 = σ and equation (19) in a more simple form: G̃ir 2,q[σ, i(ω + Ω), σ, i(ω1 − Ω)\|σ, iω1;σ, iω)] = = G̃ (0)ir 2 [σ, i(ω + Ω);σ, i(ω1 − Ω)\|σ, iω1, σ, iω] + 1 2β ∑ Ω1 G̃ (0)ir 2 [σ, i(ω + Ω);σ, i(ω1 − Ω)\|σ, i(ω1 − Ω1), σ, i(ω + Ω1)] ×ξ2,q [σ, i(ω1 − Ω1);σ, i(ω + Ω1)] ×G̃ir 2,q[σ, i(ω +Ω1);σ, i(ω1 − Ω1)\|σ, iω1;σ, iω]. (22) But in the second case σ1 = −σ,we have two possibilities to select α1 = −α2 = ±σ and equation (19) takes the form: G̃ir 2,q [σ, i(ω + Ω);−σ, i(ω1 − Ω)\| − σ, iω1;σ, iω] = = G̃ (0)ir 2 [σ, i(ω + Ω),−σ, i(ω1 −Ω)\| − σ, iω1;σ, iω]+ + 1 2β ∑ Ω1 G̃ (0)ir 2 [σ, i(ω + Ω);−σ, i(ω1 − Ω)\|σ, i(ω1 − Ω1);−σ, i(ω + Ω1)] ×ξ2,q [σ, i(ω1 − Ω1),−σ, i(ω + Ω1)] ×G̃ir 2,q [−σ, i(ω +Ω1);σ, i(ω1 − Ω1)\| − σ, iω1;σ, iω] + + 1 2β ∑ Ω1 G̃ (0)ir 2 [σ, i(ω + Ω);−σ, i(ω1 − Ω)\| − σ, i(ω1 −Ω1);σ, i(ω +Ω1)] ×ξ2,q [−σ, i(ω1 − Ω1), σ, i(ω + Ω1)] ×G̃ir 2,q [σ, i(ω + Ω1);−σ, i(ω1 − Ω1)\| − σ, iω1;σ, iω] . (23) In the second term of the right-hand side of equation (23) we can make the fol- lowing substitution of Ω1 by variable Ω1 = ω1 − ω − Ω′ 2 and take into account the antisymmetric properties of local and delocalized irreducible Green functions according to the permutation of their arguments. The result is the equality of the last two terms of (23). We can use only one of them multiplying by the coefficient 31 V.A.Moskalenko, L.Z.Kon two. We have G̃ir 2,q [σ, i(ω + Ω);−σ, i(ω1 − Ω)\| − σ, iω1;σ, iω] = = G̃ (0)ir 2 [σ, i(ω + Ω);−σ, i(ω1 − Ω)\| − σ, iω1;σ, iω]+ + 1 β ∑ Ω1 G̃ (0)ir 2 [σ, i(ω + Ω);−σ, i(ω1 − Ω)\| − σ, i(ω1 − Ω1);σ, i(ω + Ω1)] ×ξ2,q [−σ, i(ω1 − Ω1);σ, i(ω + Ω1)] ×G̃ir 2,q [σ, i(ω + Ω1);−σ, i(ω1 − Ω1)\| − σ, iω1;σ, iω)] . (24) To solve equations (22) and (24) it is necessary to know two kernels of these integral equations, namely, G̃ (0)ir 2 [σ, i(ω+Ω);σ, i(ω1−Ω)\|σ, i(ω1−Ω1);σ, i(ω+Ω1)] and G̃ (0)ir 2 [σ, i(ω +Ω),−σ, i(ω1 − Ω)\| − σi(ω1 −Ω), σi(ω + Ω1)] which were studied in papers [1-2] and will be discussed below. To find the second contribution to the irreducible function Z(x\|x′) we have to analyse diagrams d) in figure 2 and the analogous diagrams of the higher order of the perturbation theory (see figure 4).The infinite series of such kind of diagrams belong to the new renormalized and delocalized function H ir 2 [x1x2\|x2x ′ 1]. All such diagrams have a coefficient equal to one. The sign of the diagrams corresponds to a special order of arguments of all the irreducible functions. It must be of a clockwise order if we begin to count from the lower left corner of the rectangle or counter-clockwise if the count begins from the lower right corner of the rectangle. The function H ir 2 has partial local properties H ir 2 [x1, x ′ 2\|x2, x ′ 1] = δx1,x ′ 2 δx2,x ′ 1 H ir 2 [x1 − x′ 1;σ1τ1;σ ′ 2, τ2\|σ2, τ ′ 2;σ ′ 1, τ ′ 1] . (25) This new function, which depends on the relative arguments x1 − x′ 1, obeys the integral equation (see figure 4b): H ir 2 [x1 − x′ 1;σ1, τ1;σ ′ 2, τ ′ 2\|σ2, τ2;σ ′ 1τ ′ 1] = −δx1,x ′ 1 G (0)ir 2 [σ1, τ1;σ2, τ2\|σ′ 2, τ ′ 2;σ ′ 1, τ ′ 1] + + ∑ 1 ∑ α1,α2 ∫ .. ∫ dθ1dθ2dθ ′ 1dθ ′ 2G (0)ir 2 [σ1, τ1;α1θ1\|σ′ 2τ ′ 2α2θ ′ 2] ×t̃α1 (1− x1\|θ′1 − θ1)t̃α2 (x1 − 1\|θ′2 − θ2) ×H ir 2 [ 1− x′ 1;α2, θ2;α1, θ ′ 1\|σ2, τ2;σ ′ 1, τ ′ 1 ] . (26) Using the Fourier representation we obtain H ir 2,q [σ1, iω1;σ ′ 2, iω ′ 2\|σ2, iω2;σ ′ 1, ω ′ 1] = −G (0)ir 2 [σ1, iω1;σ2, iω2\|σ′ 2, iω ′ 2;σ ′ 1, iω ′ 1] + + 1 β2 ∑ ω′′ 1 ∑ ω′′ 2 ∑ α1α2 ξ2,q [α1, iω ′′ 1 ;α2, iω ′′ 2 ]G (0)ir 2 [σ1, iω1;α1, iω ′′ 1 \|σ′ 2, iω ′ 2;α2, iω ′′ 2 ] ×H ir 2,q [α2, iω ′′ 2 ;α1, iω ′′ 1 \|σ2, iω2;σ ′ 1, iω ′ 1] . (27) Here we suppose diagonality by spin indices of the renormalized hopping elements and an even symmetry of tight-binding electron energy ξ(k) = ξ(−k). 32 Diagram technique for the Hubbard model Figure 4. The summation of the second class of ladder diagrams. a) Infinite series of diagrams for the renormalized Hir 2 function. b) Graphical integral equation for the renormalized H ir 2 function. Now we shall take into account the law of frequency conservation both for the local and delocalized irreducible functions. The latter have the property H ir 2,q [σ1, iω1;σ ′ 2, iω ′ 2\|σ2, iω2;σ ′ 1, iω ′ 1] = βδ(ω1 + ω2 − ω′ 2 − ω′ 1) ×H̃ ir 2,q [ σ1, iω1;σ ′ 2, iω ′ 2\|σ2, iω2;σ ′ 1,i(ω1 + ω2 − ω′ 2 ] . (28) Then we use in (27) and (28) new frequencies which obey the law: ω′′ 1 = ω′ 2 +Ω1, ω′′ 2 = ω1 +Ω1, ω′ 2 = ω2 + Ω, ω′ 1 = ω1 − Ω. (29) 33 V.A.Moskalenko, L.Z.Kon Here Ω and Ω1 are even Matsubara frequencies. On the basis of equations (27)-(29) we obtain: H̃ ir 2,q [σ, iω;σ ′ 1, i(ω1 +Ω)\|σ1, iω1;σ ′, i(ω − Ω)] = −G̃ (0)ir 2 [σ, iω;σ1, iω1\|σ′ 1, i(ω1 + Ω), σ′, i(ω − Ω)]+ + 1 β ∑ Ω1,α1,α2 G̃ (0)ir 2 [σ, iω;α1, i(ω1 +Ω1 + Ω)\|σ′ 1, i(ω1 +Ω);α2, i(ω +Ω1)] ×ξ2,−q [α1, i(ω1 + Ω1 + Ω);α2, i(ω + Ω1)] ×H̃ ir 2q [α2, i(ω + Ω1), α1, i(ω +Ω1 + Ω)\|σ1, iω1;σ ′, i(ω −Ω)] . (30) The law of spin conservation is fulfilled here in the form σ + σ1 = σ′ + σ′ 1 and σ + α1 = σ′ 1 + α2. If we are interested in the case σ′ = σ, then we have to discuss only the equalities σ′ 1 = σ1 and σ + α1 = σ1 + α2. As a consequence, when σ1 = σ wehave α1 = α2 = ±σ, but when σ1 = −σ we have α1 = −α2 = −σ. The corresponding integral equations are the following: H̃ ir 2,q [σ, iω;−σ, i(ω1 + Ω)\| − σ, iω1;σ, i(ω − Ω)] = −G̃ (0)ir 2 [σ, iω;−σ, iω1\| − σ, i(ω1 + Ω);σ, i(ω − Ω)] + + 1 β ∑ Ω1 G̃ (0)ir 2 [σ, iω;−σ, i(ω1 − Ω1 + Ω)\| − σ, i(ω1 + Ω);σ, i(ω − Ω1)] ×ξ2−q [−σ, i(ω1 − Ω1 +Ω);σ, i(ω − Ω1)] ×H̃ ir 2,q [σ, i(ω − Ω1);−σ, i(ω1 − Ω1 + Ω)\| − σ, iω1;σ, i(ω −Ω)] (31) and H̃ ir 2,q [σ, iω;σ, i(ω1 + Ω)\|σ, iω1;σ, i(ω − Ω)] = −G̃ (0)ir 2 [σ, iω;σ, iω1; \|σ, i(ω1 +Ω), σ, i(ω − Ω)] + + 1 β ∑ Ω1 G̃ (0)ir 2 [σ, iω;σ, i(ω1 + Ω1 +Ω)\|σ, i(ω1 + Ω);σ, i(ω +Ω1)] ×ξ2,−q [σ, i(ω1 + Ω1 + Ω);σ, i(ω + Ω1)] ×H̃ ir 2,q [σ, i(ω + Ω1), σ, i(ω1 + Ω1 + Ω)\|σ, iω1, σ, i(ω − Ω)]+ + 1 β ∑ Ω1 G̃ (0)ir 2 [σ, iω;−σ, i(ω1 + Ω1 +Ω)\|σ, i(ω1 + Ω);−σ, i(ω + Ω1)] ×ξ2,−q [−σ, i(ω1 + Ω1 + Ω);−σ, i(ω + Ω1)] ×H̃ ir 2,q [−σ, i(ω + Ω1);−σ, i(ω1 +Ω1 + Ω)\|σ, iω1;σ, i(ω −Ω)] . (32) In distinction from (31) equation (32) is not closed because a new function with quite different spin indices appears in the right-hand side of it. 34 Diagram technique for the Hubbard model Returning to equation (30), we can find the lacking equation: H̃ ir 2,q [−σ, iω;−σ, i(ω1 +Ω)\|σ, iω1;σ, i(ω − Ω)] = −G̃ (0)ir 2 [−σ, iω;σiω1\| − σ, i(ω1 + Ω);σ, i(ω − Ω)] − 1 β ∑ Ω1 G̃ (0)ir 2 [−σ, iω;σ, i(ω1 + Ω1 + Ω)\| − σ, i(ω1 + Ω);σ, i(ω + Ω1)] ×ξ2,−q [σ, i(ω1 + Ω1 + Ω);σ, i(ω + Ω1)] ×H̃ ir 2,q [σ, i(ω + Ω1);σ, i(ω1 + Ω1 + Ω)\|σ, iω1;σ, i(ω − Ω)] − 1 β ∑ Ω1 G̃ (0)ir 2 [−σ, iω;−σ, i(ω1 + Ω1 +Ω)\| − σ, i(ω1 + Ω),−σ, i(ω + Ω1)] ×ξ2,−q [−σ, i(ω1 + Ω1 + Ω);−σ, i(ω + Ω1)] ×H̃ ir 2,q [−σ, i(ω + Ω1);−σ, i(ω1 +Ω1 + Ω)\|σ, iω1;σ, i(ω −Ω)] . (33) Equation (32) and (33) have to be solved together, while (31) – separately. To do that, it is necessary to know the kernels of these equations which are localized irreducible two-particle Green functions. In a special case of half-filling, when the number of electrons is equal to the lattice sites number, the chemical potential of the system µ is equal to U 2 and, supposing the independence of the on-site electron energy of a spin index, we have [1] the following values for such irreducible functions: G (0)ir 2 (σ, iω1;σ, iω2\|σ, iω3;σ, iω4) = δ(ω1 + ω2 − ω3 − ω4)) ×β2µ2 [δ(ω1 − ω4) − δ(ω1 − ω3)] (ω2 1 + µ2)(ω2 2 + µ2) (34) G (0)ir 2 [σ, iω1;−σ, iω2\| − σ, iω3;σ, iω4)] = βµδ(ω1 + ω2 − ω3 − ω4) ×{ βµδ(ω1 − ω4)(1 − eβµ) (eβµ + 1)(ω2 1 + µ2)(ω2 2 + µ2) − 2µβδ(ω1 − ω3)e βµ (eβµ + 1)(ω2 1 + µ2)(ω2 2 + µ2) + 2µβδ(ω1 + ω2) (ω2 1 + µ2)(ω2 3 + µ2)(eβµ + 1) −2 [ω1ω2ω3ω4 − µ2(ω2 1 + ω2 2 + ω2 3 + ω1ω2 − ω2ω3 − ω1ω3)− 3µ4] (ω2 1 + µ2)(ω2 2 + µ2)(ω2 3 + µ2)(ω2 4 + µ2) } (35) G (0)ir 2 [σ, iω1;−σ, iω2\|σ, iω3;−σ, iω4] = βµδ(ω1 + ω2 − ω3 − ω4) { βµδ(ω1 − ω3)(e βµ − 1) [ω2 1 + µ2] [ω2 2 + µ2] (eβµ + 1) + 2µβeβµδ(ω1 − ω4) (ω2 1 + µ2)(ω2 2 + µ2)(eβµ + 1) − 2µβδ(ω1 + ω2) (ω2 1 + µ2)(ω2 3 + µ2)(eβµ + 1) + 2 [ω1ω2ω3ω4 − µ2(ω2 1 + ω2 2 + ω2 3 + ω1ω2 − ω1ω3 − ω2ω3) − 3µ4] (ω2 1 + µ2)(ω2 2 + µ2)(ω2 3 + µ2)(ω2 4 + µ2) }}. (36) The corresponding functions with a simultaneous change of all the spin indices to the opposite value are supposed to be identically equal to the initial ones because of the independence of electron energies of a spin index. 35 V.A.Moskalenko, L.Z.Kon Really, to find contribution ZII(x\|x′) of these ladder diagrams to function Z(x\|x′) we have to extract some superfluous diagrams and, therefore, to deter- mine a new function: H2 [x1, x ′ 2\|x2, x ′ 1] = H ir 2 [x1, x ′ 2\|x2, x ′ 1] +G (0)ir 2 [x1, x2\|x′ 2, x ′ 1] + ∑ 11′22′ G (0)ir 2 [x1, 1\|x′ 2, 2 ′]G (0)ir 2 [2, x2\|1′, x′ 1] t(2′ − 2) t(1′ − 1). (37) The contribution of the second kind of the ladder diagrams to irreducible function Z(x\|x′) is ZII(x1\|x′ 1) = ∑ x2x 1 2 t̃(x′ 2 − x2)H2(x1, x ′ 2\|x2, x ′ 1)]. (38) The contribution of the both kinds of the ladder diagrams to the Z(x\|x′) function is Z(x1\|x′ 1) ≃ ZI(x1\|x′ 1] + ZII [x1\|x′ 1]. (39) There are also more complicated diagrams that must be taken into account to obtain a more correct value of this function. 3. The coherent potential theory In the given section attention is paid to the solution of the equation for one- particle Green function in the coherent potential approximation. A special case of half-filling, when the number of electrons is equal to the lattice sites number, is considered. The following approach is taken into account: the renormalized quan- tity Gir 2 is determined only by equation (22). Applying expression (34) to the localized irreducible two-particle Green func- tion, equation (22) can be solved and we obtain G̃ir 2q [σ, iω;σ, iω1\|σ, iω1;σ, iω] = β a (iω; iω1) (1− δ (ω − ω1)) 1− a (iω, iω1) ξ2,q [σ, iω;σ, iω1] , (40) where a (iω, iω1) = µ2 (ω2 + µ2) (ω2 1 + µ2) . On the basis of expressions (6), (21) and (40) function Λσ (q\|ω) in this approxi- mation can be found as Λσ (q\|iω) = −iω ω2 + µ2 − 1 N 1 β ∑ k1 ∑ ω1 ξσ (k1 − q\|iω1) β(1− δ(ω − ω1)) 1− a (iω, iω1) ξ2,k1 [σ, iω;σ, iω1] . (41) In order to solve equation (41), the coherent potential approximation is used. [11]. After analytical continuation (iω → E ) we obtain the following equation: Λσ (0\|E) = −E µ2 − E2 + Tσ(E)a(E,E) 1− T 2 σ (E)a(E,E) . (42) 36 Diagram technique for the Hubbard model Here Tσ(E) = 1 N ∑ q ξ(q) 1− ξ(q)Λσ(0\|E) . (43) It is convenient to determine function g(E) as g(E) = 1− √ 1− λ2(E) λ(E) , (44) where Λσ(0\|E)W = 2λσ(E), λ↑(E) = λ↓(E) = λ(E). W is the width of the band energy. Then, Tσ(E) = W 4 g(E) ( g2(E) + 1 ) ;λ(E) = 2g(E) g2(E) + 1 and equation (42) may be rewritten as 2g(E)[1 − µ2a(E,E)(g2(E) + 1)3/(4v2)] = −W 2 E(g2(E) + 1)/(µ2 −E2) [1− µ2a(E,E)(g2(E) + 1)2g2(E)/(4v2)] , (45) where v = 2µ W = U W . In order to obtain the condition for the Mott metal-dielectric transition we put E = 0 in (45) and then obtain 2g̃(4v2 − ( g̃2 + 1 )3 ) = 0. (46) Here g̃ = g(E = 0). From equation (46) it can be seen that the critical value of Coulomb interaction is equal to vc = 1 2 . On the condition that v > vc the Mott metal-dielectric transition takes place. This condition is reached on the basis of equation (22), taking into account only a contribution of the irreducible function Gir. 2 with parallel spins. Therefore, it is different from the one of work [2] by factor √ 3, in work [2] vc = √ 3 2 . 4. Conclusions Our main concern was to investigate the properties of the systems with strong electron correlations taking into account a special ladder kind of perturbation theory’s diagrams. We have obtained integral equations that realize the generalized random phase approximation for different spin channels of the scattering processes and have determined one-particle Green function. 37 V.A.Moskalenko, L.Z.Kon One of these equations has been solved by using the coherent potential approx- imation. The exact solution of the integral equations can be done only for special as- sumptions about the theory parameters and remains our next task. References 1. M I.Vladimir, V.A Moskalenko, Teor. Matem. Fiz. 82, 428 (1990) [Theor. Math. Phys. 82, .301 (1990)]. 2. S.I.Vacaru, M.I.Vladimir, V.A.Moskalenko, Theor. Matem. Fiz. 85, 248 (1990) [Theor. Math. Phys. 85, 1185 (1990)]. 3. S.P. Cojocaru, V.A. Moskalenko, Teor. Matem. Fiz. 97, .270 (1993)[Theor.Math.Phys. 97, 1290 (1993)]. 4. V.A.Moskalenko, S.P.Cojocaru, M.I.Vladimir, A diagram technique for strongly inter- action fermion systems. One and two-band Hubbard Models (1994), preprint ICTP, IC/94/182, Trieste, Italy. 5. V.A.Moskalenko, Teor. Matem. Fiz. 110, 308 (1997) [Theor. Math. Phys. 110 , 243 (1997)]. 6. J.Hubbard, J. Proc. Roy. Soc. A 276,.233 (1963). 7. P.A.Anderson, Phys. Rev. 124,.41 (1961). 8. P.M.Slobodyan, I.V.Stasyuk, Teor. Matem. Fiz. 19, 423 (1974) (in Russian). 9. R.O.Zaitsev, Zh.Eksp.Teor.Fiz. 68, 207 (1975), ibad 70, 1100.(1976). 10. Yu.A.Izyumov, Yu.M.Skryabin, Statistical Mechanics of magnetically ordered systems. Moscow. (1987) (in Russian). 11. V.A.Moskalenko, L.Z.Kon, 3 rd General conference of the Balkan Physical Union (BPU-3) Cluj-Napoca 25 sept., 381, 6P-041 (1997). Діаграмна техніка в моделі Хаббарда. Драбинкове наближення В.А.Москаленко 1,2 , Л.З.Кон 2 1 Об’єднаний інститут ядерних досліджень, 141980 Дубна, Росія 2 Інститут прикладної фізики, Молдова, 277028 м. Кишинів, вул. Гросул, 5 Отримано 20 березня 1998 р. На основі узагальненої теореми Віка розроблено нову діаграмну тех- ніку для систем з сильними електронними кореляціями. Кулонівське відштовхування електронів в моделі Хаббарда розглядається як го- ловна частина гамільтоніана і враховується в нульовому наближенні. Матричний елемент міжвузлового перескоку електронів враховуєть- ся як збурення. Досліджено одночастинкову мацубарівську функ- 38 Diagram technique for the Hubbard model цію Гріна й отримано рівняння Дайсона. Новими елементами теорії, що характерні для даного наближення, є багаточастинкові незвідні функції Кубо. Вони обертаються в нуль, коли кулонівська взаємодія рівна нулю. Головна мета даної роботи – це сумування драбинкових діаграм, що враховують найбільш суттєві спінові і зарядові флуктуації системи. Отримані інтегральні рівняння для двох каналів розсіяння. Метод когерентного потенціалу використовується для спрощення і розв’язування цих рівнянь. На цій основі встановлена умова перехо- ду метал-діелектрик. Ключові слова: узагальнена теорема Віка, драбинкові діаграми, модель Хаббарда PACS: 71.28.+d, 71.27.+a 39 40

Diagram technique for the Hubbard model. Ladder diagram summation

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