Asymmetric Hubbard model within generating functional approach in dynamical mean field theory

In the paper a new analytic approach to the solution of the effective single-site problem in the dynamical mean field theory is developed. The approach is based on the method of the Kadanoff-Baym generating functional in the form developed by Izyumov et al. It makes it possible to obtain a close...

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Datum:	2006
Hauptverfasser:	Stasyuk, I.V., Hera, O.B.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2006
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spelling	irk-123456789-1213652017-06-15T03:02:47Z Asymmetric Hubbard model within generating functional approach in dynamical mean field theory Stasyuk, I.V. Hera, O.B. In the paper a new analytic approach to the solution of the effective single-site problem in the dynamical mean field theory is developed. The approach is based on the method of the Kadanoff-Baym generating functional in the form developed by Izyumov et al. It makes it possible to obtain a closed equation in functional derivatives for the irreducible part of the single-site particle Green’s function; the solution is constructed iteratively. As an application of the proposed approach the asymmetric Hubbard model (AHM) is considered. The inverse irreducible part Ξ⁻¹σ of the single-site Green’s function is constructed in the linear approximation with respect to the coherent potential Jσ. Basing on the obtained result, the Green’s function of itinerant particles in the Falicov-Kimball limit of AHM is considered, and the decoupling schemes in the equations of motion approach (GH3 approximation, decoupling by Jeschke and Kotliar) are analysed. В роботi розвивається новий аналiтичний пiдхiд для розв’язання ефективної одновузлової задачi в методi динамiчного середнього поля. Пiдхiд ґрунтується на методi твiрного функцiоналу Каданова-Бейма у формi, розробленiй в роботах Iзюмова та iн. Вiн дає можливiсть отримати замкнене рiвняння у функцiональних похiдних для незвiдної частини одновузлової функцiї Грiна частинок; розв’язки будуються iтеративним способом. В ролi застосування запропонованої схеми взято асиметричну модель Хаббарда (АМХ). Побудовано обернену незвiдну частину Ξ⁻¹σ одновузлової функцiї Грiна в лiнiйному наближеннi за когерентним потенцiалом Jσ. Виходячи з отриманого результату, розглянено функцiю Грiна рухомих частинок у границi Фалiкова-Кiмбала АМХ, проаналiзовано схеми розщеплень у рiвняннях руху для одновузлової функцiї Грiна (наближення GH3, розщеплення ЄшкеКотляра). 2006 Article Asymmetric Hubbard model within generating functional approach in dynamical mean field theory / I.V. Stasyuk, O.B. Hera // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 587–602. — Бібліогр.: 40 назв. — англ. 1607-324X PACS: 71.10.Fd, 71.27.+a, 71.30.+h DOI:10.5488/CMP.9.3.587 http://dspace.nbuv.gov.ua/handle/123456789/121365 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	In the paper a new analytic approach to the solution of the effective single-site problem in the dynamical mean field theory is developed. The approach is based on the method of the Kadanoff-Baym generating functional in the form developed by Izyumov et al. It makes it possible to obtain a closed equation in functional derivatives for the irreducible part of the single-site particle Green’s function; the solution is constructed iteratively. As an application of the proposed approach the asymmetric Hubbard model (AHM) is considered. The inverse irreducible part Ξ⁻¹σ of the single-site Green’s function is constructed in the linear approximation with respect to the coherent potential Jσ. Basing on the obtained result, the Green’s function of itinerant particles in the Falicov-Kimball limit of AHM is considered, and the decoupling schemes in the equations of motion approach (GH3 approximation, decoupling by Jeschke and Kotliar) are analysed.
format	Article
author	Stasyuk, I.V. Hera, O.B.
spellingShingle	Stasyuk, I.V. Hera, O.B. Asymmetric Hubbard model within generating functional approach in dynamical mean field theory Condensed Matter Physics
author_facet	Stasyuk, I.V. Hera, O.B.
author_sort	Stasyuk, I.V.
title	Asymmetric Hubbard model within generating functional approach in dynamical mean field theory
title_short	Asymmetric Hubbard model within generating functional approach in dynamical mean field theory
title_full	Asymmetric Hubbard model within generating functional approach in dynamical mean field theory
title_fullStr	Asymmetric Hubbard model within generating functional approach in dynamical mean field theory
title_full_unstemmed	Asymmetric Hubbard model within generating functional approach in dynamical mean field theory
title_sort	asymmetric hubbard model within generating functional approach in dynamical mean field theory
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2006
url	http://dspace.nbuv.gov.ua/handle/123456789/121365
citation_txt	Asymmetric Hubbard model within generating functional approach in dynamical mean field theory / I.V. Stasyuk, O.B. Hera // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 587–602. — Бібліогр.: 40 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT stasyukiv asymmetrichubbardmodelwithingeneratingfunctionalapproachindynamicalmeanfieldtheory AT heraob asymmetrichubbardmodelwithingeneratingfunctionalapproachindynamicalmeanfieldtheory
first_indexed	2025-07-08T19:43:29Z
last_indexed	2025-07-08T19:43:29Z
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fulltext	Condensed Matter Physics 2006, Vol. 9, No 3(47), pp. 587–602 Asymmetric Hubbard model within generating functional approach in dynamical mean field theory I.V.Stasyuk, O.B.Hera Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine Received June 6, 2006 In the paper a new analytic approach to the solution of the effective single-site problem in the dynamical mean field theory is developed. The approach is based on the method of the Kadanoff-Baym generating functional in the form developed by Izyumov et al. It makes it possible to obtain a closed equation in functional derivatives for the irreducible part of the single-site particle Green’s function; the solution is constructed iteratively. As an application of the proposed approach the asymmetric Hubbard model (AHM) is considered. The inverse irreducible part Ξ −1 σ of the single-site Green’s function is constructed in the linear approximation with respect to the coherent potential Jσ . Basing on the obtained result, the Green’s function of itinerant particles in the Falicov-Kimball limit of AHM is considered, and the decoupling schemes in the equations of motion approach (GH3 approximation, decoupling by Jeschke and Kotliar) are analysed. Key words: strongly correlated systems, asymmetric Hubbard model, single-site problem, dynamical mean field theory PACS: 71.10.Fd, 71.27.+a, 71.30.+h 1. Introduction The lattice models with Hubbard correlations are used in investigating strongly-correlated materials. The one-band Hubbard model [1,2] and the spinless Falicov-Kimball model [3] can be considered as the simplest models of this kind. In the present paper those models are combined into the asymmetric Hubbard model describing a system with two sorts of mobile particles (electrons, ions, . . . ) with different hopping integrals and different values of chemical potentials. Thus, the model can be considered in investigating mixed-valence compounds [4] or ionic conductors. The model Hamiltonian has the following form: Ĥ = ∑ i Ĥi + ∑ ijσ tσija † iσajσ, (1) Ĥi = Uni↑ni↓ − ∑ σ∈{↓,↑} µσniσ. (2) Here, the electron transfer is described using fermionic creation and annihilation operators (a† iσ, ajσ) and hopping parameters tσij . The single site part Hi contains the Coulomb repulsion U (niσ = a † iσaiσ) and chemical potentials µσ. If the model is used for describing two sorts of fermionic quasiparticles the electron spin indices are replaced by sort indices σ = A,B. The hopping integrals and chemical potentials in asymmetric Hubbard model are dependent on particle sort: tAij 6= tBij , µA 6= µB . However, even such a relatively simple model cannot be solved exactly and various simplificati- ons are required in investigating the problem. As for now, only some specific cases were considered. Thus, in the ground state of the model, a possibility of a phase separation phenomenon was ana- lyzed in [5]. In the case of large on-site repulsion U , it is possible to use the effective anisotropic c© I.V.Stasyuk, O.B.Hera 587 I.V.Stasyuk, O.B.Hera Heisenberg model with antiferromagnetic interaction [4,6,7]. Also, some thermodynamic properties of the model were investigated in the case of one-dimensional space [7,8]. In infinite dimensions, the spectral functions (densities of states) in the asymmetric Hubbard model were investigated using approximations in the equation of motion approach for Green’s functions [9,10]. In this paper we consider the model within the dynamical mean field theory (DMFT) that is exact in the limit of infinite dimension. In this case the lattice model is reduced to the single- site problem where the self energy Σ(k) is independent of the wave vector k [11]. The single-site problem can be formulated constructing the effective Hamiltonian: e−βH → e−βHeff = e−βH0T exp [ − ∫ β 0 dτ ∫ β 0 dτ ′ ∑ σ Jσ(τ − τ ′)a† σ(τ)aσ(τ ′) ] , (3) where H0 = ∑ i Ĥi. A solution of the single-site problem is supplemented by the solution of the Larkin equation Gσ(ωn) = 1 Ξ−1 σ (ωn) − Jσ(ωn) (4) and the relation following from the equality of the lattice Green’s function and the single-site Green’s function: Gσ(ωn) = ∫ +∞ −∞ ρ0 σ(t)dt Ξ−1 σ (ωn) − t , (5) where ρ0 σ(t) is unperturbed particle density of states. Let us note that for solving the single-site problem (3), it can be reformulated in terms of the single impurity Anderson model [12] or using the auxiliary Fermi-field describing the effective external bath as it was done in [9,10,13,14]. The task of the single-site problem is to find the functional relation between the single-site Green’s function Gσ and the coherent potential Jσ. A number of numerical methods has been de- veloped for this purpose (for example, the quantum Monte-Carlo [15–17], the exact diagonalization [18,19], and the numerical renormalization group [20]). However, with the recent development of the ab initio methods for calculating electronic structure of real materials combining the density functional theory with the dynamical mean field theory (see reviews [21,22]), new fast methods for solving the single-site problem are required. The analytic approximations for the Green’s func- tion (self-energy) in terms of the coherent potential can be used as such fast methods [13,23–27]. Among them there are methods based on decoupling of Green’s functions in the equation of mo- tion approach [13,23] and methods based on perturbation theory expansion (for example, iterated perturbation theory [12,28]). As shown in [13], the different-time decoupling approach includes the modified alloy-analogy approximation [29] and the Hubbard-III approximation [30] as particular cases. In this paper, to systematize and consider a possibility of improving analytic methods we develop the generating functional approach as solver for the single-site problem. This method is based on the Kadanoff and Baym functional scheme [31] in the form elaborated by Izyumov and Chaschin for the lattice models with strong correlations (for example, Hubbard model and Heisenberg model) [32–35]. The approach allows one to obtain a closed equation in functional derivatives for irreducible part of the Green’s function. Solutions of the equation are constructed iteratively in a form of expansion around the atomic limit (in powers of hopping tij or Jσ) Exemplified by the asymmetric Hubbard model, it is shown that the problem can be formulated as the equation for the Larkin irreducible part Ξ(ω) or the equation for the self-energy and termi- nal part of the Green’s function. This technique allows one to construct an analytic expression for the irreducible part with arbitrary precision in powers of hopping (coherent potential). The first iteration leads after some simplification to the so-called generalized Hubbard-III (GH3) approxi- mation that was recently proposed using different-time decoupling of irreducible Green’s functions in the equations of motion approach [9,14]. In the same way, there is established a relation with the decoupling scheme of Green’s functions of higher order used by Jeschke and Kotliar [23] for the Hubbard model at U → ∞ (JK decoupling). The generating functional approach enables us to improve these approximations constructing successive iterations for the irreducible part. 588 Asymmetric Hubbard model within generating functional approach in DMFT 2. Effective single-site problem and generating functional approach Let us consider the effective single-site problem (3) using the procedure proposed in [13] where the problem was reformulated in terms of the auxiliary Fermi-field. In this case, the Hamiltonian Heff can be written explicitly Heff = Hi + ∑ σ Vσ(a† σξσ + ξ†σaσ) + Hξ , (6) where the last term (Hξ) characterizes the environment of a given site in terms of ξ-field. The Green’s function Gσ(τ − τ ′) = 〈 Tτξ†σ(τ)ξσ(τ ′) 〉 0 , (7) that is calculated by formal averaging with the zero-order Hamiltonian Hξ, determines the coherent potential Jσ(τ − τ ′) = V 2 σ Gσ(τ − τ ′) ≡ V 2 σ 〈Tτξ†σ(τ)ξσ(τ ′)〉0. (8) The main task in the given approach is to calculate the single-site Green’s function Gσ = 〈Tτa† σ(τ)aσ(τ ′)〉 (9) for the problem with the Hamiltonian (6) and defined function Gσ. As a result, such a solution gives the relation between the single-site Green’s function Gσ and coherent potential Jσ. In the Hubbard operators representation the creation and annihilation operators are expressed as aσ = X0σ + ζX σ̄2, a† σ = Xσ0 + ζX2σ̄, (10) where indices and signs are determined as follows { σ̄ = B, ζ = + for σ = A, σ̄ = A, ζ = − for σ = B. Then, the Hamiltonian of the effective single-site problem reads Heff = H0 + Hint , (11) H0 = UX22 − ∑ σ [ µσ ( Xσσ + X22 )] + Hξ , (12) Hint = ∑ σ Vσ [( Xσ0 + ζX2σ̄ ) ξσ + ξ†σ ( X0σ + ζX σ̄2 )] . (13) We use below the thermodynamic perturbation theory. For this purpose the term that describes the hopping between the given site and environment in (11) is considered as a perturbation (interaction) term Hint. The scheme of calculating the Green’s function Gσ(τ − τ ′) ≡ 〈Tτa† σ(τ)aσ(τ ′)〉 = 〈TτXσ0(τ)X0σ(τ ′)〉 + ζ〈TτXσ0(τ)X σ̄2(τ ′)〉 + 〈TτX2σ̄(τ)X σ̄2(τ ′)〉 + ζ〈TτX2σ̄(τ)X0σ(τ ′)〉 (14) is based on the Kadanoff and Baym generating functional method [31] applied and developed for the problems with strong particle correlations by Izyumov et al. [35]. To simplify the formulation of the method we consider the limit case U → ∞, when the doubly occupied site is excluded, and the problem is reduced to the calculation of the Green’s function Gσ0,0σ ≡ 〈TτXσ0(τ)X0σ(τ ′)〉. Let us introduce the time dependent fluctuating fields vγ(τ) that are conjugated to the quanti- ties described by corresponding bosonic Hubbard operators Xγ (γ = 00, σσ, σσ̄). In that case the partition function is rewritten in the following form Z → ZV = Sp ( e−βHTτe−V ) , (15) 589 I.V.Stasyuk, O.B.Hera where V = ∫ β 0 dτV (τ) = ∫ β 0 dτ ∑ γ vγ(τ)Xγ(τ). (16) The calculation of operator average values (averaging over Gibbs ensemble with Hamiltonian Heff in the presence of fluctuating fields) is performed according to the formula: 〈. . .〉V = Sp ( Tτe−βH . . . e−V ) Sp (Tτe−βHe−V ) . (17) The average values of bosonic operators Xγ are expressed as functional derivatives with respect to vγ : 〈Xγ〉 = 〈Xγ〉V \|V →0 = − 1 ZV δZ δvi(τ) ∣ ∣ ∣ ∣ V →0 , (18) where the fields vγ are directed to zero after the differentiation to obtain the operator average values for the system with initial Hamiltonian. Introducing the generating functional [35] ZV = eΦV , ΦV = lnZV , (19) we can rewrite the relation (18) in the following form 〈Xγ〉 = − δΦV δvi(τ) ∣ ∣ ∣ ∣ V →0 . (20) From the expression for the second-order derivatives δ2ΦV δvα(τ1)δvβ(τ2) = 1 ZV δ2ZV δvα(τ1)δvβ(τ2) − 1 Z2 V δZV δvα(τ1) δZV δvβ(τ2) (21) (and the similar ones for the higher-order derivatives) follows a relation between functional deriva- tives of average values of bosonic X-operators (Green’s functions) and the higher order Green’s functions on such operators: δ δvβ(τ2) 〈TτXα(τ1)〉V = −〈TτXα(τ1)X β(τ2)〉V + 〈TτXα(τ1)〉V 〈TτXβ(τ2)〉V . (22) Similarly, one can write an arbitrary Green’s function of the higher order with bosonic Hubbard operator using functional derivatives: 〈TτXα(τ1) . . . 〉V = − δ〈Tτ . . . 〉V δvα(τ1) − 〈Tτ . . . 〉V δΦV δvα(τ1) = − δ〈Tτ . . . 〉V δvα(τ1) + 〈Tτ . . . 〉V 〈TτXα(τ1)〉V . (23) The last relation makes it possible to obtain a closed set of equations in functional derivatives for the fermionic Green’s functions G σ0,0η V (τ, τ ′) ≡ 〈TτXσ0(τ)X0η(τ ′)〉V . (24) The obtained solutions give the single-site Green’s function Gσ0,0σ(τ − τ ′) after the transition to the vγ = 0 limit. 590 Asymmetric Hubbard model within generating functional approach in DMFT 2.1. Recurrent form of Wick’s theorem Let us consider the procedure of constructing the equations for the Green’s function with X- operators within the framework of thermodynamic perturbation theory. In this case ZV = Sp ( Tτ σ̂(β)e−V e−βH0 ) = Z0〈σ̂(β)e−V 〉0 , (25) where σ̂(β) = Tτ exp [ − ∫ β 0 dτ1Hint(τ1) ] , (26) and Z0 = Sp eβH0 is the zero-order partition function. Here, when U → ∞, the zero-order Hamil- tonian reads H0 = −µAXAA − µBXBB + Hξ , (27) and the interaction (perturbation) term has the form Hint = VA ( XA0ξA + ξ † AX0A ) + VB ( XB0ξB + ξ † BX0B ) . (28) Time ordered (Tτ ) operator averaging can be performed by calculating Gibbs average values with unperturbed Hamiltonian H0 〈Tτ . . .〉V = 〈 Tτ . . . σ̂(β)e−V 〉 0 〈Tτ σ̂(β)e−V 〉0 = Z0 ZV 〈Tτ . . . σ̂(β)e−V 〉0. (29) To calculate average values of time ordered (Tτ ) products of X-operators in (29), we use a form of the Wick’s theorem formulated in [36], that can be called its recurrent form. Let us write, in a closed form, the result of the pairing of one selected non-diagonal X-operator with all others 〈TτXσ0(τ)Xγ1(τ1) . . . Xγn(τn)〉 = Z0 ZV ∑ i 〈Tτ - Xσ0(τ)Xγ1(τ1) . . . Xγi(τi) . . . Xγn(τn)σ̂(β)e−V̂ 〉0 + Z0 ZV 〈Tτ - Xσ0(τ)Xγ1(τ1) . . . Xγn(τn)σ̂(β)e−V̂ 〉0. (30) According to [36], the result of pairing of two selected operators is a product of their commuta- tor (anticommutator, when both operators are of the Fermi-type) and the unperturbed Green’s function. In that case 〈 TτXσ0(τ)Xγ1(τ1) . . . Xγn(τn) 〉 = Z0 ZV ∑ i gσ0(τ − τi) × 〈 TτXγ1(τ1) . . . [Xσ0,Xγi ]±τi . . . Xγn(τn)σ̂(β)e−V̂ 〉 0 (−1)pi + Z0 ZV ∫ β 0 dτ ′gσ0(τ − τ ′)〈TτXγ1(τ1) . . . Xγn(τn)[Xσ0,Hint + V̂ ]τ ′ σ̂(β)e−V̂ 〉0, (31) where the alternating (−1)pi multiplier is defined by a number of Fermi-permutations pi of operator Xσ0(τ) from the starting position to the position directly on the left of the operator that is paired to. Going back in (31) to the averages with the full (perturbed) Hamiltonian, we have 〈TτXσ0(τ)Xγ1(τ1) . . . Xγn(τn)〉 = ∑ i gσ0(τ − τi)〈TτXγ1(τ1) . . . [Xσ0,Xγi ]±τi . . . Xγn(τn)〉(−1)pi + ∫ β 0 dτ ′gσ0(τ − τ ′)〈TτXγ1(τ1) . . . Xγn(τn)[Xσ0,Hint + V̂ ]τ ′〉. (32) 591 I.V.Stasyuk, O.B.Hera The similar procedure can be applied to the averages (Green’s functions) involving ξ-operators. If the pairing procedure is started from the operator ξ†σ, then, instead of unperturbed Green’s function gσ0(τ − τ ′), the Green’s function of auxiliary fermions Gσ(τ − τ ′) is used and the ξ†σ operator is on the first position in the commutator. Relation (32) can be called a recurrent form of the Wick’s theorem. When it is used repeatedly to the averages appearing from the right hand side, the complete structure of the expansion of initial averages in products of unperturbed Green’s functions is reproduced. However, the distinction is that the final expressions containing averages of diagonal X-operators that terminate the pairing sequence are calculated with the full Hamiltonian Heff . It is different from using the standard Wick’s theorem [36] where such averages are taken over the Gibbs ensemble with the unperturbed Hamiltonian H0. Let us note, that the formula (32) can be interpreted as an equation relating the averages (Green’s functions) with the Green’s functions of higher order. The same structure of equations is characteristic of the equation of motion approach [31] for the temperature Matsubara Green’s functions. 2.2. Equations for single-particle Green’s function The relation (32) is applied to the Green’s function 〈TτXσ0(τ)X0η(τ ′)〉V . For this purpose, we need to calculate the commutators of the Hubbard operator Xσ0 with Hint+V and anticommutator of operators of Fermi-type Xσ0, X0η. For example: [XB0,Hint] = − VAξ † AXBA − VBξ † B(X00 + XBB), [XB0, V ] = − (vB − v0)X B0 − v̄XA0, {XA0,X0A} = X00 + XAA, {XA0,X0B} = XAB , (where v0 ≡ v00; vσ ≡ vσσ; v̄ ≡ vAB ; v ≡ vBA). This leads to the appearance of Green’s func- tions such as 〈Tτ ξ†σ(τ1)[X 00 + Xσσ](τ1)X 0η(τ ′)〉V and 〈Tτ ξ † σ̄(τ1)X σσ̄(τ1)X 0η(τ ′)〉V , which can be expressed using a relation like (32). Taking into account that [ξ†A,Hint] = −VAXA0, [ξ†B ,Hint] = −VBXB0, (33) we come to Green’s functions with three Hubbard operators, where one operator is of a bosonic type. The Bose-operators are excluded using the functional differentiation (23). As a result we obtained the following equation for the Green’s function G σ0,0η V (τ, τ ′): G σ0,0η V (τ, τ ′) = −gσ0(τ − τ ′) [( δΦV δv0(τ ′) + δΦV δvσ(τ ′) ) δσ,η + δΦV δv̄σ,σ̄(τ ′) δσ̄,η ] − β ∫ 0 dτ1gσ0(τ − τ1) × β ∫ 0 dτ2V 2 σ̄ Gσ̄(τ1 − τ2) [ δΦV δv̄σ,σ̄(τ1) G σ̄0,0η V (τ2, τ ′) + δ δv̄σ,σ̄(τ1) G σ̄0,0η V (τ2, τ ′) ] − β ∫ 0 dτ1gσ0(τ − τ1) β ∫ 0 dτ2V 2 σ Gσ(τ1 − τ2) × [( δΦV δv0(τ1) + δΦV δvσ(τ1) + δ δv0(τ1) + δ δvσ(τ1) ) G σ0,0η V (τ2, τ ′) ] + β ∫ 0 dτ1gσ0(τ − τ1)[vσ(τ1) − v0(τ1)]G σ0,0η V (τ1, τ ′) + β ∫ 0 dτ1gσ0(τ − τ1)v̄σ̄,σ(τ1)G σ̄0,0η V (τ1, τ ′), (34) where δσ,η is a Kronecker delta and v̄B,A ≡ v, v̄A,B ≡ v̄. 592 Asymmetric Hubbard model within generating functional approach in DMFT Let us introduce an inverse Green’s function g−1 σ0 (τ − τ ′) according to the definition ∫ β 0 dτ ′g−1 σ0 (τ − τ ′)gσ0(τ ′ − τ ′′) = δ(τ − τ ′′). (35) Accordingly, in frequency representation, it corresponds to g−1 σ0 (ωn) = [gσ0(ωn)]−1 = iωn + µσ. (36) Let us multiply both sides of equation (34) by g−1 σ0 (τ ′′−τ) and integrate out the free time argument τ . Then, we rewrite the obtained equation in more compact form using matrix representation where the required single-site Green’s function has the following form Ĝ = ( GB0,0B GB0,0A GA0,0B GA0,0A ) . (37) Introducing a matrix representation for the differentiation operator Â′(τ, τ ′) = δ(τ − τ ′)     δ δv0B(τ) δ δv(τ) δ δv̄(τ) δ δv0A(τ)     , (38) where δ δv0σ = δ δv0 + δ δvσ , and introducing matrices for the Green’s functions of auxiliary fermions (coherent potential) Ĵ(τ, τ ′)= ( V 2 BGB(τ − τ ′) 0 0 V 2 AGA(τ − τ ′) ) = ( JB(τ − τ ′) 0 0 JA(τ − τ ′) ) (39) and the inverse unperturbed Green’s functions (combined with the linear contributions of the fluctuating field V ): ĝ−1(τ, τ ′)= ( g−1 B0(τ − τ ′) 0 0 g−1 A0(τ − τ ′) ) + ( v0(τ) − vB(τ) v̄(τ) −v(τ) v0(τ) − vA(τ) ) δ(τ − τ ′), (40) the equation (34) can be presented in the form ĝ−1Ĝ + Â0ĴĜ + Â′ĴĜ = −Â0. (41) Here, the matrix Â0 is a result of the differentiation operator Â′ action on the generating functional ΦV (Â0 = Â′ΦV ). The multiplication of arbitrary two matrices X̂(τ, τ ′) and Ŷ (τ ′, τ ′′) contains integrating out the inner time argument τ ′: X̂Ŷ ≡ ∫ β 0 dτ ′X̂(τ, τ ′)Ŷ (τ ′, τ ′′). (42) At the final stage (after exclusion of fluctuating fields vγ(τ)), multiplication (42) corresponds to the product of Fourier’s components in the frequency representation. 2.3. Irreducible part We introduce the irreducible part Ξ̂ of the single-site Green’s function Ĝ meaning Larkin irreducibility. Then, the equation (4) can be written in the following form Ĝ(ωn) = [ Ξ̂−1(ωn) − Ĵ(ωn) ]−1 . (43) 593 I.V.Stasyuk, O.B.Hera To obtain an equation defining Ξ̂ we use a property of the differentiation δ δv Ĝ = −Ĝ [ δ δv Ĝ−1 ] Ĝ, (44) and rewrite the equation (41) in the form of a functional differential equation where the differen- tiation operator Â′ acts on the inverse Green’s function matrix G−1: Ĝ−1 = −Â−1 0 ĝ−1 − Ĵ − Â−1 0 - Â′ĴĜĜ−1ĜĜ−1; (45) the arrow line points a matrix function operated on by the differentiation operators. Transforming this equation we obtain the equation for the irreducible part: Ξ̂−1 = Ĝ−1 0 + Â−1 0 - Â′ĴĜΞ̂−1, (46) where G−1 0 = −Â−1 0 ĝ−1. (47) In explicit form it reads (Ξ̂−1)pl(τ, τ ′) = (Ĝ−1 0 )pl(τ, τ ′) + ∑ sqm β ∫ 0 dτ1 . . . β ∫ 0 dτ4Ĵqq(τ1, τ4)Ĝqm(τ4, τ2)(Â −1 0 )ps(τ, τ3) ×[Â′ sq(τ3, τ1)(Ξ̂ −1)ml(τ2, τ ′)], (48) Ĝ−1 0 (τ, τ ′)=− 1 D(τ)     δΦV δv0A(τ) g−1 B0(τ − τ ′) − δΦV δv(τ) g−1 A0(τ − τ ′) − δΦV δv̄(τ) g−1 B0(τ − τ ′) δΦV δv0B(τ) g−1 A0(τ − τ ′)     −     δΦV δv0A(τ) [v0(τ)−vB(τ)]+ δΦV δv(τ) v(τ) − δΦV δv0A(τ) v̄(τ)− δΦV δv(τ) [v0(τ)−vA(τ)] − δΦV δv̄(τ) [v0(τ)−vB(τ)]− δΦV δv0B(τ) v(τ) δΦV δv̄(τ) v̄(τ)+ δΦV δv0B(τ) [v0(τ)−vA(τ)]     × 1 D(τ) δ(τ − τ ′), (49) where D(τ) = δΦV δv0B(τ) δΦV δv0A(τ) − δΦV δv(τ) δΦV δv̄(τ) . (50) Equation (46) has the structure that corresponds in the generating functional approach to the similar equations for irreducible parts of Green’s functions for models of magnetics or strongly correlated electron systems [32–35]. Its solution can be sought in the form of a series with summands involving various number of Ĵ multipliers. The iterations generate terms where operators Â′ from (46) act only on matrices Â−1 0 or Ĝ and do not affect the terminating matrix ĝ−1 (Ĝ−1 0 is taken as a zero order function). Another set is formed by summands involving derivatives of ĝ−1. Therefore, the solution can be presented in the form Ξ̂−1 = −Â−1 0 M̂ ( ĝ−1 + F̂ ) , (51) 594 Asymmetric Hubbard model within generating functional approach in DMFT where two new matrices are introduced (M̂ and F̂ ), and the mentioned terms with differentiation of ĝ−1 are collected in F̂ . Let us insert (51) into the equation (46). Collecting separately terms of the first and second type, we obtain two equations: M̂ = 1̂ + - R̂Â−1 0 M̂ + - R̂Â−1 0 M̂, (52) F̂ = - R̂Â−1 0 M̂ ĝ−1 + - R̂Â−1 0 M̂F̂ , (53) where the notation - R̂ · · · = - Â′ĴĜ . . . (54) is introduced. Iterations in the equation (52) point to a possibility of writing the matrix M̂ as M̂ = (1 − L̂)−1, (55) where the matrix L̂ is determined by the functional equation L̂ = L̂0 + - R̂Â−1 0 [1̂ − L̂]−1L̂, (56) where L̂0 = - R̂Â−1 0 ≡ - A′JGA−1 0 . (57) Accordingly, the equation (53) is rewritten as F̂ = F̂0 + - R̂Â−1 0 [1̂ − L̂]−1F̂ , (58) where F0 = - R̂Â−1 0 [1̂ − L̂]−1ĝ−1. (59) Let us note, that matrix M̂ can be connected with the so-called terminating part Λ̂ of the Green’s function Ĝ. Let us write Ĝ in the form Ĝ = Π̂Λ̂, (60) and introduce the mass operator Σ̂ according to the equation Π̂ = ĝ + ĝΣ̂Π̂, Π̂−1 = ĝ−1 − Σ̂. (61) In this case the equation for the Green’s function Ĝ (41) can be split into two equations for the mass operator Σ̂ and terminating part Λ̂, as it was done by Izyumov et al. [32–35]: Σ̂ + Â0Ĵ − - Â′ĴΠ̂ĝ−1 + - Â′ĴΠ̂Σ̂ = 0, (62) Λ̂ + - Â′ĴΠ̂Λ̂ = −A0. (63) Consider the relation between the mass operator Σ̂ and irreducible part Ξ̂ Σ̂ = ĝ−1 + Λ̂Ĵ − Λ̂Ξ̂−1, (64) 595 I.V.Stasyuk, O.B.Hera following from the relations (60), (61) and equation (46). Comparing equations (56), (58) and (62), (63), one can see that there exists the correspondence - R̂Â−1 0 [1̂ − L̂]−1 · · · = − - Â′ĴΠ̂ . . . , (65) Λ̂ = −[1̂ − L̂]Â0, Σ̂ = Λ̂Ĵ − F̂ . (66) This makes possible to write the inverse irreducible part of the Green’s function as Ξ̂−1 = Λ̂−1(ĝ−1 + F̂ ). (67) From here, it is seen that Λ̂ is also a terminating part of the matrix Ξ̂, and (−F̂ ) has the meaning of a Dyson irreducible part of Ξ̂. Summarizing, let us note that the equations (46) or (56) and (58) constructed by us allow one to find, respectively, the inverse irreducible part Ξ̂−1 or, separately, its components L̂ and F̂ . Direct iteration procedure permits to construct their solutions in the form of functionals of Ĝ functions, bosonic correlators and coherent potential Ĵ . 3. Falicov-Kimball model: Green’s function of itinerant pa rticles As an example of the simple application, we consider within the framework of the developed approach the Green’s function of itinerant particles in the Falicov-Kimball model as the simplest specific case. This problem can be exactly solved in DMFT and its solution corresponds to the so-called alloy-analogy (AA) approach. In the case of the Falicov-Kimball model, particles of one sort are localized (JB = 0; in the matrix notations Ĵ11 = 0). To find the Green’s function of itinerant particles (corresponding matrix element (Ĝ)22) the irreducible part (Ξ̂−1)22 should be calculated. The series for (Ξ̂−1)22 does not contain nonzero terms where the differentiation operator Â′ acts on the zero-order Green’s function ĝ: (F̂1)22 = 0, (F̂2)22 = 0, (68) because (Â′)22(ĝ −1)22 = 0, (69) and, in terms where the operator (Â′)12 acts on (ĝ−1)12, either matrix element (Â−1 0 )21 or Ĝ21 is present that gives a vanishing result when the fluctuating field tends to zero. Thus (Ξ̂−1)22 = −(Â−1 0 )22([1̂ − L̂]−1)22(g −1 A0), (70) and taking into account that when v → 0, the matrix L̂ is diagonal: (Ξ̂−1)22 = −(Â−1 0 )22[1 − L̂22] −1g−1 A0 . (71) To find the function L̂, we use a fact that operator X00 + XAA is an integral of motion in the case JB = 0 ([X00 + XAA,Heff + V ] = 0). It gives 〈Tτ (X00 + XAA)(τ)(X00 + XAA)(τ ′)〉 = 〈X00 + XAA〉, (72) 〈Tτ (X00 + XAA)(τ1)X A0(τ)X0A(τ ′)〉 = 〈TτXA0(τ)X0A(τ ′)〉. (73) Then, using (23) we obtain the differentiation properties in the 22 subspace (corresponding to itinerant particles): (Â′)22(Â0)22 = −(Â0)22 − (Â0) 2 22, (Â′)22(Â −1 0 )22 = (Â−1 0 )22 + 1, (74) (Â′)22Ĝ22 = −(1 + (Â0)22)Ĝ22. (75) 596 Asymmetric Hubbard model within generating functional approach in DMFT This makes possible to write the iteration series for L̂22 as a sum of geometric progression: L̂22 = [ 1 + (Â−1 0 )22 ] [ JA(ωn)GA(ωn) − [JA(ωn)GA(ωn)]2 + · · · ] = − 1 − 〈X00 + XAA〉〈X00 + XAA〉 JA(ωn)GA(ωn) 1 + JA(ωn)GA(ωn) . (76) Finally, we get the expression for the Green’s function irreducible part for itinerant particles Ξ−1 A = g−1 A0(ωn) − JA(ωn) 〈X00 + XAA〉 + JA(ωn). (77) This expression corresponds to the exact result (see for example [37–39]); it is also obtained within the different-time decoupling approach [9,13,14]. 4. Relation of the generating functional approach to other a pproximations Let us compare the results of the generating functional approach with other approximate meth- ods of calculating the Green’s function of the single-site problem. We consider, in detail, the first iteration in the series for the total irreducible part. When the fluctuating field tends to zero (vα → 0), the expression for (Ξ̂−1)11, as it follows from (46), is (Ξ̂−1)11(τ − τ ′) = − 1 D(τ) δΦV δv0A(τ) g−1 B0(τ − τ ′) + β ∫ 0 dτ2 β ∫ 0 dτ4Ĵ22(τ − τ4)Ĝ22(τ4 − τ2) δΦV δv0A(τ) δ2ΦV δv(τ)δv̄(τ2) g−1 B0(τ2 − τ ′) D(τ)D(τ2) + β ∫ 0 dτ4Ĵ22(τ − τ4)Ĝ22(τ4 − τ ′) δΦV δv0A(τ) δΦV δv0B(τ ′) δ(τ − τ ′) D(τ)D(τ ′) + β ∫ 0 dτ2 β ∫ 0 dτ4Ĵ11(τ − τ2)Ĝ11(τ2 − τ4) δΦV δv0A(τ) δ2ΦV δv0B(τ)δv0B(τ4) [ δΦV δv0B(τ4) ]−2 g−1 B0(τ4 − τ ′) D(τ) . (78) Here δ2ΦV δv(τ)δv̄(τ2) = GAB(τ − τ2) ≡ 〈TτXAB(τ)XBA(τ2)〉, (79) while the second derivative of the generating functional with respect to the field v0B can be represented as a cumulant Green’s function: G0B c (τ − τ4) ≡ δ2ΦV δv0B(τ)δv0B(τ4) ∣ ∣ ∣ ∣ v=0 = 〈Tτ (X00+XBB)τ (X00+XBB)τ4 〉 − 〈Tτ (X00+XBB)τ 〉〈Tτ (X00+XBB)τ4 〉. (80) In the frequency representation (Ξ̂−1)11(ωn) = 1 A0B { g−1 B0(ωn)[1 − QB(ωn) − NB(ωn)] + 1 A0A S̃B } , (81) where QB(ωn) = ( δΦV δv0A )−2 1 β ∑ ωm eiωm0+ JB(ωm)G0B c (ωm − ωn) (82) 597 I.V.Stasyuk, O.B.Hera NB(ωn) = 1 β ∑ ωm eiωm0+ JA(ωm)GA(ωm)GAB(ωm − ωn) 1 D , (83) S̃B = 1 β ∑ ωm eiωm0+ JA(ωm)GA(ωm). (84) Formula (81) gives a full expression for the inverse irreducible part Ξ̂−1 in the linear, with respect to Jσ, approximation. 4.1. Comparison with GH3 Basing on the obtained expression for Ξ̂−1, let us consider some of its possible simplifications. At first, we compare the result (81) of the generating functional approach with the irreducible part Ξ−1 obtained in the GH3 approximation [9]. For this purpose we consider the fermionic and bosonic Green’s functions in (81) in the zero approximation: GAB(ωm) ≈ 〈XAA − XBB〉gAB(ωm), G0B c (ωm) ≈ βδ(ωm)A0B(1 − A0B), (85) GA(ωm) ≈ A0AgA0(ωm), GB(ωm) ≈ A0BgB0(ωm), (86) gpq = 1 iω − λpq , λpq = λp − λq, (87) where λp(q) are eigenvalues of the unperturbed singe-site Hamiltonian. This approximation allows us to apply the following identity 1 iωm − λA0 1 iωm − iωn − λAB = −1 iωn − λB0 [ 1 iωm − λA0 − 1 iωm − iωn − λAB ] (88) calculating the product g−1 B0(ωn)NB(ωn) in (81). Thus, we obtain QB(ωn) = (1 − A0B)JB(ωn)gB0(ωn) (89) and (Ξ−1)11(ωn) = 1 A0B [ g−1 0B(ωn) − 1 A0B PB(ωn) + 1 A0B S̃B(ωn) − (1 − A0B)JB(ωn) ] , (90) where PB(ωn) = 1 β ∑ ωm eiωm0+ JA(ωm) 〈XAA − XBB〉 iωm − iωn − λAB (91) is the function that in diagram representation corresponds to the loop containing the lines of the coherent potential JA and unperturbed bosonic Green’s function gAB oriented in the opposite directions. On the other hand, such a loop appears from different-time decoupling of the Green’s function 〈〈XABξA\|ξ † AXBA〉〉ω that is contributing to the total irreducible part of the single-site Green’s function in the equation of motion approach in the GH3 approximation [9,14]. The coherent potential Jσ can be written in the Lehmann representation Jσ(ωn) = − 1 π lim η→0+ ∫ +∞ −∞ ImJσ(ω′ + iη)dω′ iωn − ω′ . (92) Using the relation of the (88) type as well as formulae 1 β ∑ ωm 〈XAA + XBB〉 iωm − iωn − λAB eiωm0+ = 〈XAA〉, (93) 1 β ∑ ωm 1 iωm − ω′ eiωm0+ = 1 eβω′ + 1 = 1 2 − 1 2 tanh βω′ 2 , (94) 598 Asymmetric Hubbard model within generating functional approach in DMFT we obtain the expression for PB in the form of integral over frequency. PB(ωn) = 〈XAA + XBB〉 2 JA(iωn + λAB) + 〈XAA − XBB〉 2 YB(ωn), YB(ωn) = 1 2π ∫ +∞ −∞ [−2 Im JA(ω′ + i0+)] iωn − ω′ + λAB tanh βω′ 2 dω′. (95) Considering the expression for the Green’s function GB(ωn) = A0B g−1 B0(ωn) − PB(ωn) A0B + S̃B A0B − JB(ωn) (96) one can see that (96) together with (95) corresponds to the GH3 approximation. The correspon- dence: PB(ωn) ↔ −RB(ω); S̃B ↔ −VAϕB (in the notations used in [9]) takes place. 4.2. Extension of JK decoupling to AHM Our next step is to compare the obtained expression (81) for Ξ̂−1 with the expression that can be found for the single-site problem of AHM within the framework of the equation of motion approach using the decoupling [40] used by Jeschke and Kotliar in the paper [23]. Their scheme for the model with the degeneracy and with the exclusion of the doubly occupied states can be easily extended to the case when µA 6= µB . Following the scheme proposed in [23] for solving the DMFT problem, we obtain (see Appendix) Gσ = A0σ + I1,σ̄(ω) ω − λσ0 + Jσ(ω) + I2,σ̄(ω) − Jσ(ω)I1,σ̄(ω) . (97) Let us write the corresponding expression for the irreducible part and linearize it with respect to the coherent potential (Jσ): Ξ−1 σ (ωn) = g−1 σ0 (ωn) + I2,σ̄(ωn) − Jσ(ωn)I1,σ̄(ωn) − Jσ(ωn) A0σ + I1,σ̄(ωn) + Jσ ≈ 1 A0σ [ g−1 σ0 (ωn) − (1 − A0σ)Jσ(ωn) + I2,σ̄(ωn) − g−1 σ0 I1,σ̄(ωn) A0σ ] . (98) We transform the expressions for functions I1,σ̄ (111) and I2,σ̄ (112) using the zero approxima- tion for electron Green’s function (〈〈X0A\|XA0〉〉iωm = A0[iωm −λA0] −1) and the scheme described in the previous subsection. As a result we obtain: g−1 B0(ωn)I1,A(ωn) = − 〈XAA − X00〉 2 JA(iωn + λAB) + 〈XAA + X00〉 2 YB(ωn) − S̃B , (99) I2,A(ωn) = − 1 2 JA(iωn + λAB) + 1 2 YB(ωn). (100) In its turn, it leads to the relation I2,A(ωn) − g−1 B0 I1,A(ωn) A0B = S̃B A0B − PB(ωn) A0B . (101) One can see that in this case expressions (98) and (96) coincide. Therefore, the JK decoupling scheme corresponds to the GH3 approximation after the replace- ment Gσ → G0 σ and at the linearization of Ξ−1 σ with respect to Jσ,(σ̄). Besides, such a scheme corresponds to the linear approximation in the generating functional approach, while in the latter the fermionic and bosonic Green’s functions are considered in the zero approximation. 599 I.V.Stasyuk, O.B.Hera 5. Conclusions The generating functional approach developed based on the Kadanoff-Baym idea by Izyumov et al. is adapted to solving the single-site problem in the dynamical mean field theory. To construct the equations for Green’s functions the recurrent form of Wick’s theorem for Hubbard operators is used. This approach is similar to the usual scheme where the equations are constructed using differentiation with respect to the time variable. However, when there are more than one operators with the same time that is differentiated, these methods are different. Within the framework of the generating functional approach a closed self-consistent equation in functional derivatives is obtained for constructing the irreducible part (self energy, Green’s function, etc.). To solve this equation the iterative procedure is used. It is shown that the solution for the irreducible part can be expressed in terms of two functions. These functions are formed by the different types of terms in the iteration series: one function involves the derivatives of linear terms of fluctuating fields which are present in the g−1 function; the other one can be considered as a terminating part of the Green’s function (or the irreducible part). As an example, the asymmetric Hubbard model is considered. The algorithm is given for constructing the inverse irreducible part Ξ−1 σ , σ = {A,B} (or the mass operator Fσ and terminating part Λσ) in the form of a series in powers of coherent potentials JA, JB. The irreducible part in the approximation linear with respect to Jσ is obtained and anal- ysed explicitly. Comparison is performed with the results of other approximations based on the decoupling of the single-site Hubbard operator Green’s functions of higher order in equations of mo- tion. It is established that after replacing fermionic and bosonic Green’s functions in the expression for Ξ−1 σ by their zero approximations, this expression is reduced to the corresponding expression in the GH3 approximation. Besides, at the similar simplification, the decoupling scheme used by Jeschke and Kotliar for the Hubbard-like models can be also reduced to the GH3 approximation. It should be mentioned, that such a scheme uses the zero-order bosonic Green’s functions. Therefore it is also more simple than the generating functional approach formulated in the approximation linear with respect to Jσ. It should be noted that, in the proposed approach, a full closure of equations for the Green’s function requires calculation of bosonic Green’s functions appearing in the expression for Ξ−1 (see for example (81)). It also can be done within the framework of the generating functional approach, similarly as it was done in [35] using equations of motion with functional derivatives. A. Decoupling for Anderson impurity model for AHM In the equation of motion decoupling method proposed by Jeschke and Kotliar [23], the DMFT single-site problem is interpreted as Anderson impurity model. In this case, when U → ∞, the Hamiltonian of the model reads Ĥ = ∑ kσ εkσc † kσckσ + ∑ σ EσXσσ + E0X 00 + ∑ kσ (V ∗ kσc † kσX0σ + VkσXσ0ckσ), (102) where σ = A,B; Vkσ is the hybridization parameter. Unlike the similar problem for the standard Hubbard model, here EA 6= EB , εkA 6= εkB , VkA 6= VkB . To determine the Green’s function 〈〈X0σ\|Xσ0〉〉 the equation of motion approach is used. At the first stage, the Green’s functions 〈〈(X00 + Xσσ)ckσ\|X σ0〉〉 and 〈〈X σ̄σckσ̄\|X σ0〉〉 appear; they are calculated using the equations of motion and differentiating again the left time argument. The Green’s functions of higher order appearing at the second stage are decoupled [23]. This procedure leads to the average values 〈ckσ̄X σ̄0〉 and 〈c†k′σ̄ckσ̄〉 that are determined self-consistently. As a result, the initial function 〈〈X0σ\|Xσ0〉〉 is determined from the set of equations (ω − λσ0)〈〈X 0σ\|Xσ0〉〉ω = A0σ + ∆σ(ω)〈〈X0σ\|Xσ0〉〉ω + I1,σ̄(ω) − I2,σ̄(ω)〈〈X0σ\|Xσ0〉〉ω + I1,σ̄(ω) ∑ k′ Vk′σ〈〈ck′σ\|X σ0〉〉ω , (103) (ω − εk′σ)〈〈ck′σ\|X σ0〉〉ω = V ∗ k′σ〈〈X 0σ\|Xσ0〉〉ω, (104) 600 Asymmetric Hubbard model within generating functional approach in DMFT where λσσ̄ = Eσ − Eσ̄; λσ0 = Eσ − E0; A0σ = 〈X00 + Xσσ〉, (105) ∆σ(ω) = ∑ k \|Vkσ\| 2 ω − εkσ , (106) I1,σ̄(ω) = ∑ k Vkσ̄ 〈ckσ̄X σ̄0〉 ω − λσσ̄ − εkσ̄ ; I2,σ̄(ω) = − ∑ kk′ Vkσ̄V ∗ k′σ̄ 〈c†kσ̄ckσ̄〉 ω − λσσ̄ − εkσ̄ . (107) As a result we come to Gσ = A0σ + I1,σ̄(ω) ω − λσ0 + ∆σ(ω) + I2,σ̄(ω) − ∆σ(ω)I1,σ̄(ω) . (108) In this approach, the hybridization function ∆σ(ω) corresponds to the coherent potential: ∆σ(ω) = Jσ(ω) (see [23] and references therein). To calculate the average values in (107), the following method is used 〈ckσ̄X σ̄0〉 = − 1 β ∑ ωn 〈〈ckσ̄\|X σ̄0〉〉iωn eiωn0+ (109) 〈c†k′σ̄ckσ̄〉 = 1 β ∑ ωn 〈〈ckσ̄\|c † k′σ̄〉〉iωn eiωn0+ = 1 β ∑ ωn [ δkk′ iωn − εkσ̄ + V ∗ kσ̄Vk′σ̄ (iωn − εkσ̄)(iωn − εk′σ̄) ] . (110) Following the procedure described in [23], we obtain I1,σ̄(iωn) = − 1 β ∑ ωn eiωm0+ ∆σ̄(iωm) 1 iωn − iωm − λσσ̄ 〈〈X0σ̄\|X σ̄0〉〉iωm + 1 β ∑ ωn eiωm0+ ∆σ̄(iωn − λσσ̄) 1 iωn − iωm − λσσ̄ 〈〈X0σ̄\|X σ̄0〉〉iωm , (111) I2,σ̄(iωn) = − 1 β ∑ ωn eiωm0+ [∆σ̄(iωm) − ∆σ̄(iωn − λσσ̄)] 1 iωn − iωm − λσσ̄ − 1 β ∑ ωn eiωm0+ [∆σ̄(iωm) − ∆σ̄(iωn − λσσ̄)] iωn − iωm − λσσ̄ 〈〈X0σ̄\|X σ̄0〉〉iωm ∆σ̄(iωm). (112) In comparison with [23], where the case of degeneracy (Eσ = −µ) was considered, in the expressions (111) and (112), instead of the factor [iωn − iωm]−1, the unperturbed bosonic Green’s function [iωn − iωm − λσσ̄]−1 is present. References 1. Izyumov Y.A., UFN, 1995, 165, 403. [Izyumov Y.A., Phys. Usp., 1995, 38, 385]. 2. 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Асиметрична модель Хаббарда в теорiї динамiчного середнього поля в методi твiрного функцiоналу I.В.Стасюк, О.Б.Гера Iнститут фiзики конденсованих систем НАН України, 79011 Львiв, вул. Свєнцiцького, 1 Отримано 6 червня 2006 р. В роботi розвивається новий аналiтичний пiдхiд для розв’язання ефективної одновузлової задачi в методi динамiчного середнього поля. Пiдхiд ґрунтується на методi твiрного функцiоналу Каданова- Бейма у формi, розробленiй в роботах Iзюмова та iн. Вiн дає можливiсть отримати замкнене рiвнян- ня у функцiональних похiдних для незвiдної частини одновузлової функцiї Грiна частинок; розв’язки будуються iтеративним способом. В ролi застосування запропонованої схеми взято асиметричну модель Хаббарда (АМХ). Побудовано обернену незвiдну частину Ξ −1 σ одновузлової функцiї Грiна в лiнiйному наближеннi за когерентним потенцiалом Jσ . Виходячи з отриманого результату, роз- глянено функцiю Грiна рухомих частинок у границi Фалiкова-Кiмбала АМХ, проаналiзовано схеми розщеплень у рiвняннях руху для одновузлової функцiї Грiна (наближення GH3, розщеплення Єшке- Котляра). Ключовi слова: сильноскорельованi системи, асиметрична модель Хаббарда, одновузлова задача, теорiя динамiчного середнього поля PACS: 71.10.Fd, 71.27.+a, 71.30.+h 602

Asymmetric Hubbard model within generating functional approach in dynamical mean field theory

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