Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping: general approach

The Falicov-Kimball model with correlated hopping is studied in the limit of in nite spatial dimensions. Dynamical susceptibilities are calculated using the generalization of the Dynamical Mean-Field Theory (DMFT), which is based on expansion over electron hopping around the single-site limit. A s...

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Дата:	2009
Автори:	Shvaika, A.M., Farenyuk, O.Ya.
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Опубліковано:	Інститут фізики конденсованих систем НАН України 2009
Назва видання:	Condensed Matter Physics
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Цитувати:	Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping: general approach / A.M. Shvaika, O.Ya. Farenyuk // Condensed Matter Physics. — 2009. — Т. 12, № 1. — С. 63-74. — Бібліогр.: 26 назв. — англ.

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spelling	irk-123456789-1197652017-06-09T03:04:46Z Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping: general approach Shvaika, A.M. Farenyuk, O.Ya. The Falicov-Kimball model with correlated hopping is studied in the limit of in nite spatial dimensions. Dynamical susceptibilities are calculated using the generalization of the Dynamical Mean-Field Theory (DMFT), which is based on expansion over electron hopping around the single-site limit. A special case of semi-elliptic density of states and diagonal correlated hopping is considered numerically and the absence of phase transition for all temperatures except T = 0 is demonstrated, which corresponds to the known results. Дослiджується модель Фалiкова-Кiмбала з корельованим переносом в границi безмежної розмiрностi простору. Динамiчнi сприйнятливостi розраховуються за допомогою теорiї динамiчного середнього поля, що ґрунтується на розвиненнях за електронним переносом навколо одновузлової границi. Дослiджується чисельно частковий випадок напiвелiптичної густини станiв та дiагонального корельованого переносу та показано вiдсутнiсть фазових переходiв для всiх температур крiм T = 0, що узгоджується з вiдомими результатами. 2009 Article Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping: general approach / A.M. Shvaika, O.Ya. Farenyuk // Condensed Matter Physics. — 2009. — Т. 12, № 1. — С. 63-74. — Бібліогр.: 26 назв. — англ. 1607-324X PACS: 71.10.Fd DOI:10.5488/CMP.12.1.63 http://dspace.nbuv.gov.ua/handle/123456789/119765 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	The Falicov-Kimball model with correlated hopping is studied in the limit of in nite spatial dimensions. Dynamical susceptibilities are calculated using the generalization of the Dynamical Mean-Field Theory (DMFT), which is based on expansion over electron hopping around the single-site limit. A special case of semi-elliptic density of states and diagonal correlated hopping is considered numerically and the absence of phase transition for all temperatures except T = 0 is demonstrated, which corresponds to the known results.
format	Article
author	Shvaika, A.M. Farenyuk, O.Ya.
spellingShingle	Shvaika, A.M. Farenyuk, O.Ya. Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping: general approach Condensed Matter Physics
author_facet	Shvaika, A.M. Farenyuk, O.Ya.
author_sort	Shvaika, A.M.
title	Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping: general approach
title_short	Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping: general approach
title_full	Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping: general approach
title_fullStr	Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping: general approach
title_full_unstemmed	Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping: general approach
title_sort	dynamical susceptibilities of the falicov-kimball model with correlated hopping: general approach
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/119765
citation_txt	Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping: general approach / A.M. Shvaika, O.Ya. Farenyuk // Condensed Matter Physics. — 2009. — Т. 12, № 1. — С. 63-74. — Бібліогр.: 26 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT shvaikaam dynamicalsusceptibilitiesofthefalicovkimballmodelwithcorrelatedhoppinggeneralapproach AT farenyukoya dynamicalsusceptibilitiesofthefalicovkimballmodelwithcorrelatedhoppinggeneralapproach
first_indexed	2025-07-08T16:33:37Z
last_indexed	2025-07-08T16:33:37Z
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fulltext	Condensed Matter Physics 2009, Vol. 12, No 1, pp. 63–74 Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping: general approach A.M.Shvaika, O.Ya.Farenyuk Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine Received November 21, 2008, in final form February 18, 2009 The Falicov-Kimball model with correlated hopping is studied in the limit of infinite spatial dimensions. Dynami- cal susceptibilities are calculated using the generalization of the Dynamical Mean-Field Theory (DMFT), which is based on expansion over electron hopping around the single-site limit. A special case of semi-elliptic density of states and diagonal correlated hopping is considered numerically and the absence of phase transition for all temperatures except T = 0 is demonstrated, which corresponds to the known results. Key words: Falicov-Kimball model, correlated hopping, electron susceptibilities, dynamical mean field theory, strong coupling PACS: 71.10.Fd 1. Introduction The Falicov-Kimball model was first introduced in [1] as a model of metal-insulator transition. It appears that Falicov-Kimball model is the simplest model of electron correlations which displays long-range ordered phases [2]. It was applied to the investigations of such phenomena as valence transitions [3,4], phase separation and crystallization [5]. Dynamical properties of the model, in- cluding electron susceptibilities [6,7] and Raman response [8,9] were also investigated. The model contains two types of Fermi-particles: itinerant (conductive) particles, referred to as d-electrons, and localized particles, f-electrons or “ions”. On the other hand, the model can be interpreted as an asymmetric Hubbard model with infinite mass of electrons for one of the spin orientations. Falicov-Kimball model is attractive not only due to its simplicity but also because many exact results were obtained for this model [2]. Among them there are proofs of phase separation for all spatial dimensions of the lattice from one to infinity, theorems of ground-state properties in one and two dimensions. In addition to the results for low spatial dimensions, there exists an exact solution of the model in the infinite dimensions case [10] (see for review [2]). This solution is important in applying Dynamical Mean-Field Theory (DMFT). DMFT is based on the self-consistent mapping of a lattice problem onto an auxiliary single- site problem [11]. This transformation is advantageous due to the local nature of self-energy in the limit of infinite spatial dimensions or large coordination. The development of the DMFT approach brought about a noticeable progress in studying strongly correlated systems, but in its traditional form (see, e.g. [11]) it cannot be applied to an important form of hopping, the so-called correlated hopping. Correlated hopping describes the case when the hopping integral depends on the occupation of neighboring sites. Though Falicov-Kimball model gives a reasonable physical description of many processes, it neglects all nonlocal dynamical interactions. Correlated hopping is a notable example of such an interaction. For the first time it was introduced in the well-known Schubin and Wonsowsky work [12], the value of that kind of hopping was estimated for the first time by Hubbard [13], then Hirsch pointed out that correlated hopping can play an important role in the appearance of superconductivity [14]. The importance of the correlated hopping in ferromagnetism was shown in [15–18]. Other reasons for considering the correlated hopping are based on a substantial Coulombian interaction between electrons, reported for some materials [19, c© A.M.Shvaika, O.Ya.Farenyuk 63 A.M.Shvaika, O.Ya.Farenyuk 20], dynamical part of which can be approximated by correlated hopping. When correlated hopping is present, the self-energy becomes non-local, and, as it was shown in earlier investigations [21], this non-locality can be presented as the next-nearest-neighbors hopping contribution to self-energy. But using expansions over electron hopping (Larkin approach, opposite to Dyson approach which is based on expansions over many-electron interactions and leads to the self-energy concept), it can be shown that such non-locality is just an artifact of the Dyson approach. Even for correlated hopping, the irreducible part of Green’s function, which cannot be divided by cutting a hopping line, remains local which permits to construct the DMFT for such systems [22]. In this article we develop a general scheme of calculating dynamical susceptibilities for systems with correlated hopping, which is based on generalization of DMFT. It is used to investigate electron susceptibilities of the Falicov-Kimball model with correlated hopping. This method permits to obtain explicit expressions for dynamical susceptibilities. The limit of diagonal correlated hopping on Bethe lattice is studied using the obtained general expressions. 2. The model Hamiltonian of the (spinless) Falicov-Kimball model can be written as: H = ∑ i (Unidnif − µfnif − µdnid) + Ht , (2.1) where Ht is a hopping term for the d-particles, µf and µd are the chemical potentials for the f- and d-particles, respectively, and U is the Coulomb repulsion of the particles at the same site. Correlated hopping term for the Falicov-Kimball model has the following form: Ht = 1√ D ∑ 〈i,j〉 [ t1d † i dj + t2d † idj(nif + njf ) + t3d † i djnif njf ] . (2.2) Here t1 describes the part of hopping between sites i and j, which does not depend on the occupation of the sites, t2 takes into account the hopping between sites, one of which is occupied by the f-particle, and t3 describes the hopping between the sites occupied by the f-particles. Such a Hamiltonian can be presented in terms of hopping between sites with fixed occupations: Ht = 1√ D ∑ 〈i,j〉 (t++ ij P+ i d†i djP + j + t−− ij P− i d†i djP − j +t+− ij P+ i d†i djP − j + t−+ ij P− i d†i djP + j ) = 1√ D ∑ 〈i,j〉 d † i tijdj , (2.3) where P+ i = nif , P− i = 1− nif are the projection operators and we introduce matrix annihilation operator di = [ P+ i P− i ] di ; (2.4) and hopping matrix tij = ∥ ∥ ∥ ∥ t++ ij t+− ij t−+ ij t−− ij ∥ ∥ ∥ ∥ = atij . (2.5) In fact, correlated hopping naturally leads to matrix generalization for all quantities [22]. Both representations for the Hamiltonian are related by the following equations: t−−=t1, t1= t−−, t+−=t−+ = t1 + t2, t2= t+−(−+) − t−−, t++=t1 + 2t2 + t3, t3= t++ + t−− − t+− − t−+. (2.6) 64 Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping The components of the matrix Green’s function are defined by the following formula: Gαγ ij (τ − τ ′)=− 〈 Tτdiα(τ)d†jγ (τ ′) 〉 = β δΩ δtγα ji (τ ′ − τ) , (2.7) Gij(τ − τ ′)=− 〈 Tτdiα(τ) ⊗ d † jγ(τ ′) 〉 , (2.8) where Ω is a grand canonical potential functional. The total Green’s function is a sum of all components of the matrix Green’s function: Gij(τ − τ ′) = − 〈 Tdi(τ)d+ j (τ ′) 〉 = β δΩ δtji(τ ′ − τ) = ∑ αγ Gαγ ij (τ − τ ′). (2.9) As it was shown in the reference [22], for the systems with correlated hopping, the strong- coupling approach, based on the expansions over electron hopping around the atomic limit, is more convenient. In this approach, the single-electron Green’s function is presented as a solution of the matrix generalization of the Larkin equation: Gij(ω) = Ξij(ω) + ∑ lm Ξil(ω)tlmGmj(ω), (2.10) where Ξij(ω) is an irreducible part of the Green’s function which cannot be divided into parts by cutting one hopping line. This irreducible part is local in the limit of infinite spatial dimensions even for the systems with correlated hopping [22,23]. After the Fourier transformation to the momentum representation, this equation can be solved for the Green’s function: Gk(ω) = [ Ξ−1 k (ω) − tk ]−1 , (2.11) with tk = aεk, where εk is an unperturbed band energy. Now we can apply the modified Dynamical Mean Field Theory (DMFT), which leads to exact results in the limit of infinite spatial dimen- sions D → ∞, when hopping integrals are scaled according to: tαβ → tαβ/ √ D. In this limit, an irreducible part is local Ξk(ω) = Ξ(ω) (2.12) and can be obtained from the condition that single-site Green’s function for the lattice and for the auxiliary single-site problem is the same [22]. On the other hand, matrix Green’s function of the single-site problem can be written in Larkin representation as: Gimp(ω) = [ Ξ−1(ω) − J(ω) ]−1 , (2.13) where J is the coherent potential matrix (λ-field) [22,10]. For the Falicov-Kimball model, Gimp can be calculated exactly: Gimp(ω) = ∥ ∥ ∥ ∥ G++(ω)G+−(ω) G−+(ω)G−−(ω) ∥ ∥ ∥ ∥ = ∥ ∥ ∥ ∥ ∥ 〈P+〉 ω+µd−U−J++(ω) 0 0 〈P−〉 ω+µd−J−−(ω) ∥ ∥ ∥ ∥ ∥ = ∥ ∥ ∥ ∥ 〈P+〉g+(ω) 0 0 〈P−〉g−(ω) ∥ ∥ ∥ ∥ . (2.14) Furthermore, itinerant electrons concentration can be calculated as: 〈nd〉 = 1 β ∑ ν [ G++ imp(iων) + G−− imp(iων) ] = 〈P+〉n+ + 〈P−〉n− . (2.15) 3. Susceptibilities from the DMFT point of view In general, susceptibilities are defined by: χAB ij (τ − τ ′) = 〈TτAi(τ)Bj (τ ′)〉. (3.1) 65 A.M.Shvaika, O.Ya.Farenyuk For instance, itinerant electron charge susceptibilities for the Falicov-Kimball model with correlated hopping can be obtained from the following matrix of projected susceptibilities: χnn ij (τ − τ ′)= ∥ ∥ ∥ ∥ 〈TτP+ i nid(τ)njd(τ ′)P+ j 〉〈TτP+ i nid(τ)njd(τ ′)P− j 〉〈TτP− i nid(τ)njd(τ ′)P+ j 〉〈TτP− i nid(τ)njd(τ ′)P− j 〉 ∥ ∥ ∥ ∥ . (3.2) Analyzing the diagrammatical series, obtained by using Wick theorem in the same way as in [7], one can see that many-particle Green functions can be written in the following form: χAB q (ω) = , (3.3) where t̃q = = + (3.4) is a sum of the chains of hopping and arrow denotes lattice Green’s function (2.11). Here , 〈 ∣ ∣ ∣ and L are irreducible parts that cannot be divided by cutting two hopping lines and q = − 1 N ∑ k t̃k ⊗ t̃k+q . (3.5) It should be noted, that here correlated hopping leads to a Cartesian product over α = + and − in place of ordinary multiplication in the similar equations for systems without correlated hopping [7]. The full four-vertex in equation (3.3) is connected with the irreducible one L by the Bethe- Salpeter-like equation: + . (3.6) All the introduced irreducible parts are local in the limit of infinite spatial dimensions [7] and can be calculated from the corresponding single-site many-particle Green functions: Υ ( 1 4 2 3 ) = 1 2 3 4 ≡ 〈 Tτa† 1a2 a† 3a4 〉 imp − 〈 Tτa† 1a2 〉 imp 〈 Tτa† 3a4 〉 imp + 〈 Tτa† 1a4 〉 imp 〈 Tτa† 3a2 〉 imp , (3.7) where Υ is 4 × 4 matrix, whose indices take the values “++”, “−−”, “+−”, “−+”; 1 2 0 ≡ 〈 Tτ Â0 a† 1a2 〉 imp − 〈 Â0 〉 imp 〈 Tτa† 1a2 〉 imp , (3.8) 10 A B ≡ 〈 Tτ Â0 B̂1 〉 imp − 〈 Â0 〉 imp 〈 B̂1 〉 imp . (3.9) Quantity (3.7) correspond to the following diagrammatical sequence: 1 2 3 4 = + + + + + . . . , where thin arrows correspond to the irreducible parts Ξ and + . (3.10) 66 Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping Now, using definition (3.4), one can state that: L= . (3.11) Here the thin wavy lines correspond to the coherent potential J, the thick one J̃ = = + (3.12) is the sum of chains of the wavy lines, and a line above another one should be treated as [see also (3.5)]: = −J̃⊗ J̃. (3.13) From equations (3.10) and (3.6) the four-vertex can be obtained as =L q L -1 + - q -1 [ ] . (3.14) In the same way, equations (3.8) and (3.9) for other irreducible many-particle Green functions can be written down: + , (3.15) . (3.16) After performing all the necessary substitutions into equation (3.3) one can obtain the following formula for the lattice susceptibility χAB q (ωn): χAB q (ωn) = , (3.17) where = −Gimp ⊗Gimp , (3.18) q = − 1 N ∑ k Gσk(ων) ⊗Gσk+q(ων+m). (3.19) In equation (3.17), quantity Λ = [. . . ]−1 denotes an inverse kernel of the corresponding integral equation: L . (3.20) Many-particle correlation function (3.7) contains two contributions (see Appendix): = ∥ ∥ ∥ ∥ Υ1 s 0 0 Υ2 s ∥ ∥ ∥ ∥ δνν′ − ∥ ∥ ∥ ∥ Υ1 d0 0 0 ∥ ∥ ∥ ∥ δm0 . (3.21) 67 A.M.Shvaika, O.Ya.Farenyuk As a result, the expression enclosed in square brackets in (3.20) consists of the terms diagonal on internal ν = ν′ and external ν = ν + m (m = 0) frequencies too: [ . . .] = Π ( ν ν′ ν + m ν′ + m ) δνν′ − ∥ ∥ ∥ ∥ Υ1 d 0 0 0 ∥ ∥ ∥ ∥ ( ν ν′ ν + m ν′ + m ) δm0 . (3.22) The term, diagonal on internal frequencies is equal to: Πδνν′ = δνν′ − ∥ ∥ ∥ ∥ Υ1 s 0 0 Υ2 s ∥ ∥ ∥ ∥ δνν′ . (3.23) Explicit form of irreducible many-particle Green functions of the single-site model is presented in the Appendix. Since the matrix for the 3-time irreducible Green’s function has the following structure: = [ 1 ,0 ] (3.24) and is constructed by the 2 × 2 blocks (see Appendix), we are interested only in the upper left 2 × 2 block of matrix Λ: L = [ 1 , 0 ] ∥ ∥ ∥ ∥ Λ11Λ12 Λ21Λ22 ∥ ∥ ∥ ∥   1 0   = L 11 11 . (3.25) Substituting (3.22) and (3.23) into (3.20) one can obtain: ∑ ν′   ∥ ∥ ∥ ∥ Π11Π12 Π21Π22 ∥ ∥ ∥ ∥ ( ν ν′ ν + m ν′ + m ) δνν′ − 〈P+〉〈P−〉 ∥ ∥ ∥ ∥ ∥ ∥ [ g+ ν g−ν ] ⊗ [ g+ ν′ ,g − ν′ ] 0 0 0 ∥ ∥ ∥ ∥ ∥ ∥ δm0   × ∥ ∥ ∥ ∥ Λ11Λ12 Λ21Λ22 ∥ ∥ ∥ ∥ ( ν′ ν′′ ν′ + m ν′′ + m ) = δνν′′ I(4), (3.26) which leads to the following closed system of equations for Λ11 and Λ21: Π11 ( ν ν ν ν ) Λ11 ( ν ν′′ ν ν′′ ) + Π12 ( ν ν ν ν ) Λ21 ( ν ν′′ ν ν′′ ) − δm0〈P+〉〈P−〉 ∑ ν′ { [ g+ ν g−ν ] ⊗ [ g+ ν′ g−ν′ ]} Λ11 ( ν′ ν′′ ν′ ν′′ ) = δνν′′ I(2), (3.27) Π21 ( ν ν ν ν ) Λ11 ( ν ν′′ ν ν′′ ) + Π22 ( ν ν ν ν ) Λ21 ( ν ν′′ ν ν′′ ) = 0. (3.28) Matrix Λ21 can be obtained from the equation (3.28). After its substitution into the formula (3.27), one obtains final system of equations. This system is algebraic for the dynamic susceptibilities when m 6= 0. For the static susceptibilities m = 0 one has a system of integral equations with separable kernel, which can be easily solved by multiplicating both sides of equation from the left by [g+ ν , g−ν ] and carrying out the sum over frequency ν: 1 β ∑ ν [ g+ ν g−ν ] Λ11 ( ν ν′′ ν ν′′ ) = [ g+ ν′′ , g − ν′′ ] D−1 11 (ν′′) T − Θ , (3.29) where: D11(ν) = Π11 ( ν ν ν ν ) −Π12 ( ν ν ν ν ) Π−1 22 ( ν ν ν ν ) Π21 ( ν ν ν ν ) (3.30) and Θ = 〈P+〉〈P−〉 1 β ∑ ν [ g+ ν g−ν ] D−1 11 (ν) [ g+ ν , g−ν ] (3.31) 68 Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping determines critical temperature by equation T − Θ(T, q) = 0. Finally one can find Λ11 ( ν ν′′ ν ν′′ ) = D−1 11 (ν)δνν′′ + 〈P+〉〈P−〉 T − Θ D−1 11 (ν) [ g+ ν g−ν ] ⊗ [ g+ ν′′ g−ν′′ ] D−1 11 (ν′′) (3.32) which must be substituted in equation (3.25) and then in (3.17). 4. Diagonal correlated hopping in the case of semi-elliptic density of states Equation (3.25) together with (3.17) gives an exact analytical form for the electron susceptibi- lities, but to rewrite this equation explicitly one need to calculate the inverse of the 4 × 4 matrix in (3.23). Although, this can be easily done, the resulting expressions are cumbersome. In order to proceed, we can consider a simpler case, when correlated hopping and coherent potential are both diagonal (a+− = a−+ = 0). For the Bethe lattice with semielliptic density of states ρ(t) = 2 πW 2 √ W 2 − t2 (4.1) DMFT system of equations is reduced to the matrix one [22]: J(ω) = W 2 4 aGimp(ω)a. (4.2) Solving equation (4.2) for diagonal correlated hopping, one can obtain expressions for the coherent potential [22]: J++(ω)= (a++)2W 2 4 G++(ω) = 1 2 [ ω + µd − U ± i √ W 2 + − (ω + µd − U)2 ] , J−−(ω)= (a−−)2W 2 4 G−−(ω) = 1 2 [ ω + µd ± i √ W 2 − − (ω + µd)2 ] , J+−(ω)=0, J−+(ω)=0, (4.3) where: W 2 ± = (a±±)2W 2〈P±〉. Lattice Green functions also acquire a simple form: G++ k = 〈P+〉 ω + µd − U − J++〈P−〉 − a++〈P+〉εk , G−− k = 〈P−〉 ω + µd − U − J−−〈P+〉 − a−− k 〈P−〉εk . (4.4) As a result, matrices W and D11 will be diagonal and Θ(T, q) will be equal to: Θ(T,q) = 〈P+〉〈P−〉β ∑ ν [ g+ ν Π−1 ++,++(ν,q) g+ ν + g−ν Π−1 −−,−−(ν,q) g−ν ] , (4.5) where Παα,αα(ν,q) = gα ν gα ν Gαα ν Gαα ν − 1 N ∑ k Gαα ν,kGαα ν,k+q Gαα ν Gαα ν [2〈P α〉 − 1] + 〈P α〉 1 N ∑ k Gαα ν,kGαα ν,k+q = gα ν gα ν [ Lαα ν,q ]−1 , (4.6) which leads to: Θ(T,q) = 〈P+〉〈P−〉 1 β ∑ ν ( L++ ν,q + L−− ν,q ) . (4.7) 69 A.M.Shvaika, O.Ya.Farenyuk In the case of the Bethe lattice (Kelly-tree) the sum over k cannot be correctly defined in general. But we can redefine it for the uniform (q = 0) and chess-board [q = (π, π, π, . . . )] responses. When correlated hopping is diagonal, such a sum can be easily calculated: 1 N ∑ k G αα k,νG αα k+q,ν =        1 N ∑ k G αα k,νG αα k,ν = ∫ dtρ(t) [ Ξ−1 ν − a αα〈P α〉t ] −1 [ Ξ−1 ν − a αα〈P α〉t ] −1 , q = 0, 1 N ∑ k G αα k,νG αα −k,ν = ∫ dtρ(t) [ Ξ−1 ν − a αα〈P α〉t ] −1 [ Ξ−1 ν + a αα〈P α〉t ] −1 , q = π, (4.8) which gives 1 N ∑ k Gαα k,νGαα k+q,ν =          −Gαα ν Gαα ν 1 − Gαα ν Jαα ν q = 0, +Gαα ν Gαα ν 1 + Gαα ν Jαα ν , q = π (4.9) and Lαα ν,q = ±Gαα ν Jαα ν 〈P α〉 ± (2〈P α〉 − 1)Gαα ν Jαα ν , (4.10) where “−” sign corresponds to q = 0 and “+” sign to q = (π, π, π, . . . ). As a result: χAB q (m = 0) = + ∑ ν {∥ ∥ ∥ ∥ L++ ν g+ ν g+ ν 〈P+〉 0 0 L−− ν g−ν g−ν 〈P−〉 ∥ ∥ ∥ ∥ + 〈P+〉〈P−〉 [ L++ ν n+ −L−− ν n− ] ⊗ [ g+ ν ,−g−ν ] + 〈P+〉〈P−〉 [ n+ −n− ] ⊗ [ L++ ν g+ ν ,−L−− ν g−ν ] + [ n+ −n− ] ⊗ [ n+L++ ν 〈P−〉,−n−L−− ν 〈P+〉 ] + [ n+ −n− ] ⊗ [ n−g+ ν (g−ν )−1L−− ν 〈P+〉,−n+g−ν (g+ ν )−1L++ ν 〈P−〉 ] } + 〈P+〉〈P−〉 T − Θ(T, q) ∑ νν′ {[ L++ ν g+ ν L−− ν g−ν ] ⊗ [ L++ ν′ g+ ν′ , L −− ν′ g−ν′ ] + [ L++ ν g+ ν L−− ν g−ν ] ⊗ [ n+L++ ν′ 〈P−〉, n−L−− ν′ 〈P+〉 ] − [ L++ ν g+ ν L−− ν g−ν ] ⊗ [ n−g+ ν′(g − ν′)−1L−− ν′ 〈P+〉, n+g−ν′(g + ν′)−1L++ ν′ 〈P−〉 ] + ( L++ ν 〈P−〉 − L−− ν 〈P+〉 ) [ n+ −n− ] ⊗ [ L++ ν′ g+ ν′ , L −− ν′ g−ν′ ] + ( L++ ν 〈P−〉 − L−− ν 〈P+〉 ) [ n+ −n− ] ⊗ [ n+L++ ν′ 〈P−〉, n−L−− ν′ 〈P+〉 ] − ( L++ ν 〈P−〉 − L−− ν 〈P+〉 ) [ n+ −n− ] ⊗ [ n−g+ ν′(g − ν′)−1L−− ν′ 〈P+〉, n+g−ν′(g + ν′)−1L++ ν′ 〈P−〉 ]} . (4.11) Critical temperature is determined by the divergence of susceptibility and can be obtained from the equation T −Θ(T,q) = 0. The solutions of this equation were studied numerically and revealed that it can have only one solution T = 0 and the absence of phase transition for all temperatures except T = 0 can be stated. That result conforms with the known results for binary alloys with off- diagonal randomness [24,25]. Physical reason for that is the absence of self-intersections in Bethe lattice along with diagonal correlated hopping. In our case diagonal correlated hopping means that hopping is allowed only between sites with the same occupation of f-particles. That leads the lattice to “break-down” into finite-sized domains of sites with the same f-particles occupation “diluted” by the domains of sites of the other occupation. It should be noted that in the case of a fixed total electron concentration nd + nf = const the phase transition into a mixed-valence phase can take place in the Falicov-Kimball model without divergence of charge susceptibilities [26]. Such transitions can take place only when the energy level of f-states Ef = µd − µf is negative Ef < 0 which is not the case considered in the current article. 70 Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping 5. Conclusions In this paper we have investigated the susceptibilities of the Falicov-Kimball model with corre- late hopping. The model was studied using modified DMFT, applicable to systems with correlated hopping. Diagrammatical expression for the susceptibilities was obtained and proved to be identi- cal to the corresponding diagrammatic expression for the ordinary Falicov-Kimball model. Starting from this expression, explicit formulas for susceptibilities were obtained. As a particular case we have studied electron charge susceptibilities for the case of Bethe lattice with correlated hopping, which corresponds to the hopping only between sites with the same occupation. It was shown that phase transition cannot occur in such a system for any non- zero temperature which is consistent with the earlier result in the CPA approach. 71 A.M.Shvaika, O.Ya.Farenyuk Appendix. Irreducible many-particle Green functions for the Falicov-Kimball model Four-time Green’s function contains only the following non-zero components: + + + + n+m' n n+m n' ++ ++ = 〈P+〉〈P−〉δm,0 (iων−J ++ ν −U)(iω ν′−J ++ ν′ −U) − 〈P+〉〈P−〉δ ν,ν′ (iων−J ++ ν −U)(iων+m−J ++ ν+m −U) = (Υ1 s) ++δνν′ − (Υ1 d)++δm0 , - - - - n+m' n n+m n' −− −− = 〈P+〉〈P−〉δm,0 (iων−J −− ν )(iω ν′−J −− ν′ ) − 〈P+〉〈P−〉δ ν,ν′ (iων−J −− ν )(iων+m−J −− ν+m ) = (Υ1 s) −−δνν′ − (Υ1 d) −−δm0, + - + - n+m' n n+m n' ++ −− =−G++ ν G−− ν′ δm,0 = −(Υ1 d)+−δm0, - + - + n+m' n n+m n' −− ++ =−G−− ν G++ ν′ δm,0 = −(Υ1 d)−+δm0, + + - - n +m' n n+m n‘ +− +− =+G++ ν G−− ν+mδν,ν′ = (Υ2 s) ++δνν′ , - - + + n+m' n n+m n' −+ −+ =+G−− ν G++ ν+mδν,ν′ = (Υ2 s) −−δνν′ . Three-time correlation function has the following non-zero components: n n+m m + + + + ++ =− 〈P+〉 (iων − J++ ν − U)(iων+m − J++ ν+m − U) + 〈P+〉〈P−〉n+δm,0 iων − J++ ν − U , n n+m m - - - − −− =− 〈P−〉 (iων − J−− ν )(iων+m − J−− ν+m) + 〈P+〉〈P−〉n−δm,0 iων − J−− ν , n n+m m - + + − ++ =−n− d 〈P−〉G++ ν δm,0 , n n+m m + - - + −− =−n+ d 〈P+〉G−− ν δm,0 . All components of the two-time correlation function are non-zero: + + m =− ∑ ν 〈P+〉 (iων − J++ ν − U)(iων+m − J++ ν+m − U) + δm,0〈P+〉〈P+〉n+ d n+ d , − − m =− ∑ ν 〈P−〉 (iων − J−− ν )(iων+m − J−− ν+m) + δm,0〈P−〉〈P−〉n− d n− d , + − m =−δm,0〈P+〉〈P−〉n+ d n− d , − + m =−δm,0〈P−〉〈P+〉n− d n+ d . 72 Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping References 1. Falicov L.M., Kimball J.C., Phys. Rev. Lett., 1969, 22, 2267. 2. Freericks J. K., Zlatić V., Rev. Mod. Phys., 2003, 75, 1333. 3. Ramirez R., Falicov L. M., Phys. Rev. B, 1971, 3, 2425. 4. Lawrence J.M., Riseborough P.S., Parks R.D., Rep. Prog. Phys. 1981, 44, 1. 5. Kennedy T., Lieb E.H., Physica A, 1986, 138, 320. 6. Freericks J. K., Phys. Rev. B, 1993, 48, 14797 7. Shvaika A.M., Physica C, 2000, 341-348, 177; J. Phys. Studies, 2001, 5, 349. 8. Freericks J. K., Devereaux T.P., Phys. Rev. B, 2001, 64, 125110. 9. Shvaika A.M., Vorobyov O., Freericks J.K., Devereaux T.P., Phys. Rev. Lett., 2004, 93, 147402; Phys. Rev. B, 2005, 71, 045120 10. Brandt U., Mielsch C., Z. Phys. B: Condens. Matter, 1989, 75, 365; 1990, 79, 295;1991, 82, 37. 11. Georges A., Kotliar G., Krauth W., Rozenberg M., Rev. Mod. Phys., 1996, 68, 13. 12. Schubin S., Wonsowsky S., Proc. Roy. Soc., 1934, A145, 159 13. Hubbard J., Proc. R. Soc. London, 1963, Ser. A 276, 238. 14. Hirsch J. E., Physica C, 1989, 158, 236 15. Didukh, L.D., Sov. Phys. Solid State, 1977, 19, 711 16. Vollhardt D., Blümer N., Held K., Kollar M., Schlipf J., Ulmke M., Wahle J., Advances in Solid State Physics, 1999, 38, 383 17. Vollhardt D., Blümer N., Held K., Kollar M., Lecture Notes in Physics, 2001, 580, 191 18. Kollar M., Vollhardt D., Phys. Rev. B, 2001, 63, 045107 19. Appel J. et. al., Phys. Rev. B, 1993, 47, 2812. 20. Parr R. G. et. al., J. Chem. Phys., 1950, 18, 1561. 21. Schiller A., Phys. Rev. B, 1999, 60, 15660. 22. Shvaika A.M., Phys. Rev. B., 2003, 67, 075101; 23. Metzner W., Phys. Rev. B, 1991, 43, 8549. 24. Blackman A., Esterling D.M., Berk N.F., Phys. Rev. B, 1971, 4, 2412. 25. Koepernik K., Velický B., Hayn R., Eschrig H., Phys. Rev. B, 1998, 58, 6944. 26. Chung W., Freericks J.K., Phys. Rev. Lett., 2000, 84, 2461 73 A.M.Shvaika, O.Ya.Farenyuk Динамiчнi сприйнятливостi моделi Фалiкова-Кiмбала з корельованим переносом: загальний пiдхiд А.М.Швайка, О.Я.Фаренюк Iнститут фiзики конденсованих систем НАН України, 79011 Львiв, вул. Свєнцiцького, 1 Отримано 21 листопада 2008 р., в остаточному виглядi – 18 лютого 2009 р. Дослiджується модель Фалiкова-Кiмбала з корельованим переносом в границi безмежної розмiр- ностi простору. Динамiчнi сприйнятливостi розраховуються за допомогою теорiї динамiчного се- реднього поля, що ґрунтується на розвиненнях за електронним переносом навколо одновузлової границi. Дослiджується чисельно частковий випадок напiвелiптичної густини станiв та дiагонального корельованого переносу та показано вiдсутнiсть фазових переходiв для всiх температур крiм T = 0, що узгоджується з вiдомими результатами. Ключовi слова: модель Фалiкова-Кiмбала, корельований перенос, електроннi сприйнятливостi, теорiя динамiчного середнього поля, сильноскорельованi системи PACS: 71.10.Fd 74

Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping: general approach

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