Reference approach in theory of pseudospin systems

For theoretical description of pseudospin systems with essential short-range and long-range interactions we use the method based on calculations of the free energy functional taking into account the short-range interactions within the reference approach in cluster approximation. We propose a consi...

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Дата:	2000
Автори:	Levitskii, R.R., Sorokov, S.I., Baran, O.R.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2000
Назва видання:	Condensed Matter Physics
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/120999
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Reference approach in theory of pseudospin systems / R.R. Levitskii, S.I. Sorokov, O.R. Baran // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 515-543. — Бібліогр.: 60 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1209992025-02-10T01:39:29Z Reference approach in theory of pseudospin systems Базисний підхід в теорії псевдоспінових систем Levitskii, R.R. Sorokov, S.I. Baran, O.R. For theoretical description of pseudospin systems with essential short-range and long-range interactions we use the method based on calculations of the free energy functional taking into account the short-range interactions within the reference approach in cluster approximation. We propose a consistent formulation of the cluster expansion method for quantum pseudospin systems. We develop a method allowing one to obtain within the cluster approximation an Ornstein-Zernike type equation for reference cumulant Green function of an arbitrary order. In the two-particle cluster approximation we derived an explicit expression for pair temperature cumulant Green function of the reference system. In the cluster random phase approximation we calculated and studied thermodynamic characteristics, elementary excitation spectrum, and integral intensities of the Ising model in transverse ﬁeld. Для теоретичного опису псевдоспінових систем з суттєвими короткосяжними та далекосяжними взаємодіями використовується метод, який грунтується на розрахунку функціоналу вільної енергії з базисним урахуванням короткосяжних взаємодій у кластерному наближенні. Для квантових псевдоспінових систем запропоновано послідовне формулювання методу кластерних розвинень та метод, який дає змогу в рамках кластерного наближення отримати для базисних температурних кумулянтних функцій Гріна довільного порядку рівняння типу рівнянь Орнштейна-Церніке. В наближенні двочастинкового кластера в явному вигляді отримано вираз для парної температурної кумулянтної функції Гріна базисної системи. У кластерному наближенні хаотичних фаз розраховані та досліджені термодинамічні характеристики, спектр елементарних збуджень та інтегральні інтенсивності гілок спектра моделі Ізінга в поперечному полі. 2000 Article Reference approach in theory of pseudospin systems / R.R. Levitskii, S.I. Sorokov, O.R. Baran // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 515-543. — Бібліогр.: 60 назв. — англ. 1607-324X DOI:10.5488/CMP.3.3.515 PACS: 03.65.-w, 05.30.-d https://nasplib.isofts.kiev.ua/handle/123456789/120999 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	For theoretical description of pseudospin systems with essential short-range and long-range interactions we use the method based on calculations of the free energy functional taking into account the short-range interactions within the reference approach in cluster approximation. We propose a consistent formulation of the cluster expansion method for quantum pseudospin systems. We develop a method allowing one to obtain within the cluster approximation an Ornstein-Zernike type equation for reference cumulant Green function of an arbitrary order. In the two-particle cluster approximation we derived an explicit expression for pair temperature cumulant Green function of the reference system. In the cluster random phase approximation we calculated and studied thermodynamic characteristics, elementary excitation spectrum, and integral intensities of the Ising model in transverse ﬁeld.
format	Article
author	Levitskii, R.R. Sorokov, S.I. Baran, O.R.
spellingShingle	Levitskii, R.R. Sorokov, S.I. Baran, O.R. Reference approach in theory of pseudospin systems Condensed Matter Physics
author_facet	Levitskii, R.R. Sorokov, S.I. Baran, O.R.
author_sort	Levitskii, R.R.
title	Reference approach in theory of pseudospin systems
title_short	Reference approach in theory of pseudospin systems
title_full	Reference approach in theory of pseudospin systems
title_fullStr	Reference approach in theory of pseudospin systems
title_full_unstemmed	Reference approach in theory of pseudospin systems
title_sort	reference approach in theory of pseudospin systems
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2000
url	https://nasplib.isofts.kiev.ua/handle/123456789/120999
citation_txt	Reference approach in theory of pseudospin systems / R.R. Levitskii, S.I. Sorokov, O.R. Baran // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 515-543. — Бібліогр.: 60 назв. — англ.
series	Condensed Matter Physics
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first_indexed	2025-12-02T12:51:46Z
last_indexed	2025-12-02T12:51:46Z
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fulltext	Condensed Matter Physics, 2000, Vol. 3, No. 3(23), pp. 515–543 Reference approach in theory of pseudospin systems R.R.Levitskii, S.I.Sorokov, O.R.Baran Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine Received April 18, 2000 For theoretical description of pseudospin systems with essential short- range and long-range interactions we use the method based on calcula- tions of the free energy functional taking into account the short-range in- teractions within the reference approach in cluster approximation. We pro- pose a consistent formulation of the cluster expansion method for quantum pseudospin systems. We develop a method allowing one to obtain within the cluster approximation an Ornstein-Zernike type equation for reference cumulant Green function of an arbitrary order. In the two-particle cluster approximation we derived an explicit expression for pair temperature cumu- lant Green function of the reference system. In the cluster random phase approximation we calculated and studied thermodynamic characteristics, elementary excitation spectrum, and integral intensities of the Ising model in transverse field. Key words: phase transitions, pseudospin models, reference approach, cluster approximation, Ornstein-Zernike equation, soft mode PACS: 03.65.-w, 05.30.-d 1. Introduction Modern statistical theory of condensed media pays a great attention to the stud- ies of ferroelectric and magnetic materials, described by pseudospin models with essential short-range and a long-range interactions, especially of hydrogen-bonded ferroelectrics [1–12] and low-dimensional magnets [13,14]. For an adequate descrip- tion of these objects, such an approach is required that would allow to use different techniques to take into account a short-range and a long-range interaction. This is a typical mathematical problem in theories of multiparticle systems. It has been successfully solved in the studies of equilibrium properties of classical systems [15– 21] and metals [21–25] using the approach proposed in [15–18,22,24,25]. Within this approach, the long-range and short-range interactions are described in phase spaces of collective variables and individual coordinates, respectively. The system with short-range interaction is called then the reference system. c© R.R.Levitskii, S.I.Sorokov, O.R.Baran 515 R.R.Levitskii, S.I.Sorokov, O.R.Baran Using the idea of separating the reference system [15–18], in [26–30] a method was proposed for description of pseudospin systems with essential short-range and long-range interactions. This method is based on the calculation of the free energy functional taking into account the short-range interactions within the reference ap- proach. In [26,27,30] expansions of the free energy functional and functionals of the temperature cumulant Green functions (CGF) in the inverse long-range interaction radius were studied. For the first time there has been performed a total summation of the reducible in blocks diagrams in the free energy functional and of non-compact diagrams in functionals of CGFs for quantum pseudospin models. Expressions for the free energy and temperature CGFs of the considered systems were obtained. It was shown how to obtain consistent approximations for their thermodynamic and dynamic characteristics, using classification of the approximations for free energy functional according to loop diagrams. It should be noted that the general expressions for thermodynamic and dynamic characteristics of pseudospin systems with short-range and long-range interactions obtained in [26–30] contain thermodynamic and correlation functions of the refer- ence system. Hence, to solve a general problem one needs to solve a reference one, that is, to calculate free energy and CGFs of the reference system. The maximal order of the reference CGFs depends on the order of the approximation for the long- range interactions. Depending on the reference Hamiltonian, the reference problem can be solved exactly (see, for instance, [14,31–36]) or approximately, taking into account peculiarities of the reference system. The best description of the reference system for a wide class of pseudospin models can be obtained based on the clus- ter expansions method (see [1–12,37–39]). In some papers [40–43] this method was successfully used to study disordered magnetic and ferroelectric materials. Unfor- tunately, the cluster method was correctly developed only for Hamiltonians with commuting single-particle (describing the interaction of pseudospins with external and internal fields) and multiparticle parts (describing the interaction between pseu- dospins). It was mostly used to calculate thermodynamic characteristics of pseu- dospin models. In [44–46] a problem of calculating the distribution functions for Ising models within the cluster approach was considered but not solved completely. Equations for pair correlation functions of the reference system (Ornstein-Zernike type equation) in [44,45] were not derived consistently but constructed artificially. The problem of calculating the quasimomentum-dependent pair correlation func- tions was also considered in [46] within the cluster approach. The results obtained are valid in paraphase only. Later, a method was proposed [47,48], which allows one to obtain Ornstein-Zernike type equations for arbitrary order correlation functions of Ising models. These equations for pair and three-particle correlation functions were derived and solved within the two-particle cluster approximation (TPCA). It was shown that the cluster approach to the calculation of correlation functions of the reference Ising models proposed in [47,48] yields the known exact results [31,32] for pair and three-particle correlation functions of the one-dimensional Ising model. In [48,49], using the four-particle cluster approximation, pair ~q-dependent cor- relation functions of deuterons were calculated for KD2PO4 type ferroelectrics and 516 Reference approach in theory of pseudospin systems ND4D2PO4 type antiferroelectrics. Dynamics of hydrogen-bonded ferroelectrics tak- ing into account the tunnelling effects was considered in [50–52] within the orig- inal approach proposed in [50]. For the first time it has been shown, that in the reference approach, with the short-range interactions and tunnelling taken into ac- count in cluster approximation, the dynamic properties of the studied systems are to a great extent determined by an effective tunnelling parameter, renormalized by the short-range interactions. Later, this peculiarity of the dynamic properties of hydrogen-bonded compounds was also noticed in [53]. Unfortunately, expressions for dynamic (at ~q = 0 and E = 0) and static characteristics, calculated in [50–52], turned out to be inconsistent. That results from the fact, that dynamic character- istics were obtained using the method of two-time temperature Green functions, equations for which were decoupled in the spirit of Tyablikov approximation. Thus, the intracluster Green functions of the reference system were connected only via the long-range interactions, whereas the short-range correlations were not taken into account. Thus, the method has not been developed, which would allow one to con- sistently describe thermodynamic and dynamic characteristics of reference quantum pseudospin models. In the present paper, for a theoretical description of pseudospin systems with essential short-range and long-range interactions we shall use the self-consistent reference approach developed in [26–30]. In section 2 we shall briefly consider the main results obtained within this approach. Then, a consistent formulation of cluster expansion method for reference quantum pseudospin systems will be given for the first time. We shall propose a method, allowing to obtain Ornstein-Zernike type equations for reference temperature cumulant Green functions of an arbitrary order within a cluster approximation. An Ornstein-Zernike type equation for the pair correlator will be derived and solved in the two-particle cluster approximation. The last section is devoted to the investigation of the Ising model in transverse field (IMTF) within the cluster random phase approximation (CRPA) using the results obtained in this paper. 2. Theory of pseudospin system with short-range interactio ns taken into account in reference approach We consider pseudospin systems with short-range and long-range interactions, described by the Hamiltonian H({Γ}) = −βH = N∑ ν=1 ∑ a Γa νS a ν + 1 2 ∑ ν,δ ∑ a,b KabSa νS b ν+δ + 1 2 ∑ ν,µ ∑ a,b Jab νµS a νS b µ . (2.1) Here Kab and Jab νµ are the short-range and the long-range parts of the pair inter- actions. Sa ν (a = x, y, z or +,−, z) are components of a normalized (S z = −1, 1) spin ~S. H({Γ}) means H({Γ}) = H(Γa1 1 , ...,Γ a1 N ,Γ a2 1 , ...,Γ a2 N ,Γ a3 1 , ...,Γ a3 N , ). Hereafter, the argument {Γ} will be frequently omitted. The factor β = 1/(kBT ), occurring in 517 R.R.Levitskii, S.I.Sorokov, O.R.Baran Γ, K, and J (in the above presented form of the Hamiltonian H), will be written explicitly only in some of the final expressions. After an identity transformation of the operators Sa ν = 〈Sa ν 〉 + ∆Sa ν in the last term of Hamiltonian (2.1), which describes the long-range interactions between pseu- dospins, we obtain H({Γ}) = kH({κ})− 1 2 ∑ ν,µ ∑ a,b Jab νµ〈Sa ν 〉〈Sb µ〉+ 1 2 ∑ ν,µ ∑ a,b Jab νµ∆S a ν ·∆Sb µ . (2.2) The first term in (2.2) describes the short-range interactions between pseudospins placed in a field created by the long-range interactions and by Γa ν kH({κ}) = −β ·kH({κ}) = N∑ ν=1 ∑ a κ a νS a ν + 1 2 ∑ ν,δ ∑ a,b KabSa νS b ν+δ ; (2.3) κ a ν = Γa ν + N∑ µ=1 ∑ b Jab νµ〈Sb µ〉 . (2.4) The Hamiltonian kH({κ}) is called the reference Hamiltonian [26]. The argument {κ} will be often dropped. Let us note that in the mean field approximation (MFA) over the long-range interactions, the last term in (2.2) is neglected. Our main task is to calculate the free energy F ({Γ}) = −kBT lnZ({Γ}) , Z({Γ}) = Sp eH (2.5) and pair temperature cumulant Green functions b(2)(a1ν1,τ1\a2 ν1,τ2 \) = 〈Tτ ˜̃Sa1 ν1 (τ1) ˜̃Sa2 ν2 (τ2)〉cρ (2.6) for the models, described by Hamiltonian (2.1). Here ˜̃Sa ν(τ) = e−τHSa ν eτH ; (2.7) the averaging is performed with the density matrix ρ = ρ({Γ}) = [Z({Γ})]−1 · eH . (2.8) For the sake of convenience, in our calculations we do not use the free energy F ({Γ}) but the F({Γ})-function (logarithm of the partition function). According to the theory proposed in [26], to solve the formulated problem, one should calculate the kF -function kF({κ}) = ln kZ({κ}); kZ({κ}) = Sp e kH (2.9) and CGFs kb(l)(a1ν1,τ1\ a2 ν2,τ2 \ ...\al νl,τl \) = k〈Tτ S̃a1 ν1 (τ1)S̃ a2 ν2 (τ2)...S̃ al νl (τl)〉ckρ ; (2.10) S̃a ν (τ) = e−τ ·kH Sa ν eτ · kH, kρ = kρ({κ}) = [kZ({κ})]−1 · ekH (2.11) 518 Reference approach in theory of pseudospin systems of the reference system (2.3). In the present paper the kF -function and pair CGFs will be calculated within the two-particle cluster approximation. Assuming that the reference problem is solved in [26] the expansion of the free energy functional in the inverse radius of the long-range interactions was studied, for the systems described by the Hamiltonian (2.1) and expressions for temperature Green functions were obtained. Here we present only some of their results for non- uniform fields (Γa ν = Γa) up to r−d 0 in the long-range interactions. The F -function of the considered system reads F({Γ})= kF({κ})− N 2 ∑ a,b Jab 0 〈Sa〉ρ〈Sb〉ρ − 1 2 ∑ ωn,~q ln det [ 1̂− k̂b(2)(~q, ωn)Ĵ(~q ) ] , (2.12) where Jab 0 = Jab(~q = 0), κ a = Γa + ∑ b Jab 0 〈Sb〉ρ ; (2.13) k̂b(2)(~q, ωn) and Ĵ(~q) are matrices 3 × 3 in the indices a, b; their elements are Fourier transforms kb(2)(a b ~q,ωn ) and Jab(~q) of the pair CGFs of the reference system kb(2)(aν,τ1\ b µ,τ2 \) and of the long-range interactions J ab νµ (for uniform fields (κa ν = κ a)). For the pair CGFs b(2)(aν,τ1\ b µ,τ2 \) in the frequency-momentum space, the following relation [26] holds b̂(2)(~q, ωn) = [ 1− M̂(~q, ωn)Ĵ(~q ) ]−1 M̂(~q, ωn). (2.14) Here we use the notation M(a b ~q,ωn ) = kb(2)(a b ~q,ωn ) + 1 2N ∑ {a,b} ∑ {~qi,ωni } kb(3)(aν0,τ0\ a1 ~q1,ωn1 \a2 ~q2,ωn2 \) ×kb(3)(bν0,τ0\ b1 −~q1,−ωn1 \ b2 −~q2,−ωn2 \)R(a1 b1 −~q1,−ωn1 )R(a2 b2 −~q2,−ωn2 ) ×δ(~q + ~q1 + ~q2)δ(ωn + ωn1 + ωn2 ) + 1 2N ∑ a1,a2 ∑ ~q1,ωn1 kb(4)(aν0,τ0\ b −~q,−ωn \a1 ~q1,ωn1 \a2 −~q1,−ωn1 \)R(a1 a2 −~q1,−ωn1 ) , (2.15) where R̂(~q, ωn) = Ĵ(~q ) [ 1− k̂b(2)(~q, ωn)Ĵ(~q ) ]−1 (ν0 = 0, τ0 = 0, ωn = 2πnβ−1) (2.16) is the Fourier transform of the effective interaction in the considered system, whereas kb(3)(aν0,τ0\ a1 ~q1,ωn1 \a2 ~q2,ωn2 \) and kb(4)(aν0,τ0\ b −~q,−ωn \a1 ~q1,ωn1 \a2 −~q1,−ωn1 \) are Fourier transforms of the three and four-particle CGFs of the reference system, respectively. We also present here an expression for the order parameter [26] 〈Sa〉 = 〈Sa〉ρ = k〈Sa〉kρ + 1 2N ∑ a1,a2 ∑ ~q,ωn kb(3)(aν0,τ0\ a1 ~q,ωn \a2 −~q,−ωn \)R(a1 a2 −~q,−ωn ) . (2.17) Hence, we have general expressions for the free energy (F = −kBTF) and pair CGFs, and the equation for the order parameter in the r−d 0 approximation. These expressions contain the free energy and correlation functions of the reference system. 519 R.R.Levitskii, S.I.Sorokov, O.R.Baran 3. Two-particle cluster approximation for short-range interactions 3.1. Problem formulation Our task in this section is to obtain the kF({κ})-function, parameters k〈Sa ν 〉kρ and pair cumulant Green functions kb(2)(a1ν,τ1\ a2 µ,τ2 \ ) of the reference pseudospin system, described by Hamiltonian kH({κ}) (2.3), in the two-particle cluster approximation in the short-range interactions. Let us define the functional of the partition function logarithm (the kF({ε})- functional) of the reference model as [26] kF({ε}, {κ}) = ln kZ({ε}, {κ}) ; kZ({ε}, {κ}) = Sp [ eH({ε})Tτ exp (∫ 1 0 dτ kH(τ, {κ}) )] , (3.1) where kH(τ, {κ}) = ∑ a N∑ ν=1 κ a ν,τS a ν,τ + 1 2 ∑ a,b ∑ ν,δ KabSa ν,τS b ν+δ,τ ; (3.2) H({ε}) = N∑ ν=1 Hν({εν}) ; Hν({εν}) = ∑ a εaν S a ν ; Aτ = e−τH({ε}) A eτH({ε}) . (3.3) Dependence κ a ν,τ on τ here is necessary to perform functional differentiation with respect to κ a ν,τ [26]. It should be noted, that since spin operators at different sites commute, and κ a ν is a scalar, the quantity Aτ (if A = Sa ν or κa ν) can be written as Aτ = e−τHν({εν}) A eτHν({εν}) . Starting from (3.1), we introduce functionals of CGFs of the reference system kb(l)(a1ν1,τ1\ a2 ν2,τ2 \ ...\al νl,τl \{ε}) = k〈Tτ Sa1 ν1,τ1 Sa2 ν2,τ2 ...Sal νl,τl 〉c kρ({ε}) ; kρ({ε}) = 1 kZ({ε}, {κ̃}) eH({ε}) exp [ ∫ 1 0 dτ kH(τ, {κ}) ] . (3.4) They will be found using kb(l)(a1ν1,τ1\ a2 ν2,τ2 \ ...\al νl,τl \{ε}) = δ δκa1 ν1,τ1 δ δκa2 ν2,τ2 . . . δ δκal νl,τl kF({ε}, {κ}) . (3.5) According to [26], the following relations between the kF({κ})-function (2.9) and temperature CGFs (2.10) and their functionals hold kF({κ}) = kF({ε}, {κ})∣∣∣κ a ν,τ=κ a ν εaν=0 ; (3.6) kb(l)(a1ν1,τ1\a2 ν2,τ2 \ ...\al νl,τl \) = kb(l)(a1ν1,τ1\a2 ν2,τ2 \ ...\al νl,τl \{ε})∣∣∣κ a ν,τ=κ a ν εaν=0 . (3.7) 520 Reference approach in theory of pseudospin systems That is, calculation of the kF({κ})-function and temperature CGFs is reduced to calculation of the kF({ε}, {κ})-functional. 3.2. Cluster approximation. Free energy Let us calculate now the kF({ε}, {κ})-functional in the two-particle cluster ap- proximation. We perform a cluster expansion, with the lattice being divided into the two-particle clusters [47,54,55]. As ∑ a rϕa ν,τS a ν,τ we denote an operator of the effective field created by the site r and acting on the site ν, provided that the site r is the nearest neighbour of the site ν (r ∈ πν). Obviously, the number of fields acting on an arbitrary site ν ∑ r∈πν ∑ a rϕa ν,τS a ν,τ is z (z is the nearest neighbours number). After an identity transformation, the reference Hamiltonian (3.2) takes the form kH(τ, {κ, ϕ}) = ∑ ν Hν(τ, {κ̃ν}) + ∑ (ν,r) Uνr(τ, {rϕν , νϕr}) , (3.8) where Hν(τ, {κ̃ν}) = ∑ a κ̃ a ν,τS a ν,τ ; κ̃ a ν,τ = κ a ν,τ + ∑ r∈πν rϕa ν,τ ; (3.9) Uνr(τ, {rϕν , νϕr}) = ∑ a ( − rϕa ν,τS a ν,τ − νϕa r,τS a r,τ + ∑ b KabSa ν,τS b r,τ ) . (3.10) Hν(τ, {κ̃ν}) means Hν = Hν(κ̃ a1 ν,τ , κ̃ a2 ν,τ , κ̃ a3 ν,τ ). Hereafter, the arguments {κ̃ν}, {rϕν} will be frequently omitted. Let us present the kF({ε})-functional (3.1) as kF({ε}, {κ, ϕ}) = = lnSp { eH({ε})Tτ exp [ N∑ ν=1 ∫ 1 0 dτ Hν(τ) ] exp [∑ (ν,r) ∫ 1 0 dτ Uνr(τ) ]} = ∑ ν Fν({εν}, {κ̃ν}) + ln〈Tτ exp (∑ (ν,r) ∫ 1 0 dτ Uνr(τ) ) 〉ρ0({ε}) . (3.11) Here Fν({εν}) is the so-called single-particle intracluster F({ε})-functional Fν({εν}, {κ̃ν}) = lnZν({εν}, {κ̃ν}) ; (3.12) Zν({εν}, {κ̃ν}) = Sp Sν { eHν({εν})Tτ exp [ ∫ 1 0 dτ Hν(τ, {κ̃ν}) ]} (3.13) and averaging is performed with the functional of the density matrix ρ0({ε})= ∏ ν ρν({εν}) ; (3.14) ρν({εν}) = 1 Zν({εν}, {κ̃ν}) eHν({εν}) exp [ ∫ 1 0 dτ Hν(τ, {κ̃ν}) ] . (3.15) 521 R.R.Levitskii, S.I.Sorokov, O.R.Baran We restrict our consideration by the first order of the cluster expansion [47,54, 55]; this corresponds to the two-particle cluster approximation. Then the kF({ε})- functional becomes a sum of the single- and two-particle intracluster F({ε})-fun- ctionals kF({ε}, {κ, ϕ}) = ∑ ν Fν({εν}, {κ̃ν}) + ∑ (ν,r) ln〈Tτ exp (∫ 1 0 dτ Uνr(τ) ) 〉ρ0({ε}) = (1− z) ∑ ν Fν({εν}, {κ̃ν}) + 1 2 ∑ ν,r Fνr({εν, εr}, {rκ̃ν , ν κ̃r}) . (3.16) The two-particle Fνr({εν, εr})-functional reads Fνr({εν , εr}, {rκ̃ν , ν κ̃r})= lnZνr({εν , εr}, {rκ̃ν , ν κ̃r}); (3.17) Zνr({εν, εr})= Sp S1,S2 { eHν({εν})+Hr({εr})Tτ exp [∫ 1 0 dτ Hνr(τ, {rκ̃ν , ν κ̃r}) ]} ; (3.18) Hνr(τ, {rκ̃ν , ν κ̃r}) = Hν(τ, {κ̃ν}) +Hr(τ, {κ̃r}) + Uνr(τ) = ∑ a [ r κ̃ a ν,τS a ν,τ + ν κ̃ a r,τS a r,τ + ∑ b KabSa ν,τS b r,τ ] ; (3.19) r κ̃ a ν,τ = κ̃ a ν,τ − rϕa ν,τ = κ a ν,τ + ∑ r′∈πν r′ 6=r r′ϕa ν,τ . (3.20) Putting εaν = 0 (κa ν,τ = κ a ν , rϕa ν,τ = rϕa ν , see also (3.6)), and going to the uniform fields case κa ν = κ a (rϕa ν = ϕa), from (3.16) we obtain the kF -function of the reference system in the TPCA for the uniform fields kF({κ, ϕ}) = (1− z)NF1({κ̃}) + Nz 2 F12({ ˜̃κ}) ; (3.21) F1({κ̃}) = lnZ1({κ̃}) ; Z1({κ̃}) = Sp S1 eH1({κ̃}) ; (3.22) H1({κ̃}) = ∑ a κ̃ aSa 1 ; κ̃ a = κ a + zϕa ; (3.23) F12({ ˜̃κ}) = lnZ12({ ˜̃κ}) ; Z12({ ˜̃κ}) = Sp S1,S2 eH12({ ˜̃κ}) ; (3.24) H12({ ˜̃κ}) = ∑ a [ ˜̃κa(Sa 1 + Sa 2 ) + ∑ b KabSa 1S b 2] ; ˜̃κa = κ a + (z − 1)ϕa. (3.25) 3.3. System of equations for single-particle distribution functions and vari- ational parameters Let us now find equations for functionals k〈Tτ Sa ν,τ 〉kρ({ε}) and for cluster fields rϕa ν,τ . From (3.5) we obtain 522 Reference approach in theory of pseudospin systems k〈Tτ Sa µ,τ 〉kρ({ε}) = ∂ kF({ε}) ∂ κa µ,τ + ∑ ν ∑ r∈πν ∑ b ∫ 1 0 dτ ′ ∂ kF({ε}) ∂ rϕb ν,τ ′ · δ rϕb ν,τ ′ δ κa µ,τ . (3.26) Starting from the fact that the kF({ε})-functional (3.16) is a sum of the single- and two-particle F({ε})-functionals, we get ∂ kF({ε}) ∂ κa ν,τ = (1−z)F (1) ν ( a τ \|{εν}) + ∑ r∈πν F (1,0) νr ( a τ \|{εν , εr}) ; (3.27) ∂ kF({ε}) ∂ r1ϕa ν,τ =(1−z)F (1) ν ( a τ \|{εν})+ ∑ r∈πν r 6=r1 F (1,0) νr ( a τ \|{εν, εr}), (r1∈πν). (3.28) Here we introduce the notations F (k) ν ( a1 τ1 \| a2 τ2 \| ··· ak τk \| {εν}, {κ̃ν}) = ∂ ∂ κ̃ a1 ν,τ1 · ∂ ∂ κ̃ a2 ν,τ2 ··· ∂ ∂ κ̃ ak ν,τk Fν({εν}, {κ̃ν}) , (3.29) F (k,l) νr ( a1 τ1 \| a2 τ2 \| ··· ak τk \| a′1 τ ′ 1 ‖ a′2 τ ′ 2 ‖ ··· a′ l τ ′ l ‖ {εν, εr}, {rκ̃ν , ν κ̃r}) = = ∂ ∂ rκ̃ a1 ν,τ1 ··· ∂ ∂ rκ̃ ak ν,τk · ∂ ∂ νκ̃ a′ 1 r,τ ′ 1 ··· ∂ ∂ νκ̃ a′ l r,τ ′ l Fνr({εν, εr}, {rκ̃ν , ν κ̃r}) . (3.30) From the explicit form of the intracluster F({ε})-functionals (3.12), (3.17) it follows that F (k) ν ( a1 τ1 \| a2 τ2 \| ··· ak τk \| {εν}, {κ̃ν}) = 〈Tτ Sa1 ντ1 Sa2 ντ2 ··· Sak ντk 〉cρν({εν}) ; (3.31) F (k,l) νr ( a1 τ1 \| a2 τ2 \| ··· ak τk \| a′1 τ ′ 1 ‖ a′2 τ ′ 2 ‖ ··· a′ l τ ′ l ‖ {εν, εr}, {rκ̃ν , ν κ̃r}) = = 〈Tτ Sa1 ντ1 Sa2 ντ2 ··· Sak ντk S a′1 rτ ′ 1 S a′2 rτ ′ 2 ··· Sa′ l rτ ′ l 〉cρνr({εν ,εr}) , (3.32) where the averagings are performed with the density matrix functionals (3.15) and ρνr({εν , εr}) = eHν({εν})+Hr({εr}) Zνr({εν, εr}, {rκ̃ν , νκ̃r}) exp [∫ 1 0 dτ Hνr(τ, {rκ̃ν , ν κ̃r}) ] . (3.33) Hereafter, the functionals (3.31) and (3.32) will be called the single-particle and two-particle intracluster functionals of CGF, respectively. Similar to (3.6), from (3.12) and (3.17) one can obtain expressions relating the intracluster F -functions with their functionals Fν({κ̃ν}) = Fν({εν}, {κ̃ν})∣∣∣κ̃ a ν,τ=κ̃ a ν εaν=0 ; (3.34) Fνr({rκ̃ν , ν κ̃r}) = Fνr({εν , εr}, {rκ̃ν , ν κ̃r})∣∣∣ r κ̃ a ν,τ= r κ̃ a ν , ν κ̃ a r,τ= ν κ̃ a r εaν=εar=0 . (3.35) 523 R.R.Levitskii, S.I.Sorokov, O.R.Baran One can also derive expressions relating the single-particle and two-particle intra- cluster CGFs with their functionals 〈Tτ S̄a1 ν (τ1) S̄ a2 ν (τ2) ... S̄ ak ν (τk)〉cρν = 〈Tτ Sa1 ν,τ1 Sa2 ν,τ2 ... Sak ν,τk 〉cρν({εν}) ∣∣∣κ̃ a ν,τ=κ̃a ν εaν=0 , (3.36) 〈Tτ ¯̄Sa1 ν (τ1) ¯̄Sa2 ν (τ2) ··· ¯̄Sak ν (τk) ¯̄Sa′1 r (τ ′1) ¯̄Sa′2 r (τ ′2) ··· ¯̄S a′ l r (τ ′ l )〉cρνr = = 〈Tτ Sa1 ντ1 Sa2 ντ2 ··· Sak ντk S a′ 1 rτ ′ 1 S a′ 2 rτ ′ 2 ··· Sa′ l rτ ′ l 〉cρνr({εν ,εr}) ∣∣∣ rκ̃a ν,τ= rκ̃a ν , νκ̃a r,τ= νκ̃a r εaν=εar=0 , (3.37) where S̄a ν (τ) = e−τHν Sa ν eτHν ; ρν = eHν Sp(eHν ) ; Hν({κ̃ν}) = ∑ a κ̃ a νS a ν ; (3.38) ¯̄S a ν(τ) = e−τHνr Sa ν eτHνr ; ρνr = eHνr Sp(eHνr) ; Hνr({rκ̃ν , ν κ̃r}) = ∑ a [rκ̃a νS a ν + ν κ̃ a rS a r + ∑ b KabSa νS b r ] . (3.39) From equations (3.26)–(3.28), taking into account the condition of the extremum of the kF({ε})-functional with respect to rϕa ν,τ ∂ kF({ε}, {κ, ϕ}) ∂ rϕa ν,τ = 0 , (3.40) we obtain the system of equations for the functionals k〈Tτ Sa ν,τ 〉kρ({ε}) and cluster fields rϕa ν,τ k〈Tτ Sa ν,τ 〉kρ({ε}) = F (1) ν ( a τ \| {εν}, {κ̃ν}), (3.41) F (1) ν ( a τ \| {εν}, {κ̃ν}) = F (1,0) νr ( a τ \| {εν, εr}, {rκ̃ν , ν κ̃r}). (3.42) One can see that equation (3.42) for rϕa ν,τ , obtained from the kF({ε})-functional extremum condition (3.40), is equivalent to equations 〈Tτ Sa ν,τ 〉ρν({εν}) = 〈Tτ Sa ν,τ 〉ρνr({εν ,εr}) . That is, in the present approximation (see (3.36)–(3.39)) the relations between the density matrices are not violated: 〈Sa ν 〉ρν = 〈Sa ν 〉ρνr =⇒ ρν = Sp Sr ρνr . Putting εaν = 0 (κa ν,τ = κ a ν , rϕa ν,τ = rϕa ν , see also (3.7), (3.36), (3.37)), taking into account the following relations 〈Sa ν 〉ρν = ∂ Fν({κ̃ν}) ∂ κ̃a ν , 〈Sa ν 〉ρνr = ∂ Fνr({rκ̃ν , ν κ̃r}) ∂ rκ̃a ν 524 Reference approach in theory of pseudospin systems and going to the uniform fields case κ a ν = κ a (rϕa ν = ϕa, κ̃a ν = κ̃ a, r κ̃ a ν = ˜̃κa), from (3.41), (3.42) we obtain the system of equations for the single-particle distribution functions k〈Sa〉kρ and cluster fields ϕa in the TPCA for the uniform fields case. k〈Sa〉kρ = ∂ F1({κ̃}) ∂ κ̃a , (3.43) ∂ F1({κ̃}) ∂ κ̃a = 1 2 ∂ F12({ ˜̃κ}) ∂ ˜̃κa . (3.44) Here F1({κ̃}) and F12({ ˜̃κ}) are the single-particle and two-particle intracluster F - functions for the uniform fields case (3.22), (3.24). The factor 1 2 in the right hand side of equation (3.44) arose at going from the partial derivative of the Fνr({rκ̃ν , ν κ̃r})- function with respect to r κ̃ a ν in the non-uniform fields case to the partial derivative of the F12({ ˜̃κ})-function with respect to ˜̃κa in the uniform fields case. 3.4. Pair distribution functions Let us briefly discuss the method of calculation of the pair CGFs functionals of the reference system presented in [54], based on the technique developed in [47] for the Ising model. Starting from (3.5) and (3.41), we obtain an expression for the pair CGF functional kb(2)(a1ν,τ1\ a2 µ,τ2 \{ε})= k〈Tτ Sa1 ν,τ1 Sa2 µ,τ2 〉c kρ({ε}) = ∑ a3 ∫ 1 0 dτ3 F (2) ν (a1τ1\| a3 τ3 \|{εν})· δ κ̃a3 ν,τ3 δ κa2 µ,τ2 . (3.45) Having in mind the calculations of the pair CGFs (see (3.7)) for the uniform fields case, and since for specific systems single-particle intracluster pair CGFs 〈Tτ S̄a1 ν (τ1) S̄ a2 ν (τ2)〉cρν (see (3.31), (3.36)) in uniform fields case can be calculated directly, we need to obtain an equation for δ κ̃a3 ν,τ3 /δ κa2 µ,τ2 [47,54,56]. We introduce the notations κ̃ ′ νµ( a3 τ3 \|a2τ2 ) = δ κ̃a3 ν,τ3 δ κa2 µ,τ2 ; r κ̃ ′ νµ( a3 τ3 \|a2τ2 ) = δ r κ̃ a3 ν,τ3 δ κa2 µ,τ2 ; rϕ ′ νµ( a3 τ3 \|a2τ2 ) = δ rϕa3 ν,τ3 δ κa2 µ,τ2 . (3.46) Taking the functional derivative δ /δ κa2 µ,τ2 from both sides of equation (3.42), and taking into account the relation r κ̃ ′ νµ( a3 τ3 \|a2τ2 ) = κ̃ ′ νµ( a3 τ3 \|a2τ2 )− rϕ ′ νµ( a3 τ3 \|a2τ2 ) (3.47) (see (3.20)), we obtain ∑ a3 ∫ 1 0 dτ3 F (2) ν ( a1 τ1 \| a3 τ3 \|{εν}) · κ̃ ′ νµ( a3 τ3 \|a2τ2 ) = = ∑ a3 ∫ 1 0 dτ3 F (2,0) νr ( a1 τ1 \| a3 τ3 \|{εν, εr}) [ κ̃ ′ νµ( a3 τ3 \|a2τ2 )− rϕ ′ νµ( a3 τ3 \|a2τ2 ) ] + ∑ a3 ∫ 1 0 dτ3 F (1,1) νr ( a1 τ1 \| a3 τ3 ‖{εν, εr}) [ κ̃ ′ rµ( a3 τ3 \|a2τ2 )− νϕ ′ rµ( a3 τ3 \|a2τ2 ) ] . (3.48) 525 R.R.Levitskii, S.I.Sorokov, O.R.Baran Going to a matrix form in (3.48) 1 and performing some transformations, we obtain: [ F̂ (2,0) νr ({εν, εr})− F̂ (2) ν ({εν}) ] · ̂̃κ ′ νµ + F̂ (1,1) νr ({εν , εr}) · ̂̃κ ′ rµ = = F̂ (2,0) νr ({εν , εr}) · rϕ̂ ′ νµ + F̂ (1,1) νr ({εν, εr}) · νϕ̂ ′ rµ . (3.49) Introducing the notations f̂νr = [F̂ (2,0) νr ({εν , εr})]−1 · F̂ (1,1) νr ({εν, εr}) ; v̂νr = [F̂ (2,0) νr ({εν , εr})]−1 · F̂ (2) ν ({εν , εr}) , (3.50) we rewrite equation (3.49) as rϕ̂ ′ νµ + f̂νr · νϕ̂ ′ rµ = f̂νr · ̂̃κ ′ rµ + (1̂− v̂νr)̂̃κ ′ νµ , (r ∈ πν). (3.51) We obtain an equation (3.51) with unknown rϕ ′ νµ, νϕ ′ rµ. One more linearly indepen- dent equation still should be derived. After changing indices r ⇋ ν, we get νϕ̂ ′ rµ + f̂rν · rϕ̂ ′ νµ = f̂rν · ̂̃κ ′ νµ + (1̂− v̂rν)̂̃κ ′ rµ , (ν ∈ πr). (3.52) One can easily see that (3.51) and (3.52) are a system of equations for rϕ ′ νµ, νϕ ′ rµ. Summing up over r ∈ πν in (3.51) and taking into account the fact that ̂̃κ ′ νµ = δνµ · 1̂ + ∑ r∈πν rϕ̂ ′ νµ , (3.53) from the system of equations (3.51), (3.52) one obtains a closed equation for κ̃ ′ νµ { 1̂ + ∑ r∈πν f̂νr [ 1̂− f̂rν ·f̂νr ]−1[ f̂rν − f̂−1 νr ( 1̂− v̂νr) ]} ̂̃κ ′ νµ = = δνµ ·1̂ + ∑ r∈πν f̂νr [ 1̂− f̂rν ·f̂νr ]−1 v̂rν · ̂̃κ ′ rµ . (3.54) 1Here the matrices have a block structure; for instance in terms of (x, y, z): F̂ (2) ν ({εν}) =   F (2) ν (x\| x\|{εν}) F (2) ν (x\| y\|{εν}) F (2) ν (x\| z\|{εν}) F (2) ν (y\| x\|{εν}) F (2) ν (y\| y\|{εν}) F (2) ν (y\| z\|{εν}) F (2) ν (z\| x\|{εν}) F (2) ν (z\| y\|{εν}) F (2) ν (z\| z\|{εν})   , where the submatrices F (2) ν (a1\| a2\|{εν})=   F (2) ν ( a1 0 ∣∣ a2 0 ∣∣{εν} ) F (2) ν ( a1 0 ∣∣ a2 dτ ∣∣{εν} ) F (2) ν ( a1 0 ∣∣ a2 2dτ ∣∣{εν} ) ··· F (2) ν ( a1 0 ∣∣ a2 1 ∣∣{εν} ) ... F (2) ν ( a1 1 ∣∣ a2 0 ∣∣{εν} ) F (2) ν ( a1 1 ∣∣ a2 dτ ∣∣{εν} ) F (2) ν ( a1 1 ∣∣ a2 2dτ ∣∣{εν} ) ··· F (2) ν ( a1 1 ∣∣ a2 1 ∣∣{εν} )   are the M×M matrices (M = 1 dτ + 1). At dτ −→ 0 M −→ ∞. 526 Reference approach in theory of pseudospin systems Let us put εaν = 0 and go to the uniform fields case κ a ν = κ a F̂ (2,0) νr ({εν , εr}) = F̂ (2,0) (ν−r)({εν, εr}) −→ F̂ (2,0) 12 ; F̂ (2) ν ({εν}) −→ F̂ (2) 1 ; F̂ (1,1) νr ({εν , εr}) = F̂ (1,1) (ν−r)({εν, εr}) −→ F̂ (1,1) 12 ; f̂νr = f̂(ν−r) −→ f̂ ; v̂νr = v̂(ν−r) −→ v̂ (3.55) in equation (3.54). Then it can be rewritten as [ 1̂ + (z−1)f̂ 2 − z( 1̂− v̂) ] ̂̃κ ′ νµ = [1̂− f̂ 2] δνµ + f̂ · v̂ N∑ r=1 πνr · ̂̃κ ′ rµ , (3.56) where πνr = { 1, r ∈ πν 0, r /∈ πν . (3.57) It should be remembered, that with putting εaν = 0 we go from the single-particle and two-particle intracluster CGF functionals (see (3.31), (3.32)) to the corresponding CGFs (see (3.36), (3.37)). Going to the frequency-momentum representation in (3.56) and solving the ob- tained equation, we get for ̂̃κ ′(~q, ωn) ̂̃κ ′(~q, ωn) = [ 1̂ + (z−1)f̂ 2(ωn)− z[ 1̂− v̂(ωn)]− f̂(ωn)·v̂(ωn)·π(~q ) ]−1 × [ 1̂− f̂ 2(ωn)] . (3.58) Here f̂(ωn) = [F̂ (2,0) 12 (ωn)] −1 · F̂ (1,1) 12 (ωn) ; v̂(ωn) = [F̂ (2,0) 12 (ωn)] −1 · F̂ (2) 1 (ωn) (3.59) and F̂ (2,0) 12 (ωn), F̂ (1,1) 12 (ωn), F̂ (2) 1 (ωn) are 3× 3 matrices in the indices a, b (a = x, y, z or +,−, z), their elements are Fourier transforms (F (2,0) 12 (a,bωn ), F (1,1) 12 (a,bωn ), F (2) 1 (a,bωn )) of the pair intracluster CGFs 〈Tτ ¯̄Sa 1(τ) ¯̄Sb 1(0)〉cρ12, 〈Tτ ¯̄Sa 1(τ) ¯̄Sb 2(0)〉cρ12, 〈Tτ S̄a 1 (τ)S̄ b 1(0)〉cρ1, respectively. π(~q ) is the Fourier transform of the function πνr. For simple lattices with a hypercubic symmetry, π(~q ) reads π(~q ) = 2 d∑ i=1 cos(qi · α) ; (3.60) d is the lattice dimensionality; α is the lattice constant. The obtained matrix ex- pression (3.58) can be rewritten as ̂̃κ ′(~q, ωn) = [ zv̂(ωn)− (z−1)[ 1̂+ f̂(ωn)] + z[ 1̂− f̂(ωn)] −1 f̂(ωn)·v̂(ωn)·Θ(~q ) ]−1 × [ 1̂ + f̂(ωn)] , (3.61) 527 R.R.Levitskii, S.I.Sorokov, O.R.Baran where Θ(~q ) for simple lattices with a hypercubic symmetry is Θ(~q ) = 1− π(~q ) z = 2 d d∑ i=1 sin2 (qi · α 2 ) . (3.62) Putting εaν = 0, going to the uniform fields case in relation (3.45), and going to the frequency-momentum representation, we obtain expressions for pair CGFs, which are convenient to rewrite in a matrix form in the indices a, b: k̂b(2)(~q, ωn) = F̂ (2) 1 (ωn) · ̂̃κ ′(~q, ωn) . (3.63) Hence, in order to calculate the pair CGFs of the reference system in the uni- form fields case from (3.63), we need to calculate (according to (3.61) and (3.59)) the single- and two-particle intracluster pair CGFs F (2) 1 (a,bωn ) = (〈Tτ S̄a 1 (τ)S̄ b 1(0)〉cρ1)ωn , F (2,0) 12 (a,bωn ) = (〈Tτ ¯̄Sa 1(τ) ¯̄Sb 1(0)〉cρ12)ωn , F (1,1) 12 (a,bωn ) = (〈Tτ ¯̄Sa 1(τ) ¯̄Sb 2(0)〉cρ12)ωn . 4. Ising model in transverse field 4.1. Thermodynamics. General results We consider the Ising model in transverse field with a renormalized pseudospin operator (Sz = (−1, 1)). H = − N∑ ν=1 (hSz ν + ΓSx ν )− 1 2 ∑ ν,δ KSz νS z ν+δ − 1 2 ∑ ν,µ JνµS z νS z µ . (4.1) Here K and Jνµ are the short-range and long-range pair interactions, respectively; Γ is the transverse field; the quantity h→ 0 is introduced for the sake of convenience. Hereafter, the factor β = (kBT ) −1 is written explicitly. In the framework of MFA for the long-range interactions, the Hamiltonian (4.1) can be written as H = kH + 1 2 NJ0m 2 , (4.2) where J0 = N∑ µ=1 Jνµ , m = 〈Sz〉ρ (4.3) and kH is the Hamiltonian of the reference IMTF kH = − N∑ ν=1 [ κ zSz ν + κ xSx ν ] − 1 2 ∑ ν,δ KSz νS z ν+δ ; (4.4) κ z = h+ J0m ; κ x = Γ . (4.5) 528 Reference approach in theory of pseudospin systems According to the results of previous sections, the free energy of IMTF within the TPCA for the short-range interactions, with the long-range interactions taken into account within the MFA, is F = −kBT · kF + 1 2 NJ0m 2. (4.6) The kF -function of the reference IMTF kF = (1− z)NF1 + zN 2 F12 (4.7) is expressed via the single-particle F1 = lnZ1 , Z1 = Sp S1 e−βH1 , (4.8) H1 = − ∑ a=x,z κ̃ aSa 1 ; κ̃ a = κ a + zϕa , (a = z, x), (4.9) and two-particle F12 = lnZ12 , Z12 = Sp S1,S2 e−βH12 ; (4.10) H12 = − ∑ a=x,z ˜̃κa(Sa 1 + Sa 2 )−KSz 1S z 2 ; ˜̃κa = κ a + (z−1)ϕa , (a = z, x) (4.11) intracluster F -functions. Let us show briefly how these functions can be obtained. The Hamiltonian H1 acts based on the two functions of state of a single particle 1 + 2 − (4.12) In the representation (4.12), the single-particle Hamiltonian reads H1 = − ( κ̃ z κ̃ x κ̃ x −κ̃ z ) . (4.13) Taking into account (4.8), one can easily obtain the single-particle partition function in an explicit form Z1 = 2ch(βΛ) ; Λ = √ (κ̃z)2 + (κ̃x)2 . (4.14) The two particle Hamiltonian H12 acts based on the four functions of state of a two-particle cluster 1 + + 2 + − 3 − + 4 − − (4.15) 529 R.R.Levitskii, S.I.Sorokov, O.R.Baran In the representation (4.15), the Hamiltonian H12 reads H12 = −   2 ˜̃κz +K ˜̃κx ˜̃κx 0 ˜̃κx −K 0 ˜̃κx ˜̃κx 0 −K ˜̃κx 0 ˜̃κx ˜̃κx − 2 ˜̃κz +K   . (4.16) Based on (4.10) and (4.16) we obtain the two-particle partition function Z12 = 4∑ i=1 e−β(E12)i , (4.17) where (E12)4 = K , (4.18) whereas three other eigenvalues (E12)1, (E12)2, (E12)3 of the matrix (4.16) are roots of a cubic equation E3 12 +KE2 12 − [ K2 + 4(˜̃κx)2+ 4(˜̃κz)2 ] E12 −K [ K2+ 4(˜̃κx)2− 4( ˜̃κz)2 ] =0 . (4.19) From (3.43) and (3.44), taking into account the fact that in the framework of the MFA for the long-range interactions 〈Sz〉ρ = − 1 N d F d h = k〈Sz〉kρ = − 1 N d kF d κz (4.20) (this can be obtained from the explicit expression for the free energy (4.6)), we get equations for the parameters m = 〈Sz〉ρ, η = 〈Sx〉ρ and cluster fields ϕa (a = z, x) κ̃ x Λ th(βΛ) = 4˜̃κx Z12 3∑ i=1 [−(E12)i −K]e−β(E12)i 3(E12)2 i + 2K(E12)i − [K2 + 4(˜̃κx)2 + 4(˜̃κz)2] , (4.21) κ̃ z Λ th(βΛ) = 4˜̃κz Z12 3∑ i=1 [−(E12)i +K]e−β(E12)i 3(E12)2i + 2K(E12)i − [K2 + 4(˜̃κx)2 + 4(˜̃κz)2] , (4.22) m = κ̃ z Λ th(βΛ), (4.23) η = κ̃ x Λ th(βΛ). (4.24) When the long-range interaction is absent (J0 = 0), we have a system of two equa- tions (4.21) and (4.22) for ϕx, ϕz in an implicit form ((E12)i are roots of cubic equation (4.19)) and expressions (4.23) for m and (4.24) for η. When J0 6= 0, we have a system of three equations (4.21) – (4.23) for ϕx, ϕz, and m, and an expression for η. 530 Reference approach in theory of pseudospin systems Numerical analysis of the thermodynamic characteristics and longitudinal static susceptibility χzz (which is too cumbersome to be presented here) obtained here within the TPCA in the short-range interactions, with the long-range interactions taken into account within the MFA, as well as the study of the applicability bounds of this approach to the IMTF on different types of lattices at different values of the parameters Γ, J0 will be given elsewhere. Here we shall only briefly consider the major results atK > 0, J0 > 0, Γ> 0. We shall use the terminology of ferroelectricity. For the one-dimensional IMTF at J0 =0, the two-particle cluster approximation, unlike the MFA for the short-range interactions, does not predict the existence of ferroelectric ordering (at T > 0 and arbitrary Γ the system is in the paraelectric phase). Comparison of the TPCA results for the free energy, entropy, and specific heat as functions of temperature (expressions for entropy and specific heat were obtained for the paraelectric phase only) at different values of Γ/K has shown, that this approximation yields fair results for these characteristics at all temperatures ex- cept for the low-temperature region. Thus, at high temperatures, the TPCA results accord with exact results not only qualitatively, but also well enough quantitatively. The lower is the temperature the more the results of TPCA differ from the exact ones (too low values of free energy, entropy, and specific heat), whereas in the low- temperature region T < Tl (kBTl/K < th( 4 √ 1 6 ·Γ/K) + 1 6 · Γ/K) are qualitatively incorrect (for instance, the free energy is an increasing function of temperature). For the one-dimensional model at J0 > 0, as well as for two-dimensional and three-dimensional models at J0 > 0, the TPCA for the short-range interactions with the long-range interactions taken into account within the MFA predicts that a limit- ing value (Γ/K)k exists which depends on J0 and z, and above which a ferroelectric ordering is impossible (the latter is a qualitatively correct result). At (Γ/K) a < Γ/K < (Γ/K)k (where (Γ/K)a = √ c(z,Γ, J0, K)·zJ0/K, c(z,Γ, J0, K)≈ 2) this ap- proximation predicts a phase transition from the paraelectric phase to the ferroelec- tric phase on lowering temperature and the phase transition from the ferroelectric to the paraelectric phase – the so-called anti-Curie point. At small enough values of Γ/K 6 (Γ/K)a the anti-Curie point is absent, but the temperature behaviour of the thermodynamic characteristics remains qualitatively incorrect. The low-temperature region of T <Tl, where the TPCA yields incorrect results for thermodynamic char- acteristics, is reduced when the values of Γ/K and J0/K decrease (see table 1). In the high-temperature region T > Tl at J0 = 0 and z > 2, the TPCA is much more correct than the mean field approximation for the short-range interactions. We also performed a numerical analysis of the TPCA results at neglecting the variational parameter ϕx (ϕx=0). This version of the approximation is not suitable for one-dimensional chains. Thus, at small enough values of J0/K < 0.09 and Γ/K, Curie temperature increases on increasing Γ/K. At J0 = 0 and Γ/K ∈ ]0, 1.28] a ferroelectric ordering is predicted. For two-dimensional and three-dimensional lattices, neglecting the variational parameter ϕx leads to a slight quantitative worsening of the results in a high- temperature region and to a qualitatively correct description of temperature de- pendences of thermodynamic characteristics (m(T ), χzz(T )) in a low-temperature 531 R.R.Levitskii, S.I.Sorokov, O.R.Baran Table 1. Temperature of the anti-Curie point Ta, temperature Tl below which unphysical results for m(T ) and χzz(T ) are obtained, and Curie temperature for a square lattice (z = 4) at different values of Γ and J0 within the TPCA for the short-range interactions, calculated with the long-range interactions taken into account in the MFA. kBTa/K kBTl/K kBTc/K J0/K Γ/K 0.01 0.57 2.86 0.0 0.5 0.26 0.93 2.40 0.0 2.0 0.03 0.96 3.04 0.4 2.0 region. The larger is the lattice dimensionality and the value of the long-range inter- action and the smaller is the transverse field the smaller is the mentioned worsening. 4.2. Thermodynamics and intracluster pair distribution fu nctions in para- electric phase In order to study the dynamic characterictics of the IMTF in the paraelectric phase, we write here certain relations for some thermodynamic quantities in the paraelectric phase (κz = κ̃ z = ˜̃κz = ϕz = 0, m = 0). The eigenvalues (E12)i of two-particle Hamiltonian (4.16) in the paraelectric phase are (see (4.19)): (E12)1 = −L ; (E12)2 = L ; (E12)3 = K ; (E12)4 = −K , (4.25) where L = √ K2 + 4(˜̃κx) 2 . (4.26) From (4.25) we obtain the two-particle partition function in the paraelectric phase explicitly Z12 = 2 [ ch(βL) + ch(βK) ] . (4.27) In the paraelectric phase, equations (4.22), (4.23) turn to identity, while equation (4.21) for the variational parameter ϕx can be written explicitly, using (4.25) th(βκ̃x) = 4˜̃κx LZ12 sh(βL) . (4.28) We also present here an expression for η = 〈Sx〉ρ: η = th(βκ̃x) . (4.29) To calculate the pair CGFs of the reference model (4.4) within the TPCA (see (3.63)), we need to know the single-particle and two-particle intracluster pair CGFs F (2) 1 (a,bωn ), F (2,0) 12 (a,bωn ), F (1,1) 12 (a,bωn ) (a, b = x, y, z). Let us calculate now the two-particle 532 Reference approach in theory of pseudospin systems intracluster CGFs. It is convenient to do so in the self-representation of the operator H12. Since we have explicit expressions for the eigenvalues of the two-particle Hamilto- nian (4.16) in the paraelectric phase (see (4.25)), it is easy to obtain the normalized unitary matrix, which diagonalizes the two-particle Hamiltonian Û =   r1 r2 0 1/ √ 2 r2 −r1 1/ √ 2 0 r2 −r1 −1/ √ 2 0 r1 r2 0 −1/ √ 2   . (4.30) Here we use the notations r1 = 1 2 · √ 1 +K/L ; r2 = 1 2 · √ 1−K/L . (4.31) Going from the Pauli operators to their four-row analogs [50–52] σa 1 = Sa 1 ⊗ I ; σa 2 = I ⊗ Sa 2 ; a = x, y, z (4.32) (here I is the two-row unit matrix, ⊗ is the direct product symbol; matrices σ a ν obey Pauli commutation rules) and performing a unitary transformation σ̃a ν = Û−1σa ν Û , (4.33) we obtain the pseudospin operators in the self-representation of the operator H12: σ̃z 1 = √ 2   0 0 r2 r1 0 0 −r1 r2 r2 −r1 0 0 r1 r2 0 0   ; σ̃z 2 = √ 2   0 0 −r2 r1 0 0 r1 r2 −r2 r1 0 0 r1 r2 0 0   ; σ̃x 1 =   2 ˜̃κx/L −K/L 0 0 −K/L −2 ˜̃κx/L 0 0 0 0 0 −1 0 0 −1 0   ; σ̃x 2 =   2 ˜̃κx/L −K/L 0 0 −K/L −2 ˜̃κx/L 0 0 0 0 0 1 0 0 1 0   ; σ̃y 1 = i √ 2   0 0 r1 r2 0 0 r2 −r1 −r1 −r2 0 0 −r2 r1 0 0   ; σ̃y 2 = i √ 2   0 0 −r1 r2 0 0 −r2 −r1 r1 r2 0 0 −r2 r1 0 0   . (4.34) Expanding operators σ̃a i (4.34) in finite series in the four-dimensional Hubbard op- erators [50–52], following [14,57], we easily calculate the two-particle cumulant pair intracluster Green functions: F̂ (2,0) 12 (ωn) =   F (2,0) 12 (x,xωn ) 0 0 0 F (2,0) 12 (y,yωn ) F (2,0) 12 (y,zωn ) 0 F (2,0) 12 (z,yωn ) F (2,0) 12 (z,zωn )   , F̂ (1,1) 12 (ωn) =   F (1,1) 12 (x,xωn ) 0 0 0 F (1,1) 12 (y,yωn ) F (1,1) 12 (y,zωn ) 0 F (1,1) 12 (z,yωn ) F (1,1) 12 (z,zωn )   , (4.35) 533 R.R.Levitskii, S.I.Sorokov, O.R.Baran where F (2,0) 12 (x,xωn ) = (〈Tτ σ̃x 1 (τ) σ̃ x 1 (0)〉cρ12)ωn = Asδ(ωn) + A+(ωn) ; F (2,0) 12 (y,yωn ) = 4 βLZ12ψ(ωn) [(2 ˜̃κx) 4 sh(βL) + C+ · ω2 n] ; F (2,0) 12 (y,zωn ) = −F (2,0) 12 (z,yωn ) = 8˜̃κxωn βLZ12ψ(ωn) [C− + sh(βL)ω2 n] ; F (2,0) 12 (z,zωn ) = 16( ˜̃κx) 2 βLZ12ψ(ωn) [C− + sh(βL)ω2 n] ; (4.36) F (1,1) 12 (x,xωn ) = (〈Tτ σ̃x 1 (τ) σ̃ x 2 (0)〉cρ12)ωn = Asδ(ωn) + A−(ωn) ; F (1,1) 12 (y,yωn ) = − 16BKω2 n βZ12ψ(ωn) ; F (1,1) 12 (y,zωn ) = −F (1,1) 12 (z,yωn ) = 32BK ˜̃κxωn βZ12ψ(ωn) ; F (1,1) 12 (z,zωn ) = 64BK( ˜̃κx) 2 βZ12ψ(ωn) . (4.37) Here we use the notations As = ( 4 ˜̃κx LZ12 )2 · [1 + ch(βK)ch(βL)] ; B = 1 2 [ch(βL)− ch(βK)] ; A±(ωn) = 8K βLZ12 (Ksh(βL) 4L2 + ω2 n ± Lsh(βK) 4K2 + ω2 n ) ; C± = [L2 +K2]sh(βL)± 2LKsh(βK) ; (4.38) ψ(ωn) = [(L+K)2 + ω2 n][(L−K)2 + ω2 n] . (4.39) The most convenient way of obtaining the single-particle intracluster CGFs is, by performing a rotation in a spin space, to go to such a coordinate system, where Hamiltonian (4.13) is diagonal. We present here the final result (after the inverse transformation) in the paraelectric phase in terms of a = x, y, z: F̂ (2) 1 (ωn) =   [1− η2]δ(ωn) 0 0 0 g(ωn) g′(ωn) 0 −g′(ωn) g(ωn)   . (4.40) Here we use notations g(ωn) = 4 β · η · κ̃x (2κ̃x)2 + ω2 n ; g′(ωn) = ωn 2κ̃x · g(ωn) . (4.41) At obtaining (4.40) we used the fact that within TPCA for the short-range interac- tions, taking into account the long-range interactions in the MFA, 〈S x〉ρ1 = 〈Sx〉ρ ≡ η. 534 Reference approach in theory of pseudospin systems 4.3. Dynamics in paraelectric phase. Cluster random phase a pproximation Our task is to investigate the dynamics characteristics of the IMTF in the para- electric phase within the TPCA for the short-range interactions and within the (r−d 0 )0 approximation for the long-range interactions [54,58] – the cluster random phase approximation. The first step is then to calculate the temperature CGFs. In CRPA the pair CGF Ĝ(~q, ωn) ≡ b̂(2)(~q, ωn) according to (2.14) is Ĝ(~q, ωn) = [1− kĜ(~q, ωn)βĴ(~q )] −1 kĜ(~q, ωn) , (4.42) where Ĵ(~q ) =   0 0 0 0 0 0 0 0 J(~q )   , (4.43) (in terms of a = x, y, z), and kĜ(~q, ωn) ≡ k̂b(2)(~q, ωn) is the pair CGF of the reference system, which in the TPCA reads (3.63). From (4.42), (3.63), using (4.35), (4.40), we obtain pair CGFs. For Gxx(~q, ωn), and Gzz(~q, ωn) in the paraelectric phase we have: Gxx(~q, ωn) = δ(ωn) · Gxx α (~q ) , Gzz(~q, ωn) = 4Γη[p + + ω2 n][p− + ω2 n] R(~q, ωn) . (4.44) Here we introduce the notations Gxx α (~q ) = [1−η2] { z[1−η2] dx(T ) − (z − 1) + z[1−η2] bx(T ) dx(T ) Θ(~q ) }−1 ; dx(T ) = K2 L2 · η β ˜̃κx + 2 [ 1− K2 L2 ]1 + ch(βL)ch(βK) ch(βL) + ch(βK) ; bx(T ) = 1 2sh(βK) { K3 L3 sh(βL)−sh(βK)+βK [ 1−K 2 L2 ]1+ch(βK)ch(βL) ch(βL)+ch(βK) } ; (4.45) R(~q, ωn) = [p − + ω2 n] [ ω4 n + u2ω 2 n + u0 − 4ΓηJ(~q )[p + + ω2 n] ] + 4BLΓK ˜̃κxsh(βL) ·ψ(ωn)zΘ(~q ); p ± =K2+L2+ 2LK[− sh(βK)± 2B] sh(βL) ; ps =K2+L2+ 2LK[sh(βK)−2B] sh(βL) ; u2=2z2[K2+L2]+(z−1)2[(2κ̃x)2+p + ]− 2z(z−1) { κ̃ x[(2 ˜̃κx)2+ps] 2 ˜̃κx + p + } ; u0 = 4Γ{4z( ˜̃κx)3 − (z−1)κ̃xp + } . (4.46) In calculations of (4.44) we used the relations (4.28), (4.29). 535 R.R.Levitskii, S.I.Sorokov, O.R.Baran It should be noted that from (4.44) one obtains the static longitudinal suscepti- bility (χzz = βGzz(0, 0)) of the IMTF in the paraelectric phase χzz = [ zLZ12 2{L+K L−K · eβL − L−K L+K · e−βL − 4LK L2−K2 · eβK} − (z−1)κ̃x th(βκ̃x) − J0 ]−1 , (4.47) (J0 = J(~q=0)) which accords with the one calculated from thermodynamic relations in the TPCA for the short-range interactions, with the long-range interactions taken into account in the MFA. To explore the dynamic properties of the IMTF we needn’t know the temperature CGFs (4.44), but the retarded CGFs. We can calculate them [59] by performing analytical continuation of the temperature CGFs G ab(~q, ωn) (iωn → E + iE ′) and going to the limit E ′ → 0. The final results for spectral densities J xx(~q, E) and J zz(~q, E), defined as J ab(~q, E) = lim E′→0 [ 2~β eβE − 1 ImGab(~q, ωn)\|ωn→−iE+E′ ] , (4.48) and for the pair cumulant correlation function 〈S z ~qS z −~q〉c are the following. The spectral density J xx(~q, E) of the IMTF in the paraelectric phase within the CRPA reads J xx(~q, E) = δ(E) · Gxx α (~q ) . (4.49) Let us note, that an exact expression for J xx(~q, E) of the one-dimensional IMTF with the short-range interactions only [31] has not only the central peak (∼ δ(E)) but also two symmetrical resonance zones. The absence of the resonance zones within the CRPA for J xx(~q, E) results from neglecting the fluctuations of cluster fields in this approximation. The spectral density J zz(~q, E) (in the paraelectric phase) can be presented as 1 ~ J zz(~q, E) = ∑ i=−,+,r kJi (~q ) [ δ(E −Ei(~q )) + eβEi(~q )δ(E + Ei(~q )) ] , (4.50) where Ei(~q ) (i = −,+, r) are the elementary excitation spectrum modes, determined from the equation (see (4.44), (4.46)) R(~q, ωn)\|ωn→−iE = 0 , (4.51) and kJi (~q ) are the integral intensities of the elementary excitations spectrum modes kJi (~q ) = 4πT Γη Ai(~q )· [p + −E2 i (~q )][p− − E2 i (~q )] (eβEi(~q ) − 1)Ei(~q ) . (4.52) Here we use the notations: Ai1(~q ) = 1 [E2 i1 (~q )− E2 i2 (~q )][E2 i1 (~q )− E2 i3 (~q )] , i1, i2, i3 = (−,+, r) , i1 6= i2, i2 6= i3, i3 6= i1 . (4.53) 536 Reference approach in theory of pseudospin systems It should be noted that at ~q = 0 (see (4.44), (4.46)) there are only two modes E±(0) = 1√ 2 √ U2 ± √ U2 2 − 4U0 , (4.54) where U2 = u2 − 4ηΓJ0; U0 = u0 − 4ηΓJ0 p+ . (4.55) The mode E−(~q ) is soft (E−(0) → 0, T → Tc). For the pair cumulant correlation function 〈S z ~qS z −~q〉c from 〈Sa ~qS b −~q〉c\|t−→0 = 1 ~ ∫ ∞ −∞ dE J ab(~q, E) , (4.56) we obtain 〈Sz ~qS z −~q〉c\|t−→0 = 2T Γη ∑ i=−,+,r Ai(~q )·cth(12βEi)· [p + −E2 i (~q )][p− − E2 i (~q )] Ei(~q ) . (4.57) Let us briefly consider the results of numerical analysis of the longitudinal char- acteristics of the IMTF at z = 2, Jνµ = 0 (K = 1). As we have already mentioned, within CRPA the spectrum of the longitudinal characteristics of the model contains three nondamping modes E−(q), E+(q), Er(q) with the integral intensities kJ −(q), kJ+(q) and kJr (q) in the spectral density J zz(q, E). The calculated dependences of Ei(q), k J i (q) at different transverse fields and temperatures are presented in figure 1. On increasing Γ, temperature, and q, redistribution of the intensities from low fre- quencies to higher frequencies is observed. At large Γ and at low temperatures the redistribution on increasing q takes place, first, mainly from E−(q) to Er(q), and then from Er(q) to E+(q). At large Γ and at high temperatures the redistribution takes place mainly from E−(q) to E+(q). At small Γ the redistribution is practically absent. In figure 2 we present exact and approximate (CRPA) results for the static corre- lator 〈Sz νS z ν+n〉 at kBT = 0.6 and kBT = 1.0 at different Γ. The CRPA gives too low values of 〈Sz νS z ν+n〉, especially at low temperatures. Thus, the autocorrelator 〈S z νS z ν〉 at low temperatures is essentially smaller than unity. The higher is the temperature and the smaller is Γ the better the CRPA results accord with the exact ones. Let us also discuss the redistribution of the modes intensities E−(0), E+(0) on changing Γ and temperature. At temperatures kBT > √ Γ, change of the modes positions and integral intensities (obtained within the CRPA) on changing Γ and T qualitatively describes the change in the frequency dependences of the real part of the relaxation function ReΨzz(0, E), calculated numerically [35,36] or exactly. Thus, for instance, at Γ = 1 and kBT = 0.8 (see figure 3) Re Ψzz(0, E) has a prominent resonance zone at E close to zero. Smearing of this resonance zone on increasing T is qualitatively described by increasing kJ +(0) and E−(0) and by decreasing kJ−(0). On the other hand, for instance, at T → ∞ (see figure 4), a shift of the resonance zone ReΨzz(0, E) to higher frequencies region on increasing Γ is described by increasing 537 R.R.Levitskii, S.I.Sorokov, O.R.Baran 0 0.5 1.0 1.5 2.0 2.5 0 1 2 3 4 Γ = 2.0Γ = 1.5Γ = 0.5 + r _ Ei q 0 0.5 1.0 1.5 2.0 2.5 1 2 3 4 5 + r _ Ei q 0 0.5 1.0 1.5 2.0 2.5 2 3 4 5 6 + r _ Ei q 0 0.5 1.0 1.5 2.0 2.5 0 1 2 3 4 10 15 + − − kJ q 0 0.5 1.0 1.5 2.0 2.5 0 1 2 3 4 10 15 r + r − − kJ q 0 0.5 1.0 1.5 2.0 2.5 0 1 2 3 4 10 15 rr + − − kJ q 0 1 2 3 0 0.2 0.4 r r + + − 0 1 2 3 0 0.2 0.4 r r + + − 0 1 2 3 0 0.2 0.4 − + r r + − Figure 1. Elementary excitations spectrum modes Ei(q) and their integral inten- sities kJi (q) (within CRPA) as functions of quasimomentum q at different tem- peratures (thick lines – kBT = 1.0, thin lines – kBT = 4.0) for Γ = 0.5, 1.5, 2.0. 538 Reference approach in theory of pseudospin systems 〈Sz νS z ν+n〉〈Sz νS z ν+n〉 0 2 4 6 8 10 12 0.0 0.2 0.4 0.6 0.8 1.0 Γ=0.25 IM 1.5 1 0.5 kBT = 0.6 n 0 2 4 6 8 10 12 0.0 0.2 0.4 0.6 0.8 1.0 IM 1.5 2.0 1.0 Γ=0.5 kBT = 1 n Figure 2. Static correlation function 〈Sz νS z ν+n〉 at different values of Γ and tem- perature (kBT = 0.6, 1.0) calculated within CRPA (solid lines) and exactly [60] (dash lines). Exact and approximate (CRPA) results for Ising model (short dash lines) coincide. 2Γ·ReΨzz(0, E), kJi (0) 0 1 2 3 4 5 0 2 4 6 8 0 _ kBT = 0.8 E / Γ 0 1 2 3 4 5 0 2 4 6 8 _ 0 + kBT = 4 E / Γ 0 1 2 3 4 5 0 2 4 6 8 → 0 _ + kBT ∞ E / Γ 0 2 4 6 0 0.1 0.2 + Figure 3. Frequency dependence of 2Γ ·Re Ψzz(0, E) for Γ = 1.0 at different temperatures (kBT = 0.8, 4, ∞) calculated numerically [35,36]. Vertical lines correspond to the mode integral intensities kJ −(0), kJ+(0) within CRPA (solid line) and kJ0 (0) within RPA (dash line). 539 R.R.Levitskii, S.I.Sorokov, O.R.Baran 2Γ·ReΨzz(0, E), kJi (0) 0 2 4 0 1 2 3 0_ + Γ = 0.5 E / Γ 0 2 4 0 1 2 3 0 _ + E / Γ Γ = 1.0 0 2 4 0 1 2 3 0 _ + E / Γ Γ = 1.5 0 2 4 0 1 2 3 0 _ + E / Γ Γ = 2.0 Figure 4. Exact results for frequency dependence of 2Γ·ReΨzz(0, E) at T → ∞ at different values of Γ (Γ = 0.5, 1.0, 1.5, 2.0). Vertical lines correspond to the mode integral intensities kJ −(0), k J +(0) within CRPA (solid line) and kJ 0 (0) within RPA (dash line). E−(0) and k J +(0). The fact that the resonance zone at Γ = 0.5 is more prominent than at Γ = 1.0 is described by decreasing kJ−(0) and increasing kJ+(0) at Γ increasing. The fact that at Γ = 2.0 the resonance zone ReΨzz(0, E) is more prominent than at Γ = 1.5 is described by closing in the positions of E−(0), E+(0) modes at Γ increasing. In figures 3, 4 we also depicted the results of the random phase approximation (RPA) for the short-range interactions. This approximation describes the frequency dependences of the real part of the relaxation function in the cases presented in figures 3, 4 qualitatively well only at T → ∞ Γ = 1.5, 2.0. We restricted the presented numerical analysis of the longitudinal characteristics of the one-dimensional IMTF (at Jνµ = 0) by the values of the transverse field 0.2 6 Γ 6 2. A more detailed analysis both at Jνµ = 0 and at Jνµ 6= 0 and with the values of the microparameters corresponding to the CsH2PO4 crystal will be performed elsewhere. References 1. Vaks V.G. Introduction to the Microscopic Theory of Ferroelectrics. Moskva, Nauka, 1973 (in Russian). 2. Blinc R., Žekš B. 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System., IFKS–92–18U, Lviv, 1992 (in Ukrainian). 56. Sorokov S.I., Levitskii R.R., Baran O.R. // Ukr. J. Phys., 1996, vol. 41, p. 490 (in Ukrainian). 57. Slobodyan P.M., Stasyuk I.V. Teor. Mat. Fiz., 1974, vol. 19, p. 423 (in Russian). 58. Levitskii R.R. Sorokov S.I. Dynamics and thermodynamic properties of the Ising model in a transverse field. The two-particle cluster approximation. Preprint of the Inst. Teor. Fiz., ITF–88–34R, Kiev, 1988 (in Russian). 59. Izyumov Yu.A., Kassan-Ogly F.A., Skriabin Yu.N. Field Method in Theory of Ferro- magnetism. Moskva, Nauka, 1974 (in Russian). 60. Pfeuty P. // Ann. Phys., 1970, vol. 57, p. 79. 542 Reference approach in theory of pseudospin systems Базисний підхід в теорії псевдоспінових систем Р.Р.Левицький, С.І.Сороков, О.Р.Баран Інститут фізики конденсованих систем НАН Укpаїни, 79011 Львів, вул. Свєнціцького, 1 Отримано 18 квітня 2000 р. Для теоретичного опису псевдоспінових систем з суттєвими корот- косяжними та далекосяжними взаємодіями використовується ме- тод, який грунтується на розрахунку функціоналу вільної енергії з базисним урахуванням короткосяжних взаємодій у кластерному на- ближенні. Для квантових псевдоспінових систем запропоновано по- слідовне формулювання методу кластерних розвинень та метод, який дає змогу в рамках кластерного наближення отримати для ба- зисних температурних кумулянтних функцій Гріна довільного поряд- ку рівняння типу рівнянь Орнштейна-Церніке. В наближенні двоча- стинкового кластера в явному вигляді отримано вираз для парної температурної кумулянтної функції Гріна базисної системи. У кла- стерному наближенні хаотичних фаз розраховані та досліджені тер- модинамічні характеристики, спектр елементарних збуджень та інте- гральні інтенсивності гілок спектра моделі Ізінга в поперечному полі. Ключові слова: фазові переходи, псевдоспінові моделі, базисний підхід, кластерне наближення, рівняння Орнштейна-Церніке, м’яка мода PACS: 03.65.-w, 05.30.-d 543 544

Reference approach in theory of pseudospin systems

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