On the spectral relations for multitime correlation functions

A general approach to the derivation of the spectral relations for the multitime correlation functions is presented. A special attention is paid to the consideration of the non-ergodic (conserving) contributions and it is shown that such contributions can be treated in a rigorous way using multiti...

Повний опис

Збережено в:

Бібліографічні деталі
Дата:	2006
Автор:	Shvaika, A.M.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2006
Назва видання:	Condensed Matter Physics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/121370
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	On the spectral relations for multitime correlation functions / A.M. Shvaika // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 447–458. — Бібліогр.: 39 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-121370
record_format	dspace
spelling	irk-123456789-1213702017-06-15T03:02:52Z On the spectral relations for multitime correlation functions Shvaika, A.M. A general approach to the derivation of the spectral relations for the multitime correlation functions is presented. A special attention is paid to the consideration of the non-ergodic (conserving) contributions and it is shown that such contributions can be treated in a rigorous way using multitime temperature Green functions. Representation of the multitime Green functions in terms of the spectral densities and solution of the reverse problem, i.e., finding the spectral densities from the known Green functions, are given for the case of the three-time correlation functions. Запропоновано загальний пiдхiд до отримання спектральних спiввiдношень для багаточасових кореляцiйних функцiй. Особлива увага звертається на розгляд неергодичних (збережних) внескiв i показано, що такi внески можна послiдовно отримати використовуючи багаточасовi температурнi функцiї Ґрiна. Для випадку тричасових кореляцiйних функцiй знайдено представлення багаточасових функцiй Ґрiна через спектральнi густини i розв’язано обернену задачу – вираження спектральних густин через вiдомi функцiї Ґрiна. 2006 Article On the spectral relations for multitime correlation functions / A.M. Shvaika // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 447–458. — Бібліогр.: 39 назв. — англ. 1607-324X PACS: 05.30.-d DOI:10.5488/CMP.9.3.447 http://dspace.nbuv.gov.ua/handle/123456789/121370 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	A general approach to the derivation of the spectral relations for the multitime correlation functions is presented. A special attention is paid to the consideration of the non-ergodic (conserving) contributions and it is shown that such contributions can be treated in a rigorous way using multitime temperature Green functions. Representation of the multitime Green functions in terms of the spectral densities and solution of the reverse problem, i.e., finding the spectral densities from the known Green functions, are given for the case of the three-time correlation functions.
format	Article
author	Shvaika, A.M.
spellingShingle	Shvaika, A.M. On the spectral relations for multitime correlation functions Condensed Matter Physics
author_facet	Shvaika, A.M.
author_sort	Shvaika, A.M.
title	On the spectral relations for multitime correlation functions
title_short	On the spectral relations for multitime correlation functions
title_full	On the spectral relations for multitime correlation functions
title_fullStr	On the spectral relations for multitime correlation functions
title_full_unstemmed	On the spectral relations for multitime correlation functions
title_sort	on the spectral relations for multitime correlation functions
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2006
url	http://dspace.nbuv.gov.ua/handle/123456789/121370
citation_txt	On the spectral relations for multitime correlation functions / A.M. Shvaika // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 447–458. — Бібліогр.: 39 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT shvaikaam onthespectralrelationsformultitimecorrelationfunctions
first_indexed	2025-07-08T19:44:01Z
last_indexed	2025-07-08T19:44:01Z
_version_	1837109185416790016
fulltext	Condensed Matter Physics 2006, Vol. 9, No 3(47), pp. 447–458 On the spectral relations for multitime correlation functions A.M.Shvaika Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine Received April 17, 2006 A general approach to the derivation of the spectral relations for the multitime correlation functions is pre- sented. A special attention is paid to the consideration of the non-ergodic (conserving) contributions and it is shown that such contributions can be treated in a rigorous way using multitime temperature Green functions. Representation of the multitime Green functions in terms of the spectral densities and solution of the reverse problem, i.e., finding the spectral densities from the known Green functions, are given for the case of the three-time correlation functions. Key words: multitime correlation functions, Green functions, spectral relations, non-ergodicity PACS: 05.30.-d 1. Introduction One of the main tasks of the quantum statistic physics is calculation of the correlation functions for many-body systems of different kind because they contain the most important information about the observable quantities and system properties. As it was first noticed by Kubo [1], linear transport coefficients are expressed in terms of the Fourier transforms of appropriate correlation functions, which relate by spectral relations to the two-time Green functions. Since that time the Green’s function method has been noticeable and extensively developed [2–4], providing equations that permit to calculate the expectation values of operators and observable quantities. But very soon it was noticed that spectral relations should be completed by a special treatment of the pole at zero frequency, which gives additional contribution connected with the presence of the conserving quantities [5–7]. Later, it was shown that such contributions describe the difference between the isothermal and isolated response of the many-body system and are specific for the non-ergodic systems [1,6–8] where the regions exist in the phase space which cannot be achieved by the trajectory of the point that describes an evolution of the many-body system. In the Green’s function formalism, the issue of ergodicity appears as a difficulty in determining the zero-frequency bosonic propagators [5–7,9–18]. Nevertheless, even now many textbooks on the quantum statistics and many-body theory do not provide complete discussion of the spectral relations and special treatment of the zero-frequency functions. The equations of motion do not uniquely determine the causal and retarded Green’s functions, but only up to δ-function of frequency with some unknown coefficients which produce additional contributions in the zero-frequency functions (see, section 2). Usually, these zero-frequency func- tions are fixed by being assigned their ergodic values, but this cannot be justified a priori. A wrong determination of them dramatically affects the values of directly measurable quantities like compressibility, specific heat, and magnetic susceptibility. In order to handle these zero-frequency functions, different approaches were developed, e.g. anti-commutator bosonic Green functions [14], direct algebraic method [19], singular-value decomposition [20], algebra constraints in the compos- ite operator method [21,22], etc. On the other hand, temperature Green functions [23,24] are free from these issues and make it possible to avoid all complications connected with the presense of the non-ergodic terms [13], c© A.M.Shvaika 447 A.M.Shvaika e.g. they permit to calculate isothermal susceptibilities for the Ising model, where Kubo response is equal to zero, as well as for the more complicated spin and electron [26,27] and pseudospin- electron [28] systems, and for the infinite-dimensional Falicov-Kimball model [29,30], where the exact expression for the isothermal charge susceptibility [31,32] contains both Kubo response, which is finite at all temperatures, and non-ergodic contribution, whose divergencies give the phase transition points. The above mentioned issues concerning a special treatment of the zero-frequency functions are mostly investigated for the two-time correlation and Green functions, but nobody has cosidered this for a multitime correlation. In the original Kubo’s paper [1], the formulation of the transport theory is not limited to the linear phenomena, and the solution of the Liouville equation for the density matrix is given to the arbitrary order in the strength of disturbance. Resulting multitime correlation and Green functions can be used for the description of the nonlinear transport phe- nomena and resonances [33–35]. Besides, multitime correlation functions also appear as puzzles in different orders of the perturbation theories for many-body systems [36]. Moreover, cross-sections of the inelastic scattering processes can be expressed in terms of the multitime correlation func- tions too, e.g. for the electronic inelastic light (Raman) scattering. The nonresonant, mixed, and resonant responses are connected with the two-time, three-time, and four-time temperature Green functions [37,38], respectively, and can be rewritten in terms of the multitime correlation functi- ons. Nonresonant contribution is connected with the spectral density of the two-time correlation function RN (q,Ω) = 2πg2(ki)g 2(ko)Iγ̃γ̃(Ω,−Ω), (1.1) mixed contribution is connected with the spectral density of the three-time correlation functions RM (q,Ω) = 2πg2(ki)g 2(ko) +∞∫ −∞ dω [ Iγ̃j(o)j(i)(Ω, ω − Ω,−ω) ω − ωi + iδ + Iγ̃j(i)j(o)(Ω, ω − Ω,−ω) ω + ωo − iδ + Ij(i)j(o)γ̃(ω,Ω − ω,−Ω) ω − ωi − iδ + Ij(o)j(i)γ̃(ω,Ω − ω,−Ω) ω + ωo + iδ ] , (1.2) and resonant contribution is connected with the spectral density of the four-time correlation functions RR(q,Ω) = 2πg2(ki)g 2(ko) +∞∫ −∞ dω +∞∫ −∞ dω̃ × [ Ij(i)j(o)j(o)j(i)(ω,Ω − ω, ω̃ − Ω,−ω̃) (ω − ωi − iδ)(ω̃ − ωi + iδ) + Ij(o)j(i)j(i)j(o)(ω,Ω − ω, ω̃ − Ω,−ω̃) (ω + ωo + iδ)(ω̃ + ωo − iδ) + Ij(o)j(i)j(o)j(i)(ω,Ω − ω, ω̃ − Ω,−ω̃) (ω + ωo + iδ)(ω̃ − ωi + iδ) + Ij(i)j(o)j(i)j(o)(ω,Ω − ω, ω̃ − Ω,−ω̃) (ω − ωi − iδ)(ω̃ + ωo − iδ) ] , (1.3) where IAB...(ω1, ω2, . . .) are spectral densities of multitime correlation functions (see below) and ˆ̃γ and ĵ(i,o) are expressed in terms of the stress tensor and current operator, respectively. In general, spectral relations include (a) representation of the observable quantities in terms of the spectral densities of the multitime correlation functions [like equations (1.1), (1.2), and (1.3) for the inelastic light scattering], (b) extraction of the non-ergodic contributions in the spectral densities, (c) representation of the multitime temperature Green functions in terms of the spectral densities, and (d) solution of the reverse problem, i.e., finding the spectral densities from the known multitime temperature Green functions and based on this calculating the observable quantities. Spectral relations for multitime correlation functions were formulated in [4,39] only for the case of the multitime generalizations of the retarded and advanced functions without considering the zero-frequency non-ergodic contributions. The main purpose of this paper is to show how non- ergodic contributions enter the multitime correlation functions and how a complete set of spectral relations for the multitime correlation and Green functions can be derived. 448 On the spectral relations for multitime correlation functions 2. Two-time correlation functions Before we start considering the spectral relations for the multitime correlation functions, let us remind basic relations and the main results for the two-time correlation functions. In general, a two-time correlation function is defined by the following expression KAB(t1 − t2) = 〈Â(t1)B̂(t2)〉, (2.1) where Â(t) = eiHtÂe−iHt is an operator Â in the Heisenberg representation, and angular brackets denote statistical average with total Hamiltonian of the many-body system. Here and below we shall consider the case of the bosonic operators only, generalization for the fermionic one being obvious. We can perform Fourier transformation of equation (2.1) that gives a spectral density for the two-time correlation function IAB(ω1,−ω1) = 1 2π +∞∫ −∞ d(t1 − t2)e iω1(t1−t2)KAB(t1 − t2) = ĨAB(ω1,−ω1) + δ(ω1)ĨAB(◦, ◦), (2.2) where we have separated two contributions ĨAB(ω1,−ω1) = 1 Z ∑ jl ′ e−βεj AjlBljδ(εjl + ω1), (2.3) ĨAB(◦, ◦) = 1 Z ∑ jl εj=εl e−βεj AjlBlj (2.4) with different frequency and time dependences. The first one is time-dependent and includes sum over states with different energies (εjl ≡ εj−εl 6= 0) denoted by prime (2.3). We shall call it regular contribution, and the second one is purely static and includes sum over the states with the same energy (2.4). We shall call it non-ergodic contribution. Here, Ajl = 〈j\|Â\|l〉 are matrix elements of the operator Â, Z = ∑ l e −βεl is partition function, and ◦ denotes that given contribution does not depend on the respective frequency. Besides, spectral densities (2.2) satisfy the following permutation relation IAB(ω1,−ω1) = IBA(−ω1, ω1)e βω1 (2.5) and for the given operators Â and B̂ there is only one nonidentical spectral density. At this point there is no special need separating regular and non-ergodic contributions, and, as a rule, in the textbooks nobody does it. But such separation becomes important when one is going to find spectral densities using spectral relations for the two-time retarded Green’s function G (r) AB(t1 − t2) = 〈〈 Â(t1)\|B̂(t2) 〉〉 = −iΘ(t1 − t2) 〈[ Â(t1), B̂(t2) ]〉 . (2.6) Its Fourier transformation, from the formal point of view, GAB(ω1,−ω1) = 1 2π ∞∫ −∞ d(t1 − t2)e iω1(t1−t2)GAB(t1 − t2) = G̃AB(ω1,−ω1) + δ(ω1)G̃c(◦, ◦) (2.7) also includes two contributions with different time dependences G̃AB(ω1,−ω1) = +∞∫ −∞ dω̃1ĨAB(ω̃1,−ω̃1) 1 − e−βω̃1 ω̃1 − ω1 ± iδ , (2.8) G̃AB(◦, ◦) = ĨAB(◦, ◦), (2.9) but only the first one can be derived using an equation of motion techniques ω1GAB(ω1,−ω1) = ω1G̃AB(ω1,−ω1) (2.10) 449 A.M.Shvaika and the second one is omitted in the equations of motion ω1δ(ω1)G̃AB(◦, ◦) ≡ 0. As a result, a corresponding static contribution into spectral density cannot be handled directly by the spectral theorem for the retarded Green functions. On the other hand, a method of the temperature or Matsubara Green functions permits to avoid all the complications connected with the presense of the non-ergodic terms and obtain all contri- butions in the spectral density in a straightforward way [13]. In general, a two-time temperature Green’s function can be defined as Kc(τ1 − τ2) = 〈T Â(τ1)B̂(τ2)〉, (2.11) where Â(τ) = eHτ Âe−Hτ , τ is imaginary time (inverse temperature), and T is operator of the imaginary time chronological ordering. Its Fourier transform Kc(iν1,−iν1) = β∫ 0 d(τ1 − τ2)e iν1(τ1−τ2)Kc(τ1 − τ2) = K̃c(iν1,−iν1) + β∆(iν1)K̃c(◦, ◦), (2.12) where iν1 ≡ iων1 = 2πiTν1 are Matsubara’s frequency and ∆(z) = { 1, z = 0, 0, z 6= 0 (2.13) is generalization of the Kronecker symbol, also contains two contributions with different time dependences K̃c(iν1,−iν1) = +∞∫ −∞ dω̃1ĨAB(ω̃1,−ω̃1) 1 − e−βω̃1 ω̃1 − iν1 , (2.14) K̃c(◦, ◦) = ĨAB(◦, ◦). (2.15) As a rule, temperature Green functions are handled using different kinds of diagrammatic techni- ques which permit to calculate contributions with ∆-symbols and according to (2.15) they can be identified as non-ergodic contributions in spectral densities, which is the main difference from the case of the retarded Green functions. For the non-ergodic systems such contributions are the main ones that determine the critical behavior. A typical example is the Ising model for which there are only non-ergodic contributions whose divergencies give Curie points and there are no regular ones. Another example of the non-ergodic fermionic system is the Falicov-Kimball model [29,30] whose isothermal charge susceptibility contains both non-ergodic and regular contributions but only the first one determines the critical point [31,32]. Regular contribution in spectral density is connected with nonanalyticy (imaginary part) of the Green’s function at real axis and can be obtained by performing an analytic continuation of the Matsubara Green’s function from the imaginary to complex frequencies and then to the real one iν1 → z1 → ω1 ± iδ K̃c(z1,−z1) ∣∣∣∣ 1 ≡ 1 2πi K̃c(z1,−z1) ∣∣∣∣ z1=ω1+iδ z1=ω1−iδ = ĨAB(ω1,−ω1)(1 − e−βω1), (2.16) that completes spectral relations for the two-time correlation functions. 3. Three-time correlation functions Now let us proceed to the consideration of the three-time correlation functions. Generalization for the case of the higher-order multitime correlation functions can be done in the same way, but it is much more cumbersome and will not be considered here. 450 On the spectral relations for multitime correlation functions Three-time correlation function can be defined in a usual way as KABC(t1, t2, t3) = 〈Â(t1)B̂(t2)Ĉ(t3)〉. (3.1) Here, we shall consider only the case of the equilibrium many-body systems which are time-shift invariant KABC(t1, t2, t3) = KABC(t1 − t, t2 − t, t3 − t). (3.2) Spectral density is defined as its Fourier transform IABC(ω1, ω2, ω3) = 1 (2π)2 +∞∫ −∞ d(t1 − t3) +∞∫ −∞ d(t2 − t3)e i(ω1t1+ω2t2+ω3t3)KABC(t1, t2, t3) = [ ĨABC(ω1, ω2, ω3) + δ(ω1)ĨABC(◦,−ω3, ω3) + δ(ω2)ĨABC(ω1, ◦,−ω1) + δ(ω3)ĨABC(−ω2, ω2, ◦) + δ(ω1)δ(ω2)ĨABC(◦, ◦, ◦) ] ∆(ω1 + ω2 + ω3) (3.3) and includes five different contributions with different time dependences ĨABC(ω1, ω2,−ω1 − ω2) = 1 Z ∑ jlf ′ e−βεj AjlBlfCfjδ(εjl + ω1)δ(εlf + ω2), (3.4) ĨABC(◦,−ω3, ω3) = 1 Z ∑ jlf εj=εl 6=εf e−βεj AjlBlfCfjδ(εfj + ω3), (3.5) ĨABC(ω1, ◦,−ω1) = 1 Z ∑ jlf εl=εf 6=εj e−βεj AjlBlfCfjδ(εjl + ω1), (3.6) ĨABC(−ω2, ω2, ◦) = 1 Z ∑ jlf εf=εj 6=εl e−βεj AjlBlfCfjδ(εlf + ω2), (3.7) ĨABC(◦, ◦, ◦) = 1 Z ∑ jlf εf=εj=εl e−βεj AjlBlfCfj . (3.8) Besides, the total spectral density (3.3) as well as each contribution satisfy the following cyclic permutation identities (ω1 + ω2 + ω3 = 0) [4,36] IABC(ω1, ω2, ω3) = IBCA(ω2, ω3, ω1)e βω1 = ICAB(ω3, ω1, ω2)e −βω3 (3.9) and for the given operators Â, B̂, and Ĉ there are only two nonidentical spectral densities, e.g. IABC(ω1, ω2, ω3) and ICBA(ω3, ω2, ω1). Now we introduce three-time temperature Green’s function Kc(τ1, τ2, τ3) = 〈T Â(τ1)B̂(τ2)Ĉ(τ3)〉, Kc(τ1, τ2, τ3) = Kc(τ1 − τ, τ2 − τ, τ3 − τ). (3.10) Due to the imaginary time ordering its Fourier transform contains 3! = 6 terms which can be collected into two groups of three terms connected by the cyclic permutations Kc(iν1, iν2, iν3) = 1 β β∫ 0 dτ1 β∫ 0 dτ2 β∫ 0 dτ3e (iν1τ1+iν2τ2+iν3τ3)Kc(τ1, τ2, τ3) = 1 Z ∑ jlf [ AjlBlfCfjP(j, iν1, l, iν2, f, iν3) + CjfBflAljP(j, iν3, f, iν2, l, iν1) ] , (3.11) 451 A.M.Shvaika where P(j, iν1, l, iν2, f, iν3) = 1 β [ e−βεj β∫ 0 dτ1 τ1∫ 0 dτ2 τ2∫ 0 dτ3 + e−βεl β∫ 0 dτ2 τ2∫ 0 dτ3 τ3∫ 0 dτ1 + e−βεf β∫ 0 dτ3 τ3∫ 0 dτ1 τ3∫ 0 dτ2 ] × exp[(εjl + iν1)τ1 + (εlf + iν2)τ2 + (εfj + iν3)τ3]. (3.12) In the general case, when all Matsubara frequencies are nonzero or when there are no eigenstates with the same energy value, function (3.12) is equal to P̃(j, iν1, l, iν2, f, iν3) = ∆(iν1 + iν2 + iν3) × [ e−βεj (εlj − iν1)(εfj + iν3) + e−βεl (εfl − iν2)(εjl + iν1) + e−βεf (εif − iν3)(εlf + iν2) ] . (3.13) Besides, we must consider several special cases, when we have levels with the same energy value: case εj = εl 6= εf and iν1 = −iν2 − iν3 = 0, when P(j, 0, l, iν2, f,−iν2) = ∆(iν2 + iν3) [ βe−βεl εfl − iν2 + e−βεf − e−βεl (εfl − iν2)2 ] , (3.14) case εl = εf 6= εj and iν2 = −iν3 − iν1 = 0, when P(j,−iν3, l, 0, f, iν3) = ∆(iν3 + iν1) [ βe−βεf εjf − iν3 + e−βεj − e−βεf (εjf − iν3)2 ] , (3.15) case εf = εj 6= εl and iν3 = −iν1 − iν2 = 0, when P(j, iν1, l,−iν1, f, 0) = ∆(iν1 + iν2) [ βe−βεj εlj − iν1 + e−βεl − e−βεj (εlj − iν1)2 ] , (3.16) and the case εj = εl = εf and iν1 = iν2 = iν3 = 0, when P(j, 0, l, 0, f, 0) = β2 2 e−βεj . (3.17) The second term in the r.h.s. of equation (3.14) can be derived from equation (3.13) by the analytic continuation of the Matsubara frequencies to the complex one iν1 → z1 and iν3 → −iν2−z1 followed by the limit z1 → 0, but the first one cannot be derived from equation (3.13) by manipulating by frequencies only and corresponds to the additional non-ergodic or conserving contribution, which originates from the presence of the states with the same energies, and appears only at zero frequency. Such non-ergodic contributions can be derived from equation (3.13) by [ lim εjl→0 lim z1→0 − lim z1→0 lim εjl→0 ] P̃(j, z1, l, z2, f,−z1 − z2) = βe−βεl εfl − z2 , (3.18) but such derivation involves manipulations with the many-body quantum states energies, that cannot be, in general, reproduced by the quantum statistics many-body methods. A similar analysis can be done for equations (3.15) and (3.16) and, after analytic continuation from the imaginary axis to the complex plane iνi → zi using a constraint ∑ i zi = 0, (3.19) 452 On the spectral relations for multitime correlation functions equation (3.12) can be rewritten as P(j, z1, l, z2, f, z3) = ∆(z1 + z2 + z3) [ β2 2 ∆(z1)∆(z2)∆εj ,εl,εf e−βεj + β∆(z1)∆εj ,εl e−βεl εfl − z2 + β∆(z2)∆εl,εf e−βεf εjf − z3 + β∆(z3)∆εf ,εj e−βεj εlj − z1 + e−βεj (εlj − z1)(εfj + z3) + e−βεl (εfl − z2)(εjl + z1) + e−βεf (εif − z3)(εlf + z2) ] , (3.20) where ∆εj ,...,εf = { 1, εj = . . . = εf 0, other case . (3.21) Now, after substitution of equation (3.20) in equation (3.11), we get the following representation for the analytically continued three-time temperature Green’s function Kc(z1, z2, z3) = β2 2 ∆(z1)∆(z2)∆(z3)K̃c(◦, ◦, ◦) + β∆(z1)∆(z2 + z3)K̃c(◦, z2, z3) + β∆(z2)∆(z3 + z1)K̃c(z1, ◦, z3) + β∆(z3)∆(z1 + z2)K̃c(z1, z2, ◦) + ∆(z1 + z2 + z3)K̃c(z1, z2, z3). (3.22) It includes five contributions which always can be distinguished by the different ∆ factors. The first contribution is expressed directly by the spectral densities of the (3.8) type K̃c(◦, ◦, ◦) = ĨABC(◦, ◦, ◦) + ĨCBA(◦, ◦, ◦), (3.23) the next three contributions are expressed in terms of the spectral densities of the (3.5)–(3.8) type K̃c(◦, z2,−z2) = +∞∫ −∞ dx2 ĨABC(◦, x2,−x2) − ĨCBA(−x2, x2, ◦) x2 − z2 − 1 z2 [ ĨABC(◦, ◦, ◦) − ĨCBA(◦, ◦, ◦) ] , (3.24) K̃c(−z3, ◦, z3) = +∞∫ −∞ dx3 ĨABC(−x3, ◦, x3)e βx3 − ĨCBA(x3, ◦,−x3)e −βx3 x3 − z3 + 1 z3 [ ĨABC(◦, ◦, ◦) − ĨCBA(◦, ◦, ◦) ] , (3.25) and K̃c(z1,−z1, ◦) = +∞∫ −∞ dx1 ĨABC(x1,−x1, ◦) − ĨCBA(◦,−x1, x1) x1 − z1 − 1 z1 [ ĨABC(◦, ◦, ◦) − ĨCBA(◦, ◦, ◦) ] , (3.26) 453 A.M.Shvaika and the last contribution is expressed in terms of the spectral densities of the (3.4)–(3.7) type K̃c(z1, z2, z3) = +∞∫ −∞ dx2 ĨABC(◦, x2,−x2)(1 − e−βx2) + ĨCBA(−x2, x2, ◦)(1 − eβx2) (z2 − x2)(z3 + x2) + +∞∫ −∞ dx3 ĨABC(−x3, ◦, x3)(e βx3 − 1) + ĨCBA(x3, ◦,−x3)(e −βx3 − 1) (z3 − x3)(z1 + x3) + +∞∫ −∞ dx1 ĨABC(x1,−x1, ◦)(1 − e−βx1) + ĨCBA(◦,−x1, x1)(1 − eβx1) (z1 − x1)(z2 + x1) − +∞∫ −∞ dx3 +∞∫ −∞ dx1 ĨABC(x1,−x3 − x1, x3) + ĨCBA(x3,−x3 − x1, x1) (x3 − z3)(x1 − z1) − +∞∫ −∞ dx1 +∞∫ −∞ dx2 ĨABC(x1, x2,−x1 − x2)e −βx1 + ĨCBA(−x1 − x2, x2, x1)e βx1 (x1 − z1)(x2 − z2) − +∞∫ −∞ dx2 +∞∫ −∞ dx3 ĨABC(−x2 − x3, x2, x3)e βx3 + ĨCBA(x3, x2,−x1 − x2)e −βx3 (x2 − z2)(x3 − z3) . (3.27) Here, we have used the cyclic permutation identities (3.9) according to which there are only two nonidentical three-time correlation functions, e.g. IABC(ω1, ω2, ω3) and ICBA(ω3, ω2, ω1). Equati- ons (3.22)–(3.27) give a complete representation of the three-time temperature Green functions in terms of the spectral densities. Now we pass to the solution of the reverse problem – finding the spectral densities from the known multitime temperature Green functions. According to equations (3.22)–(3.27) each of five contributions in the three-time Green’s function (3.22) has different set of branch cuts, which permits to extract all ten spectral densities that enter. First of all we perform an analytic continuation of equation (3.24) (z2 → ω2 ± iδ, z3 = −z2, z1 = 0): K̃c(◦, z2,−z2) ∣∣∣∣ 2 = −K̃c(◦,−z3, z3) ∣∣∣∣ 3 = δ(ω2)K0(◦, ◦, ◦) + K1(◦, ω2,−ω2), (3.28) where K0(◦, ◦, ◦) = ĨABC(◦, ◦, ◦) − ĨCBA(◦, ◦, ◦), (3.29) K1(◦, ω2,−ω2) = ĨABC(◦, ω2,−ω2) − ĨCBA(−ω2, ω2, ◦). (3.30) Next, we perform an analytic continuation of equation (3.25) (z3 → ω3 ± iδ, z1 = −z3, z2 = 0) K̃c(−z3, ◦, z3) ∣∣∣∣ 3 = −K̃c(z1, ◦,−z1) ∣∣∣∣ 1 = −δ(ω3)K0(◦, ◦, ◦) + K2(−ω3, ◦, ω3), (3.31) K2(−ω3, ◦, ω3) = ĨABC(−ω3, ◦, ω3)e βω3 − ĨCBA(ω3, ◦,−ω3)e −βω3 , (3.32) and of equation (3.26) (z1 → ω1 ± iδ, z2 = −z1, z3 = 0) K̃c(z1,−z1, ◦) ∣∣∣∣ 1 = −K̃c(−z2, z2, ◦) ∣∣∣∣ 2 = δ(ω1)K0(◦, ◦, ◦) + K3(ω1,−ω1, ◦), (3.33) K3(ω1,−ω1, ◦) = ĨABC(ω1,−ω1, ◦) − ĨCBA(◦,−ω1, ω1). (3.34) One can see, that in equations (3.28), (3.31), and (3.33) all factors at δ-functions are the same. 454 On the spectral relations for multitime correlation functions The next step is more complicated and requires a two-stage procedure. First of all we perform an analytic continuation of equation (3.27) over the first frequency z1 → ω1 ± iδ (z3 = −ω1 − z2) K̃c(z1, z2, z3) ∣∣∣∣ 1 = 1 z2 [ ĨABC(ω1, ◦,−ω1)(e −βω1 − 1) + ĨCBA(−ω1, ◦, ω1)(e βω1 − 1) ] + 1 z2 + ω1 [ ĨABC(ω1,−ω1, ◦)(e −βω1 − 1) + ĨCBA(◦,−ω1, ω1)(e βω1 − 1) ] + +∞∫ −∞ dx2 ĨABC(ω1, x2,−ω1 − x2)(e −βω1 − 1) + ĨCBA(−ω1 − x2, x2, ω1)(e βω1 − 1) z2 − x2 (3.35) and then an analytic continuation over the second one z2 → ω2 ± iδ (z3 → −ω1 − ω2 ∓ iδ) K̃c(z1, z2, z3) ∣∣∣∣ 1 ∣∣∣∣ 2 = −K̃c(z1, z2, z3) ∣∣∣∣ 1 ∣∣∣∣ 3 = δ(ω2)K (2) 1,2(−ω3, ◦, ω3) + δ(ω3)K (3) 1,2(ω1,−ω1, ◦) + K1,2(ω1, ω2, ω3), (3.36) where K (2) 1,2(−ω3, ◦, ω3) = −ĨABC(−ω3, ◦, ω3)(e βω3 − 1) − ĨCBA(ω3, ◦,−ω3)(e −βω3 − 1), (3.37) K (3) 1,2(ω1,−ω1, ◦) = −ĨABC(ω1,−ω1, ◦)(e −βω1 − 1) − ĨCBA(◦,−ω1, ω1)(e βω1 − 1), (3.38) K1,2(ω1, ω2, ω3) = −ĨABC(ω1, ω2, ω3)(e −βω1 − 1) − ĨCBA(ω3, ω2, ω1)(e βω1 − 1). (3.39) We can perform another sequence of the analytic continuations: z2 → ω2 ± iδ (z1 = −ω2 − z3) and z3 → ω3 ± iδ (z1 → −ω2 − ω3 ∓ iδ), that gives a different set of relations K̃c(z1, z2, z3) ∣∣∣∣ 2 ∣∣∣∣ 3 = −K̃c(z1, z2, z3) ∣∣∣∣ 2 ∣∣∣∣ 1 = δ(ω3)K (3) 2,3(ω1,−ω1, ◦) + δ(ω1)K (1) 2,3(◦, ω2,−ω2) + K2,3(ω1, ω2, ω3), (3.40) where K (3) 2,3(ω1,−ω1, ◦) = −ĨABC(ω1,−ω1, ◦)(1 − e−βω1) − ĨCBA(◦,−ω1, ω1)(1 − eβω1), (3.41) K (1) 2,3(◦, ω2,−ω2) = −ĨABC(◦, ω2,−ω2)(e −βω2 − 1) − ĨCBA(−ω2, ω2, ◦)(e βω2 − 1), (3.42) K2,3(ω1, ω2, ω3) = −ĨABC(ω1, ω2, ω3)(e βω3 − e−βω1) − ĨCBA(ω3, ω2, ω1)(e −βω3 − eβω1). (3.43) One can see that K̃c(z1, z2, z3) ∣∣∣∣ 1 ∣∣∣∣ 2 6= K̃c(z1, z2, z3) ∣∣∣∣ 2 ∣∣∣∣ 1 , (3.44) but there are common elements in final expressions, e.g. K (3) 1,2(ω1,−ω1, ◦) + K (3) 2,3(ω1,−ω1, ◦) = 0, (3.45) and the following Jacobi type identity is fullfilled K̃c(z1, z2, z3) ∣∣∣∣ 1 ∣∣∣∣ 2 + K̃c(z1, z2, z3) ∣∣∣∣ 2 ∣∣∣∣ 3 + K̃c(z1, z2, z3) ∣∣∣∣ 3 ∣∣∣∣ 1 = 0. (3.46) Due to this identity the procedure is unambiguous and the resulting set of equations for the spectral densities is not overdetermined. In general, the three-time temperature Green’s functions (3.22), as well as multitime Green functions of higher order, have very complicated analytic properties as a function of the complex 455 A.M.Shvaika frequencies zi, but they can be separated into contributions with different ∆-factors and different frequency dependences. Analytic properties of these contributions can be also complicated. Never- theless, we can always derive a complete set of equations for the multitime spectral densities from the multitime temperature Green functions by different sequences of the analytic continuations zi → ωi ± iδ accompanied by the constraint (3.19). Finally, we find from equations (3.23) and (3.29): ĨABC(◦, ◦, ◦) = 1 2 [ K̃c(◦, ◦, ◦) + K0(◦, ◦, ◦) ] , ĨCBA(◦, ◦, ◦) = 1 2 [ K̃c(◦, ◦, ◦) − K0(◦, ◦, ◦) ] , (3.47) from equations (3.30) and (3.42): ĨABC(◦, ω2,−ω2) = K1(◦, ω2,−ω2) 1 − e−βω2 + K (1) 2,3(◦, ω2,−ω2)e −βω2 (1 − e−βω2)2 , ĨCBA(−ω2, ω2, ◦) = K1(◦, ω2,−ω2) eβω2 − 1 + K (1) 2,3(◦, ω2,−ω2)e βω2 (eβω2 − 1)2 , (3.48) from equations (3.32) and (3.37): ĨABC(−ω3, ◦, ω3) = K2(−ω3, ◦, ω3) eβω3 − 1 + K (2) 1,2(−ω3, ◦, ω3) (eβω3 − 1)2 , ĨCBA(ω3, ◦,−ω3) = K2(−ω3, ◦, ω3) 1 − e−βω3 + K (2) 1,2(−ω3, ◦, ω3) (1 − e−βω3)2 , (3.49) from equations (3.34) and (3.38) or (3.41): ĨABC(ω1,−ω1, ◦) = K3(ω1,−ω1, ◦) 1 − e−βω1 − K (3) 1,2(ω1,−ω1, ◦)e −βω1 (1 − e−βω1)2 , ĨCBA(◦,−ω1, ω1) = K3(ω1,−ω1, ◦) eβω1 − 1 − K (3) 1,2(ω1,−ω1, ◦)e βω1 (eβω1 − 1)2 , (3.50) and from equations (3.39) and (3.43): ĨABC(ω1, ω2, ω3) = K2,3(ω1, ω2, ω3) (eβω2 − 1)(eβω3 − 1) − K1,2(ω1, ω2, ω3) (eβω3 − 1)(1 − e−βω1) , ĨCBA(ω3, ω2, ω1) = K2,3(ω1, ω2, ω3) (1 − e−βω2)(1 − e−βω3) − K1,2(ω1, ω2, ω3) (1 − e−βω3)(eβω1 − 1) , (3.51) that complete our main task and find spectral densities for multitime correlation functions from the known multitime temperature Green functions. 4. Summary In conclusion, we have presented a general approach to the derivation of the spectral rela- tions for the multitime correlation functions. An analysis of the frequency dependences of their spectral densities is performed with special attention paid to the consideration of the non-ergodic (conserving) contributions. It is shown that such contributions can be treated in a rigorous way using multitime temperature Green functions: representation of the Green functions in terms of the spectral densities and solution of the reverse problem, i.e., finding the spectral densities from the known Green functions are given for the case of the three-time bosonic correlation functions. Generalization for the case of the higher-order multitime correlation functions and for the functions constructed from the fermionic operators can be done in the same way, but this was not 456 On the spectral relations for multitime correlation functions considered here. Here we can only note that for the higher-order functions, besides non-ergodic terms, which appear at zero value of one frequency and are connected with the conserving of one operator, the one connected with the conserving of products of operators, which appear at zero value of the sum of corresponding frequencies, always exist and they will be the only “non- ergodic” contributions for the pure fermionic correlation functions resulting from the presence of the matrix elements like 〈f \|c†\|l〉〈l\|c\|f〉. Moreover, for some models, the fermionic correlation and Green functions will contain only such “non-ergodic” contributions, e.g. local four-time two- electron Green’s function for the Falicov-Kimball model [32] for which only contributions with zero sum of two frequencies exist. This publication is based on work supported by Award No. UKP2–2697–LV–06 of the U.S. Civilian Research & Development Foundation (CRDF). I am grateful to Professor I.V. Stasyuk for useful and stimulating discussions. References 1. Kubo R., J. Phys. Soc. Japan, 1957, 12, 570. 2. Bogolyubov N.N., Tyablikov S.V., Dokl. Acad. Nauk SSSR, 1959, 1, 53 [Soviet Phys. – Doklady, 1959, 4, 589]. 3. Zubarev D.N., Usp. Fiz. Nauk, 1960, 71, 71 [Soviet Phys. – Uspekhi, 1960, 3, 320]. 4. Bonch-Bruevich V.L., Tyablikov S.V. Metod Funktsiy Grina v Statisticheskoy Mekhanike, Moscow, Fizmatgiz, 1961 [The Green Function Method in Statistical Mechanics, Amsterdam, Noth-Holland Publishing Company, 1962]. 5. Stevens K.W.H., Toombs G.A., Proc. Phys. Soc. London, 1965, 85, 1307. 6. Suzuki M., Prog. Theor. Phys., 1970, 43, 882. 7. Suzuki M., Physica, 1971, 51, 277. 8. Wilcox R.M., Phys. Rev., 1968, 174, 624. 9. Fernandez J.F., Gersh H.A., Proc. Phys. Soc. London, 1967, 91, 505. 10. Callen H., Swendsen R.H., Tahir-Kheli R., Phys. Lett. A, 1967, 25, 505. 11. Lucas G.L., Horwitz G., J. Phys. A: General Phys., 1969, 2, 503. 12. Morita T., Katsura S., J. Phys. C: Solid State Phys., 1969, 2, 1030. 13. Kwok P.C., Schultz T.D., J. Phys. C: Solid State Phys., 1969, 2, 1196. 14. Ramos J.G., Gomes A.A., Nuovo Cimento A, 1971, 3, 441. 15. Huber D.L., Physica A, 1977, 87, 199. 16. Aksenov V.L., Konvent H., Schreiber J., phys. status solidi (b), 1978, 88, K43. 17. Aksenov V.L., Schreiber J., Phys. Lett. A, 1978, 69, 56. 18. Aksenov V.L., Bobeth M., Plakida N.M., Schreiber J., J. Phys. C: Solid State Phys., 1987, 20, 375. 19. Sarry M.F., Usp. Fiz. Nauk, 1991, 161, 47 [Soviet Phys. – Uspekhi, 1991, 34, 958]. 20. Fröbrich P., Kuntz P.J., Phys. Rev. B, 2003, 68, 014410. 21. Mancini F., Avella A., Eur. Phys. J. B, 2003, 36, 37. 22. Mancini F., Avella A., Adv. Phys., 2004, 53, 537. 23. Matsubara T., Prog. Theor. Phys., 1955, 14, 351. 24. Abrikosov A.A., Gorkov L.P., Dzyaloshinski I.E., Metody kvantovoy teorii polya v statisticheskoy fizike, Moscow, Fizmatgiz, 1962 [Methods of Quantum Field Theory in Statistical Physics, New York, Dover Publications Inc., 1975]. 25. Izyumov Yu.A., Kasan-Ogly F.A., Skryabin Yu.N., Polevye metody v teorii ferromagnetizma [Field Methods in the Theory of Ferromagnetism], Moscow, Nauka, 1974. 26. Baryakhtar V.G., Krivoruchko V.V., Yablonskii D.A., Funktsii Grina v teorii magnetizma [Green Functions in the Theory of Magnetism], Kyiv, Naukova Dumka, 1984. 27. Izyumov Yu.A., Skryabin Yu.N. Statisticheskaya mekhanika magnitouporyadochennykh sistem. Moskow, Nauka, 1987 [Statistical Mechanics of Magnetically Ordered Systems, New York, Consul- tants Bureau, 1989]. 28. Stasyuk I.V., Shvaika A.M., Condens. Matter Phys., 1994, 3, 134. 29. Falicov L.M., Kimball J.C., Phys. Rev. Lett., 1969, 22, 997. 30. Freericks J.K., Zlatić V., Rev. Mod. Phys., 2003, 75, 1333. 31. Shvaika A.M., Physica C, 2000, 341–348, 177. 32. Shvaika A.M., J. Phys. Studies, 2001, 5, 349. 33. Tyablikov S.V., Pu Fu-Cho, Fiz. Tverd. Tela, 1961, 3, 142 [Soviet Phys. – Solid State, 1961, 3, 102]. 457 A.M.Shvaika 34. Tanaka T., Moorjani K., Morita T., Phys. Rev., 1967, 155, 388. 35. Harper C., Phys. Rev. B, 1972, 5, 1613. 36. Shvets V.T., Teor. Mat. Fiz., 1996, 106, 438 [Theor. Math. Phys., 1996, 106, 359]. 37. Shvaika A.M., Vorobyov O., Freericks J.K., Devereaux T.P., Phys. Rev. Lett., 2004, 93, 137402. 38. Shvaika A.M., Vorobyov O., Freericks J.K., Devereaux T.P., Phys. Rev. B., 2005, 71, 045120. 39. Bonch-Bruevich V.L., Doklady Akad. Nauk SSSR, 1959, 129, 529 [Soviet Phys. – Doklady, 1959, 4, 1275]. Про спектральнi спiввiдношення для багаточасових кореляцiйних функцiй А.М.Швайка Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 Львiв Отримано 17 квiтня 2006 р. Запропоновано загальний пiдхiд до отримання спектральних спiввiдношень для багаточасових ко- реляцiйних функцiй. Особлива увага звертається на розгляд неергодичних (збережних) внескiв i показано, що такi внески можна послiдовно отримати використовуючи багаточасовi температурнi функцiї Ґрiна. Для випадку тричасових кореляцiйних функцiй знайдено представлення багаточасо- вих функцiй Ґрiна через спектральнi густини i розв’язано обернену задачу – вираження спектраль- них густин через вiдомi функцiї Ґрiна. Ключовi слова: багаточасовi кореляцiйнi функцiї, функцiї Ґрiна, спектральнi спiввiдношення, неергодичнiсть PACS: 05.30.-d 458

On the spectral relations for multitime correlation functions

Репозитарії

Схожі ресурси