Quantum effects in an anharmonic crystal

A model of quantum particles performing D -dimensional anharmonic oscillations around their equilibrium positions which form the d -dimensional simple cubic lattice Zd is considered. The model undergoes a structural phase transition when the fluctuations of displacements of particles become macr...

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Datum:	2002
1. Verfasser:	Kozitsky, Yu.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2002
Schriftenreihe:	Condensed Matter Physics
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spelling	irk-123456789-1206812017-06-13T03:03:43Z Quantum effects in an anharmonic crystal Kozitsky, Yu. A model of quantum particles performing D -dimensional anharmonic oscillations around their equilibrium positions which form the d -dimensional simple cubic lattice Zd is considered. The model undergoes a structural phase transition when the fluctuations of displacements of particles become macroscopic. This phenomenon is described by susceptibilities depending on Matsubara frequencies ωn , n ∈ Z . We prove two theorems concerning the thermodynamic limits of these susceptibilities. The first theorem states that the susceptibilities with nonzero ωn remain bounded at all temperatures, which means that the macroscopic fluctuations in the model are always non-quantum. The second theorem gives a sufficient condition for the static susceptibility (i.e. corresponding to ωn = 0 ) to be bounded at all temperatures. This condition involves the particle mass, the anharmonicity parameters and the interaction intensity. The physical meaning of this result is that, for all D and all values of the temperature, strong quantum effects suppress critical points and the long range order. The proof is performed in the approach where the susceptibilities are represented as functional integrals. A brief description of the main features of this approach is delivered. Розглядається модель квантових частинок, які виконують D -вимірні коливання довкола їх положень рівноваги, що утворюють d -вимірну просту кубічну ґратку Zd . Ця модель зазнає фазового переходу, коли флуктуації зміщень частинок стають макроскопічними. Таке явище описується сприйнятливостями, залежними від мацубарівських частот ωn , n є Z . Ми доводимо дві теореми, що описують термодинамічні властивості цих сприйнятливостей. Перша теорема стверджує, що сприйнятливості з ненульовими ωn залишаються обмеженими при всіх температурах, а це означає, що макроскопічні флуктуації в даній моделі є завжди неквантові. Друга теорема дає достатню умову на те, щоб і статична сприйнятливість (яка відповідає ωn = 0 ) теж була обмеженою при всіх температурах. Ця умова включає в себе масу частинки, параметри ангармонізму та інтенсивність взаємодії. Фізичний сенс цього результату полягає в тому, що для всіх D і для всіх значень температури сильні квантові ефекти унеможливлюють виникнення критичних точок і далекого порядку. Доведення проводиться в рамках підходу, у якому сприйнятливості представляються за допомогою функціональних інтегралів. Дається короткий опис головних аспектів цього підходу. 2002 Article Quantum effects in an anharmonic crystal / Yu. Kozitsky // Condensed Matter Physics. — 2002. — Т. 5, № 4(32). — С. 601-616. — Бібліогр.: 41 назв. — англ. 1607-324X PACS: 05.50.-d, 64.60.-i DOI:10.5488/CMP.5.4.601 http://dspace.nbuv.gov.ua/handle/123456789/120681 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 model of quantum particles performing D -dimensional anharmonic oscillations around their equilibrium positions which form the d -dimensional simple cubic lattice Zd is considered. The model undergoes a structural phase transition when the fluctuations of displacements of particles become macroscopic. This phenomenon is described by susceptibilities depending on Matsubara frequencies ωn , n ∈ Z . We prove two theorems concerning the thermodynamic limits of these susceptibilities. The first theorem states that the susceptibilities with nonzero ωn remain bounded at all temperatures, which means that the macroscopic fluctuations in the model are always non-quantum. The second theorem gives a sufficient condition for the static susceptibility (i.e. corresponding to ωn = 0 ) to be bounded at all temperatures. This condition involves the particle mass, the anharmonicity parameters and the interaction intensity. The physical meaning of this result is that, for all D and all values of the temperature, strong quantum effects suppress critical points and the long range order. The proof is performed in the approach where the susceptibilities are represented as functional integrals. A brief description of the main features of this approach is delivered.
format	Article
author	Kozitsky, Yu.
spellingShingle	Kozitsky, Yu. Quantum effects in an anharmonic crystal Condensed Matter Physics
author_facet	Kozitsky, Yu.
author_sort	Kozitsky, Yu.
title	Quantum effects in an anharmonic crystal
title_short	Quantum effects in an anharmonic crystal
title_full	Quantum effects in an anharmonic crystal
title_fullStr	Quantum effects in an anharmonic crystal
title_full_unstemmed	Quantum effects in an anharmonic crystal
title_sort	quantum effects in an anharmonic crystal
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2002
url	http://dspace.nbuv.gov.ua/handle/123456789/120681
citation_txt	Quantum effects in an anharmonic crystal / Yu. Kozitsky // Condensed Matter Physics. — 2002. — Т. 5, № 4(32). — С. 601-616. — Бібліогр.: 41 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT kozitskyyu quantumeffectsinananharmoniccrystal
first_indexed	2025-07-08T18:23:22Z
last_indexed	2025-07-08T18:23:22Z
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fulltext	Condensed Matter Physics, 2002, Vol. 5, No. 4(32), pp. 601–616 Quantum effects in an anharmonic crystal Yu.Kozitsky Institute of Mathematics, Maria Curie-Skłodowska University, 20-031 Lublin, Poland Received August 30, 2002 A model of quantum particles performing D -dimensional anharmonic os- cillations around their equilibrium positions which form the d -dimensional simple cubic lattice Z d is considered. The model undergoes a structural phase transition when the fluctuations of displacements of particles be- come macroscopic. This phenomenon is described by susceptibilities de- pending on Matsubara frequencies ωn , n ∈ Z . We prove two theorems concerning the thermodynamic limits of these susceptibilities. The first the- orem states that the susceptibilities with nonzero ωn remain bounded at all temperatures, which means that the macroscopic fluctuations in the model are always non-quantum. The second theorem gives a sufficient condition for the static susceptibility (i.e. corresponding to ωn = 0 ) to be bounded at all temperatures. This condition involves the particle mass, the anhar- monicity parameters and the interaction intensity. The physical meaning of this result is that, for all D and all values of the temperature, strong quan- tum effects suppress critical points and the long range order. The proof is performed in the approach where the susceptibilities are represented as functional integrals. A brief description of the main features of this approach is delivered. Key words: displacements, structural phase transition, critical point PACS: 05.50.-d, 64.60.-i 1. Introduction We consider the following model of anharmonic crystal. To each point l of the d-dimensional simple cubic lattice Zd there is attached a quantum particle of mass m performing isotropic D-dimensional anharmonic oscillations in the crystalline field. The Hamiltonian of our model has the form H = 1 2 ∑ l,l′∈Zd dll′(ql,ql′) + ∑ l∈Zd Hl , (1) Hl = 1 2m (pl,pl) + k 2 (ql,ql) + V ((ql,ql)), k > 0, c© Yu.Kozitsky 601 Yu.Kozitsky where (. . . , . . .) stands for the scalar product in RD and dll′ is a dynamical matrix of the model. We shall suppose that dll′ is: • invariant under translations on Zd; • ferroelectric, which means dll′ 6 0; • such that ∑ l∈Zd \|dll′\| <∞. These assumptions imply that there exists a function φ : R+ → R+ def = [0,+∞), such that dll′ = −φ(\|l − l′\|), (2) where \|l− l′\| = √ (l1 − l′1) 2 + . . .+ (ld − l′d) 2 is the Euclidean distance on the lattice. The anharmonicity potential V is supposed to be of the following form V (ξ) = aξ + b2ξ 2 + . . .+ bpξ p, a ∈ R, b2, . . . , bp−1 > 0, bp > 0. (3) The simplest case of such a potential is where p = 2 [18,27,34]. If the parameter a is negative and its absolute value is big enough, the equilibria at ql = 0, l ∈ Zd become unstable (a double-well profile for D = 1) and macroscopic displacements of particles from these unstable positions may occur. Such objects have been studied for many years as quite realistic models of crystalline substance undergoing structural phase transitions [14–16,26,27,34,38]. They, and similar models, are also employed as parts of the models describing strong electron-electron correlations caused by the interaction of electrons with vibrating ions [18,35,36]. Starting from the pioneering paper [31], many efforts were made to show that “the more quantum is the model, the less possible is the phase transition”. To confirm this physical common wisdom, different methods and approaches were applied. The first fully rigorous proof of the suppression of the long range order in models of this type with one-dimensional oscillations was done in [40]. This effect was also demonstrated on certain exactly solvable models of anharmonic crystals [29,39]. In [4] it was shown (also for the scalar case) that not only the long range or- der but any critical anomaly is suppressed if the model is strongly quantum. Later in [22–24] his result was obtained for the vector case with arbitrary D ∈ N. The present paper gives an extended and complete presentation of the results announced in the latter papers. In section 2 below we formulate two theorems. The physical meaning of the first theorem is that the large fluctuations of displacements of parti- cles, which appear at the critical point, are always non-quantum. This means that the only static susceptibility may diverge. The second theorem states that also the static susceptibility is bounded (i.e. non-quantum large fluctuations are suppressed as well) at all temperatures if the model is strongly quantum. This occurs if either the particles are light and/or the energy levels of the one-particle Hamiltonian of the corresponding scalar model (see below) are sufficiently separated from each other. Both these mechanisms are observed experimentally as isotopic effects (light par- ticles) [1,38] and external hydrostatic pressure effects (big separation) [13,37]. The proof of the second theorem is performed in the approach which employs functional 602 Quantum effects in an anharmonic crystal integrals, briefly described in section 3. This approach, initiated in [2], now is well elaborated in [3,8–11,20,25]. A complete description of this approach in application to the model considered in the present paper and an extended related bibliography are given in the review article [6]. In section 4 we discuss some consequences of our theorems. 2. The results The description of the Gibbs states of lattice models is performed with the help of local Gibbs states, which are constructed by means of local Hamiltonians (see [8,9,19,20]). For our purposes, it is enough to consider periodic states only. Let Λ stand for a finite subset of the lattice Zd, which we may choose as a cubic box of finite length. For such a box, we define the periodic dynamical matrix dΛ ll′ = −φ(\|l − l′\|Λ), (4) where the function φ is the same as in (2) and the periodic distance \|l− l′\|Λ is taken on the torus which one obtains identifying the opposite walls of the cube Λ. The local Hamiltonian HΛ corresponding to the periodic boundary conditions is constructed according to (1) and has the following form HΛ = ∑ l,l′∈Λ dΛ ll′ (ql,ql′) + ∑ l∈Λ Hl . (5) This Hamiltonian defines a density matrix %β,Λ = exp (−βHΛ) /Zβ,Λ , Zβ,Λ = trace exp (−βHΛ) , (6) where β−1 is the temperature. For the Hamiltonian (5), a general theory of the Schrödinger operators [12] implies that the latter trace exists. The local periodic Gibbs state γβ,Λ in the box Λ is defined as γβ,Λ(A) = trace {A%β,Λ} , (7) where A is an observable, which is a self-adjoint operator representing a certain physical quantity. The state γβ,Λ maps the set of such observables into the real numbers. The physical properties of the system in Λ are described by Matsubara functions, which, for the observables A1, . . . , An and τ1, . . . , τn ∈ [0, β], are defined as follows Γ β,Λ A1,...,An (τ1, . . . , τn) = γβ,Λ { Tτh Λ τ1(A1) . . . h Λ τn (An) } , (8) where hΛ τk (Ak) = exp (−τkHΛ)Ak exp (τkHΛ) , k = 1, 2, . . . , n. (9) is the Heisenberg representation of the operator Ak taken at the imaginary time tk = iτk and Tτ stands for the τ -ordering. By (8), each Matsubara function has the property Γ β,Λ A1,...,An (τ1 + θ, . . . , τn + θ) = Γ β,Λ A1,...,An (τ1, . . . , τn), θ ∈ [0, β], (10) 603 Yu.Kozitsky where addition is understood to be of modulo β. A significant role in the Euclidean approach is played by multiplication operators. Let \|Λ\| be the number of lattice points in the box Λ. By xΛ we denote the vector xΛ = (xl)l∈Λ, which has \|Λ\| com- ponents xl, each of which is a D-dimensional vector. For a wave function ψ = ψ(xΛ) and for a continuous complex valued function A(xΛ), the corresponding multiplica- tion operator, denoted also as A, acts (Aψ)(xΛ) = A(xΛ)ψ(xΛ). (11) Since such multiplication operators commute, the Matsubara functions constructed with them are continuous as functions of τ1, . . . , τn on the whole cube [0, β]n. In the model considered the phase transition is connected with the appearance of the macroscopic displacements of particles breaking O(D)-symmetry, thus the operator describing fluctuations of these displacements will play an important role in our consideration. This operator is Q (α) Λ = 1√ \|Λ\| ∑ l∈Λ q (α) l , α = 1, 2, . . . , D. (12) It is a multiplication operator. The Matsubara function (8) constructed with n = 2 and A1 = A2 = Q (α) Λ will be in the center of our study. To simplify notations we set G (α) β,Λ(τ, τ ′) def = Γ β,Λ Q (α) Λ ,Q (α) Λ (τ, τ ′), α = 1, 2, . . . , D. (13) By (10), this function depends on the distance \|τ−τ ′\|β def = min{\|τ−τ ′\|, β−\|τ−τ ′\|}, which makes it possible to write its Fourier transformation in the following form G (α) β,Λ(τ, τ ′) = 1 β +∞∑ n=−∞ eiωn(τ−τ ′)Ĝ (α) β,Λ(ωn), (14) Ĝ (α) β,Λ(ωn) = ∫ β 0 G (α) β,Λ(τ, τ ′)eiωn(τ−τ ′)dτ ′, where ωn = (2π/β)n, n ∈ Z are Matsubara’s frequencies. The Fourier coefficients Ĝ (α) β,Λ(ωn) have direct physical meaning – they are susceptibilities of the system in the box Λ at the temperature β−1. For every box Λ and every β, these quantities are well defined. Let {Λl}l∈N be a sequence of boxes such that, for every l ∈ N, Λl ⊂ Λl+1 and ⋃ l∈N Λl = Zd. In the sequel we deal with such sequences of boxes only, thus we shall assume that every sequence {Λl}l∈N has the above properties. The phase transition in the model considered may be detected as follows. If, for a given β and an arbitrary sequence of boxes {Λl}l∈N, all the sequences {Ĝ (α) β,Λl (ωn)}l∈N, n ∈ Z are bounded, then there is no symmetry breaking at this β. One can show 604 Quantum effects in an anharmonic crystal (see below) that all G (α) Λ (τ, τ ′) and all Ĝ (α) β,Λ(ωn) are nonnegative, thus by (14), one has Ĝ (α) β,Λ(ωn) 6 Ĝ (α) β,Λ(0), (15) for all ωn. This yields that all the sequences {Ĝ (α) β,Λl (ωn)}l∈N, n ∈ Z are bounded if {Ĝ (α) β,Λl (0)}l∈N is bounded. In this case one says that the fluctuations of displacements of particles are normal. The abnormal fluctuations appear at the critical point of the model or below it (with respect to the temperature, for more details see [39]), they occur if some of the above sequences of susceptibilities become unbounded (divergent). By (15), in all such cases the sequence {Ĝ (α) β,Λl (0)}l∈N should diverge. Such divergent sequences may be made bounded by multiplying by elements of an infinitely small sequence of numbers, e.g. by λl = \|Λl\| −δ/2, δ > 0. Suppose that, for a divergent sequence {Ĝ (α) β,Λl (ωn)}l∈N, n ∈ N, one finds δn > 0 and l0 ∈ N such that {\|Λl\| −δ/2Ĝ (α) β,Λl (ωn)}l>l0 ⊂ [a, b], (16) with certain 0 < a < b <∞. Then the exponent δn > 0 gives the rate of divergence of the sequence {Ĝ (α) β,Λl (ωn)}l∈N, and may be considered as a critical exponent. One can show that the largest value of all such exponents is δ = 1. If δ0 = 1, the spontaneous O(D)-symmetry breaking occurs at this β and the long range order appears. The smallest limiting point of the sequence (16) with ωn = 0 is treated as an order parameter. A detailed discussion of phase transitions, order parameters etc, in quantum spin models may be found in [17]. If, for a given β, (16) holds with n = 0 and a certain δ0 ∈ (0, 1), the fluctuations are not strong enough to destroy the symmetry but, being abnormal, they correspond to the critical point of the model and such β is the critical inverse temperature for this model. Thus, to prove that the model considered undergoes a phase transition, connected with the appearance of macroscopic displacements of particles, one has to show that (16) holds true (for sufficiently large β) with n = 0 and δ0 = 1. This was done, for models of the type of (1), in a number of papers [10,11,16,21,27]. Here one has to remark that, for the isotropic ferromagnetic Heisenberg model, the appearance of a long range order has not been proven so far. Another remark is that the existence of the long range order ((16) with ωn = 0 and δ0 = 1) does not mean the existence of the critical point ((16) with a certain ωn and δn ∈ (0, 1)). In this article we are going to show that, for the models considered: (a) (16) never holds for ωn 6= 0 (quantum fluctuations are always normal); (b) (16) does not hold for any β, if the model is “strongly quantum”. To make the latter condition more precise we have to introduce the scalar version of the model (1). Namely, let H̃ = 1 2 ∑ l,l′∈Zd dll′qlql′ + ∑ l∈Zd H̃l , (17) H̃l = 1 2 p2 l + k 2 q2 l + V (q2 l ), and the anharmonicity potential V is given by (3). The one-particle Hamiltonian H̃l has a pure point spectrum, each element of which εs, s ∈ N is its non-degenerate 605 Yu.Kozitsky eigenvalue (see e.g. [12]). We set ∆ = min{εs+1 − εs \| s ∈ N}. (18) One can expect that, for the double-well potentials, the above ∆ is just the frequency of the tunnelling motion between the wells. Since the model (17) in an evident way is connected with the model (1), the parameter ∆ determines the properties of the latter. In what follows, we will say that the model (1) is strongly quantum if the following condition is satisfied m∆2 > − ∑ l′∈Zd dll′ = ∑ l∈Zd φ(\|l\|) def = J. (19) Its physical interpretation could be as follows: the total force acting on a given quantum particle from all other particles is less than the energy of its quantum zero oscillations (we use the units with ~ = 1). Now we are at a position to formulate our results. Theorem 1 For the model considered, the sequences {Ĝ (α) β,Λl (ωn)}l∈N with ωn 6= 0 are bounded at all temperatures. Theorem 2 For the model considered, let the condition (19) be satisfied. Then, for all d,D ∈ N, for any β > 0, the sequence {Ĝ (α) β,Λl (0)}l∈N is bounded, which means that the model has no critical points and no symmetry breaking at all temperatures. The proof of the first theorem is quite simple and we give it right now. The proof of the second one will be performed in the framework of the Euclidean approach based on the representation of the Matsubara functions (8) as functional integrals. Proof of theorem 1. In fact we are going to prove that, for an arbitrary box Λ and for any β, 0 6 Ĝ (α) β,Λ(ωn) 6 1 mω2 n , ωn 6= 0, (20) where m is the particle mass. One can show that also the Hamiltonian (9) has a pure point spectrum, but this time the eigenvalues Es, s ∈ N are degenerate due to the O(D)-symmetry (and the corresponding parameter ∆ would be zero in this case). Let Ψs, s ∈ N be the corresponding eigenfunctions of HΛ. We set Qss′ = 〈Ψs\|Q (α) Λ \|Ψs′〉. (21) Then by (8), (12)–(14), one has Ĝ (α) β,Λ(ωn) = 1 Zβ,Λ ∑ s,s′∈N Q2 ss′ Es − Es′ (Es − Es′)2 + ω2 n [exp(−βEs′) − exp(−βEs)] . This immediately yields Ĝ (α) β,Λ(ωn) > 0. Further, one may estimate the denominator in the above expression (Es − Es′)2 + ω2 n > ω2 n > 0, 606 Quantum effects in an anharmonic crystal and obtain Ĝ (α) β,Λ(ωn) 6 1 ω2 nZβ,Λ ∑ s,s′∈N Q2 ss′ (Es − Es′) × [exp(−βEs′) − exp(−βEs)] = 1 ω2 n γβ,Λ {[ Q (α) Λ , [ HΛ, Q (α) Λ ]]} = 1 mω2 n , where [. . . , . . .] stands for commutator. 3. Euclidean approach The proof of our second theorem is not so elementary. To prove the boundedness stated we have to have instruments for estimating Matsubara functions (8). These instruments are provided by the Euclidean approach. Its basic element is the repre- sentation of the functions (8) constructed with multiplication operators as functional integrals. We begin the description of these integrals with introducing the sets of functions on which we define them. Let Wβ stand for the set of continuous functions w = w(τ) = (w(1)(τ), . . . , w(D)(τ)) defined on [0, β], taking values in RD and such that w(0) = w(β). These functions are called periodic paths. The set of all vectors wΛ = (wl)l∈Λ, each component of which is a periodic path at the point l ∈ Λ is denoted by Wβ,Λ. Now we have to construct the measure on Wβ,Λ which will define the functional integral. First we introduce the Gaussian measure dχβ which we shall use as a reference measure. This measure in determined by the first two terms in Hl (1) and describes a vector harmonic oscillator. Since our integration sets Wβ, Wβ,Λ are infinite dimensional vector spaces, on which Lebesgue’s measure dw does not exist, we cannot write our Gaussian measure in the way density × dw, as is usual for the finite dimensional case. Nevertheless, the Gaussian measure is determined by its Fourier transformation, which for our measure dχβ reads ∫ Wβ exp ( i ∫ β 0 (w(τ),x(τ))dτ ) dχβ(w) = = exp { − 1 2 ∫ β 0 ∫ β 0 (S(τ, τ ′)x(τ),x(τ ′))dτdτ ′ } , (22) where (S(τ, τ ′)x(τ),x(τ ′)) = D∑ α,α′=1 Sαα′ (τ, τ ′)x(α)(τ)x(α′)(τ ′), (23) Sαα′ (τ, τ ′) = δαα′Ω0 exp[Ω0(β − \|τ − τ ′\|)] + exp[Ω0\|τ − τ ′\|] exp(Ω0β) − 1 , (24) and Ω0 = √ k/m is the standard parameter of the harmonic oscillator. All proper- ties of the measure dχβ are known (see e.g. subsection 2.2 in [6]). By means of it 607 Yu.Kozitsky we construct the Gaussian measure which describes the system of non-interacting harmonic oscillators located in Λ: dχβ,Λ(wΛ) = ∏ l∈Λ dχβ(wl). (25) Now we take into account the anharmonicity potential and the interaction between the particles. It is performed using the Feynman-Kac formula, the detailed descrip- tion of which the reader may find in [30] and [32]. In what follows, we introduce the measure dνβ,Λ(wΛ) = 1 Zβ,Λ exp { − 1 2 ∑ l,l′∈Λ dΛ ll′ ∫ β 0 (wl(τ),wl′(τ))dτ − ∑ l∈Λ ∫ β 0 V ((wl(τ),wl(τ))dτ } dχβ,Λ(wΛ). (26) This measure is called a periodic local Euclidean Gibbs measure. As usual, the factor in front of the exp is the normalization constant, which provides that the normal- ization condition ∫ Wβ,Λ dνβ,Λ(w) = 1, (27) is satisfied. On the other hand, Zβ,Λ is the partition function of the system in the box Λ and has the representation following from (27) Zβ,Λ = ∫ Wβ,Λ exp { − 1 2 ∑ l,l′∈Λ dΛ ll′ ∫ β 0 (wl(τ),wl′(τ))dτ − ∑ l∈Λ ∫ β 0 V ((wl(τ),wl(τ))dτ } dχβ,Λ(wΛ). (28) Now we are at a position to write the main formula derived in [2]. It is Γ β,Λ A1,...,An (τ1, . . . , τn) = ∫ Wβ,Λ A1(wΛ(τ1)) · · ·A1(wΛ(τn))dνβ,Λ(wΛ), (29) where A1, . . . , An are the multiplication operators by the functions A1(xΛ), . . . , An(xΛ) respectively. This representation is employed to obtain information regard- ing mathematical properties of the Matsubara functions and their sequences. It is worth noting that representations of this kind may be used (and are used) as a base in the study of such models by means of computer simulations (see e.g. [33] and the references therein). Obviously, the same construction may be also performed for the scalar Hamilto- nian (17). This will yield the local Euclidean Gibbs measure dν̃β,Λ(wΛ) dν̃β,Λ(wΛ) = 1 Z̃β,Λ exp { − 1 2 ∑ l,l′∈Λ dΛ ll′ ∫ β 0 wl(τ)wl′(τ)dτ 608 Quantum effects in an anharmonic crystal − ∑ l∈Λ ∫ β 0 V (w2 l (τ))dτ } dχ̃β,Λ(wΛ), (30) where the Gaussian measure dχ̃β,Λ(wΛ) is defined by (25) with dχβ(wl) replaced by dχ̃β(wl). The latter measure is defined by (22)–(24) with α = α′ = 1. The corre- sponding space of scalar paths will be denoted W̃β,Λ. Accordingly, the Matsubara functions given by the integrals (29) with ν replaced by ν̃ will be written Γ̃ . Let us consider the set of continuous real valued functions F̃Λ defined on the set R\|Λ\| = {xΛ = (xl)l∈Λ \| xl ∈ R, l ∈ Λ} which have the following properties. A function A : R\|Λ\| → R belongs to F̃Λ if: (a) there exist l ∈ Λ such that A depends on xl only; (b) this dependence is either odd monotone or even positive and monotone on R+. The basic element of the proof of theorem 2 is the following assertion, proved in [23], see also [6], section 7. Proposition 3 [Scalar Domination] Let the continuous functions A1, . . . , An : RD\|Λ\| → R have the following property: there exist α ∈ {1, 2, . . . , D} and the func- tions Ã1, . . . , Ãn ∈ F̃Λ such that Aj(xΛ) = Ãj(x (α) Λ ), j = 1, 2, . . . , n. Then for arbi- trary D ∈ N and for all τ1, . . . , τn ∈ [0, β], 0 6 Γ β,Λ A1,...,An (τ1, . . . , τn) 6 Γ̃ β,Λ Ã1,...,Ãn (τ1, . . . , τn). (31) Let us stress that all Aj, j = 1, . . . , n above are supposed to depend on their x (α) Λ with one and the same α. As a corollary of this statement we have 0 6 G (α) β,Λ(τ, τ ′) 6 G̃β,Λ(τ, τ ′), (32) where the function G̃β,Λ(τ, τ ′) is defined by (13) but with Q (α) Λ (12) replaced by Q̃Λ = 1√ \|Λ\| ∑ l∈Λ ql , and with the Matsubara function Γ replaced by Γ̃ . Thus, in order to prove the boundedness of {G (α) β,Λl }l∈N it is enough to prove it for {G̃β,Λl }l∈N. To this end we introduce the following local Gibbs measure. For t ∈ [0, 1], we set dν̃ (t) β,Λ(wΛ) = 1 Z̃β,Λ(t) exp { − t 2 ∑ l,l′∈Λ dΛ ll′ ∫ β 0 wl(τ)wl′(τ)dτ − ∑ l∈Λ ∫ β 0 V (w2 l (τ))dτ } dχ̃β,Λ(wΛ), (33) where Z̃β,Λ(t) is the corresponding normalization constant defined by (27), which may be written similarly to (28). By means of this measure we define (see (29), (13), (12), (9)) G̃β,Λ(τ, τ ′\|t) = ∫ W̃β,Λ ( 1√ \|Λ\| ∑ l∈Λ wl(τ) )( 1√ \|Λ\| ∑ l∈Λ wl(τ ′) ) dν̃ (t) β,Λ(wΛ). (34) 609 Yu.Kozitsky Clearly G̃β,Λ(τ, τ ′\|1) = G̃β,Λ(τ, τ ′), (35) and (see (8) and (7)) G̃β,Λ(τ, τ ′\|0) = 1 \|Λ\| ∑ l∈Λ 1 Z̃β,l trace { ql exp ( −(τ − τ ′)H̃l ) × ql exp ( −(β − τ + τ ′)H̃l )} def = U(τ, τ ′). (36) Here we have taken into account that the traces in the latter formula do not depend on l due to the translation invariance of the one-particle Hamiltonian (17). We also set Rβ,Λ l1l2l3l4 (τ1, τ2, τ3, τ4\|t) = = ∫ W̃β,Λ wl1(τ1)wl2(τ2)wl3(τ3)wl4(τ4)dν̃ (t) β,Λ(wΛ) − ∫ W̃β,Λ wl1(τ1)wl2(τ2)dν̃ (t) β,Λ(wΛ) × ∫ W̃β,Λ wl3(τ3)wl4(τ4)dν̃ (t) β,Λ(wΛ) − ∫ W̃β,Λ wl1(τ1)wl3(τ3)dν̃ (t) β,Λ(wΛ) × ∫ W̃β,Λ wl2(τ)wl4(τ4)dν̃ (t) β,Λ(wΛ) − ∫ W̃β,Λ wl1(τ1)wl4(τ4)dν̃ (t) β,Λ(wΛ) × ∫ W̃β,Λ wl2(τ2)wl3(τ3)dν̃ (t) β,Λ(wΛ). (37) One can show that all G̃β,Λ and Rβ,Λ, as functions of t ∈ [0, 1], are differentiable on t ∈ (0, 1) and continuous on t ∈ [0, 1]. Furthermore, Rβ,Λ l1l2l3l4 (τ1, τ2, τ3, τ4\|t) 6 0, (38) for all values of its arguments. This estimate is known as Lebowitz’ inequality, its validity for the measure (30) was proved in section 6 of [6]. Differentiating both sides of (34) with respect to t after some calculations we obtain ∂ ∂t G̃β,Λ(τ1, τ2\|t) = − 1 2\|Λ\| ∑ l1,...,l4∈Λ dΛ l1l2 ∫ β 0 Rβ,Λ l1l2l3l4 (τ1, τ2, τ, τ \|t)dτ + ( − ∑ l∈Λ dΛ ll′ )∫ β 0 G̃β,Λ(τ1, τ \|t)G̃β,Λ(τ, τ2\|t)dτ. (39) Here we have taken into account the translation invariance of the moments of the measure (33), which is connected with the invariance of the dynamic matrix dΛ ll′ . The above formula may be considered as an integro-differential equation with respect to G̃β,Λ subject to the initial condition (36). By (15), which holds also for the scalar model, to prove the boundedness we need it to be sufficient to control the sequences of the following functions gβ,Λ(t) def = ∫ β 0 G̃β,Λ(τ1, τ2\|t)dτ2 = ∫ β 0 G̃β,Λ(0, τ \|t)dτ. (40) 610 Quantum effects in an anharmonic crystal For this function, we obtain from (39) the following equation ∂ ∂t gβ,Λ(t) = − 1 2\|Λ\| ∑ l1,...,l4∈Λ dΛ l1l2 ∫ β 0 ∫ β 0 Rβ,Λ l1,l2l3,l4 (τ1, τ2, τ, τ \|t)dτdτ1 + ( − ∑ l∈Λ dΛ ll′ ) [gβ,Λ(t)]2 , (41) subject to the initial condition (see (36)) gβ,Λ(0) = u def = ∫ β 0 U(0, τ)dτ. (42) Taking into account the assumed properties of the dynamical matrix dll′ one con- cludes that, for every box Λ, ( − ∑ l∈Λ dΛ ll′ ) 6 ( − ∑ l∈Zd dll′ ) = J > 0. (43) Let us consider the following differential equation dh(t) dt = J [h(t)]2 , t ∈ [0, 1], h(0) = u. (44) If the condition uJ < 1, (45) is satisfied, it has a unique solution on t ∈ [0, 1], which has the form h(t) = u 1 − tuJ . (46) Now let us compare the equations (44) and (41). By (38), the first term in (41) is nonpositive since all dΛ ll′ are nonpositive. Both our equations are subject to the same initial condition. Applying standard comparison methods of the theory of differential equations (see e.g. [41]) one obtains ∂ ∂t gβ,Λ(t) 6 dh(t) dt , for all t ∈ (0, 1). This immediately yields gβ,Λ(t) 6 h(t), for t ∈ [0, 1], in particular ∫ β 0 G̃β,Λ(0, τ)dτ = gβ,Λ(1) 6 h(1) = u 1 − uJ , (47) which holds for all boxes Λ. Here we have also taken into account (35). Proof of theorem 2. By (14), (32) and (47), the sequence {Ĝ (α) β,Λl (0)}l∈N is bounded provided (45) holds. Thus, to complete the proof one has to show that the 611 Yu.Kozitsky condition (19) does imply (45). To this end we employ the method already used in the proof of our first theorem. By (42), (36) and (18), one has u = 1 Z̃β ∫ β 0 trace { qle −τH̃lqle −(β−τ)H̃l } dτ = 1 Z̃β ∑ s,s′∈N [〈ψs\|ql\|ψs′〉]2 e−βεs − e−βεs′ εs − εs′ 6 1 ∆2 1 Z̃β ∑ s,s′∈N [〈ψs\|ql\|ψs′〉]2 [ e−βεs − e−βεs′ ] [εs − εs′ ] = 1 ∆2 · trace ([ ql, [ H̃l, ql ]] e−βH̃l ) trace e−βH̃l = 1 m∆2 . (48) Here ψs, s ∈ N are the eigenfunctions of H̃l corresponding to the eigenvalues εs. Above, in the sums ∑ s,s′ the terms with coinciding s, s′ give zero contribution be- cause of the corresponding matrix elements of ql vanishing. In deriving the above estimate we have taken into account that the eigenvalues εs are simple, which is not the case forD > 1. The latter fact gives the reason why we use our scalar domination arguments. Since the right-hand side in (47) is a monotone function of u, we apply there the latter estimate and obtain Ĝ (α) β,Λ(0) 6 ∫ β 0 G̃β,Λ(0, τ)dτ 6 1 m∆2 − J , which completes the proof. 4. Discussion First let us turn to the estimate (20). Since ωn = (2π/β)n, n ∈ Z, the series ∑ n∈Z\{0} Ĝ (α) β,Λ(ωn), (49) converges for any box Λ and for every β > 0. This fact has a deep mathematical consequence – by Ascoli’s theorem (see e.g. [28], p. 72), the sequences of Matsub- ara functions {Γ β,Λl A1,...,An }l∈N (for all multiplication operators A1, . . . , An, for which they make sense) converge uniformly on [0, β]n provided the sequence {Ĝ (α) β,Λl (0)}l∈N is bounded. This convergence of the Matsubara functions imply in turn that the limiting Gibbs measure, which describes the equilibrium state of the whole anhar- monic crystal, exists in this case. It is called periodic Euclidean Gibbs measure (see [9]). The boundedness of the sequence {Ĝ (α) β,Λl (0)}l∈N means the absence of abnor- mal fluctuations, it may be obtained by satisfying the condition (19), as follows from theorem 2. The above fact, however, does not imply that there exists the only 612 Quantum effects in an anharmonic crystal one limiting Gibbs measure (the latter would mean that there exists just one phase and no phase transitions may occur). In fact, the uniqueness of the limiting Gibbs measures may be proved based on the more sophisticated methods. Here one has to mention that such a uniqueness was proved in [5] to hold for D = 1 and for the val- ues of the particle mass from the interval (0, m∗(β)), where the bound m∗(β) tended to zero as β → +∞. In a very recent paper [7] it is stated that the uniqueness of the limiting Gibbs measures (again for D = 1 only) may be guaranteed by a condition similar to (19), which holds for m ∈ (0, m∗), where m∗ is independent of β. In the vector case D > 1, no uniqueness results have been obtained so far. Now, let us say some words about the condition (19). In the harmonic case V = 0, ∆ = Ω0, thus m∆2 = k and this condition means k > J . For the potential V given by (3), one can show that there exists C > 0 such that m∆2 ∼ Cm−(p−1)/(p+1) as m→ 0, which means that the condition (19) is satisfied for m ∈ (0, m∗), where the bound m∗ depends on the parameters J and a, b2, . . . , bp only. 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Quantum effects in an anharmonic crystal

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