Quantum anharmonic crystal in functional integral approach
A lattice model of interacting light quantum particles of mass m oscillating in a crystalline field is considered in the framework of an approach based on functional integrals. The main aspects of this approach are described on an introductory level. Then a mechanism of the stabilization of this...
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irk-123456789-1207592017-06-13T03:05:04Z Quantum anharmonic crystal in functional integral approach Kondratiev, Yu. Kozitsky, Yu. Pasurek, T. Ockner, M.R. A lattice model of interacting light quantum particles of mass m oscillating in a crystalline field is considered in the framework of an approach based on functional integrals. The main aspects of this approach are described on an introductory level. Then a mechanism of the stabilization of this model by quantum effects is suggested. In particular, a stability condition involving m , the interaction intensity, and the parameters of the crystalline field is given. It is independent of the temperature and is satisfied if m is small enough and/or the tunnelling frequency is big enough. It is shown that under this condition the infinite-volume correlation function decays exponentially; hence, no phase transitions can arise at all temperatures В рамках підходу, що базується на функціональних інтегралах, розглядається ґраткова модель взаємодіючих квантових частинок маси m , які осцилюють у кристалічному полі. Головні аспекти такого підходу описуються у спосіб доступний для неспеціалістів. У результаті запропоновано механізм стабілізації моделі, спричиненої квантовими ефектами. Зокрема, дається умова стабілізації, що включає в себе m , інтенсивність взаємодії та параметри кристалічного поля. Ця умова не залежить від температури, вона задовільняється, якщо m є достатньо малою та/або частота тунелювання є достатньо великою. Показано, що за цієї умови кореляційні функції, обчислені у термодинамічній границі, спадають експоненційно, що унеможливлює фазові переходи при всіх температурах. 2003 Article Quantum anharmonic crystal in functional integral approach / Yu. Kondratiev , Yu. Kozitsky , T. Pasurek , M.R. Ockner // Condensed Matter Physics. — 2003. — Т. 6, № 4(36). — С. 647-674. — Бібліогр.: 42 назв. — англ. 1607-324X PACS: 05.50.-d, 64.60.-i DOI:10.5488/CMP.6.4.647 http://dspace.nbuv.gov.ua/handle/123456789/120759 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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| description |
A lattice model of interacting light quantum particles of mass m oscillating
in a crystalline field is considered in the framework of an approach based
on functional integrals. The main aspects of this approach are described on
an introductory level. Then a mechanism of the stabilization of this model
by quantum effects is suggested. In particular, a stability condition involving
m , the interaction intensity, and the parameters of the crystalline field
is given. It is independent of the temperature and is satisfied if m is small
enough and/or the tunnelling frequency is big enough. It is shown that under
this condition the infinite-volume correlation function decays exponentially;
hence, no phase transitions can arise at all temperatures |
| format |
Article |
| author |
Kondratiev, Yu. Kozitsky, Yu. Pasurek, T. Ockner, M.R. |
| spellingShingle |
Kondratiev, Yu. Kozitsky, Yu. Pasurek, T. Ockner, M.R. Quantum anharmonic crystal in functional integral approach Condensed Matter Physics |
| author_facet |
Kondratiev, Yu. Kozitsky, Yu. Pasurek, T. Ockner, M.R. |
| author_sort |
Kondratiev, Yu. |
| title |
Quantum anharmonic crystal in functional integral approach |
| title_short |
Quantum anharmonic crystal in functional integral approach |
| title_full |
Quantum anharmonic crystal in functional integral approach |
| title_fullStr |
Quantum anharmonic crystal in functional integral approach |
| title_full_unstemmed |
Quantum anharmonic crystal in functional integral approach |
| title_sort |
quantum anharmonic crystal in functional integral approach |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120759 |
| citation_txt |
Quantum anharmonic crystal in
functional integral approach / Yu. Kondratiev , Yu. Kozitsky , T. Pasurek , M.R. Ockner // Condensed Matter Physics. — 2003. — Т. 6, № 4(36). — С. 647-674. — Бібліогр.: 42 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT kondratievyu quantumanharmoniccrystalinfunctionalintegralapproach AT kozitskyyu quantumanharmoniccrystalinfunctionalintegralapproach AT pasurekt quantumanharmoniccrystalinfunctionalintegralapproach AT ocknermr quantumanharmoniccrystalinfunctionalintegralapproach |
| first_indexed |
2025-07-08T18:31:57Z |
| last_indexed |
2025-07-08T18:31:57Z |
| _version_ |
1837104646659768320 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 4(36), pp. 647–674
Quantum anharmonic crystal in
functional integral approach
Yu.Kondratiev 1,3,4 , Yu.Kozitsky 2 , T.Pasurek 1,3 , M.Röckner 1,3
1 Fakultät für Mathematik, Universität Bielefeld, D 33615 Bielefeld, Germany
2 Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej w Lublinie,
PL 20-031 Lublin, Poland
3 Forschungszentrum BiBoS, Bielefeld, Germany
4 Institute of Mathematics, Kiev, Ukraine
Received October 15, 2003
A lattice model of interacting light quantum particles of mass m oscillating
in a crystalline field is considered in the framework of an approach based
on functional integrals. The main aspects of this approach are described on
an introductory level. Then a mechanism of the stabilization of this model
by quantum effects is suggested. In particular, a stability condition involv-
ing m , the interaction intensity, and the parameters of the crystalline field
is given. It is independent of the temperature and is satisfied if m is small
enough and/or the tunnelling frequency is big enough. It is shown that un-
der this condition the infinite-volume correlation function decays exponen-
tially; hence, no phase transitions can arise at all temperatures.
Key words: quantum stabilization, Gibbs states, soft mode
PACS: 05.50.-d, 64.60.-i
1. Introduction
Structural phase transitions in ionic crystals, connected with macroscopic dis-
placements of ionic complexes, may be triggered by the ordering of interacting local-
ized quantum particles. A typical example here could be the ferroelectric phase tran-
sition in KDP-type compounds, in which the quantum particles are the protons os-
cillating along hydrogen bonds [1]. The simplest way of describing this phenomenon
is to consider the subsystem of quantum particles moving in a static external field
generated by heavy ions. Then the motion of the latter objects is neglected (the
so-called adiabatic approximation) and each quantum particle oscillates around its
equilibrium position determined by the external field. These positions are supposed
to form a crystalline lattice L. The motion of the particle with the equilibrium posi-
tion at a given ` ∈ L is described by the displacement operator q` and the momentum
c© Yu.Kondratiev, Yu.Kozitsky, T.Pasurek, M.Röckner 647
Yu.Kondratiev et al.
operator p`, which satisfy the commutation relation
[p`, q`] = p`q` − q`p` = 1/i. (1.1)
The quantum particles carry an electric charge, hence, their displacements cre-
ate dipole moments and the inter-particle interaction potential should contain the
dipole-dipole component J``′q`q`′ . The Hamiltonian of the system of such particles
which takes into account the dipole-dipole interaction only may be written in the
form
H = −1
2
∑
`,`′
J``′q`q`′ +
∑
`
H` , (1.2)
where the sums run through the lattice L and
H` =
1
2m
p2
` +
a
2
q2
` + V (q`), m, a > 0, (1.3)
is the one-particle Hamiltonian describing the particle with the equilibrium position
at q` = 0. The third term in (1.3), in which V : R → R, gives an anharmonic
correction to the potential energy. In order for the particle to possess equilibrium
at q` = 0 and to be localized in the vicinity of this point, the potential energy
(a/2)ξ2 + V (ξ) ought to grow to +∞ as |ξ| → +∞, to have a local extremum at
ξ = 0 and to be below bounded, i.e., to obey
(a/2)ξ2 + V (ξ) > v0, for all ξ ∈ R, (1.4)
with a certain real constant v0. On the other hand, the system undergoes a structural
phase transition if and only if the potential energy in (1.3) has at least two minima
along with the maximum at q` = 0. If V ≡ 0, the particle is a harmonic oscillator
of rigidity a and reduced mass m = mphys/~2, where mphys is the physical mass of
the particle. We have included the Planck constant ~ into m in order to have the
commutation relation in the form of (1.1). If the function V is even, the oscillations
are symmetric with respect to the origin and the phase transition in the system of
such particles breaks this symmetry. A typical example of V is
V (ξ) = b1ξ
2 + b2ξ
4 + · · ·+ brξ
2r, r ∈ N, r > 2. (1.5)
The most known case r = 2 has been studied for many years, see e.g., [2,3]. A
detailed presentation of arguments in favor of the model (1.2) with such V may be
found in [4]. The model (1.1) with V in the form (1.5) with r = 2 was also used as
a base of the model proposed in [5] to describe strong electron-electron correlations
caused by the interaction of electrons with vibrating ions. The model (1.2) with a
nonzero V is called quantum anharmonic crystal.
The operators p`, q` act in the physical Hilbert space H` = L2(R) of square
integrable functions with respect to Lebesgue’s measure on the real line R. These are
wave functions describing the states of the particle oscillating around the equilibrium
position at q` = 0. Unlike the case of spin models (see e.g., [6,7]), the space H`
648
Quantum anharmonic crystal
is infinite dimensional, which produces serious complications in the mathematical
study of the model (1.1). In particular, the notion of the Gibbs state based on
the so-called Kubo-Martin-Schwinger condition (see below), commonly used in the
theory of quantum spin models, cannot be generalized to this case. Methods based on
functional integrals have proved to be very effective in various problems of quantum
theory (see e.g., [8–10]). A method of constructing Gibbs states of models like (1.1),
based on functional integrals, was initiated in [11,12]. Now it is well elaborated,
see [13–27]. The aim of this paper is to give a brief introduction into this method,
accessible also to physicists. As an application, we give a detailed description of
quantum effects in the model (1.1) based on the results published in [19,21–24,28].
2. Gibbs states
2.1. Basic notions
For simplicity reasons, we consider the model (1.1) on a d-dimensional simple cu-
bic lattice of unit spacing, i.e., L = Z
d. In this case the lattice sites are d-dimensional
integer vectors ` = (`1, . . . , `d) with `j ∈ Z, j = 1, . . . , d. Regarding the interaction
potential J``′ we suppose that it is:
• invariant under translations on L;
• ferroelectric, i.e., J``′ > 0 and J`` = 0;
• sufficiently rapidly decaying, so that
∑
`′∈L
J``′ <∞.
The Hamiltonian (1.2) has no direct mathematical meaning. Usually, it is “represent-
ed” by local Hamiltonians – indexed by finite subsets Λ ⊂ L essentially self-adjoint
operators (see below) HΛ acting in the complex Hilbert space HΛ = L2(R|Λ|), where
|Λ| stands for the number of lattice sites in Λ. Elements of this space are wave func-
tions ψ(xΛ), where xΛ = (x`)`∈Λ is the |Λ|-dimensional vector, which components
x` ∈ R are numbered by the lattice sites ` belonging to Λ. The scalar product in HΛ
is defined as follows:
〈ψ|φ〉HΛ
= 〈φ|ψ〉HΛ
=
∫
R|Λ|
ψ(xΛ)φ(xΛ)
∏
`∈Λ
dx`,
where z means complex conjugation. Suppose that a linear operator Q : HΛ → HΛ
acts on a given φ ∈ HΛ. We write the scalar product of Qφ and a given ψ ∈ HΛ in
the form
〈ψ|Qφ〉HΛ
= 〈ψ|Q|φ〉HΛ
, 〈Qψ|φ〉HΛ
= 〈φ|Q|ψ〉HΛ
.
In infinite dimensional spaces linear operators need not be defined on the whole
space. Given a linear operator Q, by D(Q) we denote its domain, i.e., the set of
functions ψ ∈ HΛ on which the action of Q is defined. A linear operator Q : HΛ →
HΛ is called bounded if there exists a constant C > 0 such that
〈ψ|Q|ψ〉HΛ
6 C〈ψ|ψ〉HΛ
, (2.1)
649
Yu.Kondratiev et al.
for all ψ ∈ D(Q). The least such C characterizes the operator – it is its norm ‖Q‖.
We suppose that, for every bounded operator, D(Q) = HΛ. The operators which
fail to obey (2.1) are called unbounded. Each unbounded operator considered in this
article is densely defined, i.e., the closure of its domain coincides with HΛ. For a
bounded operator Q, its adjoint operator Q∗ : HΛ → HΛ is defined by the relation
〈ψ|Q|φ〉HΛ
= 〈φ|Q∗|ψ〉HΛ
,
which has to hold for all ψ, φ ∈ HΛ. It is also a bounded operator and ‖Q‖ = ‖Q∗‖.
A bounded operator Q is called self-adjoint if Q∗ = Q. For unbounded operators,
the definition of self-adjointness is a bit more complicated. Given Q,Q′, suppose
D(Q) ⊂ D(Q′) and, for every ψ ∈ D(Q), one has Qψ = Q′ψ. Then the operator Q′
is called an extension of Q to the set D(Q′). Given a densely defined operator Q,
the operator S : HΛ → HΛ with the maximal domain obeying the condition
〈ψ|Q|φ〉HΛ
= 〈φ|S|ψ〉HΛ
, for all φ ∈ D(Q) and ψ ∈ D(S),
is called adjoined to Q, i.e., S = Q∗. The operator Q is called symmetric if Q∗ is
its extension; it is called self-adjoint if Q = Q∗. Finally, such Q is called essentially
self-adjoined if it possesses a unique self-adjoint extension.
By CΛ we denote the set of all bounded operators Q : HΛ → HΛ. This set is
equipped with the above introduced norm ‖ ·‖, with the addition and multiplication
by complex numbers and with the multiplication QQ′ such that
‖QQ′‖ 6 ‖Q‖‖Q′‖, ‖Q‖ = ‖Q∗‖.
Then CΛ is a C∗-algebra (see [29] for more details). Its unity is the identity operator
I which maps each ψ ∈ HΛ into itself. A self-adjoint operator Q ∈ CΛ is called
positive if,
〈ψ|Q|ψ〉HΛ
> 0, for every ψ ∈ HΛ.
Let ω : CΛ → C be a bounded linear functional on the algebra CΛ, i.e., such that
there exists a constant ω0 > 0 for which and for any Q ∈ CΛ,
|ω(Q)| 6 ω0‖Q‖.
A linear functional ω : CΛ → C is called a state on CΛ if: (a) it is normalized, i.e.,
ω(I) = 1; (b) it is positive, i.e., ω(Q) > 0 for any positive Q ∈ CΛ.
Each base of HΛ – a complete orthonormal system of wave functions – should
be an infinite countable set. Let {ψn}n∈N be such a base. For a linear operator
T : HΛ → HΛ, suppose that {ψn}n∈N is contained in its domain. Set
trace(T ) =
∞
∑
n=1
〈ψn|T |ψn〉HΛ
. (2.2)
The above series need not be convergent. If it is absolutely convergent, the operator
T is bounded, hence D(T ) = HΛ, and (2.2) has the same value for any orthonormal
650
Quantum anharmonic crystal
base of HΛ. In this case we say that T is of trace class and the number trace(T ) is
called its trace. In the case of spin models the Hilbert spaces are finite dimensional
and all linear operators are of trace class. For any bounded operator Q and any
trace class operator T , the operators QT and TQ are of trace class and
|trace(QT )| 6 ‖Q‖|trace(T )|, |trace(TQ)| 6 ‖Q‖|trace(T )|. (2.3)
The latter estimate follows from the former one and from the cyclicity
trace[ABC] = trace[BCA] = trace[CAB]. (2.4)
For a finite subset Λ ⊂ L, the local Hamiltonian corresponding to the zero
boundary conditions outside of Λ is set as follows:
HΛ(0) = −1
2
∑
`,`′∈Λ
J``′q`q`′ +
∑
`∈Λ
H`, (2.5)
where J``′ and H` are the same as in (1.2), (1.3). A standard fact from the theory
of such operators (see e.g., [30]) is that, for any β > 0, the operator exp(−βHΛ(0))
is of trace class. At the same time, the operators q`, p`, as well as the Hamiltonians
H`, HΛ(0), defined in the Hilbert space HΛ, are unbounded. Usually, such opera-
tors are defined on the sets of differentiable functions with a sufficiently fast falloff
(Schwartz spaces of test functions, see e.g., [31]). Then one proves that they are
essentially self-adjoint.
In a standard situation, it is enough to consider local Hamiltonians indexed by
boxes. Given L ∈ N, the box ΛL is
ΛL = (−L, L]d
⋂
L. (2.6)
We shall often drop the subscript L assuming that a given Λ is of the form (2.6)
with some L ∈ N. The assumed properties of J``′ imply that there exists a function
Φ : R+ → R+
def
= [0,+∞) such that
J``′ = Φ(|`− `′|), (2.7)
where |`−`′| =
√
|`1 − `′1|2 + · · · + |`d − `′d|2 is the Euclidean distance on the lattice
L. Given ΛL, we set
|`− `′|ΛL
=
√
|`1 − `′1|2L + · · ·+ |`d − `′d|2L, `, `′ ∈ ΛL, (2.8)
where |`j − `′j|L
def
= min{|`j − `′j|, 2L− |`j − `′j|} is the distance on the torus which
one obtains by identifying the opposite walls of the box ΛL. It is invariant under
translations on this torus. For a box Λ (2.6) and `, `′ ∈ Λ, we set
JΛ
``′ = Φ(|`− `′|Λ), (2.9)
651
Yu.Kondratiev et al.
where Φ is the same as in (2.7). Then the potential JΛ
``′ is invariant under translations
on the torus. The local Hamiltonian corresponding to the periodic conditions on the
boundaries of Λ is
HΛ(p) = −1
2
∑
`,`′∈Λ
JΛ
``′q`q`′ +
∑
`∈Λ
H`. (2.10)
As above, the Hamiltonian HΛ(p) is defined on the set of test functions, it is essen-
tially self-adjoint. The operator exp(−βHΛ(p)) is of trace class for any β > 0, which
allows us to set
Zβ,Λ(0) = trace[exp(−βHΛ(0))], Zβ,Λ(p) = trace[exp(−βHΛ(p))]. (2.11)
These are the local partition functions corresponding to the zero and periodic bound-
ary conditions respectively. Furthermore, we set
Rβ,Λ(0) = exp(−βHΛ(0))/Zβ,Λ(0), Rβ,Λ(p) = exp(−βHΛ(p))/Zβ,Λ(p). (2.12)
Both these operators are of trace class and
trace(Rβ,Λ(0)) = trace(Rβ,Λ(p)) = 1. (2.13)
They are density matrices corresponding to the zero and periodic boundary condi-
tions respectively. The local Gibbs states of our model, corresponding to the zero
and periodic boundary conditions, appear to be the following linear functionals on
the algebra CΛ
%
(0)
β,Λ(Q) = trace[QRβ,Λ(0)], %
(p)
β,Λ(Q) = trace[QRβ,Λ(p)], Q ∈ CΛ. (2.14)
By (2.13), they are normalized; by (2.3) and (2.13), they are bounded with ω0 = 1
in both cases. The elements of CΛ are called local observables and CΛ itself is called
the algebra of local observables. Here local means related with a finite subset of the
lattice L. Thermodynamic quantities are obtained in the limit Λ → L, which can be
reached by letting L→ +∞ (see (2.6)).
In the Heisenberg approach to quantum mechanics, the dynamics of the system
described by the local Hamiltonians (2.5), (2.10) is generated by the following time
maps acting on the algebra of observables CΛ
a
t
0,Λ(Q) = U t
0,ΛQ[U t
0,Λ]−1, a
t
p,Λ(Q) = U t
p,ΛQ[U t
p,Λ]−1, Q ∈ CΛ, (2.15)
where t ∈ R is time and
U t
0,Λ = exp(itHΛ(0)), U t
p,Λ = exp(itHΛ(p)). (2.16)
Sometimes, (2.15) are called Heisenberg representations ofQ. Since the Hamiltonians
HΛ(0), HΛ(p) are essentially self-adjoint, all U t
0,Λ, U t
p,Λ, t ∈ R are unitary operators
on the Hilbert space HΛ. Therefore, the time maps (2.15) are norm-preserving
‖at
0,Λ(Q)‖ = ‖Q‖, ‖at
p,Λ(Q)‖ = ‖Q‖, Q ∈ CΛ, (2.17)
652
Quantum anharmonic crystal
and for any t, s ∈ R,
a
t
0,Λ
(
a
s
0,Λ(Q)
)
= a
t+s
0,Λ (Q), a
t
p,Λ
(
a
s
p,Λ(Q)
)
= a
t+s
p,Λ (Q), Q ∈ CΛ.
By (2.15), (2.16), a0
0,Λ(Q) = a0
p,Λ(Q) = Q, which means that such maps consti-
tute one-parameter groups. These groups generate unitary time evolutions of the
corresponding algebras of observables. Given Q1, . . . , Qn ∈ CΛ, n ∈ N, we set
G0,β,Λ
Q1,...,Qn
(t1, . . . , tn) = %
(0)
β,Λ
{
a
t1
0,Λ(Q1) · · ·atn
0,Λ(Qn)
}
, t1, . . . , tn ∈ R, (2.18)
Gp,β,Λ
Q1,...,Qn
(t1, . . . , tn) = %
(p)
β,Λ
{
a
t1
p,Λ(Q1) · · · atn
p,Λ(Qn)
}
, t1, . . . , tn ∈ R. (2.19)
One can consider the left-hand sides of (2.18), (2.19) as functions of real variables
t1, . . . , tn. They are called Green functions corresponding to the operators (observ-
ables) Q1, . . . , Qn and to the zero and periodic boundary conditions, respectively.
The analytic properties of these functions play a crucial role in our theory. They are
described by the following statement which was proved in [22], see Lemma 2.1. For
a complex number z = x + iy, we write x = <(z), y = =(z). Set
Oβ
n = {(t1, . . . , tn) ∈ C
n | 0 < =(t1) < · · · < =(tn) < β}. (2.20)
Proposition 1 For every Q1, . . . , Qn ∈ CΛ,
(a) G0,β,Λ
Q1,...,Qn
can be extended to a function analytic on Oβ
n;
(b) this extension (which will also be denoted by G0,β,Λ
Q1,...,Qn
) is
continuous on the closure Oβ
n of Oβ
n; moreover, for all
(t1, . . . , tn) ∈ Oβ
n,
|G0,β,Λ
Q1,...,Qn
(t1, . . . , tn)| 6 ‖Q1‖ · · · ‖Qn‖; (2.21)
(c) for every ξ1, . . . , ξn ∈ R, the set
Oβ
n(ξ1, . . . , ξn)
def
= {(t1, . . . , tn) ∈ Oβ
n | <(tj) = ξj, j = 1, . . . , n}, (2.22)
is such that, for any two functions F,G analytic on Oβ
n,
the equality F = G for all (t1, . . . , tn) ∈ Oβ
n(ξ1, . . . , ξn)
implies that these functions are equal on the whole Oβ
n.
The functions (2.19) possess the same properties.
The property described by claim (c) may be also interpreted as follows: every func-
tion, analytic on Oβ
n, can be uniquely determined by the values which it takes at
fixed real parts of its arguments. Let us set them to be zero and consider
Γ 0,β,Λ
Q1,...,Qn
(τ1, . . . , τn) = G0,β,Λ
Q1,...,Qn
(iτ1, . . . , iτn), (2.23)
Γ p,β,Λ
Q1,...,Qn
(τ1, . . . , τn) = Gp,β,Λ
Q1,...,Qn
(iτ1, . . . , iτn), (2.24)
where we suppose that (c.f., (2.20))
0 6 τ1 6 · · · 6 τn 6 β. (2.25)
653
Yu.Kondratiev et al.
The left-hand sides of (2.23), (2.24) are called Matsubara functions corresponding
to the observables Q1, . . . , Qn and to the zero and periodic boundary conditions
respectively. By claim (b), the Matsubara functions are continuous on their do-
mains; by claim (c), they completely determine the corresponding Green functions
(2.18), (2.19). In what follows, the Green functions possess analytic continuations
to complex values of time and their values at imaginary time determine uniquely
the values at real time. The latter fact also takes place in quantum field theory
where the points of the Minkowski space corresponding to imaginary time may be
interpreted as points of a Euclidean space. The approach which exploits this fact is
known as Euclidean quantum field theory.
The Matsubara functions possess one more property implied by Proposition 1.
Given τ, τ ′ ∈ [0, β], we set
τ
◦
+ τ ′ =
{
τ + τ ′, τ + τ ′ 6 β,
τ + τ ′ − β, τ + τ ′ > β.
(2.26)
This operation is called addition of modulo β. It may be interpreted as addition on
the circle of length β obtained by identifying the endpoints of [0, β]. For t = θ + iτ ,
θ ∈ R, τ ∈ [0, β] and some τ ′ ∈ [0, β], we write t
◦
+ iτ ′ meaning θ + i(τ
◦
+ τ ′).
Proposition 2 The Green functions possess the following property
G0,β,Λ
Q1,...,Qn
(t1, . . . , tn) = G0,β,Λ
Q1,...,Qn
(t1
◦
+ iβ, . . . , tn
◦
+ iβ),
Gp,β,Λ
Q1,...,Qn
(t1, . . . , tn) = Gp,β,Λ
Q1,...,Qn
(t1
◦
+ iβ, . . . , tn
◦
+ iβ). (2.27)
In particular,
Γ 0,β,Λ
Q1,...,Qn
(τ1, . . . , τn) = Γ 0,β,Λ
Q1,...,Qn
(τ1
◦
+ β, . . . , τn
◦
+ β),
Γ p,β,Λ
Q1,...,Qn
(τ1, . . . , τn) = Γ p,β,Λ
Q1,...,Qn
(τ1
◦
+ β, . . . , τn
◦
+ β). (2.28)
2.2. KMS condition and Høegh-Krohn theorem
Let us consider the property (2.27) in the case of n = 2 and the zero boundary
conditions. For any A,B ∈ CΛ and any t ∈ R, one has
G0,β,Λ
A,B (0, t
◦
+ iβ) = trace {A exp(itHΛ(0) − βHΛ(0))B
× exp(−itHΛ(0) + βHΛ(0)) exp(−βHΛ(0))} /Zβ,Λ(0)
= trace {A exp(−βHΛ(0)) exp(itHΛ(0))B exp(−itHΛ(0)} /Zβ,Λ(0)
= trace {exp(itHΛ(0))B exp(−itHΛ(0)A exp(−βHΛ(0))} /Zβ,Λ(0)
= %
(0)
β,Λ(at
0,Λ(B)A) = %
(0)
β,Λ(at
0,Λ(B)a0
0,Λ(A))
= G0,β,Λ
B,A (t, 0).
Here we have used (2.4) and (2.14)–(2.17). The above computations may be repeated
for the Green functions corresponding to the periodic boundary conditions. In what
follows, Proposition 1 and the above computations yield the next statement.
654
Quantum anharmonic crystal
Proposition 3 The local Gibbs states (2.14) possess the following property. For
any A,B ∈ CΛ, there exist functions F
(0)
A,B(z), F
(p)
A,B(z), analytic in the open strip
{t+iτ | t ∈ R, τ ∈ (0, β)} and continuous on its closure, which satisfy the conditions
F
(0)
A,B(t) = %
(0)
β,Λ(Aa
t
0,Λ(B)), F
(0)
A,B(t
◦
+ iβ) = %
(0)
β,Λ(at
0,Λ(B)A),
F
(p)
A,B(t) = %
(p)
β,Λ(Aa
t
p,Λ(B)), F
(p)
A,B(t
◦
+ iβ) = %
(p)
β,Λ(at
p,Λ(B)A). (2.29)
Proof: Take F
(0)
A,B(z) = G0,β,Λ
A,B (0, z) and F
(p)
A,B(z) = Gp,β,Λ
A,B (0, z). �
The property described by Proposition 3 reflects a consistency between the dy-
namics and the states %
(0)
β,Λ, %
(p)
β,Λ. It comes from the fact that the dynamics, defined
by (2.15), (2.16), and the density matrices, defined by (2.12)–(2.14), are determined
by one and the same Hamiltonian. Such a consistency, natural for local Gibbs states,
cannot be transferred automatically to the infinite-volume case. First of all due to
the fact that in this case one has no Hamiltonians (as operators) to determine the
dynamics and the states. Moreover, to describe such phenomena as phase transi-
tions one has to allow multiplicity of infinite-volume Gibbs states corresponding to
one and the same heuristic Hamiltonian (1.1). If the infinite-volume dynamics can
be constructed as the limit of the time maps (2.15), (2.16), it would be quite nat-
ural to expect that the limiting Gibbs states and time maps are consistent in the
sense of Proposition 3. Therefore, one can define an infinite-volume Gibbs state as
the state satisfying a condition like (2.29). The latter is called the Kubo-Martin-
Schwinger condition, see [29] for detailed explanations, and [32] for the latest devel-
opment. Let us give a definition of a Kubo-Martin-Schwinger state. For Λ,Λ′ such
that Λ ⊂ Λ′, one may consider the elements of CΛ as elements of CΛ′. Indeed, for
Q ∈ CΛ, we set Q′ = Q ⊗ I, where I ∈ CΛ′\Λ is the identity operator. Then, for
ψ′(xΛ′) = ψ(xΛ) · φ(xΛ′\Λ),
Q′ψ′(xΛ′) = (Qψ(xΛ)) · φ(xΛ′\Λ).
One can ignore the difference between the above Q′ and Q and consider Q as an
element of the “bigger” algebra CΛ′ , which acts on “its own” variables. This allows
us to consider the “smaller” algebra CΛ as a subalgebra of the “bigger” one. Now,
for a sequence of boxes L, which exhausts the lattice L, we set
Cloc =
⋃
Λ∈L
CΛ, (2.30)
which is called the local algebra of observables. This algebra contains all bounded
operators on the Hilbert spaces HΛ, Λ ∈ L. It need not be complete – some funda-
mental sequences in Cloc may have no limits in this algebra. The completion C of the
algebra Cloc is called the quasi-local algebra of observables. It is a Banach algebra and
contains the latter one as a dense subset. The dynamics of the model is defined by
a group A = {at}t∈R of time maps, which have to be obtained as limits of the maps
(2.15), if those ones do exist. We recall that a state on C is a positive normalized
functional on this algebra. Since each state is norm-continuous, it may be uniquely
determined by its values on a norm-dense subset of C, i.e., on the local algebra Cloc.
655
Yu.Kondratiev et al.
Definition 4 A state ω on the quasi-local algebra C is called a β-KMS state corre-
sponding to the group of time maps A, if for any A,B ∈ C, there exists a function
FA,B(z), analytic in the open strip {t + iτ | t ∈ R, τ ∈ (0, β)} and continuous on
its closure, which satisfies the β-KMS condition
FA,B(t) = ω(Aa
t(B)), FA,B(t
◦
+ iβ) = ω(at(B)A).
Such β-KMS states are consistent with the dynamics defined by A. If a state is
unique for a given β, then there exists exactly one phase (the state itself) at this
temperature. For an observable A ∈ C, its value in this phase is just ω(A). If, for a
given β, one has more than one β-KMS state, then a phase transition occurs. Let ω
and ω′ be two different β-KMS states. Then their convex combination
ω′′ = θω + (1 − θ)ω′, θ ∈ [0, 1], (2.31)
is also a β-KMS state. Such a state is called a mixture of the states ω and ω ′ if
θ ∈ (0, 1). A β-KMS state is called a pure β-KMS state if it is not a mixture of
any other β-KMS states. Only pure β-KMS states have physical meaning, they are
called phases. In this setting the phase transition corresponds to non-uniqueness of
pure β-KMS states. The above scheme does work in the case of spin models, where
one can construct, by taking the limit of the maps (2.15), the group of time maps A.
The situation in the case of the model (1.2) is different – there are no infinite-volume
dynamics, hence, there are no β-KMS states, at least in the sense of Definition 4.
This difficulty is a principle one, it demands new concepts in defining the notion
of equilibrium state. The approach we describe in this publication presents such
a concept; it may be outlined as follows. Like in classical statistical physics, one
defines local Gibbs states as probability measures, but now on infinite dimensional
spaces. Then one defines the phases by means of the so-called equilibrium equation,
also known as the Dobrushin-Lanford-Ruelle (DLR) equation – the central notion
of classical statistical physics [33]. This approach is based on the Høegh-Krohn
theorem which establishes the one-to-one correspondence between the local Gibbs
states (2.14) and the mentioned measures. A crucial role here is played by the
Matsubara functions corresponding to multiplication operators and by Proposition 1.
Given Λ and a bounded function F : R|Λ| → R, we define the operator
(Fψ)(xΛ) = F (xΛ)ψ(xΛ), (2.32)
which action on ψ ∈ HΛ is multiplication by the function F . This operator is
bounded since the function F is so. The displacement operator q` is a multiplication
operator but it is unbounded. Given Λ, by MΛ we denote the set of all multiplication
operators by bounded continuous functions F : R|Λ| → R. For such a function, we
set
‖F‖∞ = sup
xΛ∈R|Λ|
|F (xΛ)|. (2.33)
Then for every F ∈ MΛ, the norm ‖F‖ of this operator is equal to the above
supremum norm of the corresponding function.
656
Quantum anharmonic crystal
For our purposes, there is no need in employing the Høegh-Krohn theorem in its
entirety, so we formulate only a simplified version. Take n ∈ N and F1, . . . , Fn ∈ MΛ.
Proposition 5 The set of operators
a
t1
0,Λ(F1) · · · atn
0,Λ(Fn), F1, . . . , Fn ∈ MΛ, t1, . . . , tn ∈ R, (2.34)
with all possible choices of n, the operators and time moments, is a state determining
set. This means if two states on CΛ have the same values on all such operators, then
they coincide. The same property have the operators (2.34) corresponding to the
periodic boundary conditions.
The essence of the above statement is that a state on CΛ is completely determined by
its values on the operators (2.34). But, the values of the local Gibbs states (2.14) on
such operators are the Green functions (2.18), (2.19) corresponding to multiplication
operators. Thus, the states (2.14) are completely determined by the Green functions
corresponding to multiplication operators by bounded continuous functions. Finally,
in view of Proposition 1, such Green functions are completely determined by the
Matsubara functions (2.23), (2.24) corresponding to the mentioned multiplication
operators. In what follows, we have the following corollary of the Høegh-Krohn
theorem.
Corollary 6 The Matsubara functions (2.23), (2.24) corresponding to all multi-
plication operators by bounded continuous functions completely determine the local
Gibbs states (2.14).
In the functional integral approach the Matsubara functions corresponding to mul-
tiplication operators are written as moments of certain probability measures, which
we describe in the next subsection.
2.3. Euclidean Gibbs measures
Given β > 0, by E we denote the real Hilbert space L2[0, β] of all real valued
functions which are square integrable on [0, β]. For x, y ∈ E , the scalar product and
the norm are
(x, y)E =
∫ β
0
x(τ)y(τ)dτ, ‖x‖E =
√
(x, x)E . (2.35)
The following set of functions
ek(τ) =
√
2/β cos(kτ), k > 0,
−
√
2/β sin(kτ), k < 0,
√
1/β, k = 0
(2.36)
is a base of the space E . Here
k ∈ K = {k = (2π/β)κ | κ ∈ Z}, (2.37)
657
Yu.Kondratiev et al.
are the so-called Matsubara frequencies. Let Pk, k ∈ K be the orthogonal projector
on ek. By S we denote the linear operator S : E → E , given by
S =
∑
k∈K
1
mk2 + a
Pk, (2.38)
where m and a are the same as in (1.3). For this operator,
Sek = skek, sk
def
=
1
mk2 + a
, trace(S) =
∑
k∈K
sk <∞. (2.39)
In the construction of the Euclidean Gibbs measures the central role is played by
the symmetric Gaussian measure χ, determined by its Fourier transformation
∫
E
exp(i(x, y)E)dχ(y) = exp {(Sx, x)E/2} . (2.40)
Hence, (2.38) is the covariance operator for the measure χ and
∫
E
y(τ)y(τ ′)dχ(y) = S(τ, τ ′)
def
=
∑
k∈K
skek(τ)ek(τ
′). (2.41)
A complete account of the properties of χ is given in subsection 2.2 of [22]. It appears,
that this measure is concentrated on a subset of the space E . Set
C = {x : [0, β] → R | x is continuous and x(0) = x(β)}. (2.42)
The elements of C are called continuous loops. They are square integrable on [0, β],
hence, C ⊂ E . At the same time, the set C may be equipped with its own norm
‖x‖C = sup
τ∈[0,β]
|x(τ)|, (2.43)
which turns it into a Banach space. The measure χ is concentrated on this space,
i.e., χ(C) = 1. This fact, together with some properties of the spaces E and C, allow
us also to consider χ as a measure on the Banach space C. Given a box Λ, we set
CΛ = {xΛ = (x`)`∈Λ | x` ∈ C}, (2.44)
which is the set of vectors with the components labelled by the elements of Λ, each
component being a continuous loop. This set is equipped with the norm
‖xΛ‖CΛ
= sup
`∈Λ
‖x`‖C,
which turns it into a Banach space. Now we set
dχΛ(xΛ) =
∏
`∈Λ
dχ(x`), (2.45)
658
Quantum anharmonic crystal
which is a symmetric Gaussian measure on the Banach space (2.44). The measure
χ describes an isolated harmonic oscillator of mass m and rigidity a. Its two-point
Matsubara function corresponding to the displacement operators is given by (2.41).
The measure (2.45) describes a system of non-interacting harmonic oscillators in Λ.
In order to also take into account the inter-particle interaction and the anharmonicity
we introduce the energy functionals
EV
Λ
(xΛ|0) = −1
2
∑
`,`′∈Λ
J``′
∫ β
0
x`(τ)x`′(τ)dτ +
∑
`∈Λ
∫ β
0
V (x`(τ))dτ,
EV
Λ
(xΛ|p) = −1
2
∑
`,`′∈Λ
JΛ
``′
∫ β
0
x`(τ)x`′(τ)dτ +
∑
`∈Λ
∫ β
0
V (x`(τ))dτ, (2.46)
which are continuous maps from the Banach space CΛ into R. Both are below bound-
ed in view of (1.4). Set
dνΛ(xΛ|0) =
1
Zβ,Λ(0)
exp
(
−EV
Λ
(xΛ|0)
)
dχΛ(xΛ),
dνΛ(xΛ|p) =
1
Zβ,Λ(p)
exp
(
−EV
Λ
(xΛ|p)
)
dχΛ(xΛ), (2.47)
where
Zβ,Λ(0) =
∫
CΛ
exp
(
−EV
Λ
(xΛ|0)
)
dχΛ(xΛ),
Zβ,Λ(p) =
∫
CΛ
exp
(
−EV
Λ
(xΛ|p)
)
dχΛ(xΛ), (2.48)
are the normalizing constants the role of which is to assure the measures (2.47) to
be probabilistic. The crux of the Euclidean approach is contained in the following
statement.
Proposition 7 The normalizing constants (2.48) coincide with the corresponding
local partition functions (2.11). Moreover, the Matsubara functions (2.23), (2.24)
corresponding to multiplication operators F1, . . . Fn have the representations
Γ 0,β,Λ
F1,...,Fn
(τ1, . . . , τn) =
∫
CΛ
F1(xΛ(τ1)) · · ·Fn(xΛ(τn))dνΛ(xΛ|0),
Γ p,β,Λ
F1,...,Fn
(τ1, . . . , τn) =
∫
CΛ
F1(xΛ(τ1)) · · ·Fn(xΛ(τn))dνΛ(xΛ|p). (2.49)
For more details, including the proof, the reader is referred to the article [12] and
to the references in [22]. Since, by Corollary 6, the functions (2.49) corresponding
to multiplication operators by continuous functions F1, . . . , Fn completely determine
the local Gibbs states, the measures (2.47) also possess this property. In what follows,
we have established the one-to-one correspondence between the local Gibbs states
(2.14) and the probability measures (2.47). This is the reason to call them local
659
Yu.Kondratiev et al.
Euclidean Gibbs measures. This correspondence allows us to translate the description
of the model from the “algebraic” language to the language of probability measures,
which paves the way to employing a plenty of powerful techniques developed in
stochastic analysis and classical statistical physics [33].
2.4. Infinite-volume Euclidean Gibbs states
As was mentioned above, there is no way of getting the group of time maps
describing infinite-volume dynamics of our model, and hence, to define its infinite-
volume Gibbs states as β-KMS states. Therefore, the only way of defining such states
is to use the language of probability measures. To this end we introduce conditional
local Gibbs measures. To make the things as easy as we can from now on we suppose
that the interaction potential is of finite range. This means that there exists R > 0
such that J``′ = 0 whenever |`− `′| > R. A typical example is the nearest-neighbor
interaction. Consider
Ω = {x = (x`)`∈L | x` ∈ C}. (2.50)
This set consists of vectors x – configurations – which have infinitely many compo-
nents labelled by lattice sites running through the lattice L, each component being
a continuous loop. The Euclidean Gibbs states we are going to construct are proba-
bility measures on Ω . We do not give a complete description of these measures. The
reader who would desire to get into the details is referred to [12,22]. Given ξ ∈ Ω ,
we set
dνΛ(xΛ|ξ) =
1
Zβ,Λ(ξ)
exp
(
−EV
Λ
(xΛ|ξ)
)
dχΛ(xΛ), (2.51)
Zβ,Λ(ξ) =
∫
CΛ
exp
(
−EV
Λ
(xΛ|ξ)
)
dχΛ(xΛ),
where
EV
Λ
(xΛ|ξ) def
= EV
Λ
(xΛ|0) −
∑
`∈Λ, `′∈Λc
J``′
∫ β
0
x`(τ)ξ`′(τ)dτ, (2.52)
Λc = L \ Λ is the complement of Λ and the energy functional EV
Λ
(xΛ|0) is given
by (2.46). The energy functional (2.52), along with the interaction inside Λ also
describes the interaction of the particles in Λ with the configuration ξ fixed outside
Λ. The measure (2.51) is a conditional local Euclidean Gibbs measure, conditioned
by the configuration ξ fixed outside Λ. Given ` ∈ Λ, let dδξ`
(x`) be the Dirac measure
concentrated at ξ`, i.e.,
dδξ`
(x`)
def
= δ(x` − ξ`)dx`. (2.53)
Here the δ on the right-hand side is “the usual Dirac delta-function”, which, along
with dx`, is only a heuristic expression. However, the left-hand side of (2.53) is a
well-defined probability measure on C. Furthermore, we set
dπΛ(x|ξ) = dν(xΛ|ξ)
∏
`∈Λc
dδξ`
(x`). (2.54)
660
Quantum anharmonic crystal
In contrast to the measure (2.51) the above πΛ is a measure on the space Ω . It defines
a probability distribution on the space of configurations on the whole lattice, but
it takes nonzero values only for those configurations which outside Λ coincide with
the fixed configuration ξ. Let µ be a probability measure of Ω . Then
dµ̃(x) =
∫
Ω
dπΛ(x|ξ)dµ(ξ), (2.55)
is again a probability measure on Ω . The difference between the latter one and
the measure (2.54) is as follows. Both measures are built up with the Hamiltonian
HΛ(0), but the former measure describes the interaction of the particles in Λ with
a fixed configuration outside Λ whereas in the latter measure, boundary conditions
are averaged with the measure µ. This means that it defines the probability distri-
bution of configurations in Λ interacting with configurations outside Λ, distributed
according to the measure µ. If µ̃ = µ, both configurations – inside and outside Λ –
are in equilibrium. These arguments lead to the following definition.
Definition 8 A probability measure µ on the set of all configurations Ω is said to
be a Euclidean Gibbs measure at temperature β−1 if, for any box Λ,
dµ(x) =
∫
Ω
dπΛ(x|ξ)dµ(ξ). (2.56)
The set of all Euclidean Gibbs measures is denoted by Gβ.
In view of the above arguments (2.56) is called the equilibrium equation or the
Dobrushin-Lanford-Ruelle (DLR) equation. More on this subject one may find in
[33]. In order to exclude configurations with no physical meaning certain restrictions
on Ω are imposed. We recall that for a given configuration ξ ∈ Ω , its components
ξ` are continuous loops, hence, they are elements of the Hilbert space E . Set
Ω t = {ξ ∈ Ω | ∀ α > 0 :
∑
`∈L
‖ξ`‖Ee−α|`| <∞}. (2.57)
The elements of this set are called tempered configurations, which means that their
norms grow with |`| → +∞ not very rapidly. A Euclidean Gibbs measure µ is called
tempered if
µ(Ω t) = 1, (2.58)
i.e., it is concentrated on the set of tempered configurations. The set of all tempered
Euclidean Gibbs measures is denoted by Gt
β. Like in the case of KMS states, a con-
vex combination (2.31) of tempered Euclidean Gibbs measures is again a tempered
Euclidean Gibbs measure. A priori it is not clear whether the sets Gt
β, or even Gβ,
contain at least one element. A complete answer to this question was found only
recently in [34], where it was proven that, for the model considered, the set G t
β is
nonempty and a number of properties of its elements were established. The elements
of Gt
β which are not the mixtures of its other elements are called extreme elements of
Gt
β. If the latter set consists of just one element, it is extreme. Only extreme elements
correspond to physical phases.
661
Yu.Kondratiev et al.
3. Quantum stabilization
3.1. Quantum effects
The question as to whether or not the ordering in the system of light parti-
cles, which we describe by the Hamiltonian (1.2), (1.3), is indeed responsible for
the macroscopic displacements of ionic complexes may be answered by means of
experiments in which the properties of the light particles are changed. In the case of
the KDP-type compounds there were performed two types of experiments changing
the properties of hydrogen bonds. In the first group of experiments a part of hydro-
gen nuclei, i.e., protons, were replaced by deuterium nuclei, which have the same
electric charge but approximately two times bigger mass. Among the effects there
was a substantial increase of the Curie temperature, depending on the concentra-
tion of deuterons (the so-called isotopic effect). Another group of experiments dealt
with changing the potential energy of the crystalline field by applying high hydro-
static pressure, which makes the minima of the double wells closer to each other
and increases the tunnelling of protons. As a consequence, the Curie temperature
decreases. These results are described in the book [35], see the table on page 188,
see also [36] for further development. Clearly, the isotopic effect and the effect of
pressure on the phase transition are of the quantum mechanical origin. Thus, the
question as to whether the model describes, at least qualitatively, such quantum ef-
fects becomes very important – a positive answer would confirm the adequacy of the
model. First the problem of quantum effects in models like (1.2), (1.3) was discussed
in the pioneering paper [37]. In that paper arguments in favor of quantum effects
were obtained by means of the Bogoliubov inequalities. Later some exactly soluble
models of interacting quantum anharmonic oscillators manifesting quantum effects
were proposed [38,39]. The first rigorous proof that the long range order in the model
(1.2), (1.3) may be suppressed at all temperatures in the limit m → 0 was given in
[40]. At the same time, no explanation of the effect of pressure was given and the
quantum mechanical origin of this effect has remained unclear. In this article in the
framework of the Euclidean approach outlined above we propose a mathematical-
ly rigorous description of the mechanism of quantum stabilization, which explains
both these effects and the role of the anharmonism. Its physical essence may be
explained as follows. If the tunnelling between the wells gets intensive (smaller mass
or closer minima), the particle “forgets” about the details of the potential energy
in the vicinity of the origin (i.e., of the point q` = 0), including the instability of
its equilibrium at this point. It oscillates as if this equilibrium were stable. If the
potential energy grows at infinity faster than ξ2, the oscillations become more and
more rigid as m → 0. When this effective rigidity exceeds the total force
∑
`′ J``′,
the system gets stable at all temperatures, which means, in particular, that the
spatial decay of the two-point correlation function is exponential. This results in
preventing the model from any critical point anomaly, all the more from the long
range ordering. In a similar model, the exponential decay of correlations for small
values of the mass was proven in [41]. The proof was based on cluster expansions.
No stability arguments were used. A deeper result of such stabilization turns out to
662
Quantum anharmonic crystal
be the uniqueness of the tempered Euclidean Gibbs measures, which was proven in
[21,23].
3.2. Decay of correlations and consequences
Given a box Λ and `, `′ ∈ Λ, we set
K``′(τ, τ
′|Λ, 0) = Γ0,β,Λ
q`,q`′
(τ, τ ′),
K``′(τ, τ
′|Λ, p) = Γp,β,Λ
q`,q`′
(τ, τ ′), (3.1)
where τ, τ ′ ∈ [0, β] and the Matsubara functions are defined by (2.23), (2.24). Then
by (2.49),
K``′(τ, τ
′|Λ, 0) =
∫
CΛ
x`(τ)x`′(τ
′)dνΛ(xΛ|0),
K``′(τ, τ
′|Λ, p) =
∫
CΛ
x`(τ)x`′(τ
′)dνΛ(xΛ|p). (3.2)
We recall that the interaction potential in our model has a finite range. Set
Ĵ(Λ) =
∑
`′∈L
J``′ exp(i(λ, `− `′)), λ ∈ (−π, π]d, (3.3)
and
I(λ) = Ĵ(0) − Ĵ(λ). (3.4)
We suppose that the anharmonic potential V is an even differentiable function sat-
isfying (1.4). Moreover, we suppose that the function v(t), t ∈ [0,+∞), determined
by the relation
V (ξ) = v(ξ2), ξ ∈ R,
which holds since V is even, is such that
d2v(t)
dt2
> 0, for all t ∈ [0,+∞). (3.5)
This implies that V (ξ) grows faster than ξ2 as ξ → ∞. The polynomial (1.5) with
nonnegative b2, . . . , br−1 certainly satisfies (3.5). Under these assumptions the Hamil-
tonianH` has a pure point spectrum consisting of simple eigenvalues En, n = 0, 1, . . .
(see e.g., [30]). Thus, we may set
∆ = min
n∈N
{En − En−1}. (3.6)
For d > 3, the long range order in the model is possible if V has two deep wells
separated by a sufficiently high barrier, see [15,16]. In this case ∆ is the frequency
of the tunnelling between the wells. Thus, the long range order appears if ∆ is
small. If it is big enough, the correlation functions (3.2) decay exponentially, which
is described by the following statement.
663
Yu.Kondratiev et al.
Theorem 9 Let the conditions
m∆2 > Ĵ(0), (3.7)
and (3.5) be satisfied. Then, for any β > 0 and any box Λ, for all `, `′ ∈ Λ and
τ, τ ′ ∈ [0, β], the following estimates hold
0 6 K``′(τ, τ
′|Λ, 0)
6
1
β(2π)d
∑
k∈K
∫
(−π,π]d
exp(ik(τ − τ ′) + i(λ, `− `′))
Ξβ(k) + I(λ)
dλ, (3.8)
where Ξβ(k) is such that uniformly in β
Ξβ(k) > m∆2 +mk2 − Ĵ(0). (3.9)
Let us fix `, `′ ∈ L and τ, τ ′ ∈ [0, β]. Then one can find L0 ∈ N such that `, `′
belong to the box ΛL0
, (2.6), and hence, to all elements of the sequence of boxes
L = {ΛL}L>L0
The next statement is a corollary of Theorem 9.
Theorem 10 Under the conditions of Theorem 9 the limit
lim
L→+∞
K``′(τ, τ
′|ΛL, 0)
def
= K``′(τ, τ
′|0), (3.10)
exists, it obeys (3.8).
The proof of both theorems will be performed in several steps in the concluding
subsection. Now we describe their certain corollaries.
One may show (see [25–27]) that under the condition a + V ′′(0) > Ĵ(0) the
Euclidean Gibbs state is unique; hence, no phase transitions and no critical points
occur. However, for d > 3 and a + V ′′(0) < Ĵ(0), a structural phase transition
becomes possible if the mass m and the inverse temperature β are big enough (see
[16] for the general case and [2,3] for the case of r = 2). This phase transition
is of displacive type if 0 6 a + V ′′(0) < Ĵ(0), and it is of order-disorder type if
a+ V ′′(0) < 0 (see [4,35,37]). In both cases the instability which leads to the phase
transition causes the appearance of soft-mode collective excitations (see page 11–18
of [35]).
Corollary 11 Under the conditions of Theorem 9 the soft-mode excitations are
absent (suppressed) at all temperatures.
Proof: By (3.9), the denominator of the expression on the right-hand side of (3.8)
is positive for all λ ∈ (−π, π]d and k ∈ K if the condition (3.7) holds. Thus, the
energy of elementary excitations ε(λ) has a positive gap ε(0) > m∆2 − Ĵ(0), hence,
no soft-mode phenomena occur. �
Set
$Λ =
1
|Λ|
∑
`,`′∈Λ
∫ β
0
K``′(0, τ |Λ, 0)dτ, (3.11)
which is the static susceptibility in the box Λ. If at a given β, the sequence {$ΛL
}L∈N
is bounded, then no critical point anomaly at this β occurs.
664
Quantum anharmonic crystal
Corollary 12 Under the conditions of Theorem 9 the sequence {$ΛL
}L∈N is con-
vergent at all temperatures to a certain $∞ <∞.
The proof will be performed in the concluding subsection.
Let us look at the above statements from the following point of view. In the
harmonic case the Hamiltonian (1.2), (1.3) may be rewritten in the form
Hhar =
1
2m
∑
`
p2
` +
1
2
∑
`,`′
[aδ``′ − J``′] q`q`′ , (3.12)
where δ``′ is the Kronecker delta. The system described by this Hamiltonian is stable
if a > Ĵ(0). This condition is also applicable to the classical analog of (3.12). In the
quantum case one may write a = m∆2
har, where ∆har is defined as in (3.6) but with
the eigenvalues of the Hamiltonian Hhar
` = (p2
`/2m) + (a/2)q2
` . Thus, the system
(3.12) is stable if
m∆2
har > Ĵ(0). (3.13)
Under this condition the infinite-volume Euclidean Gibbs state of (3.12) exists and is
unique at all temperatures. Moreover, the corresponding infinite volume correlation
function can be calculated explicitly. It is
Khar
``′ (τ, τ ′|0) =
1
β(2π)d
∑
k∈K
∫
(−π,π]d
exp(ik(τ − τ ′) + i(λ, `− `′))
m∆2
har +mk2 − Ĵ(0) + I(λ)
dλ. (3.14)
Comparing the conditions (3.13) and (3.7) one comes to the following conclusion.
The parameter m∆2 may be considered as the measure of the due-to-quantum-effects
rigidity of the anharmonic oscillator which has an unstable equilibrium at q` = 0.
One can show that m∆2 → 0 as m → +∞. For V being of the form (1.5), it is a
continuous function of m such that m∆2 ∼ Cm−(r−1)/(r+1) as m → 0 (see [19,22]).
Thus, in the quasi-classical limit m → +∞ (see [20,22]) the quantum rigidity m∆2
disappears due to the instability at q` = 0. On the other hand, this rigidity becomes
large either if m gets small or if the tunnelling ∆ gets large. If the condition (3.7) is
satisfied, the spatial decay of the correlation function (3.10) is not less than it is for
its harmonic analog (3.14). It is exponential if I(λ) ∼ j|λ|2 as λ→ 0 (see e.g., claim
(c) of Proposition 7.2.1 page 162 of [42]), which is obviously the case for interaction
potentials of finite range.
3.3. The proof of theorems 9, 10 and corollary 12
For a one-point box Λ = {`}, we have
K``(τ, τ
′|{`}, 0) =
1
Z`
trace {q` exp(−(τ − τ ′)H`)q` exp(−(β − τ + τ ′)H`)} ,
Z` = trace {exp(−βH`)} . (3.15)
Set
U(k) =
∫ β
0
K``(τ, τ
′|{`}, 0) cos [k(τ − τ ′)] dτ ′, k ∈ K. (3.16)
Obviously, U(k) does not depend on ` and τ .
665
Yu.Kondratiev et al.
Lemma 13 For every β > 0 and k ∈ K,
0 6 U(k) 6
1
m(∆2 + k2)
. (3.17)
Proof: Let ψn, n = 0, 1, 2, . . . be the eigenfunction of the Hamiltonian H` which
corresponds to the eigenvalue En. Set
Qnn′ = 〈ψn|q`|ψn′〉.
Clearly, Qnn′ = Qn′n does not depend on ` and Qnn = 0. By (3.15), (3.16), one has
U(k) =
1
Z`
∞
∑
n,n′=0
|Qnn′|2 En − En′
[En − En′]2 + k2
{exp(−βEn′) − exp(−βEn)}
6
1
∆2 + k2
· 1
Z`
∞
∑
n,n′=0
|Qnn′|2 [En − En′ ] {exp(−βEn′) − exp(−βEn)}
=
1
∆2 + k2
· 1
Z`
trace {[q`, [H`, q`]] exp(−βH`)}
=
1
m(∆2 + k2)
,
where to get the latter equality we have used (1.1), (1.3) and (3.15). The lower
bound in (3.17) follows from the first two lines of the above formula. �
Along with (2.47) we use the following measure. For t ∈ [0, 1], we set
dν
(t)
Λ
(xΛ|p) =
1
Z
(t)
β,Λ(p)
exp
{
t
2
∑
`,`′∈Λ
JΛ
``′
∫ β
0
x`(τ)x`′(τ)dτ
−
∑
`∈Λ
∫ β
0
V (x`(τ))dτ
}
dχΛ(xΛ), (3.18)
where
Z
(t)
β,Λ(p) =
∫
CΛ
exp
{
t
2
∑
`,`′∈Λ
JΛ
``′
∫ β
0
x`(τ)x`′(τ)dτ
−
∑
`∈Λ
∫ β
0
V (x`(τ))dτ
}
dχΛ(xΛ), (3.19)
By means of the Gaussian upper bound estimate (see Theorem 6.4 in [22]), one can
show that Z
(t)
β,Λ(p), as well as similar integrals, can be analytically continued to entire
functions of t ∈ C. Clearly, Z
(t)
β,Λ(p) is positive for t ∈ [0, 1]. Set
S`1`2`3`4(τ1, τ2, τ3, τ4|t) =
∫
CΛ
x`1(τ1)x`2(τ2)x`3(τ3)x`4(τ4)dν
(t)
Λ
(xΛ|p),
X``′(τ, τ
′|t) =
∫
CΛ
x`(τ)x`′(τ
′)dν
(t)
Λ
(xΛ|p), (3.20)
666
Quantum anharmonic crystal
and
R`1`2`3`4(τ1, τ2, τ3, τ4|t) =
= S`1`2`3`4(τ1, τ2, τ3, τ4|t) −X`1`2(τ1, τ2|t)X`3`4(τ3, τ4|t)
−X`1`3(τ1, τ3|t)X`2`4(τ2, τ4|t) −X`1`4(τ1, τ4|t)X`2`3(τ2, τ3|t). (3.21)
By the Lebowitz inequality (see Theorem 6.3 in [22]), which holds for the moments
of the measure (3.18) with the anharmonicity potential V satisfying the condition
(3.5), one has
R`1`2`3`4(τ1, τ2, τ3, τ4|t) 6 0, (3.22)
which holds for all `1, . . . , `4 ∈ Λ, τ1, . . . , τ4 ∈ [0, β] and t ∈ [0, 1]. As it has been
mentioned above, X``′(τ, τ
′|t) is a differentiable function of t ∈ (0, 1) for any fixed
`, `′ ∈ Λ and τ, τ ′ ∈ [0, β]. By direct calculation,
∂
∂t
X``′(τ, τ
′|t) =
1
2
∑
`1,`2∈Λ
JΛ
`1`2
∫ β
0
R``′`1`2(τ, τ
′, τ1, τ1|t)dτ1
+
∑
`1,`2∈Λ
JΛ
`1`2
∫ β
0
X``1(τ, τ1|t)X`′`2(τ
′, τ1|t)dτ1. (3.23)
This is an integro-differential equation subject to the following boundary conditions
(see (3.2), (3.18)–(3.20))
X``′(τ, τ
′|1) = K``′(τ, τ
′|Λ, p); X``′(τ, τ
′|0) = δ``′K``(τ, τ
′|{`}, 0). (3.24)
The solution of (3.23), (3.24) exists, is unique and has the form (3.20). We will
compare it with the solution of the following problem
∂
∂t
Y``′(τ, τ
′|t) =
1
2
∑
`1,`2∈Λ
[
JΛ
`1`2
+
ε
|Λ|
]
∫ β
0
Y``1(τ, τ1|t)Y`′`2(τ
′, τ1|t)dτ1,
Y``′(τ, τ
′|0) = X``′(τ, τ
′|0) = δ``′K``(τ, τ
′|{`}, 0), (3.25)
where ε > 0 is a parameter.
Lemma 14 Let U(k), defined by (3.16), and Ĵ(λ), defined by (3.3), obey the con-
dition
U(0)Ĵ(0) < 1. (3.26)
Then there exists ε0 > 0 such that, for all ε ∈ [0, ε0), the Cauchy problem (3.25) has
a unique solution on t ∈ [0, 1], which has the form
Y``′(τ, τ
′|t) =
1
β|Λ|
∑
λ∈Λ∗
∑
k∈K
exp [ik(τ − τ ′) + i(λ, `− `′)]
[U(k)]−1 − t(ĴΛ(0) + εδλ,0) + tIΛ(λ)
, (3.27)
where δλ,0 is the Kronecker delta, IΛ(λ) = ĴΛ(0) − ĴΛ(λ) and, for Λ = ΛL having
the form (2.6)
Λ∗ = {λ = (λ1, . . . , λd) | λj = −π+(π/L)nj, nj = 1, . . . , 2L, j = 1, . . . , d} (3.28)
667
Yu.Kondratiev et al.
Proof: By means of the Fourier transformation
Y``′(τ, τ
′|t) =
1
β|Λ|
∑
λ∈Λ∗
∑
k∈K
Ŷ (k, λ|t) exp [ik(τ − τ ′) + i(λ, `− `′)] ,
Ŷ (k, λ|t) =
∑
`′∈Λ
∫ β
0
Y``′(τ, τ
′|t) exp [ik(τ − τ ′) + i(λ, `− `′)] dτ ′,
we bring (3.25) into the form
∂
∂t
Ŷ (k, λ|t) =
[
ĴΛ(λ) + εδλ,0
] [
Ŷ (k, λ|t)
]2
, Ŷ (k, λ|0) = U(k). (3.29)
Clearly, ĴΛ(λ) 6 ĴΛ(0) 6 Ĵ(0) and U(k) 6 U(0), thus, in view of (3.26), one finds
ε0 > 0 such that, for all ε ∈ [0, ε0), the following holds
[
ĴΛ(λ) + εδλ,0
]
U(k) < 1,
for all λ ∈ Λ∗ and k ∈ K. Thus, the problem (3.29) may be solved explicitly, which
via the Fourier transformation yields (3.27). �
Given θ ∈ (0, 1), we set
Y
(θ)
``′ (τ, τ ′|t) = Y``′(τ, τ
′|t + θ), t ∈ [0, 1 − θ]. (3.30)
Obviously, this function of t obeys the equation (3.25) on t ∈ [0, 1 − θ] with the
following initial condition
Y
(θ)
``′ (τ, τ ′|0) = Y``′(τ, τ
′|θ) > Y``′(τ, τ
′|0) = X``′(τ, τ
′|0). (3.31)
This implies
Y
(θ)
``′ (τ, τ ′|t) > 0, (3.32)
for all `, `′ ∈ Λ, τ, τ ′ ∈ [0, β] and t ∈ [0, 1 − θ].
Lemma 15 Let the condition (3.26) be satisfied. Then for all θ ∈ (0, 1) and ε ∈
[0, ε0), the solutions of the problems (3.23), (3.24) and (3.25) obey
X``′(τ, τ
′|t) < Y
(θ)
``′ (τ, τ ′|t), (3.33)
for all `, `′ ∈ Λ, τ, τ ′ ∈ [0, β] and t ∈ [0, 1 − θ].
Proof: Set
Z±
``′(τ, τ
′|t) = Y
(θ)
``′ (τ, τ ′|t) ±X``′(τ, τ
′|t), t ∈ [0, 1 − θ]. (3.34)
Then one has
∂
∂t
Z−
``′(τ, τ
′|t) =
=
1
2
∑
`1,`2∈Λ
JΛ
`1`2
∫ β
0
[
Z+
``1
(τ, τ1|t)Z−
`′`2
(τ ′, τ1|t) + Z−
``1
(τ, τ1|t)Z+
`′`2
(τ ′, τ1|t)
]
dτ1
+
ε
|Λ|
∑
`1,`2∈Λ
∫ β
0
Y θ
``1
(τ, τ1|t)Y θ
`′`2
(τ ′, τ1|t)dτ1 − T``′(τ, τ
′|t), (3.35)
668
Quantum anharmonic crystal
where T``′(τ, τ
′|t) 6 0 stands for the first term on the right-hand side of (3.23). By
(3.34) and (3.31), we have
Z−
``′(τ, τ
′|0) = Y``′(τ, τ
′|θ) −X``′(τ, τ
′|0) > 0, (3.36)
which has to hold for all `, `′ ∈ Λ, τ, τ ′ ∈ [0, β]. For every `, `′ ∈ Λ, both Y``′(τ, τ
′|t),
X``′(τ, τ
′|t), and hence Z±
``′(τ, τ
′|t) are continuous functions of their arguments. For
Y``′, this follows from (3.27). For X``′ , this follows from its above-mentioned analyt-
icity as a function of t and its analyticity as a function of τ, τ ′, which was proven in
Lemma 2.1 in [22]. Set
ζ(t) = inf{Z−
``′(τ, τ
′|t) | `, `′ ∈ Λ, τ, τ ′ ∈ [0, β]}.
By (3.36), one has ζ(0) > 0. Suppose now that ζ(t0) = 0 at some t0 ∈ [0, 1 − θ]
and ζ(t) > 0 for t ∈ [0, t0). Then, by the continuity of Z−
``′, there exists `, `′ ∈ Λ,
τ, τ ′ ∈ [0, β] such that
Z−
``′(τ, τ
′|t0) = 0, Z−
``′(τ, τ
′|t) > 0, for t < t0.
For these `, `′ ∈ Λ, τ, τ ′ ∈ [0, β], the derivative (∂/∂t)Z−
``′(τ, τ
′|t) at t = t0 is positive
since on the right-hand side of (3.35) the third term is positive by (3.32) and the rest
of them are nonnegative. But a differentiable function, which is positive at t ∈ [0, t0)
and zero at t = t0, cannot increase at t = t0. Thus, ζ(t) > 0 for all t ∈ [0, 1 − θ],
which yields (3.33). �
Corollary 16 Let Λ be of the form (2.6) and let the condition (3.26) be satisfied.
Then for all `, `′ ∈ Λ, τ, τ ′ ∈ [0, β], one has
K``′(τ, τ
′|Λ, p) 6
1
β|Λ|
∑
λ∈Λ∗
∑
k∈K
exp [ik(τ − τ ′) + i(λ, `− `′)]
[U(k)]−1 − ĴΛ(0) + IΛ(λ)
. (3.37)
Proof: For t = 1 − θ, one has from (3.33), (3.30), (3.27)
X``′(τ, τ
′|1 − θ) < Y``′(τ, τ
′|1) =
=
1
β|Λ|
∑
λ∈Λ∗
∑
k∈K
exp [ik(τ − τ ′) + i(λ, `− `′)]
[U(k)]−1 − (ĴΛ(0) + εδλ,0) + IΛ(λ)
.
All the functions above depend on θ and ε continuously. Hence, one may pass to the
limit θ = ε→ 0 and get (3.37). �
Given `, `′ ∈ L, one can find L0 ∈ N such that `, `′ belong to the box ΛL0
(2.6).
Corollary 17 Let the condition (3.26) be satisfied. Then for the above `, `′ and for
all τ, τ ′ ∈ [0, β], the sequence {K``′(τ, τ
′|ΛL, p)}L>L0
is bounded.
669
Yu.Kondratiev et al.
Proof: As it will be shown below, every correlation function (3.1) is nonnegative
for all `, `′ ∈ Λ and τ, τ ′ ∈ [0, β]. On the other hand, the right-hand side of (3.37),
converges as L→ +∞ (i.e., as Λ → L) to the following limit
lim
L→+∞
RHS(3.37) =
=
1
β(2π)d
∑
k∈K
∫
(−π,π]d
exp [ik(τ − τ ′) + i(λ, `− `′)]
[U(k)]−1 − Ĵ(0) + I(λ)
dλ, (3.38)
hence, it is bounded. �
Given a box Λ, let the matrix M = (M``′)`,`′∈Λ be such that M``′ = M`′` > 0.
For two such matrices, we write M ′ > M meaning that M ′
``′ > M``′ for all `, `′ ∈ Λ.
Set
dµM
Λ
(xΛ) =
1
ZM
β,Λ
exp
{
1
2
∑
`,`′∈Λ
M``′
∫ β
0
x`(τ)x`′(τ)dτ
−
∑
`∈Λ
∫ β
0
V (x`(τ))dτ
}
dχΛ(xΛ), (3.39)
where 1/ZM
β,Λ is the normalizing factor. Set also
KM
``′(τ, τ
′|Λ) =
∫
CΛ
x`(τ)x`′(τ
′)dµM
Λ
(xΛ). (3.40)
The next statement is a direct corollary of the second GKS inequality, which for
measures like (3.39), was proven in [22], Theorem 6.2.
Proposition 18 If M 6 M ′, then
KM
``′(τ, τ
′|Λ) 6 KM ′
``′ (τ, τ ′|Λ), (3.41)
which holds for all `, `′ ∈ Λ and τ, τ ′ ∈ [0, β].
Corollary 19 For every box Λ, for any `, `′ ∈ Λ and τ, τ ′ ∈ [0, β],
K``′(τ, τ
′|Λ, 0) 6 K``′(τ, τ
′|Λ, p). (3.42)
For any two boxes Λ, Λ′, such that Λ ⊂ Λ′,
K``′(τ, τ
′|Λ, 0) 6 K``′(τ, τ
′|Λ′, 0), (3.43)
which holds for all `, `′ ∈ Λ and τ, τ ′ ∈ [0, β].
Proof: By (2.8), |`− `′|ΛL
6 |`− `′|, which by (2.7), (2.9) yields J``′ 6 JΛ
``′. Hence,
(3.42) follows from (3.41). For Λ ⊂ Λ′ and `, `′ ∈ Λ′, we set
J̃``′ =
{
J``′, if `, `′ ∈ Λ,
0, otherwise.
670
Quantum anharmonic crystal
Then J̃``′ 6 J``′ for all `, `′ ∈ Λ′. On the other hand,
K``′(τ, τ
′|Λ, 0) = K J̃
``′(τ, τ
′|Λ′, 0),
then (3.43) also follows from (3.41). �
Proof of Theorem 9: The nonnegativity of the correlation functions (3.1)
follows form the first GKS inequality, proven in [22], Theorem 6.2. By (3.17), the
condition (3.26) will be satisfied if (3.7) holds. Given `, `′ ∈ L, we pick up L0 ∈ N
such that `, `′ ∈ Λ = ΛL0
. Then by (3.42), (3.43)
K``′(τ, τ
′|Λ, 0) 6 K``′(τ, τ
′|Λ′, p)
which holds for any Λ′ = ΛL, L > L0, for all `, `′ ∈ Λ and τ, τ ′ ∈ [0, β]. Passing here
to the limit L → +∞ and taking into account (3.37), (3.38) one arrives at (3.8)
with Ξβ(k) = [U(k)]−1 − Ĵ(0). �
Proof of Theorem 10: By (3.8), the sequence {K``′(τ, τ
′|ΛL, 0)}L>L0
is bounded
and by (3.43), it is monotone increasing, hence convergent. Its limit does not exceed
the bound (3.8). �
Proof of Corollary 12: Given Λ, one has by (3.37)
1
|Λ|
∑
`,`′∈Λ
∫ β
0
K``′(0, τ |Λ, p)dτ 6
1
[U(0)]−1 − ĴΛ(0)
6
1
[U(0)]−1 − Ĵ(0)
<∞, (3.44)
which holds in view of (3.26). By (3.11), (3.42), (3.43), one has
$Λ 6
1
|Λ|
∑
`,`′∈Λ
∫ β
0
K``′(0, τ |Λ, p)dτ 6
1
[U(0)]−1 − Ĵ(0)
,
and $Λ 6 $Λ′ for Λ ⊂ Λ′. Therefore, the sequence {$ΛL
}L∈N is monotone increasing
and bounded, hence convergent to a certain $∞ 6 [[U(0)]−1 − Ĵ(0)]−1. �
As it has been mentioned earlier, the spatial decay of the right-hand side of (3.8)
is exponential if I(λ) ∼ j|λ|2 as λ → 0. Let us prove this directly in the case of
J``′ = Jδ|`−`′|,1, J > 0. Namely, we are going to show that
RHS(3.8) 6 Cβ exp(−α0|`− `′|), (3.45)
with
α0 = d−1/2 ln
(
1 +
m∆2 − 2dJ
2J
)
,
Cβ =
1
β
∑
k∈K
1
m∆2 − 2dJ −mk2
. (3.46)
671
Yu.Kondratiev et al.
To this end we use a known formula
1
2π
∫ π
−π
eitldt
a− cos t
=
1√
a2 − 1
exp(−α|l|),
α = ln(a+
√
a2 − 1), (3.47)
which holds for all a > 1 and l ∈ Z. Now we choose the component of ` − `′ with
the biggest absolute value. Suppose it is `j − `′j. Set ˆ̀, ˆ̀′ and λ̂ to be the d − 1-
dimensional vectors which one obtains from the corresponding d-dimensional vectors
by excluding the j-ths components. Then by (3.47)
RHS(3.8) =
1
2Jβ(2π)d−1
∑
k∈K
∫
(−π,π]d−1
exp[ik(τ − τ ′) + i(λ̂, ˆ̀− ˆ̀′)]dλ̂
× 1√
a2 − 1
exp(−α|`j − `′j|),
where α is as in (3.47) with
a =
1
2J
[U(k)]−1 −
d
∑
i=1, i6=j
cosλi > 1,
which holds in view of (3.26). Applying here the estimate (3.17) and the following
evident facts
|`j − `′j| > d−1/2|`− `′|,
√
a2 − 1 > a− 1, for a > 1,
one arrives at (3.45), (3.46).
4. Acknowledgements
Yuri Kozitsky was financially supported by the Komitet Badań Naukowych
through the Grant 2 P03A 020 25, which is cordially acknowledged.
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Квантовий ангармонічний кристал у підході, що
базується на функціональних інтегралах
Ю.Кондратьєв 1,3,4 , Ю.Козицький 2 , Т.Пазурек 1,3 ,
М.Рекнер 1,3
1 Математичний факультет університету Білефельд,
Німеччина, D 33615 Білефельд
2 Інститут математики,
Університет Марії Кюрі-Склодовської в Любліні,
Польща, PL 20-031 Люблін
3 Дослідницький центр BiBoS, Німеччина, Білефельд
4 Інститут математики НАН України, Київ
Отримано 15 жовтня 2003 р.
В рамках підходу, що базується на функціональних інтегралах, роз-
глядається ґраткова модель взаємодіючих квантових частинок маси
m , які осцилюють у кристалічному полі. Головні аспекти такого під-
ходу описуються у спосіб доступний для неспеціалістів. У результа-
ті запропоновано механізм стабілізації моделі, спричиненої кванто-
вими ефектами. Зокрема, дається умова стабілізації, що включає в
себе m , інтенсивність взаємодії та параметри кристалічного поля.
Ця умова не залежить від температури, вона задовільняється, якщо
m є достатньо малою та/або частота тунелювання є достатньо ве-
ликою. Показано, що за цієї умови кореляційні функції, обчислені у
термодинамічній границі, спадають експоненційно, що унеможлив-
лює фазові переходи при всіх температурах.
Ключові слова: квантова стабілізація, стани Гіббса, м’яка мода
PACS: 05.50.-d, 64.60.-i
674
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