Long-range order in quantum lattice systems of linear oscillators
The existence of the ferromagnetic long-range order is proved for equilibrium quantum lattice systems of linear oscillators whose potential energy contains a strong ferromagnetic nearest-neighbor (nn) pair interaction term and a weak nonferromagnetic term under a special condition on a superstabilit...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509650330320896 |
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| author | Skrypnik, W. I. Скрипник, В. І. |
| author_facet | Skrypnik, W. I. Скрипник, В. І. |
| author_sort | Skrypnik, W. I. |
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| description | The existence of the ferromagnetic long-range order is proved for equilibrium quantum lattice systems of linear oscillators whose potential energy contains a strong ferromagnetic nearest-neighbor (nn) pair interaction term and a weak nonferromagnetic term under a special condition on a superstability bound. It is shown that the long-range order is possible if the mass of a quantum oscillator and the strength of the ferromagnetic nn interaction exceed special values. A generalized Peierls argument and a contour bound, proved with the help of a new superstability bound for correlation functions, are our main tools. |
| first_indexed | 2026-03-24T02:44:28Z |
| format | Article |
| fulltext |
UDC 517.9
W. I. Skrypnik (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
LONG-RANGE ORDER IN QUANTUM
LATTICE LINEAR OSCILLATOR SYSTEMS
DALEKYJ PORQDOK U KVANTOVYX
ÌRATKOVYX SYSTEMAX LINIJNYX OSCYLQTORIV
Existence of ferromagnetic long-range order (lro) is proved for equilibrium quantum lattice systems of linear
oscillators whose potential energy contains a strong ferromagnetic nn (nearest neighbor) pair interaction term
and a weak nonferromagnetic term under a special condition on a superstability bound. It is shown that lro is
possible if a mass of a quantum oscillator and a strength of a ferromagnetic nn interaction exceed special values.
A generalized Peierls argument and a contour bound, proved with the help of a new superstability bound for
correlation functions, are our main tools.
Dlq rivnovaΩnyx kvantovyx ©ratkovyx system linijnyx oscylqtoriv, potencial\na enerhiq qkyx mis-
tyt\ syl\nu feromahnitnu çastynu parno] vza[modi] blyz\kyx susidiv i slabku neferomahnitnu çastynu,
dovedeno isnuvannq feromahnitnoho dalekoho porqdku pry pevnij umovi na nerivnist\ superstijkosti.
Pokazano, wo dalekyj porqdok moΩe maty misce, qkwo masa kvantovoho oscylqtora ta syla feromah-
nitno] vza[modi] blyz\kyx susidiv perevywugt\ pevni znaçennq. Pry c\omu vykorystano uzahal\nenyj
pryncyp Paj[rlsa ta konturnu nerivnist\, dovedenu z dopomohog novo] nerivnosti superstijkosti dlq
korelqcijnyx funkcij.
1. Introduction and main result. Let’s consider Gibbs quantum systems of one-dimen-
sional oscillators on the d-dimensional hypercubic lattice Z
d, with the potential energy
U(qΛ) = U(−qΛ) on a set Λ with the finite cardinality |Λ|, where qΛ is an array of
(qx, x ∈ Λ), qx is the oscillator coordinate taking value in R. The potential energy is
assumed to be a growing function at infinity. It is invariant under the simplest discrete
symmetry, namely Z2, which is realized as a transformation of changing of all the signs
of the oscillator variables.
The Hamiltonian of the considered quantum system is given by
HΛ = − 1
2m
∑
x∈Λ
∂2
x + U(qΛ),
where ∂x is the partial derivative in qx and U is the potential energy
U(qΛ) =
∑
x∈Λ
u(qx) +
∑
〈x,y〉∈Λ
φ(qx, qy) + U ′(qΛ), (1.1)
〈x, y〉 means nearest neighbors, the external potential u is a bounded below polynomial of
the 2n-th degree, the pair potential φ is a polynomial equal to zero at coinciding arguments
and a finite-range U satisfies a special superstability and regularity conditions [1 – 4],
written down below, which allow to pass to the thermodynamic limit.
If F̂X is the operator of multiplication by the function FX(qX) then the quantum
Gibbs (equilibrium) average is given by
〈FX〉Λ = Z−1
Λ Tr
(
F̂Xe
−βHΛ)
= Z−1
Λ
∫
FX(qX)e−βH
Λ
(qΛ; qΛ)dqΛ =
=
∫
FX(qX)ρΛ(qX |qX)dqX , ZΛ = Tr
(
e−βH
Λ)
,
where the second function under the sign of the first integral is the kernel of the semi-
group generated by the Hamiltonian and the integration is performed over R
|Λ|, R
|X|,
respectively.
c© W. I. SKRYPNIK, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10 1407
1408 W. I. SKRYPNIK
Ferromagnetic long-range order (lro) occurs in the systems if for a′ > 0 independent
of Λ either
〈σxσy〉Λ ≥ a′, σx(qΛ) = qx, x, y ∈ Λ, (1.2)
or
〈sxsy〉Λ ≥ a′, sx(qΛ) =
qx
|qx|
. (1.3)
It is believed that lro occurs if minima of the potential u are sufficiently deep and rare.
In this paper we prove (1.3) for d ≥ 2 with the help of the generalized Peierls argument
based on an application of a new version of the Ruelle superstability bound (ferromagnetic
superstability bound) given below.
For pair short-range potentials and high temperatures there is a decrease of corre-
lations [5]. This should mean existence of phase transitions for such the systems. For
d ≥ 3 and the case of the pair nn(nearest neighbor) quadratic interaction potential poten-
tials was derived in [6] with the help of the swc(spin wave condensation) method. For
translation invariant U ′ having a small nonferromagnetic term and the quadratic nn-pair
interaction (1.3) was claimed to hold in [7]. The technique of this paper was based on the
conventional superstability bound.
The proposed here superstability Peierls argument is very general (the potential energy
is more general than in [7]) and was introduced for nonequilibrium systems of interact-
ing Brownian oscillators close to or far from an equilibrium [8] whose time dependent
correlation functions have the structure of reduced density matrices of quantum oscillator
systems with a three-body and pair interaction potentials. In this paper we reformulated
and generalized our arguments from [7] proving the abstract ferromagnetic superstability
bound. Its formulation and proof for quantum oscillator systems does not differ essen-
tially from the case of nonequlibrium systems of interacting Brownian oscillators since
the Winer integrals over unclosed paths in the latter have to be changed by Wiener inte-
grals over closed paths (loops) in the former. In both cases Feynman – Kac formula has to
be employed for its proof.
For the quadratic nearest neighbor pair interaction it was proved in [9] that lro dis-
appears at sufficiently small mass of the oscillators m and appropriate general external
potential and this implies a convergence of the cluster expansion for the Gibbs state at an
arbitrary temperature [10].
We assume that
u(q) = ηq2n − gq2n0 +
n−1∑
j=1
ηjq
2j = u0(q) − gq2n0 , n > n0,
η ≥ 1, ηj ∈ R.
If the coefficients η, ηj depend on g > 0 then they should be bounded functions. For
existence of lro in our approach the following conditions are needed
m ≥ g
n−2
2(n−n0)m′(g), lim
g→∞
m′(g) = ∞. (1.4)
That is, the mass has to be sufficiently large. The pair nearest neighbor potential φ is
chosen as follows
φ(qx, qy) = g0(qx − qy)2n1Q′(qx, qy), g0 = g
ξ
2(n−n0) = z−ξ, ξ ∈ R,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
LONG-RANGE ORDER IN QUANTUM LATTICE LINEAR OSCILLATOR SYSTEMS 1409
where Q′ is an even positive function such that the following inequalities hold:
Q′(qx, qy) ≤ α0(q2n2
x + q2n2
y ), n1 + n2 < n,
Q′(qx, qy) ≥ q
2n′
2
x q
2(n2−n′
2)
y + q
2n′
2
y q
2(n2−n′
2)
x ,
the positive constants α0 does not depend on g and n2 − n′
2 ≤ n′
2 ≤ n2. For such the
choice of Q′ the interaction constant g0 can be interpreted as the strength of the nn-pair
interaction. It’ll be seen that the condition ξ > 0 is needed for occurrence of lro only if
n2 is comparable with n1. For translation invariant U ′ the parameter ξ may be negative.
The above expressions for u with ηj = 0 and φ can be obtained if one starts from the
following expressions (see [11] where this fact is proved):
u(q) = u0(q), φ(qx, qy) = −g(qn0+k
x qn0−k
y + qn0+k
y qn0−k
x ), 0 ≤ k ≤ n0.
Our method is based on a control of asymtotics of the reduced density matrices at
large g and all the conditions in our main Theorem 1.1 are aimed at that. In order to do
this we have to deal with bounded functions ug , u∗, U∗, φ∗ in g and re-scale and translate
variables by zn−1 and e0, respectively, where e0 is the deepest minimum of the potential
ug(q). We put
ug(q) = u(zn−1q), φg(q, q′) = φ(zn−1q, zn−1q′), Ug(qX) = U(zn−1qX),
u∗(q) = ug(q + e0) − ug(e0), φ∗(q, q′) = φg(q + e0, q
′ + e0),
U∗(qX) = Ug(qX + e0) − |X|ug(e0), U ′
∗(qX) = U ′
g(qX + e0).
Let W ′
∗(qX ; qY ) = U ′
∗(qΛ)−U ′
∗(qY )−U ′
∗(qX), Λ = X∪Y. The same relation will hold,
also, between U∗ and W∗. We require the superstability and regularity conditions to hold
U∗(qΛ) ≥
∑
x∈Λ
u−∗ (qx), u−∗ (q) = u∗(q) − ζv0(znq) − ζ0, (1.5)
∣∣W ′
∗(qX ; qY )
∣∣ ≤ ∑
x∈X,y∈Y
Ψ′(|x− y|
)(
v0(qx) + v0(qy)
)
, v0(q) =
n−1∑
j=1
q2j , (1.6)
where the positiveL1-function Ψ′, and numbers ζ, ζ0 ≥ 0 do not depend on g. The second
and third terms in the expression for u−∗ is a contribution of the negative (nonferromag-
netic) term in U ′ which shows that the latter is always small, that is, it depends on positive
powers of g−1. Formulae (1.5), (1.6) allow positive (ferromagnetic) interaction terms in
U to be large. Interaction is stronger for translation invariant potentials. For translation
invariant interaction two conditions for the functions U ′,W ′ may be postulated in such
the way that they will imply (1.5), (1.6) (see [7]). The following condition
ξ ≤ 2(n− 1)n1 − 2n2 (1.7)
guarantees, also, that the nn-pair potential part of U∗ and U∗ itself satisfy (1.6) with an
appropriate Ψ (see the appendix A in [11]). Let ρΛ
∗ (qX |qX) be the correlation functions
corresponding to the potential energy U∗. Then the ferromagnetic superstability bound is
given by
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1410 W. I. SKRYPNIK
ρΛ
∗ (qX |qX) ≤ e−β(HX
∗ε−|X|c∗(ε))(qX ; qX),
(1.7a)
HX
∗ε = (2mg)−1∆X + U+
∗ (qX) +
∑
x∈X
u+
∗ (qx), mg = mg−
n−1
n−n0 ,
u+
∗ (q) = u−∗ (q) − 3εv0(q),
whereU+
∗ is the part ofU∗, which is absent in the ordinary superstability bound, generated
by a positive pair interaction potential. Function c∗(ε) does not depend of the oscillator
variables and is a positive continuous function tending to infinity in the limit of vanishing
ε. Its analytical structure can be found [8].
Let χ+
x (qΛ) = χ(0,∞)(qx), χ−
x (qΛ) = χ(−∞,0)(qx), where χ(a,b) is the characteristic
function of the open interval (a, b). The corner stone of the Peierls principle (argument)
is the contour bound whose universal character was demonstrated in [12] for classical
oscillator systems with pair nn interaction (see also [13]) and Heisenberg models〈 ∏
〈x,y〉∈Γ
χ+
x χ
−
y
〉
Λ
≤ e−|Γ|E , (1.8)
where Γ is the set of pairs of nearest neighbors adjacent to a contour, i.e., an external
boundary of the connected union of unit cubes centered at the sites of the bounded subset
of the hyper-cubic lattice and E can be made arbitrary large either at low temperatures
or by varying a parameter in an expression of a potential energy (see Remark 4.1). The
generalized Peierls principle proves (1.3) if (1.8) holds. It is not obvious that (1.3) implies
(1.2) (see [12]). Our main result is formulated in the following theorem.
Theorem 1.1. Let (1.4) – (1.7) be satisfied and for the constant in the superstability
bound, corresponding to the potential energy U∗, the following equality hold:
lim
g→∞
znc∗(z2(n−1)n) = 0. Let, also, the following inequality hold:
ξ > n− 2(n1 + n2) + 2(n2 − n′
2)n. (1.9)
Then the contour bound (1.8) holds with E being a function which is growing in g as
β
(
gnn0
nη
) 1
2(n−n0)
at infinity and (1.3) holds if g is sufficiently large.
The proof that the condition for c∗(ε) in this theorem is true for finite range interaction,
determined by U ′, can be given following arguments from [8] and bounds given in this
paper. The growth at infinity of β−1E coincides with the minimum z−nµ0 of the external
potential ug with ηj = 0. Remark that zn and z2(n−1)n behave at large g like e−1
0 and
e
−2(n−1)
0 , respectively, Conditions (1.7), (1.9) mean that the nontranslation invariant part
of the nn-pair potential is not arbitrary and that the strength g0 of the nn-pair interaction
has to be correlated with the depth of the deepest minimum e0 of the external potential u.
Inequality (1.9) arises from an application of (2.1) for the product of the characteristic
functions in (1.8) and the necessity of a suppression of the first unbounded term in the
expression Qg in it (Qg in (2.10) does not allow the strength of the nn ferromagnetic
interaction g0 be arbitrary small). This suppression is more complicated for quantum than
for classical systems (see [11]). In our method an important role is played by the integral∫
e−β(u∗(q)−av0(znq)−bz2n(n−1)v0(q))dq. (1.10)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
LONG-RANGE ORDER IN QUANTUM LATTICE LINEAR OSCILLATOR SYSTEMS 1411
We show that its asymptotics are the same for v0 = 0 and ηj = 0 (this fact is more
profound than the statement of the one-dimensional Laplace method). The proof is based
on application of the inequalities (4.2), (4.3) or (3.6), (3.7), easily verified for ηj = 0. Our
result can be generalized for the potential u obeying these inequalities. The bound for the
integral is given in the third section and it is used later in a proof of our main bounds.
The expression for E is given in Theorem 2.1. The main term in it is given by the
integral
I∗(2; r) =
∫
µ(dqxdq′x)µ(dqydq′y) exp{−βrΦ∗(qx,y, q′x,y)},
where µ(dqdq′) = exp
{
− βu+
∗ (q′) − (mg/2β)(q − q′)2
}
dqdq′, Φ∗(qx,y, q′x,y) =
= −Q∗(qx, qy) + φ∗(q′x, q
′
y),
Q∗(qx, qy) = e−1
0
[
(qx − qy)2 +
4
3
(
|qx(qx + 2e0)| + |qy(qy + 2e0)|
)]
.
The polynomial Q∗ is determined in the inequality (2.1) for the product of the characteris-
tic functions in the contour bound (1.8) and mg is the rescaled mass. The last polynomial
has the remarkable property: Q∗(−qx,−qy) = Q∗(qx − 2e0, qy − 2e0), which will be
employed by us in a special way in the proof of Theorem 4.1.
Estimates of the above integrals are based on the control of the behavior of all the
functions in the neighborhod of the deepest minimum e0 of u∗ based on application of
(4.2), (4.3).
Our paper is organized as follows. In the second section we derive the expression for
E in the contour bound (1.8) in terms of c∗ and I∗(2; r) (Theorem 2.1). In Theorem 2.2
we establish the character of asymptotics of this integral proving Theorem 1.1 in this
way. In the third section some properties of the minima of the re-scaled external potential
ug are established (its higher derivatives tend to zero in the limit of infinite g) and the
estimate of the integral in (1.10) is obtained. In the fourth section Theorem 2.2 is proved.
The proofs of the Peierls principle can be found in [8, 9].
2. Contour bound and superstability. Derivation of the contour bound (1.8) is based
on an application of the ferromagnetic superstability bound and the bound for the product
of the characteristic functions whose more complicated version appeared for the first time
in [14] for a continuum (classical) system of oscillators, corresponding to an interacting
two-dimensional Euclidean quantum boson field. This bound (its proof can be found in
[8, 11]) is given by∏
〈x,x′〉∈Γ
χ+(qx)χ−(qx′) ≤ exp
{
−β
[
e0|Γ| −Qg(qΓ)
]}
, (2.1)
where
Qg(qΓ) =
∑
〈x,x′〉∈Γ
Qg(qx, qy),
Qg(qx, qy) = e−1
0
[
(qx − qy)2 +
4
3
(
|q2
x − e20| + |q2y − e20|
)]
.
The exponent with the ferromagnetic part of the potential energy in the ferromagnetic
superstability bound has to compensate a contribution of the first translation-invariant
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1412 W. I. SKRYPNIK
term in the expression for Qg . The mechanism of a compensation leads to (1.9) and
in quantum oscillator systems is more complicated than in classical oscillator systems
and nonequilibrium systems of interacting Brownian oscillators explained in [11] and
[8], respectively. In order to realize it we have to generalize the Golden – Thompson
inequality. The main result of this section is formulated as follows.
Theorem 2.1. Let the conditions of Theorem 1.1 be satisfied and
e∗(g) = c∗(z2n(n−1)) be determined by the superstability bound (1.7a). Then E in (1.8)
is represented by
E = βe0 − e∗(g) − (2d− 1)−1 ln I∗(2; 2d− 1) + ln
2πβI∗(2; 0)
e2mg
if I∗(2; 0) ≥ 1. If I∗(2; 0) ≤ 1 then the inequality holds without I∗(2; 0).
Proof. Using the fact that the scaling does not change the product of the characteristic
functions and that Laplacian is translation invariant we derive〈 ∏
〈x,y〉∈Γ
χ+
x χ
−
y
〉
Λ
=
〈 ∏
〈x,y〉∈Γ
χ+
x χ
−
y
〉
∗Λ
, (2.2)
where the Gibbs average in the right-hand side corresponds to U∗. Let’s insert (2.1) into
the right-hand side of (2.2). Then (2.1) and the ferromagnetic superstability bound yield〈 ∏
〈x,y〉∈Γ
χ+
x χ
−
y
〉
Λ
≤ e−(βe0−e∗(g))|Γ|Î∗(Γ). (2.3)
It’s clear that
Î∗(Γ) = Tr
eβQ̂∗Γ exp
{
−β
[
−(2mg)−1∆Γ +
∑
〈x,y〉∈Γ
(
û+
∗x + û+
∗y + φ̂∗(x,y)
)]}
,
(2.4)
where the operators with the hats correspond to the operators of multiplication by the
functions depending on the variables indexed by the indices of the operators and ∆Γ is
the Laplacian in qx,y, x, y ∈ Γ. Now we have to apply the generlized Golden – Thompson
inequality formulated and proved in the following proposition.
Proposition 2.1. Let ∆ be the d-dimensional Laplacian and F̂ , v̂ the operators of
multiplication by real valued continuous functions F ≥ 0, v such that v is bounded form
below, v ∈ L2(Rd, e−‖q‖2
dq) and
∫
F (q)etv(q)dq < ∞. Then
Tr
(
F̂ et((2m)−1∆+v̂)
)
≤
(
m(2πt)−1e2
)d ∫
dqF (q)
∫
dq1e
−m
2t‖q−q1‖
2
etv(q1), (2.5)
where ‖q‖ is the Euclidean norm of the vector q.
Proof. Let H denote the operator under the sign of the exponent in the left-hand side
of (2.5). The first condition for v implies that the Trotter formula holds in L2(Rd): etH =
= st lim
n→∞
(
e
t
2mn ∆e
t
n v̂
)n
(see Theorem X.51, Exercise 3.X in [15, 16]). This implies
that the function
(
e
t
2mn ∆e
t
n v̂
)n
f, f ∈ L2(Rd), converges almost everywhere to the limit
function on a subsequence of positive integers. This means that It(F |v) =
= lim
n→∞
It(F |v, n), where the limit is achieved on some subsequence of positive inte-
gers and
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
LONG-RANGE ORDER IN QUANTUM LATTICE LINEAR OSCILLATOR SYSTEMS 1413
It(F |v, n) =
∫
dqF (q)
∫
dq1 . . . dqnp
tn−1
0 (q − q1)etn
−1v(q1)×
×
n∏
j=2
ptn
−1
0 (qj − qj−1)etn
−1v(qj)δ(q − qn),
where δ(q) is the point measure concentrated at zero,
pt0(q) =
√
md(2πt)−de−m
‖q‖2
2t .
Now, lets apply the generalized Hölder inequality for a measure µ on arbitrary mea-
sure space (the usual Hölder inequality and induction are used to prove it)
∫ n∏
j=1
fjdµ ≤
n∏
j=1
[∫
|fj |ndµ
] 1
n
, (2.6)
taking R
d(n+1) as the measure space and putting fj(q, q1, . . . , qn) = e
t
nv(qj),
dµ(q, q1, . . . , qn) = F (q)dqdq1 . . . dqnptn
−1
0 (q − q1)
n∏
j=2
ptn
−1
0 (qj − qj−1)δ(q − qn).
From the semigroup property of pt0 it follows that (p0
0(q) = δ(q))
It(F |v, n) ≤
n∏
j=1
[∫
dqF (q)
∫
dq1p
t j
n
0 (q − q1)etv(q1)p
tn−j
n
0 (q1 − q)
] 1
n
=
= md(2πt)−d
n−1∏
j=1
[(
n2
j(n− j)
)d ∫
dqF (q)
∫
dq1e
−( nm
2tj + nm
2t(n−j) )‖q−q1‖
2
etv(q1)
] 1
n
×
×
[∫
dqF (q)etv(q)pt0(q, q)
] 1
n
≤
≤
[
m(2πt)−1
(
n2n
(n!)2
) 1
n
]d [∫
dqF (q)
∫
dq1e
−m
t ‖q−q1‖2
etv(q1)
]n−1
n
×
×
[∫
dqF (q)etv(q)pt0(q, q)
] 1
n
.
Hence
lim
n→∞
It(F | v, n) ≤ (m(2πt)−1e2)d
∫
dqF (q)
∫
dq1e
−m
t ||q−q1‖2
etv(q1).
Here we used the elementary inequalities
n
j
≥ n
n− 1
,
n
n− j
≥ n
n− 1
n
n− 1
≥ 1 and
the formula lim
n→∞
(
n2n
(n!)2
) 1
n
≤ e2. This formula follows from the Stirling inequality
n! ≥ nne−n
√
2πn.
The proposition is proved.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1414 W. I. SKRYPNIK
Remark that (2.5) is useful if its right-hand side is finite. The additional condition∫
e−
m
2t‖q‖
2
F (q)dq < ∞, t ∈ R
+, guarantee this (see (4.7)). For F = 1 (2.5) results in
Tr
(
et((2m)−1∆+v̂)
)
≤
(
m(2πt)−1e2
)d ∫
dq
∫
dq1e
−m
2t‖q−q1‖
2
etv(q1) =
=
(
m(2πt)−1e4
) d
2
∫
dqetv(q) = e2d Tr(et(2m)−1∆etv̂).
This bound coincide with the Golden – Thompson inequality if e2d is dropped in the right-
hand side. We’ll apply this proposition for the case of an unbounded F .
As a result (2.3) – (2.5) for d = |Γ|
Î∗(Γ) ≤
(
mg(2πβ)−1e2
)|Γ|
I∗(Γ), I∗(Γ) = I∗(Γ; 1),
(2.7)
I∗(Γ; r) =
∫ ∏
〈x,y〉∈Γ
µ(dqxdq′x)µ(dqydq′y) exp
{
−βrΦ∗(qx,y, q′x,y)
}
.
Now, we show how to bound this integral in terms of the integral I∗(2; r) coinciding with
I∗(Γ; r) when Γ is one pair adjacent to a corresponding face. In other words we have
to decouple the variables in the integral. The contour Γ, which the set of near neighbor
(nn)-pairs adjacent to the contour, contains overlapping pairs. The corners of the con-
tour creates the crossing. Let’s recollect that the contour is the external boundary of the
connected (by faces) union of unit cubes centered at the sites of the bounded subset of
the hyper-cubic lattice. A set of the nn-pairs is a union of the nonoverlapping set Γ− of
nonintersecting nn-pairs and the set Γ+ which is a disjoint union of connected sets Γ+
l of
intersecting nn-pairs. Its clear that
I∗(Γ−) =
(
I∗(2; 1)
)|Γ−|
, (2.8)
I∗(Γ) = I∗(Γ−)
∏
l
I∗(Γ+
l ). (2.9)
Our main tool of the decoupling is the generalized Helder inequality (2.6). In the case of
the d-dimensional hypercubic lattice no more than 2d− 1 contour nn-pairs can intersect.
Let’s describe the decoupling inductive process for it. Let the nn-pair (1.2) be the bound-
ary nn-pair and 1 is the site which does not belong to other nn-pairs associated to Γ+
l .
Let’s apply (2.6) for n = 2d− 1 in s variables: a variable, indexed by 2, and the variables
j = 1, 3, s, s ≤ 2d, which are linked with the site 2 in other nn-pairs. 2d − 1 − (s − 1)
functions coincide with 1 and the other with exp{−βΦ∗(2,j)}, j = 1, 3, . . . , s. Hence,
instead of the integral in (q, q′)1,...,s, we obtain the product
(∫
µ
(
d(q, q′)
)) s(2d−s)+(s−2)(s−1)
2d−1 (
I∗(2; 2d− 1)
) s−1−|Γ′
2|
2d−1 ×
×
∏
j∈Γ′
2
[∫
µ(d(q, q′)2)e−(2d−1)βΦ∗((q,q′)2,j)
] 1
2d−1
,
where Γ′
2 ⊆ (3, . . . , s) is the set of nn-sites which are linked with other nearest neighbors.
Then this operation one has to employ for any site from Γ′
2 once more. Among the
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LONG-RANGE ORDER IN QUANTUM LATTICE LINEAR OSCILLATOR SYSTEMS 1415
functions fj in (2.6) there will be functions represented by the integrals in the right-hand
side of the last inequality. Thus moving from one site to the next we’ll fulfil the procedure
of the decoupling of all nn-pairs for Γ+
l . After that we have to use (2.8), (2.9) and the
Hölder inequality for I∗(2; 1) with n = 2d− 1
∫
|f(x)|µ(dx) ≤
(∫
|f |n(x)µ(dx)
)1
n
(∫
µ(dx)
)n−1
n
. (2.10)
Since
s(2d− s) + (s− 2)(s− 1)
2d− 1
= s− 2(s− 1)
2d− 1
≤ 2(s− 1)
we proved the following proposition.
Proposition 2.2. The following formula is valid for the integral defined in (2.7)
I∗(Γ) ≤
(
I∗(2; 0)
)|Γ|(
I∗(2; 2d− 1)
) |Γ|
2d−1 (2.11)
if I∗(2; 0) ≥ 1. If I∗(2; 0) ≤ 1 then
I∗(Γ) ≤
(
I∗(2; 2d− 1)
) |Γ|
2d−1 .
Inequality (2.3) and (2.8), (2.9) and the last proposition complete the proof of the
theorem. It is not difficult to see that
I∗(2; 0) =
( ∫
µ(dqdq′)
)2
=
2πβ
mg
∥∥e−βu+
∗
∥∥2
1
. (2.12)
In the next section we’ll show that the norm in the right-hand side of the last inequality
is bounded in g (see (3.12)). Then Theorem 2.1 and the following theorem together will
prove Theorem 1.1 since e∗(g) grows at infinity weaker than e0.
Theorem 2.2. Let the conditions of Theorem 1.1. Then for arbitrary positive δ∗
there exists a positive constant C∗ independent of g such that the following inequality
holds for sufficiently large g
I∗(2; r) ≤ C∗e
βδ∗e0 .
3. Potential minima. The main result of this section are formulas (3.8), (3.12). In
order to prove them we have to introduce the new potential h and describe properties of
the minima of ug and u in terms of its minima.
u(z−1q) = z−2nh(q), ug(q) = u(zn−1q) = z−2nh(znq)
where
h(q) = ηq2n − q2n0 + h1(q), h1(q) =
n−1∑
j=1
ηjz
2(n−j)q2j = z2nu1(z−1q).
Let e′(j), µ(j), e(j) be the minima of u, h, ug , respectively. The deepest among the
sequences will be denoted by e+, µ, e0, respectively. Then
e′(j) = z−1µ(j), e(j) = z−nµ(j). (3.1)
It is so since for the equations for the minima we have
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1416 W. I. SKRYPNIK
∂u(z−1q) = ∂h(q) = 0, ∂ug(q) = ∂h(znq) = 0, ∂ =
∂
∂q
.
It is not difficult to check that
∂sug(q) = z−2n+sn∂sh(znq), ∂2ug(q) = ∂2h(znq). (3.2)
Equalities (3.1), (3.2) yield
∂sug(e(j)) = z−2n+sn∂sh(µ(j)), ∂2ug(e(j)) = ∂2h(µ(j)). (3.3)
This implies that the derivatives of ∂sug at the minima of ug tend to zero if g tends to
infinity and s > 2. This fact will play a significant role in our further consideration.
There is only one root µ of ∂h which converges to the unique positive root µ0 of ∂h0
h0(q) = ηq2n − q2n0 , ∂h0(q) = 2q2n0−1(ηnq2(n−n0) − n0)
when z tends to zero. It is simple and an analytical function of z in the neighborhod of
zero (see the first section in [15]), that is
µ = µ0 +
∑
k≥1
zkµk, µ0 =
(
n0
nη
) 1
2(n−n0)
. (3.4)
It is the deepest minimum of h since other roots µj of ∂h converge to zero for vanishing
z. For these roots there is the following convergent expansion in the neighborhod of
zero [15]
µ(j) =
∑
k≥1
z
k
lj µk,j ,
∑
s
ls = 2n0 − 1, 1 ≤ lj ≤ 2n0 − 1. (3.5)
We’ll restrict z to the neighborhod of zero, i.e., the interval [0, z0], z0 ≤ 1, where
1
2
µ0 ≤
≤ µ ≤ 2µ0, θ =
1
4
min
z∈[0,z0]
∂2h(µ) > 0. The last inequality is possible since ∂2h0(µ0) >
> 0 and h, µ are continuous in z.
The Taylor expansion for h is given by
h(q + µ) = h(µ) +
2n∑
s=2
qs
s!
∂hs(µ).
Then one can choose δ < 1 such that (h is symmetric function)
h(q ± µ) − h(µ) ≥ θq2, |q| ≤ δµ. (3.6)
Both numbers µj and h(µj) tend to zero for vanishing z, that is µj are shallow minima
with respect to the deepest minimum µ, and for sufficiently small δ we have
h(q) − h(µ) ≥ θ∗, q ∈ R\
[
(1 − δ)µ, (1 + δ)µ
]
\
[
− (1 + δ)µ,−(1 − δ)µ
]
,
(3.7)
θ∗ = min
z∈[0,z0]
(
h
(
(1 + δ)µ
)
− h(µ), h
(
(1 − δ)µ
)
− h(µ)
)
> 0.
This is obvious for the case h1 = 0. We end this section my deriving a bound for the
integral which will be used in the next section.
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LONG-RANGE ORDER IN QUANTUM LATTICE LINEAR OSCILLATOR SYSTEMS 1417
Proposition 3.1. Let ε = z2n(n−1) and z ∈ [0, z0] then for arbitrary positive num-
bers a, b there exists a positive number R independent of z such that∫
e−β(u∗(q)−av0(znq)−bεv0(q))dq ≤
≤
[√
πβ− 1
2 + (θ∗β)−
1
2nRκ+
√
π(βθ)−
1
2
]
eβ(a+b)v0(6µ0+R), (3.8)
where κ = sup
a≥0
√
ae−a.
Proof. Let us rescale the variable in the integral by z−n, use the inequality
εv0(z−nq) ≤ v0(q) and translate the variable by −µ. As result the integral is less than
z−n
∫
e−βz
−2n(h(q)−h(µ)−(a+b)z2nv0(q−µ))dq. (3.8a)
Let’ put ε instead of δ in (3.6), (3.7) and let R > 0 be such that
h(q) − h(µ) − (a+ b)v0(q − µ) ≥ q2, |q| ≥ R ≥ 1.
Let’s decompose the positive half-line into three sets (the first set is the set in the round
brackets)
R
+ = ([0, (1 − ε)µ]
⋃
[(1 + ε)µ,R])
⋃
[(1 − ε)µ, (1 + ε)µ]
⋃
[R,∞]
change sign of q in the integral in (3.8a) and make estimates of the integral over these
sets. For the integral over the second set we have
(1+ε)µ∫
(1−ε)µ
e−βz
−2n[h(q)−h(µ)−z2n(a+b)v0(q−µ)]dq =
=
εµ∫
−εµ
e−βz
−2n[h(q+µ)−h(µ)−z2n(a+b)v0(q+2µ)]dq ≤
≤ eβ(a+b)v0(6µ0)
εµ∫
−εµ
e−βz
−2n(h(q+µ)−h(µ))dq,
where we used µ ≤ 2µ0. Inequality (3.6) implies that the last integral is less than
εµ∫
−εµ
e−βθz
−2nq2dq ≤
∫
e−βθz
−2nq2dq =
√
π(βθ)−
1
2 zn. (3.9)
The integral in (3.8a) over the first set is less (due to (3.7)) than
Reβ(a+b)v0(R+2µ0)e−βθ∗z−2n
. (3.10)
The integral in (3.8a) over the third set is less than
z−n
∫
e−βz
−2nq2dq = β− 1
2
√
π. (3.11)
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1418 W. I. SKRYPNIK
Combining (3.9) – (3.11) we obtain the needed bound. Making the above decomposition
for the right half-line and repeating the same arguments we prove the needed bound.
For a = ζ, b = 3 we have
‖e−βu+
∗ ‖1 ≤ eβζ0Ceβ(ζ+3)v(6µ0+R). (3.12)
4. Proof of Theorem 2.2. In this section we derive a bound for the integral I∗(2; r).
The main result is formulated in the following theorem which proves Theorem 2.2.
Theorem 4.1. Let all the conditions, except the condition for the mass m, of The-
orem 1.1 be satisfied, g be such that z ∈ [0, z0], R1 = βθµ − 12re−1
0 > 0, R2 =
= 1 − 24rβ2(mge0)−1 > 0, θµ = µ−2θ and θ determined in (3.6). Then there exists
positive numbers Cs, s = 1, 2, 3, independent of g, m, such that the following inequality
holds for an arbitrary positive number δ∗
I∗(2; r) ≤ Ĩ(m, g)eβδ∗e0eR0 , R0 = 16m−1
g r2β3R−1
2 ,
Ĩ(m, g) =
{
C1 +m−1
g
[
C2(1 +R−1
1 ) + C3R
−1
2
]}
.
From the equalities
m−1
g ≤ g
2−n
2(n−n0) g
n−1
n−n0 m′(g)−1 = g
n
2(n−n0)m′(g)−1 = e0(µm′(g))−1,
(e0mg)−1 = (µm′(g))−1,
where µ is bounded in g, it is seen that R2 is uniformly bounded for large g, e−1
0 R0 tends
to zero in the limit of infinite g, that is the conditions of Theorem 1.1 are satisfied for
sufficiently large g (R1 is also uniformly bounded at large g since e−1
0 is small). This
concludes the proof of Theorem 2.2.
Proof of Theorem 4.1. Let the integral in prime variables in in the expression for
I∗(2; r) in (2.7) be denoted by I(q1, q2). The main idea of our bounds is to determine
the behavior of the function I(q1, q2) in a neighborhod of the critical point (−2e0, 0),
(0,−2e0) of u∗(q1) + u∗(q2) and outside. We decompose the range of integration in this
integral into several subsets Sj,l covering R
2 which are related to the behavior of u∗ near
the critical points (see (4.2), (4.3)). The restriction of the integral to the sets is denoted
by ISj,l
. We’ll show that these integrals are bounded by the products Gl(q1)Gk(q2),
Gl(q1 + 2e0)Gk(q2 + 2e0), G0(q1 + 2e0)G′(q2 + e0), k, l = 0, 1, 2, 3, of exponentially
decreasing at infinity functions Gk, G
′ with special coefficients (the coefficient before the
last product decrease exponentially in g at infinity). Then after translating the variables
by −2e0 (in the last two cases) we apply the bounds
eβre
−1
0 (q1−q2)2 ≤ e2βre
−1
0 (q21+q22), eβre
−1
0 |qj(qj±2e0)| ≤ eβre
−1
0 (q2j +2e0|qj |)|, (4.1)
and estimate the integrals
I ′j =
∫
Gj(q)eβr(3e
−1
0 q2+2|q|)dq, I ′ =
∫
G′(q)eβr(3e
−1
0 q2+2|q|)dq.
In this way we derive the left-hand side of the condition for ξ in Theorem 1.1 which is
needed to make a contribution of the set S2,1 in the asymptotics of I∗(2; r) in g moderate.
In order to simplify integrals we’ll omit r in them keeping in mind that g0 is propor-
tional to r ≥ 1. From (3.6), (3.7) (h is a symmetric function), the equalities
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LONG-RANGE ORDER IN QUANTUM LATTICE LINEAR OSCILLATOR SYSTEMS 1419
u∗(z−nq) = z−2n
(
h(q + µ) − h(µ)
)
, e0 = µz−n
we derive
u∗(q) ≥ θ∗µe
2
0,
(
|q + 2e0| ≥ δe0
) ⋂ (
|q| ≥ δe0
)
, θ∗µ = µ−2θ∗, (4.2)
u∗(q) ≥ θµq
2, |q| ≤ δe0. (4.3)
For brevity we’ll omit the index µ in θ∗, θ in the formulae of this section.
Let’s decompose the space of integration in q′1, q
′
2 into four sets Sj , j = 1, . . . , 4,
|q′1 + 2e0| ≤ δe0, |q′2 + 2e0| ≤ δe0; |q′1 + 2e0| ≥ δe0, |q′2 + 2e0| ≤ δe0,
|q′1 + 2e0| ≤ δe0, |q′2 + 2e0| ≥ δe0; |q′1 + 2e0| ≥ δe0, |q′2 + 2e0| ≥ δe0.
Every set Sj , j ≥ 2, will be, also, decomposed into subsets Sj,l with the help of the
inequalities
j = 2, l = 1, |q′1 + e0| ≥ δ; l = 2, |q′1 + e0| ≤ δ;
j = 3, l = 1, |q′2 + e0| ≥ δ; l = 2, |q′2 + e0| ≤ δ;
j = 4, l = 1, |q′1| ≤ δe0, |q′2| ≤ δe0; l = 2, |q′1| ≥ δe0, |q′2| ≤ δe0;
j = 4, l = 3, |q′1| ≤ δe0, |q′2| ≥ δe0; l = 4, |q′1| ≥ δe0, |q′2| ≥ δe0.
The integral I∗(2; r) will be denoted by I∗Sj,l
(2; r) if the integral I(q1, q2) is replaced
by ISj,l
(q1, q2). The most simple estimate is obtained for S1 neglecting a dependence on
positive φ∗
IS1 ≤
∫
S1
2∏
j=1
e−(2β)−1mg(q′j−qj)
2
e−βu
+
∗ (q′j)dq′j ≤
≤
2∏
j=1
G0(qj + 2e0)
(∫
e−βu
+
∗g(q)dq
)2
, (4.4)
where G0(qj) = e−(2β)−1mg(|qj |−δe0)2 + χ+(δe0 − |qj |). Here we used the inequalities
(qj − q′j)
2 =
(
(qj + 2e0) − (q′j + 2e0)
)2 ≥ (|qj + 2e0| − δe0)2,
|q′j + 2e0| ≤ δe0, |qj + 2e0| ≥ δe0.
Formulae (4.4) and (4.1) lead to after translating variables by −2e0
I∗S1(2; r) ≤
∥∥e−βu+
∗
∥∥2
1
I ′0
2,
(4.4a)
I ′0 ≤
∫
e3rβe
−1
0 q2e2rβ|q|e−(2β)−1mg(|q|−δe0)2dq + δe0e
βre0(3δ
2+2δ) ≤
≤ 2erβe0(3δ
2+2δ)
∫
e3rβe
−1
0 q2e2rβqe−(2β)−1mgq
2
dq + δe0.
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1420 W. I. SKRYPNIK
Here we shifted the variables in the integrals over the right half-line by δe0. Hence
I∗S1(2; r) ≤ 4
∥∥e−βu+
∗
∥∥2
1
e2rβe0(3δ
2+2δ)
[
2πβm−1
g R−1
2 eR0 + δ2e20
]
, (4.5)
where we used the elementary inequality (a+ b)2 ≤ 2(a2 + b2) and the formula
∫
e−bq
2
eaqdq =
√
b−1πe
a2
4b , a, b > 0. (4.5a)
That is, I∗S1(2; r) satisfies the main theorem bound (see (3.12)). Inequality (4.3) and the
definition of u+
∗ , v
+ (see Theorem 2.1) for the case j = 4, l = 1 yield
IS4,1 ≤
2∏
j=1
∫
|q′j |≤δe0
e−(2β)−1mg(q′j−qj)
2
e−βu
+
∗ (q′j)dq′j ≤
≤ eβv
+(e0δ)
2∏
j=1
∫
e−(2β)−1mg(q′j−qj)
2
e−βθq
′
j
2
dq′j ≤ eβv
+(δe0)
2∏
j=1
G1(qj),
(4.6)
G1(q) = e−(8β)−1mgq
2
∫
e−βθq
2
dq + e−4−1βθq2
∫
e−(2β)−1mgq
2
dq.
Here we used the inequality for two monotonic functions f, h∫
f
(
|q − q′|
)
h
(
|q′|
)
dq′ ≤ f
(
2−1|q|
)
‖h‖1 + h
(
2−1|q|
)
‖f‖1. (4.7)
To derive it we decomposed the range of integration into two sets |q′| ≥ 2−1|q|, |q′| ≤
≤ 2−1|q| and took into consideration that the functions are monotone.
For j = 4, l = 4 let’s add and subtract in the exponent the polynomial 16rβe−1
0 [σ2 +
+σ′2], σ(q) = q, and then apply (4.2) together with the Schwartz inequality in primed
variables. This leads to
IS4,4 ≤ e−βθ∗e
2
0
2∏
j=1
∫
e−(2β)−1mg(q−qj)
2
e−β(u+
∗ (q)− 1
2u∗(q))dq ≤
≤ ‖e−2β(u+
∗ − 1
2u∗−16re−1
0 σ2)‖1e
−βθ∗e20
2∏
j=1
∫
e−β
−1mg(q−qj)
2
e−32rβe−1
0 q2dq
1
2
.
From (4.7) we deduce that
IS4,4 ≤
∥∥e−2β(u+
∗ − 1
2u∗−16re−1
0 σ2)
∥∥
1
e−βθ∗e
2
0
2∏
j=1
G2(qj),
G2
2(qj) = e−8rβe−1
0 q2j
∫
e−β
−1mgq
2
dq + e−(4β)−1mgq
2
j
∫
e−32rβe−1
0 q2dq.
(4.8)
For S4,l, l = 2, 3, we have to use both (4.2), (4.3)
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LONG-RANGE ORDER IN QUANTUM LATTICE LINEAR OSCILLATOR SYSTEMS 1421
IS4,2(q1, q2) ≤ eβv
+(e0δ)eβθ∗e
2
0
∫
e−(2β)−1mg(q′j−q1)2e−β(u+
∗ (q′j)− 1
2u∗(q))dq′1×
×
∫
e−(2β)−1mg(q′2−q2)2e−βθq
′2
jdq′2 ≤
≤ eβv
+(e0δ)e−βθ∗e
2
0
∥∥e−2β(u+
∗ − 1
2u∗−16re−1
0 σ2)
∥∥ 1
2
1
G2(q1)G1(q2). (4.9)
Here we applied the same arguments as in the case j = 4, l = 4. The same inequality
holds for IS4,3(q1, q2) with the permuted variables. From the definition of φ∗ we derive
putting
φ∗(q′1, q
′
2) = φ(zn−1
(
q′1 + e0), zn−1(q′2 + e0)
)
≥
≥ z2(n−1)(n1+n2)(q′1 − q′2)
2n1×
×
[
(q′1 + e0)2n
′
2(q′2 + e0)2(n2−n′
2) + (q′2 + e0)2n
′
2(q′1 + e0)2(n2−n′
2)
]
.
As a result for q′1, q
′
2 ∈ S2,1 we have (1 − δ > δ)
φ∗(q′1, q
′
2) ≥
≥ (|q′1 + 2e0| − δe0)2n1
[
δ2(n2−n′
2)
(
(1 − δ)e0
)2n′
2 + δ2n′
2
(
(1 − δ)e0
)2(n2−n′
2)
]
≥
≥
(
|q′1 + 2e0| − δe0
)2n1
z1,
z1 = z2(n−1)(n1+n2)δ2n2e
2n′
2
0 .
This yields after applying the Schwartz inequality
IS2,1(q1, q2) ≤
≤
∥∥e−βu+
∗
∥∥
1
G0(q2 + 2e0)
∫
e−(2β)−1mg(q′−q1)2e−βrz1(|q
′+2e0|−δe0)2n1
e−βu
+
∗ (q′)dq′ ≤
≤
∥∥e−βu+
∗
∥∥
1
∥∥e−βu+
∗
∥∥
2
G0(q2 + 2e0)×
×
(∫
e−β
−1mg(q′−q1)2e−βrz1(|q
′+2e0|−δ)2n1
dq′
)1
2
≤
≤
∥∥e−βu+
∗
∥∥
1
∥∥e−βu+
∗
∥∥
2
G0(q2 + 2e0)G3(q1 + 2e0),
(4.10)
G2
3(q) =
(
e−βrz1(2
−1|q|−δe0)2n1 + χ+(δe0 − 2−1|q|
) ∫
e−(2β)−1mgq
′2
dq′+
+e−(8β)−1mgq
2
(∫
e−βrz1(|q
′|−δe0)2n1
dq′ + δe0
)
.
Here we applied (4.7) once more and put
f(q) = e−(2β)−1mg(q′−q)2 ,
h(q) = e−βrz1(|q|−δe0)
2n1
χ+(|q| − δe0) + χ+(δe0 − |q|).
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1422 W. I. SKRYPNIK
Here h is a monotonic function. For S2,2 we derive
IS2,2(q1, q2) ≤
≤ ‖e−βu+
∗ ‖1G0(q2 + 2e0)
∫
|q′|≤δ
e−(2β)−1mg(q′−q1−e0)2e−βu
+
∗ (q′−e0)dq′ ≤
≤ ‖e−βu+
∗ ‖1e
β(ug(e0)+v
+(e0+δ))
( ∫
|q′|≤δ
e−βug(q′)dq′
)
G0(q2 + 2e0)G′(q1 + e0),
(4.11)
G′(q) = (e−(2β)−1mg(|q|−δ)2 + χ+(−|q + e0| + δ)).
Function ug is a bounded in g, hence the integral in the second line in (4.11) is bounded
in g ≥ 1. Here one has to check that ug tends to a finite limit when g tends to infinity.
This follows from the fact that the coefficient before q2n0 in the expression for ug is equal
to g1−n0(n−1)
n−n0 = g−
(n0−1)n
n−n0 , n0 ≥ 1.
The same bounds are obtained for IS3,1 , IS3,2 by permuting variables in the bounds
(4.10), (4.11) for IS2.1 , IS2,2 . Now we can estimate I∗Sj,l
(2; r). For j = 4, l = 1 from
(4.6), (4.5a) and
−12re−1
0 + βθ = R1 > 0, −24β2r(e0mg)−1 + 1 = R2 > 0 (4.12)
it follows that
I∗S4,1(2; r) ≤ I ′1
2 ≤ 416π2(θmg)−1R−1
2 eR0 + 2π2βm−1
g R−1
1 e4r
2β2R−1
1 . (4.13)
For j = 4, l = 4 from (4.8) and the first inequality in (4.12) it follows that
I∗S4,4(2; r) ≤
∥∥e−2β(u+
∗ − 1
2u∗−16re−1
0 σ2)
∥∥
1
e−βθ∗e
2
0I ′2
2 ≤
≤ 2
∥∥e−2β(u+
∗ − 1
2u∗−16re−1
0 σ2)
∥∥
1
e−βθ∗e
2
0×
×
[
4πβ
mg
(∫
e−rβ(e−1
0 q2−2|q|)dq
)2
+ π2e0m
−1
g (8rR2)−1eR0
]
. (4.14)
Here we used the inequality
√
a+ b ≤ √
a +
√
b, (a + b)2 ≤ 2(a2 + b2), a, b ≥ 0 and
(4.5a). By Proposition 3.1 the norm in this inequality behaves as exp{ae0}, a > 0. The
same is true for the integral inside the square brackets. This and the dependence of mg
on g mean that I∗S4,4(2; r) is bounded in g. The analog of (4.14) is easily obtained for
I∗S4,l
(2; r), l = 2, 3,
I∗S4,l
(2; r) ≤ 2e−βθ∗e
2
0
[
I ′1
2 + I ′2
2
]
.
From (4.13), (4.14) it follows that the integrals I∗S4,l
(2; r), l = 2, 3, are bounded in g.
For j = 2, l = 1 we have (see (4.10))
I∗S2,1(2; r) ≤ 2‖e−βu∗g‖1‖e−βu∗g‖2
[
I ′0
2 + I ′3
2
]
,
I ′3
2 ≤ 2βπ
mg
[( ∫
|q|≤2δe0
erβ(3e−1
0 q2+2|q|)dq
)2
+
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
LONG-RANGE ORDER IN QUANTUM LATTICE LINEAR OSCILLATOR SYSTEMS 1423
+e2β(3δ2+2δ)e0
(( ∫
e−βrz1(2
−1q)2n1
eβ(3e−1
0 q2+2|q|)dq
)2
+
+R−1
2 eR02(βrz1)
− 1
n1
∫
e−q
′2n1
dq′
)]
. (4.15)
Here we performed the operations, given at the beginning of the proof, and applied the
formulas
√
a1 + a2 + a3 ≤ √
a1 +
√
a2 +
√
a3,
(a1 + a2 + a3)2 ≤ 23(a2
1 + a2
2 + a2
3)
translated, rescaled and translated the variable by ±2δe0 in the integrals (over R
+) cor-
responding to the last and first terms, respectively, in the expression for G3. The second
integral in the right-hand side of (4.15) can be estimated by taking the product of maxi-
mums of two exponents, depending on the quadratic and linear terms, multiplied by the
root of the third order of the first exponent. Hence, it is less than
2
(
max
q≥0
e−3−1βrz1(2
−1q)2n1
e2βq
)
×
×
(
max
q≥0
e−3−1βrz1e
n1
0 (2−1q)n1
e3βq
) (
3−1βrz1
)− 1
n1
∫
e−q
′2n1
dq′.
In the second round bracket we rescaled the variable by
√
e0. The firsts and the second
terms grow in g as the exponent of z
− 1
2n1−1
1 , (z1en1
0 )−
1
n1−1 , respectively. This growth has
to be weaker than the growth of e0, i.e. z−n. That is, −ξ + 2(n− 1)(n1 + n2)− 2n′
2n <
< (2n1 − 1)n,
−ξ + 2(n− 1)(n1 + n2) − 2n′
2n− n1n < (n1 − 1)n.
Both conditions are identical to the condition of Theorem 4.1. The second and last terms
in (4.15) grow as e0 to some power. Thus I∗S2,1(2; r) satisfies the theorem main bound
(provided (4.12) holds). For the case j = 2, l = 2 we obtain with the help of (4.1)
and (4.11)
I∗S2,2(2; r) ≤
∥∥e−βu+
∗
∥∥
1
eβ(ug(e0)+v
+(e0+δ))
( ∫
|q′|≤δ
e−βug(q′)dq′
)
×
×2
[
I ′0
2 +
∫
e−(2β)−1mg(|q+e0|−δ)2erβ(3e−1
0 q2+2|q|)dq)2
]
. (4.16)
After translations of the variables in two integrals the term in the square brackets is less
than
e32rβe0
{
I ′0
2 + 4erβ(6δ2+2δ)
[(∫
e−(2β)−1mgq
2
erβ(3e−1
0 q2+2q)dq
)2
+ δ2
]}
. (4.17)
Both integrals were estimated considering the case j = 1. From (4.5) and (4.16), (4.17)
it follows that I∗S2,2(2; r) is bounded in g ≥ 1 since ug(e0) = z−2nh(µ) = e20µ
−2h(µ),
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1424 W. I. SKRYPNIK
h(µ) < 0 and v+(e0 + δ), v+(e0δ) are bounded in g. Moreover, the norms in (4.10) are
bounded in g >> 1. It implies, also, that the norm in (4.14) grows at in infinity in g as
eβ32(6µ0)2e0 . Hence all I∗Sj,l
(2; r) satisfy the theorem main bound. This implies that the
theorem is proven since I∗(2; r) =
∑
j,l
I∗Sj,l
(2; r) and it is not difficult to calculate
Cs from (4.5), (4.13), (4.15) – (4.17): (4.11) and the bound after (4.14) , i.e., bounds for
I∗Sj,2(2), j = 2, 3, and I∗S4,l
(2; r), l = 2, 3, 4, generate the expression for C1; (4.5), i.e.,
the bound for I∗S1(2) gives expressions for C1, C3 (e20 is less than eδ∗e0 multiplied by a
coefficient depending on δ∗); (4.13) and (4.15), i.e., bounds for I∗S4,1(2; r) I∗S2,1(2; r),
respectively, give expressions for C2, C3.
Remark 4.1. In a general case a depth of the polynomial external potential is con-
trolled by n− 1 parameters. The different subsets of this space can be described with the
help of one parameter g and the representation u(q) = ηq2n −
∑n−1
s=1
ηsg
nsq2s. To deal
with this case one has to find an appropriate re-scaling reducing the polynomial to a sum
of a polynomial with easily calculated minima and an additional polynomial depending
on g−1.
1. Ruelle D. Probability estimates for continuous spin systems // Communs Math. Phys. – 1976. – 50. –
P. 189 – 194.
2. Ruelle D. Statistical mechanics. Rigorous results. – W. A. Benjamin Inc., 1969. – 219 p.
3. Ruelle D. Super-stable interactions in classical statistical mechanics // Communs Math. Phys. – 1970. –
18. – P. 127 – 159.
4. Petrina D. Ya., Gerasimenko V. I., Malyshev P. V. Mathematical foundations of classical statistical me-
chanics. – Holland: Gordon and Breach, 1989 (Kyiv: Naukova Dumka, 1985).
5. Park Y. M., Yoo H. J. Uniqueness and clustering properties of Gibbs states for classical and quantum
unbounded spin systems // J. Stat. Phys. – 1995. – 80, # 1/2. – P. 223 – 271.
6. Khoruzhenko B., Pastur L. Phase transitions in quantum models of rotators and ferroelectrics // Teor. i
Mat. Fiz. – 1987. – 73. – P. 111 – 124.
7. Skrypnik W. LRO in lattice systems of linear classical and quantum oscillators. Strong nearest-neighbor
pair quadratic interaction // J. Stat. Phys. – 2000. – 100, # 5/6. – P. 853 – 870.
8. Skrypnik W. Long-range order in non-equilibrium systems of interacting Brownian linear oscillators //
Ibid. – 2002. – 111, # 1/2. – P. 291 – 321.
9. Verbeure A., Zagrebnov V. No-go theorem for quantum structural phase transitions // J. Phys. A. – 1995. –
28. – P. 5415 – 5421.
10. Minlos R., Verbeure A., Zagrebnov V. A quantum crystal model in the light-mass limit: Gibbs states // Rev.
Math. Phys. – 2000. – 12. – P. 981 – 1032.
11. Skrypnik W. I. Long-range order in Gibbs lattice classical linear oscillator systems // Ukr. Math. J. – 2006.
– 58, # 3. – P. 388 – 405.
12. Frohlich J., Lieb E. Phase transitions in anisotropic lattice spin systems // Communs Math. Phys. – 1978.
– 60. – P. 233 – 267.
13. Skrypnik W. I. Long-range order in linear ferromagnetic oscillator systems. Strong pair quadratic n-n
potential // Ukr. Math. J. – 2004. – 56, # 6. – P. 810 – 817.
14. Glimm J., Jaffe A., Spencer T. Phase transitions for ϕ4
2 quantum fields // Communs Math. Phys. – 1975. –
45. – P. 203 – 216.
15. Reed M., Simon B. Methods of modern mathematical physics. – New York etc.: Acad. Press, 1975. –
Vol. 2.
16. Petrina D. Ya. Mathematical foundations of quantum statistical mecanics. Continuous systems // Math.
Phys. Stud. – Kluwer Acad. Publ., 1995. – 17.
Received 23.08.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
|
| id | umjimathkievua-article-3542 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:44:28Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/08/fb9443e6a238ac053dd0e10bcf9bec08.pdf |
| spelling | umjimathkievua-article-35422020-03-18T19:57:10Z Long-range order in quantum lattice systems of linear oscillators Далекий порядок у квантових граткових системах лінійних осциляторів Skrypnik, W. I. Скрипник, В. І. The existence of the ferromagnetic long-range order is proved for equilibrium quantum lattice systems of linear oscillators whose potential energy contains a strong ferromagnetic nearest-neighbor (nn) pair interaction term and a weak nonferromagnetic term under a special condition on a superstability bound. It is shown that the long-range order is possible if the mass of a quantum oscillator and the strength of the ferromagnetic nn interaction exceed special values. A generalized Peierls argument and a contour bound, proved with the help of a new superstability bound for correlation functions, are our main tools. Для рівноважних квантових ґраткових систем лінійних осциляторів, потенціальна енергія яких містить сильну феромагнітну частину парної взаємодії близьких сусідів і слабку неферомагнітну частину, доведено існування феромагнітного далекого порядку при певній умові на нерівність суперстійкості. Показано, що далекий порядок може мати місце, якщо маса квантового осцилятора та сила феромагнітної взаємодії близьких сусідів перевищують певні значення. При цьому використано узагальнений принцип Пайєрлса та контурну нерівність, доведену з допомогою нової нерівності суперстійкості для кореляційних функцій. Institute of Mathematics, NAS of Ukraine 2006-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3542 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 10 (2006); 1407–1424 Український математичний журнал; Том 58 № 10 (2006); 1407–1424 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3542/3820 https://umj.imath.kiev.ua/index.php/umj/article/view/3542/3821 Copyright (c) 2006 Skrypnik W. I. |
| spellingShingle | Skrypnik, W. I. Скрипник, В. І. Long-range order in quantum lattice systems of linear oscillators |
| title | Long-range order in quantum lattice systems of linear oscillators |
| title_alt | Далекий порядок у квантових граткових системах лінійних осциляторів |
| title_full | Long-range order in quantum lattice systems of linear oscillators |
| title_fullStr | Long-range order in quantum lattice systems of linear oscillators |
| title_full_unstemmed | Long-range order in quantum lattice systems of linear oscillators |
| title_short | Long-range order in quantum lattice systems of linear oscillators |
| title_sort | long-range order in quantum lattice systems of linear oscillators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3542 |
| work_keys_str_mv | AT skrypnikwi longrangeorderinquantumlatticesystemsoflinearoscillators AT skripnikví longrangeorderinquantumlatticesystemsoflinearoscillators AT skrypnikwi dalekijporâdokukvantovihgratkovihsistemahlíníjnihoscilâtorív AT skripnikví dalekijporâdokukvantovihgratkovihsistemahlíníjnihoscilâtorív |