Generalized inequalities for the Bogoliubov-Duhamel inner product with applications in the Approximating Hamiltonian Method
Infinite sets of inequalities which generalize all the known inequalities that can be used in the majorization step of the Approximating Hamiltonian method are derived. They provide upper bounds on the difference between the quadratic fluctuations of intensive observables of a N-particle system and...
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Інститут фізики конденсованих систем НАН України
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irk-123456789-1198012019-08-30T23:26:25Z Generalized inequalities for the Bogoliubov-Duhamel inner product with applications in the Approximating Hamiltonian Method Brankov, J.G. Tonchev, N.S. Infinite sets of inequalities which generalize all the known inequalities that can be used in the majorization step of the Approximating Hamiltonian method are derived. They provide upper bounds on the difference between the quadratic fluctuations of intensive observables of a N-particle system and the corresponding Bogoliubov-Duhamel inner product. The novel feature is that, under sufficiently mild conditions, the upper bounds have the same form and order of magnitude with respect to N for all the quantities derived by a finite number of commutations of an original intensive observable with the Hamiltonian. The results are illustrated on two types of exactly solvable model systems: one with bounded separable attraction and the other containing interaction of a boson field with matter. Отримано нескiнченнi набори нерiвностей, як i узагальнюють всi вiдомi нерiвностi, що можуть бути використанi на етапi мажорування методу апроксимуючого гамiльтонiану. Вони забезпечують верхнi границi на рiзницю мiж квадратичними флуктуацiями iнтенсивних спостережуваних N-частинкової системи i вiдповiдного внутрiшнього добутку Боголюбова-Дюамеля. Новою рисою є те, що при достатньо м’яких умовах верхнi границi мають однакову форму i порядок величини по вiдношенню до N для всiх величин, отриманих шляхом скiнченного числа перестановок початкової iтенсивної спостережуваної з гамiльтонiаном. Результати iлюструються на двох типах точно розв’язуваних моделей: однiєї з обмеженим сепарабельним притяганням та iншої, що мiстить взаємодiю бозонного поля з матерiєю. 2011 Article Generalized inequalities for the Bogoliubov-Duhamel inner product with applications in the Approximating Hamiltonian Method / J.G. Brankov, N.S. Tonchev // Condensed Matter Physics. — 2011. — Т. 14, № 1. — С. 13003: 1-17. — Бібліогр.: 37 назв. — англ. 1607-324X PACS: 05.30.Rt, 64.60.-i, 64.60.De, 64.70.Tg DOI:10.5488/CMP.14.13003 arXiv:1101.2882 http://dspace.nbuv.gov.ua/handle/123456789/119801 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Infinite sets of inequalities which generalize all the known inequalities that can be used in the majorization step of the Approximating Hamiltonian method are derived. They provide upper bounds on the difference between the quadratic fluctuations of intensive observables of a N-particle system and the corresponding Bogoliubov-Duhamel inner product. The novel feature is that, under sufficiently mild conditions, the upper bounds have the same form and order of magnitude with respect to N for all the quantities derived by a finite number of commutations of an original intensive observable with the Hamiltonian. The results are illustrated on two types of exactly solvable model systems: one with bounded separable attraction and the other containing interaction of a boson field with matter. |
| format |
Article |
| author |
Brankov, J.G. Tonchev, N.S. |
| spellingShingle |
Brankov, J.G. Tonchev, N.S. Generalized inequalities for the Bogoliubov-Duhamel inner product with applications in the Approximating Hamiltonian Method Condensed Matter Physics |
| author_facet |
Brankov, J.G. Tonchev, N.S. |
| author_sort |
Brankov, J.G. |
| title |
Generalized inequalities for the Bogoliubov-Duhamel inner product with applications in the Approximating Hamiltonian Method |
| title_short |
Generalized inequalities for the Bogoliubov-Duhamel inner product with applications in the Approximating Hamiltonian Method |
| title_full |
Generalized inequalities for the Bogoliubov-Duhamel inner product with applications in the Approximating Hamiltonian Method |
| title_fullStr |
Generalized inequalities for the Bogoliubov-Duhamel inner product with applications in the Approximating Hamiltonian Method |
| title_full_unstemmed |
Generalized inequalities for the Bogoliubov-Duhamel inner product with applications in the Approximating Hamiltonian Method |
| title_sort |
generalized inequalities for the bogoliubov-duhamel inner product with applications in the approximating hamiltonian method |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119801 |
| citation_txt |
Generalized inequalities for the Bogoliubov-Duhamel inner product with applications in the Approximating Hamiltonian Method / J.G. Brankov, N.S. Tonchev // Condensed Matter Physics. — 2011. — Т. 14, № 1. — С. 13003: 1-17. — Бібліогр.: 37 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT brankovjg generalizedinequalitiesforthebogoliubovduhamelinnerproductwithapplicationsintheapproximatinghamiltonianmethod AT tonchevns generalizedinequalitiesforthebogoliubovduhamelinnerproductwithapplicationsintheapproximatinghamiltonianmethod |
| first_indexed |
2025-07-08T16:37:56Z |
| last_indexed |
2025-07-08T16:37:56Z |
| _version_ |
1837097474917924864 |
| fulltext |
Condensed Matter Physics 2011, Vol. 14, No 1, 13003: 1–17
DOI:10.5488/CMP.14.13003
http://www.icmp.lviv.ua/journal
Generalized inequalities for the Bogoliubov-Duhamel
inner product with applications in the Approximating
Hamiltonian Method ∗
J.G. Brankov1,2 †, N.S. Tonchev3
1 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,
141980 Dubna, Russian Federation
2 Institute of Mechanics, Bulgarian Academy of Science, 4 G. Bonchev Str., 1113 Sofia, Bulgaria
3 Institute of Solid State Physics, Bulgarian Academy of Science, 72 Tzarigradsko Chaussee Blvd.,
1784 Sofia, Bulgaria
Received November 29, 2010
Infinite sets of inequalities which generalize all the known inequalities that can be used in the majorization step
of the Approximating Hamiltonian method are derived. They provide upper bounds on the difference between
the quadratic fluctuations of intensive observables of a N -particle system and the corresponding Bogoliubov-
Duhamel inner product. The novel feature is that, under sufficiently mild conditions, the upper bounds have
the same form and order of magnitude with respect to N for all the quantities derived by a finite number of
commutations of an original intensive observable with the Hamiltonian. The results are illustrated on two types
of exactly solvable model systems: one with bounded separable attraction and the other containing interaction
of a boson field with matter.
Key words: correlation functions, Bogoliubov-Duhamel inner product, statistical-mechanical inequalities,
approximating Hamiltonian method, exactly solved models
PACS: 05.30.Rt, 64.60.-i, 64.60.De, 64.70.Tg
1. Introduction
The Approximating Hamiltonian Method (AHM) provides a rigorous approach to the study of
some classes of statistical mechanical systems in the thermodynamic limit. The method consists of
the following interrelated ingredients:
(i) Description of classes of model systems which admit a rigorous treatment in terms of a more
simple approximating Hamiltonian;
(ii) Rules according to which the approximating Hamiltonian is constructed from the original
one;
(iii) Mathematical techniques for derivation of bounds which prove the thermodynamic equiva-
lence of the approximating and original Hamiltonians;
(iv) Investigation of the thermodynamic and statistical properties of the system described by the
approximating Hamiltonian.
For the first time, the idea of the Approximating Hamiltonian method (AHM) has been sug-
gested on a heuristic level by N.N. Bogoliubov in his paper on the theory of the weakly non-ideal
Bose-gas [1]. He conjectured that, under the existence of Bose condensate in the system, the nor-
malized (by the square root of the volume) creation/annihilation operators for bosons with zero
∗The paper is dedicated to the 70th jubilee of N.N. Bogoliubov Jr.
†E-mail: brankov@theor.jinr.ru
c© J.G. Brankov, N.S. Tonchev 13003-1
http://dx.doi.org/10.5488/CMP.14.13003
http://www.icmp.lviv.ua/journal
J.G. Brankov, N.S. Tonchev
momentum can be replaced by c-numbers in the model Hamiltonian. The value of these complex-
conjugate numbers was determined by thermodynamic arguments. Thus, the initial model Hamil-
tonian was replaced by an approximating (trial) one, which has the advantage to be easily diago-
nalized by a canonical u-v transformation. The further development of the AHM took place in the
framework of the Bardeen-Cooper-Schrieffer (BCS) reduced model of superconductivity [2]. In 1957
the corresponding approximating Hamiltonian was suggested and diagonalized by means of a u-v
canonical transformation [3]. A little later, N.N. Bogoliubov [4] rigorously proved that the ground
state energy density and the zero-temperature Green functions for the model and approximating
Hamiltonians coincide in the thermodynamic limit.
The foundations of the modern formulation of the AHM have been laid down by N.N. Bo-
goliubov (Jr.), see [5] and references therein. The essential generalization to classes of quantum
systems with separable interaction, considered at nonzero temperatures, has been achieved by us-
ing the Bogoliubov variational principle for the free energy density and a special majorization
technique based on integration over external sources. Some major restrictions on the applicability
of the method, such as the quadratic form of the interaction Hamiltonian and its boundedness,
were removed by further extensions of the AHM, see the review article [6] and the books [7, 8].
In this paper we generalize all the known inequalities which have been, or could have been,
used in the majorization step (iii) of the AHM. They set different lower and upper bounds on
the quadratic fluctuations, proportional to the difference between the original and approximating
Hamiltonians, by terms involving new functionals of which the Bogoliubov-Duhamel inner product
is a special case. Their place and role (not only in the AHM) is illustrated by the derivation of
some new consequences pertaining to the infinitely coordinated anisotropic Heisenberg model and
the Dicke superradiance model. These inequalities can be used for estimating the closeness of the
Gibbs average values to the corresponding Bogoliubov-Duhamel inner product for a special class
of observables. In some sense they are complementary to the Bogoliubov inequality [9] which has
been used to exclude conventional superfluid and superconducting long-range order in one- and two-
dimensional systems with gage invariant interactions. By exploiting it for setting upper bounds
on the spontaneous magnetization or sublattice magnetization, Mermin and Wagner [10] have
rigorously proved the absence of long-range order in one- and two-dimensional isotropic Heisenberg
models with finite-range interactions. Harris [11] has derived a lower bound on the symmetrized
average value of the product of an operator A and its conjugate A†, which is a special case of
the Bogoliubov inequality, but turns out to be sufficient for the derivation of the result of Mermin
and Wagner. Alternative inequalities, setting a lower bound on the Bogoliubov-Duhamel inner
product, have been used in the proof of the existence of spontaneous magnetization in a variety of
quantum spin systems [12–15]. In the physical literature, the Bogoliubov-Duhamel inner product
is also known as the Bogoliubov-Kubo-Mori scalar product. In addition, it plays an important role
in the linear response theory [16], Kondo problem [12], the so-called parameter estimation problem
in quantum statistical mechanics and noncommutative probability theory [17], see also [18] and
references therein.
2. Systems with bounded separable attraction
Here we consider a special class of quantum statistical models which are defined initially in
a finite region Λ of the d-dimensional Euclidean space Rd or the integer lattice Zd. By |Λ| we
denote the volume of Λ in the first case, or the number of lattice sites in the latter case. The
Hamiltonian HΛ is defined as a self-adjoint operator in a separable Hilbert space H, and the
corresponding free energy density fΛ[HΛ] is assumed to exist. For the sake of simplicity, we do
not explicitly distinguish between a Hamiltonian HΛ, describing a system with fixed number of
particles N in Λ, and the statistical operator HΛ−µN in the grand canonical ensemble, where µ is
the chemical potential and N is the number operator of particles. The density of the corresponding
thermodynamic potential is given by
fΛ[HΛ] = −(β|Λ|)−1 lnZ[HΛ], (2.1)
13003-2
Inequalities in the AHM
where Z[HΛ] is the partition function; in all cases the thermodynamic limit is denoted by t− lim.
The norm of a bounded operator A is ‖A‖, the symbol A# remains for both the operator A and
its adjoint A†. As usual, [A,B] = AB − BA is the commutator of two operators. Average values
in the Gibbs ensemble with the Hamiltonian HΛ are defined as
〈· · · 〉HΛ
≡ Tr(e−βHΛ · · · )/Z[HΛ]. (2.2)
Let the Hamiltonian of the system in Λ be defined as a sum of two self-adjoint operators,
HΛ = TΛ + UΛ , (2.3)
where TΛ = T †
Λ is a trace-class operator which generates the Gibbs semigroup {exp(−βTΛ)}β>0. In
addition, we impose the condition that the density of the thermodynamic potential corresponding
to TΛ is bounded uniformly with respect to the volume of the system |Λ|,
|fΛ[TΛ]| 6 M0 . (2.4)
A distinguishing feature of the models with bounded separable attraction is that the interaction
Hamiltonian can be written as an extensive self-adjoint operator of the form [5]
UΛ = −|Λ|
n
∑
s=1
gsAΛ,sA
†
Λ,s . (2.5)
Here gs > 0, s = 1, . . . , n, are interaction parameters and the intensive observables AΛ,s, A
†
Λ,s
represent uniformly bounded local operators averaged over a region of the space (real space or
conjugate momentum space). The uniform boundedness of these operators,
‖AΛ,s‖ = ‖A†
Λ,s‖ 6 M1 , s = 1, . . . , n, (2.6)
where the constant M1 is independent of the volume |Λ|, is essential for the applicability of the
method in the case under consideration. In the general framework one does not need explicit
expressions for the operators TΛ and AΛ,s. It suffices to impose, in addition to (2.4) and (2.6), the
following general constraints (s, s′ = 1, . . . , n):
‖[AΛ,s, TΛ]‖ 6 M1,T , (2.7)
‖[AΛ,s, A
#
Λ,s′ ]‖ 6 M2|Λ|
−1. (2.8)
The heuristic rule for construction of the approximating Hamiltonian H
(0)
Λ (a) consists in lin-
earization of the original interaction Hamiltonian with respect to the deviations of the intensive
operators AΛ,s, A
†
Λ,s from some complex numbers:
H
(0)
Λ (a) = TΛ − |Λ|
n
∑
s=1
gs(asA
†
Λ,s + a⋆sAΛ,s − asa
⋆
s ), (2.9)
where a = (a1, . . . , an) ∈ C
n, and a⋆s is the complex conjugate of as. The latter numbers are
considered to be variational parameters, chosen so as to minimize the contribution in the free
energy density of the residual interaction Hamiltonian
H
(1)
Λ (a) ≡ HΛ −H
(0)
Λ (a) = −|Λ|
n
∑
s=1
gs(AΛ,s − as)(A
†
Λ,s − a⋆s ) 6 0. (2.10)
The main result of the AHM for this class of model systems is summarized in the absolute
minimum principle for the approximating free energy density:
0 6 min
a
fΛ[H
(0)
Λ (a)]− fΛ[HΛ] 6 ǫ(|Λ|), (2.11)
13003-3
J.G. Brankov, N.S. Tonchev
where ǫ(|Λ|) → 0 as |Λ| → ∞. This establishes the thermodynamic equivalence of the free energy
densities for the model Hamiltonian (2.3), (2.5), and the approximating one (2.9).
The proof of (2.11) is carried out with the majorization technique developed by Bogoliubov Jr. [5].
It starts with the introduction of auxiliary external fields νs and ν⋆s conjugate to the operators A†
Λ,s
and AΛ,s, respectively,
HΛ(ν) ≡ HΛ−|Λ|
n
∑
s=1
(νsA
†
Λ,s+ν⋆sAΛ,s), H
(0)
Λ (a, ν) ≡ H
(0)
Λ (a)−|Λ|
n
∑
s=1
(νsA
†
Λ,s+ν⋆sAΛ,s). (2.12)
Next, lower and upper bounds on the difference in the free energy densities for the original and
approximating Hamiltonians are set by the Bogoliubov variational principle and the inequalities
following from it. In view of the non-positive definiteness of the residual interaction Hamiltonian
H
(1)
Λ (a) ≡ HΛ(ν)−H
(0)
Λ (a, ν), the application of the Bogoliubov inequalities yields the bounds
0 6 min
a
fΛ[H
(0)
Λ (a, ν)]− fΛ[HΛ(ν)]
6
n
∑
s=1
gs〈(AΛ,s − 〈AΛ,s〉HΛ(ν))(A
†
Λ,s − 〈A†
Λ,s〉HΛ(ν))〉HΛ(ν) , (2.13)
valid for all complex fields νs, s = 1, 2, . . . n.
The fact that the absolute minimum of the free energy density fΛ[H
(0)
Λ (a, ν))] is attained at a
finite value a = āΛ(ν), which depends on the thermodynamic parameters of the Gibbs ensemble
as well as on the size and shape of the domain Λ, has been proved by using the conditions (2.4)
and (2.6).
Here it is in place to mention that the attempt to prove directly that the correlation functions
in the right-hand side of inequalities (2.13) tends to zero as |Λ| → ∞ may happen to be an
impossible task. An efficient means for solving the problem provides the majorization technique
of Bogoliubov (Jr.), see [5]. Instead of direct evaluation of the above correlation function, it uses
inequalities which set upper bounds to averages of the form
〈δAΛ,sδA
†
Λ,s〉HΛ(ν) , δAΛ,s ≡ AΛ,s − 〈AΛ,s〉HΛ(ν) , (2.14)
in terms involving second derivatives of the free energy density with respect to external fields νs:
∂2fΛ[HΛ(ν)]
∂ν⋆s νs
= −β|Λ|(δAΛ,s; δAΛ,s). (2.15)
Here (A;B) denotes the Bogoliubov-Duhamel inner product on the algebra of observables A, B, . . . ,
defined as
(A;B)H ≡ (Z[H])−1
1
∫
0
dτ Tr
[
e−β(1−τ)HA†e−βτHB
]
. (2.16)
In the remainder we consider the given observables A,B,C, . . . and fixed Hamiltonian H per-
taining to a quantum system in a finite region Λ. Whenever no confusion could arise, for brevity
of notation we will omit the subscripts H and Λ and the argument of the partition function Z.
2.1. The Bogoluibov-Duhamel inner product
The general properties of the functional (2.16) are considered in the book [19] and in the arti-
cles [12–14, 17, 18]. We warn the reader that some authors use definitions which differ from (2.16)
by a factor of β and/or by involving the operator A instead of its adjoint A†. For our purposes
it suffices to mention the following. The inner product (A;B) is conjugate symmetric, (A;B) =
(B†;A†), antilinear (linear) in the first (second) argument, (A + αC;B) = (A;B) + α⋆(C;B)
((A;B + αC) = (A;B) + α(A;C)), it satisfies the relationship
β(A; [H, B])H = 〈[A†, B]〉H . (2.17)
13003-4
Inequalities in the AHM
The inner product (A;A) is nonnegative, (A;A) > 0, and convex,
(A;A) 6 (1/2)〈AA† +A†A〉. (2.18)
Finally, in the case when either A or B (or both) commute with the Hamiltonian, the inner product
(A;B) reduces to the bilinear complex-valued functional
〈A;B〉H = 〈A†B〉H . (2.19)
Note that in some works the operator metric in the algebra of physical observables defined
by (2.19) is called Kubo-Martin-Schwingermetric, and the one defined by (2.16) is called Bogoliubov-
Kubo-Mori metric [18, 20].
The further considerations are conveniently carried out by using the spectral representation of
the Bogoliubov-Duhamel inner product. We assume that the Hamiltonian H has a simple discrete
spectrum only, {En, n = 1, 2, 3 . . .} and denote the corresponding eigenfunctions by |n〉, i.e.,H|n〉 =
En|n〉, n = 1, 2, 3 . . . . By Amn = 〈m|A|n〉 we denote the corresponding matrix element of an
operator A. Then, the right-hand side of (2.16) can be written as
(A;B)H = (ZΛ[H])−1
∑
m,n
′A∗
mnBmn
e−βEm − e−βEn
β(En − Em)
+ (ZΛ[H])−1
∑
n
e−βEnA∗
nnBnn , (2.20)
where the prime in the double sum means that the term n = m is excluded.
Our aim is to majorize the quadratic fluctuations
〈δA†δA〉 = 〈A†A〉H − |〈A〉|2 (2.21)
by terms proportional to some power of the inner product
(δA; δA) = (A;A) − |〈A〉|2
= Z−1
Λ
∑
m,n
′|Amn|
2 e
−βEm − e−βEn
β(En − Em)
+ (ZΛ[H])−1
∑
n
e−βEn |Ann|
2 − |〈A〉|2. (2.22)
Instead of (2.21) it is more convenient to consider the symmetrized form
1
2
〈δA†δA+ δAδA†〉 =
1
2
〈A†A+AA†〉 − |〈A〉|2. (2.23)
Here we have
1
2
〈A†A+AA†〉 = Z−1
∑
m,n
e−βEn
1
2
(|Amn|
2 + |Anm|2)
= Z−1
∑
m,n
′|Amn|
2 1
2
(e−βEn + e−βEm) + Z−1
∑
n
|Ann|
2e−βEn . (2.24)
By comparing equations (2.22), (2.23) and (2.24) we obtain
1
2
〈A†A+AA†〉 − (A;A) = Z−1
∑
m,n
′|Amn|
2
{
1
2
(e−βEn + e−βEm)−
e−βEm − e−βEn
β(En − Em)
}
. (2.25)
Now, by using the identity
e−βEm + e−βEn = (e−βEn − e−βEm) coth
β(Em − En)
2
(2.26)
we can express the difference (2.25) in two equivalent forms:
1
2
〈A†A+AA†〉 − (A;A) = Z−1
∑
m,n
′|Amn|
2 e
−βEm − e−βEn
β(En − Em)
(Xmn cothXmn − 1), (2.27)
13003-5
J.G. Brankov, N.S. Tonchev
and
1
2
〈A†A+AA†〉 − (A;A) = Z−1
∑
m,n
′|Amn|
2 1
2
(e−βEn + e−βEm)
(
1−
1
Xmn cothXmn
)
, (2.28)
where
Xmn =
1
2
β(Em − En). (2.29)
The application of the elementary inequality x cothx > 1 to the right-hand side of (2.27)
immediately yields the convexity property (2.18).
Different choices of the upper bound on the right-hand side of (2.27) generate different inequal-
ities. Thus, the inequality of Brooks Harris [11],
(A;A) 6
1
2
〈AA+ +A+A〉 6 (A;A) +
β
12
〈[[A+,H], A]〉 (2.30)
is obtained by setting
1 6 x cothx 6 1 +
1
3
x2. (2.31)
On the other hand, if one uses another elementary inequality,
1 6 x cothx 6 1 + |x|, (2.32)
and subsequently applies the Hölder inequality, one obtains the result due to Ginibre [22]:
(A;A) 6
1
2
〈AA+ +A+A〉 6 (A;A) +
1
2
{(A;A)β〈[[A+,H], A]〉}
1
2 . (2.33)
A different choice of the parameters in the Hölder inequality, followed by the implementation
of the upper bound
|e−βEl − e−βEm | < |e−βEl + e−βEm |, (2.34)
generates a symmetric version of the inequality due to Bogoliubov (Jr.) [5]:
1
2
〈AA+ +A+A〉 6 (A;A) +
1
2
[(A;A)β]2/3{〈[A+,H][H, A] + [H, A][A+,H]〉}1/3. (2.35)
Due to relation (2.15), each of the above inequalities can be used in the AHM to majorize the
quadratic fluctuations (2.14) in terms involving second derivatives of the free energy density with
respect to the external fields νs. However, note that in the zero temperature limit β → ∞ the
right-hand side of the simplest inequality (2.30) diverges.
3. Main inequalities
To derive generalizations of the known inequalities involving the Bogoliubov-Duhamel inner
product, a set of new functionals Fk(J ; J), k = 0, 1, 2, . . . , is defined by their spectral representation:
Fk(J ; J) := Z−1
∑
ml
|Jml|
2|e−βEl − (−1)ke−βEm |(β|Em − El|)
k−1. (3.1)
The specific choice of k = 0, 1, 2, . . . is motivated by the relation of functionals (3.1) to the Gibbs
average values of some commutators and anticommutators involving the given operators J and
Hamiltonian H. Indeed,
(i) If k = 2n, n = 0, 1, 2, 3, . . . , then
F2n(J ; J) ≡ Z−1
∑
ml
|Jml|
2|e−βEl − e−βEm |(β|Em − El|)
2n−1
= β2n(Rn;Rn) = β2n−1〈[R+
nRn−1 −Rn−1R
+
n ]〉, (3.2)
13003-6
Inequalities in the AHM
where, by definition, R−1 ≡ XJH is a solution of the operator equation J = [XJH,H], and
R0 ≡ R0(J) = J, R1 ≡ R1(J) = [H, J ], Rn ≡ Rn(J) = [H, Rn−1(J)], n = 1, 2, 3, . . .
(3.3)
These observables have been introduced in [23].
(ii) If k = 2n+ 1, n = 0, 1, 2, 3, . . . , then
F2n+1(J ; J) ≡ Z−1
∑
ml
|Jml|
2(e−βEl + e−βEm)[β(Em − El)]
2n
= β2n〈[RnR
+
n +R+
nRn]〉. (3.4)
In particular,
F0(J ; J) = (J ; J), F1(J ; J) = 〈JJ+ + J+J〉, F2(J ; J) = β〈[[J+,H], J ]〉,
F3(J ; J) = β2〈[J+,H][H, J ] + [H, J ][J+,H]〉. (3.5)
The functionals (3.1) will be used to generalize all the known inequalities used in the AHM.
3.1. Generalization of the Harris inequality
By using the identity (2.26), one can rewrite the equality (3.4) in the form
F2n+1(J ; J) ≡ Z−1
∑
ml
|Jml|
2(e−βEl − e−βEm) coth
β(Em − El)
2
[β(Em − El)]
2n. (3.6)
Now, from the elementary inequalities (2.31) it follows that
F2n(J ; J) 6
1
2
F2n+1(J ; J) 6 F2n(J ; J) +
1
12
F2n+2(J ; J). (3.7)
This is a generalization of the Brooks Harris inequality (2.30), since the latter is recovered in
the particular case of n = 0.
3.2. Generalization of the Plechko inequalities
The application of the elementary inequalities (2.32) to the right-hand side of (3.6) leads to
F2n(J ; J) 6
1
2
F2n+1(J ; J) 6 F2n(J ; J) + (2Z)−1
∑
ml
|Jml|
2|e−βEl − e−βEm |[β(Em − El)]
2n. (3.8)
The problem now is that, due to the absolute value of the difference of the two Gibbs exponents,
there is no apparent interpretation of the sum in the right-hand side of (3.8) in terms of average
values. A known way to overcome this difficulty is based on the application of the Hölder inequality
∑
k
|xkyk| 6
(
∑
k
|xk|
p
)1/p(
∑
k
|yk|
q
)1/q
, p, q > 1, 1/p+ 1/q = 1. (3.9)
By setting k = (m, l) and
|xml| =
{
|Jml|
2 e
−βEl − e−βEm
β(Em − El)
}1/p
,
|yml| =
{
|Jml|
2 e
−βEl − e−βEm
β(Em − El)
[β|Em − El)|]
(2n+1)q
}1/q
, (3.10)
13003-7
J.G. Brankov, N.S. Tonchev
we obtain
(2Z)−1
∑
ml
|Jml|
2|e−βEl − e−βEm |[β(Em − El)]
2n
6
1
2
(J ; J)1/p
{
Z−1
∑
ml
|Jml|
2 e
−βEl − e−βEm
β(Em − El)
[β|Em − El|]
(2n+1)q
}1/q
. (3.11)
One of the possible choices of p and q here, namely even integer q = 2k (hence, p = 2k/(2k−1))
leads to the set of generalized Ginibre inequalities (k = 1, 2, 3, . . . ):
F2n(J ; J) 6
1
2
F2n+1(J ; J) 6 F2n(J ; J) +
1
2
(J ; J)(2k−1)/2k[F2k(2n+1)(J ; J)]
1/2k. (3.12)
At n = 0 the above set reduces to a symmetric version of the inequalities obtained by Plechko [21]:
(J ; J) 6
1
2
〈JJ+ + J+J〉 6 (J ; J) +
1
2
(J ; J)(2k−1)/2kβ(Rk;Rk)
1/2k, (k = 1, 2, 3, . . . ). (3.13)
Hence, in the particular case of k = 1 one obtains the Ginibre inequality (2.33).
3.3. Generalization of the Bogoliubov (Jr.)-Plechko-Repn ikov inequalities
If in (3.11) one chooses odd q = 2k + 1, hence, p = (2k + 1)/2k, then for the sum in the
right-hand side one can use the upper bound
Z−1
∑
ml
|Jml|
2 e
−βEl − e−βEm
β(Em − El)
[β|Em − El)|]
(2n+1)(2k+1)
6 Z−1
∑
ml
|Jml|
2|e−βEl + e−βEm |[β|Em − El)|]
2(2nk+n+k) = F2(2nk+n+k)+1(J ; J). (3.14)
Thus one obtains the set of inequalities (k = 1, 2, 3, . . . )
1
2
F2n+1(J ; J) 6 F2n(J ; J) +
1
2
(J ; J)2k/(2k+1)[F2(2nk+n+k)+1(J ; J)]
1/(2k+1). (3.15)
At n = 0 these reduce to a symmetric version of the set of inequalities obtained by
Bogoliubov Jr., Plechko and Repnikov [23]:
1
2
〈JJ+ + J+J〉 6 (J ; J) +
1
2
(J ; J)2k/(2k+1){β2k〈RkR
+
k +R+
k Rk〉}
1/(2k+1). (3.16)
The symmetric version of the Bogoliubov’s (Jr.) inequality (2.35) follows from here in the
particular case of k = 1.
3.4. Alternative sets of inequalities
By applying the Hölder inequality (3.9) to the second term in the right-hand side of (3.7) under
the substitution
|xml| =
{
|Jml|
2 e
−βEl − e−βEm
β(Em − El)
[β(Em − El)]
2n
}1/p
,
|yml| =
{
|Jml|
2|e−βEl − e−βEm |[β|Em − El|]
2n+q/p
}1/q
, (3.17)
instead of (3.10), one can in parallel derive two new sets of inequalities. Thus we obtain first
(2Z)−1
∑
ml
|Jml|
2|e−βEl − e−βEm |[β(Em − El)]
2n
6
1
2
[F2n(J ; J)]
1/p
{
Z−1
∑
ml
|Jml|
2 e
−βEl − e−βEm
β(Em − El)
[β|Em − El|]
2n+1/(p−1)
}(p−1)/p
. (3.18)
13003-8
Inequalities in the AHM
Now there are two choices of p, one of which yields 1/(p− 1) odd integer, and the other – even
integer. In the first case we set 1/(p− 1) = 2k − 1, k = 1, 2, 3, . . . , which implies p = 2k/(2k − 1),
q = 2k. Then, from (3.18) and (3.7) the following set of inequalities follows
1
2
F2n+1(J ; J) 6 F2n(J ; J)+
1
2
[F2n(J ; J)]
(2k−1)/2k [F2(n+k)(J ; J)]
1/2k, (k = 1, 2, 3, . . . ). (3.19)
In the second case we set 1/(p − 1) = 2k, k = 1, 2, 3, . . . , which implies p = (2k + 1)/2k,
q = 2k + 1. Then we can use the upper bound
Z−1
∑
ml
|Jml|
2|e−βEl − e−βEm |[β|Em − El)|]
2(n+k)
6 Z−1
∑
ml
|Jml|
2|e−βEl + e−βEm |[β|Em − El)|]
2(n+k) = F2(n+k)+1(J ; J), (3.20)
to obtain another set of inequalities
1
2
F2n+1(J ; J) 6 F2n(J ; J) +
1
2
[F2n(J ; J)]
2k/(2k+1)[F2(n+k)+1(J ; J)]
1/(2k+1), (k = 1, 2, 3, . . . ).
(3.21)
Note that (3.19) is different from (3.12) but at n = 0 it reduces to the Plechko inequali-
ties (3.13). Therefore, this set of inequalities can also be considered as a generalization of the
Ginibre inequality (2.33).
Similarly, for general n, equation (3.21) differs from (3.15) but reduces to the Bogoliubov (Jr.)-
Plechko-Repnikov inequalities (3.16) at n = 0. Hence, the set of inequalities (3.21) is also a gener-
alization of the Bogoliubov (Jr.) inequality (2.35).
A comment is in order here. Due to the property Rn+k(J) = Rk(Rn(J)), in terms of the
operators Bn ≡ Rn(J) one has
F2n(J ; J) = β2n(Bn;Bn), F2n+1(J ; J) = β2n〈BnB
+
n +B+
n Bn〉,
F2(n+k)(J ; J) = β2(n+k)(Rk(Bn);Rk(Bn)),
F2(n+k)+1(J ; J) = β2(n+k)〈Rk(Bn)R
†
k(Bn) +R†
k(Bn)Rk(Bn)〉. (3.22)
Therefore, inequalities (3.7) take exactly the form of the inequality of Brooks Harris (2.30) with
A replaced by Bn, inequalities (3.19), respectively (3.21), take exactly the form of a symmet-
ric version of the Plechko inequalities (3.13), respectively, the Bogoliubov (Jr.)-Plechko-Repnikov
inequalities (3.16), with J replaced by Bn.
Notably, under the above substitution, the generalized Ginibre inequalities (3.12) and the gen-
eralized Bogoliubov (Jr.)-Plechko-Repnikov inequalities (3.15) do not reduce to any of the known
types of inequalities, except in the particular case of n = 0.
3.5. General features and comparison of upper bounds
Obviously, the main inequalities have been derived under different approximations. The most
direct is the derivation of generalized Harris inequalities - it is based upon the single elementary
upper bound (2.31). Next, the generalized Ginibre inequalities are derived by first using the el-
ementary upper bound (2.32), followed by the application of the Hölder inequality (3.9) under
a special choice of the parameters: p = 2k/(2k − 1) and q = 2k. An alternative choice of these
parameters, p = (2k + 1)/2k and q = 2k+ 1, requires the use of an additional, rather crude upper
bound (2.34), in order to derive the Bogoliubov (Jr.)-Plechko-Repnikov inequalities (3.15).
The characteristic feature of our generalized inequalities is that, under sufficiently mild condi-
tions on the Hamiltonian HN of the N -particle system and the bounded intensive observable JN ,
they provide the upper bounds on the non-negative difference
0 6 ∆n(JN ) ≡
1
2
〈R+
n (JN )Rn(JN ) +Rn(JN )R+
n (JN )〉HN
− (Rn(JN );Rn(JN ))HN
(3.23)
13003-9
J.G. Brankov, N.S. Tonchev
of equal form and equal order of magnitude (with respect to N) for all finite n = 0, 1, 2, . . . ,
i.e., for any finite set of observables JN , R1(JN ) = [HN , JN ], R2(JN ) = [HN , R1(JN )], . . . . Note
that the left-hand side inequality in (3.23) is a generalization of the convexity property of the
Bogoliubov-Duhamel inner product (2.18).
The required conditions are (n = 1, 2, 3, . . . ):
|〈JN 〉HN
| 6 O(1),
F2n(JN ; JN ) = β2n−1〈[R+
n (JN ), Rn−1(JN )]〉HN
6 O(N−1),
F2n+1(JN ; JN ) = β2n〈R+
n (JN )Rn(JN ) +Rn(JN )R+
n (JN )〉HN
6 O(1). (3.24)
They are generally satisfied for extensive Hamiltonians HN with bounded interaction and bounded
intensive observables JN , which are arithmetic averages over the particles of the system of some
local observables. An example will be given in the end of this section.
The above conditions may also hold in some cases when HN and Rn(JN ), n = 0, 1, 2, . . . ,
contain unbounded operators. As an example in section 4.2 we will consider the Dicke model of
superradiance [25], for which the proper unbounded counterpart of JN will be found.
Under conditions (3.24), our generalized inequalities yield the following upper bounds for all
finite n = 0, 1, 2, . . . :
1. Generalized Harris inequality (3.7)
∆n(JN ) 6 O(N−1). (3.25)
2. Generalized Plechko inequalities (3.12)
∆n(JN ) 6 (JN ; JN)(2k−1)/2kO(N−1/2k), (k = 1, 2, 3, . . . ), (3.26)
which at k = 1 reduce to the Ginibre inequality
∆n(JN ) 6 (JN ; JN )1/2O(N−1/2). (3.27)
3. Generalized Bogoliubov (Jr.)-Plechko-Repnikov inequalities (3.15)
∆n(JN ) 6 (JN ; JN )2k/(2k+1)O(1), (k = 1, 2, 3, . . . ), (3.28)
which at k = 1 reduce to the Bogoliubov (Jr.) inequality
∆n(JN ) 6 (JN ; JN )2/3O(1). (3.29)
Due to the relationship between the Bogoliubov-Duhamel inner product and the susceptibility
of the system with respect to the external field conjugate to the observable JN , see (2.15), we
can compare the different upper bounds in the region of parameters in which the susceptibility is
bounded. We see that the generalized Harris, Plechko and Ginibre inequalities yield upper bounds
of the order O(N−1), while the generalized Bogoliubov (Jr.)-Plechko-Repnikov inequalities yield
upper bounds of the order O(N−2k/(2k+1)), k = 1, 2, 3, . . . .
Finally, to illustrate the validity of conditions (3.24) and the explicit form of the first generalized
observables Rn(JN ), we consider a simple model with separable attraction built upon bounded
operators. Let the model system contain N spins and have the Hamiltonian (normalized by kBT )
βHN = −Ngx (J
x
N )
2 −Ngy (J
y
N )
2
−Nh · JN , (3.30)
where gx, gy > 0 are dimensionless coupling constants, h = {hx, hy, hz} is the vector of the external
magnetic field, JN = {Jx
N , Jy
N , Jz
N} is the vector operator of the average spin with uniformly
bounded in N components Jα
N ,
Jα
N =
1
N
N
∑
i=1
σα
i , α = x, y, z, (3.31)
13003-10
Inequalities in the AHM
where σα are the standard Pauli matrices.
By using the commutation relations for the spin operators, we obtain the observables
R1(J
x
N ) ≡ [βHN , Jx
N ] = 2i[gy(J
y
NJz
N + Jz
NJy
N ) + hyJ
z
N − hzJ
y
N ], (3.32)
and
R2(J
x
N ) ≡ [βHN , R1(J
x
N )] = 4gy(gy − gx)(J
y
NJy
NJx
N + 2Jy
NJx
NJy
N + Jx
NJy
NJy
N )
+ 4gxgy(J
z
NJz
NJx
N + 2Jz
NJx
NJz
N + Jx
NJz
NJz
N )
+ 4(gy − gx)hy(J
x
NJy
N + Jy
NJx
N )− 4gxhz(J
x
NJz
N + Jz
NJx
N )
+ 4(h2
y + h2
z)J
x
N − 4hxhyJ
y
N − 4hxhzJ
z
N , (3.33)
which, by definition, have zero average values. Next,
F2(J
x
N ; Jx
N ) ≡ (R1(J
x
N );R1(J
x
N )) = 〈[R+
1 (J
x
N ), Jx
N ]〉
=
4
N
{
2gy[〈(J
y
N )2〉 − 〈(Jz
N )2〉] + hy〈J
y
N 〉+ hz〈J
z
N 〉
}
, (3.34)
and
F3(J
x
N ; Jx
N ) ≡ 〈[R+
1 (J
x
N )R1(J
x
N ) +R1(J
x
N )R+
1 (J
x
N )]〉
= 〈[gy(J
yJz + Jz
NJy
N ) + hyJ
z
N − hzJ
y
N ]2〉. (3.35)
Rather lengthy but straightforward calculations show that
|F4(J
x
N ; Jx
N)| = |〈[R+
2 (J
x
N ), R1(J
x
N )]〉| 6 O(N−1). (3.36)
One can readily extend the above results and show that for all finite n the following inequalities
hold: |〈Rn(J
x
N )〉| 6 ||Rn(J
x
N )|| 6 O(1), |F2n(J
x
N ; Jx
N )| 6 O(N−1), and |F2n+1(J
x
N ; Jx
N )| 6 O(1).
Thus, conditions (3.24) and, hence, the upper bounds (1)–(3) are fulfilled for the considered model.
4. Systems of matter interacting with Boson fields
4.1. Models and their treatment by the AHM
Here are two examples of models in solid state physics, which belong to this class: (a) The
Dicke model of superradiance [25], solved exactly in [26, 27], and by the AHM in [28]. The model
has been generalized to include interactions with both electromagnetic field and phonons [29, 30],
and to the case of infinitely many modes of the electromagnetic field [31]. A recent review of the
thermodynamic properties of the original Dicke model and its generalizations is given in [32]. (b)
The Mattis-Langer model of structural instability [33], solved exactly by the AHM in [34]. A class
of models, including as a special case the Dicke model, has been considered in the framework of
the AHM by Bogoliubov (Jr.) and Plechko [35]. The one-dimensional case with countably infinite
set of phonon modes has been solved by means of theta-function integration in [36].
The model Hamiltonian is defined on the tensor product of two Hilbert spaces, one for the
subsystem describing matter (e.g., electrons in a solid, spins on a lattice), and the other for the
Boson field (lattice vibrations, electromagnetic field). In the second quantization representation,
the creation, b†s , and annihilation, bs, operators of the Boson field modes (labeled by the subscript s)
satisfy the canonical commutation relations
[bs, b
†
l ] = δs,l, [bs, bl] = [b†s , b
†
l ] = 0, (4.1)
for all the allowed s and l. Since the Boson operators are unbounded, it is not possible to obtain
easy bounds on their average values in terms of Hilbert-space norm.
13003-11
J.G. Brankov, N.S. Tonchev
Another characteristic feature of this class of models is that exact solvability by the AHM is
possible when the interaction with only a finite (or growing slower than the volume V , as V → ∞)
number of Boson modes is taken into account. The typical model Hamiltonian has the form
HΛ = TΛ +
n
∑
s=1
ωs b
†
sbs + V 1/2
n
∑
s=1
λs(bsA
†
s,Λ + b†sAs,Λ). (4.2)
Here the operators TΛ = T †
Λ and As,Λ, s = 1, . . . , n, refer to the matter subsystem and satisfy the
general conditions (2.4) and (2.6). The second term in the right-hand side of (4.2) describes a finite
number of free Boson modes, s = 1, . . . , n, in the space domain Λ of volume V . For the sake of
simplicity, the energies ωs > 0 and the interaction constants λs ∈ R are taken to be independent
of the volume |Λ|. The mathematical definition of the above Hamiltonian is given in [37].
The corresponding approximating Hamiltonian depends on a set of complex numbers η =
(η1, . . . , ηn) ∈ C
n and has the form, see [6, 28],
H
(0)
Λ (η) =
n
∑
s=1
ωs b̃
†
s b̃s + TΛ − V
n
∑
s=1
(λ2
s/ωs)(ηsA
†
s,Λ + η⋆sAs,Λ − η⋆s ηs), (4.3)
where
b̃†s = b†s + V 1/2 λs
ωs
η⋆s , b̃s = bs + V 1/2 λs
ωs
ηs , (4.4)
are the creation/annihilation operators for a subsystem of free shifted bosons. The application
of the Bogoliubov inequalities to the difference of the free energy densities for the model and
approximating Hamiltonians yields
f [H
(0)
Λ (η)]− f [HΛ] > 0, (4.5)
for all η ∈ C
n. Therefore, the best approximation is reached at η = η̄Λ, where η̄Λ satisfies the
absolute minimum condition
f [H
(0)
Λ (η̄Λ)] = min
η
f [H
(0)
Λ (η)]. (4.6)
Note that the free energy density of the free bosons,
f
[
n
∑
s=1
ωs b
†
sbs
]
=
1
βV
n
∑
s=1
ln
(
1− e−βωs
)
, (4.7)
is independent of the parameters η = (η1, . . . , ηn) ∈ C
n and vanishes in the thermodynamic limit
as O(V −1).
The proof of the thermodynamic equivalence of the free energy densities f [H
(0)
Λ (η̄Λ)] and f [HΛ]
goes again through the introduction of sources of the boson fields,
HΛ(ν) = HΛ − V 1/2
n
∑
s=1
(ν⋆s bs + νsb
†
s),
H
(0)
Λ (η, ν) = H
(0)
Λ (η)− V 1/2
n
∑
s=1
(ν⋆s bs + νsb
†
s), (4.8)
where ν = (ν1, . . . , νn) ∈ C
n.
Now, from the Bogoliubov inequalities and a subsequent use of an elementary upper bound,
one obtains
0 6 min
η
f [H
(0)
Λ (η, ν)]− f [HΛ(ν)] 6 −V −1/2
n
∑
s=1
λs〈δbsδAs,Λ + δb†sδAs,Λ〉HΛ(ν)
6 V −1/2
n
∑
s=1
V −γ λ
2
s
ωs
〈δA†
s,ΛδAs,Λ〉HΛ(ν) + V −1/2
n
∑
s=1
V γωs〈δb
†
sδbs〉HΛ(ν), (4.9)
13003-12
Inequalities in the AHM
where
δAs,Λ = As,Λ − 〈As,Λ〉HΛ(ν), δbs = bs − 〈bs〉HΛ(ν), (4.10)
and γ ∈ (0, 1) is a free parameter.
Due to the boundedness of the operators As,Λ, A
†
s,Λ, see conditions (2.6), the first sum in the
right-hand side of the last inequality (4.9) is bounded from above by
V −1/2−γnM2
1 max
s
(λ2
s/ωs). (4.11)
The quadratic fluctuations of the boson fields, 〈δb†sδbs〉HΛ(ν), are to be majorized by terms
proportional to powers of the Bogoliubov-Duhamel inner product
(δbs; δbs)HΛ(ν) = −
1
β
∂2f [HΛ(ν)]
∂νs∂ν⋆s
. (4.12)
By using the Ginibre inequality (2.33) and choosing γ = 1/3 in (4.9), the subsequent applica-
tion of the majorization technique due to Bogoliubov Jr. with the following bounds on the first
derivatives (s = 1, . . . , n)
∣
∣
∣
∣
∂f [HΛ(ν)]
∂ν#s
∣
∣
∣
∣
= V −1/2|〈b#s 〉HΛ(ν)| = ω−1
s |λs〈A
#
s,Λ〉HΛ(ν) − ν#s | 6 ω−1
s (λsM1 + |ν#s |), (4.13)
made it possible to prove that
|min
η
f [H
(0)
Λ (η)]− f [HΛ]| 6 ǫV , (4.14)
where ǫV = O(V −1/3) → 0 as V → ∞. Under additional conditions on the double commutators
between the different {A#
s,Λ : s = 1, . . . , n} and on the commutator of R1(A
#
s,Λ) with A#
s′,Λ, the
above estimate was improved up to ǫV = O(V −1/2) [7].
4.2. Application of the generalized inequalities to the Dic ke model
In the remainder, by using the equality [14, 22]:
β(X ; [H, B])H = 〈[X†, B]〉H , (4.15)
we shall derive a variety of explicit relationships between Bogoliubov-Duhamel inner products
and usual thermal averages for different observables of the basic single-mode Dicke model in the
rotating wave approximation. In the latter case, the Hamiltonian has the form (4.2) with n = 1,
T =
1
2
ǫ
N
∑
j=1
σz
j , A =
1
V
N
∑
j=1
σ+
j , A† =
1
V
N
∑
j=1
σ−
j , (4.16)
where σ±
j = 1
2 (σ
x
j ± iσy
j ) and σz
j are the Pauli matrices. From the above definitions the following
commutation relations follow
[T,A] = ǫA, [A†, A] =
2
ǫV 2
T. (4.17)
By direct computation we obtain
R1(b) ≡ [H, b] = −(ωb+ V 1/2λA), (4.18)
and, since 〈[H, b]〉H = 0, we obtain a well known equality between average values of observables
pertaining to different subsystems (boson field and matter):
〈b〉H = −V 1/2 λ
ω
〈A〉H . (4.19)
13003-13
J.G. Brankov, N.S. Tonchev
Therefore, in the case of non-vanishing polarization in the matter subsystem, 〈A〉H 6= 0, the average
value of the boson annihilation (as well as creation) operator will behave as 〈b〉H = O(V 1/2). Hence,
we deduce that the unbounded counterpart of the observable JN in conditions (3.24) should be
the normalized operator V −1/2b. Keeping this in mind, we further calculate
R2(V
−1/2b) ≡ [H,R1(V
−1/2b)] = ω2V −1/2b+ λ(ω − ǫ)A−
2λ2
ǫV 3/2
bT, (4.20)
and
F2(V
−1/2b;V −1/2b) = β〈[R1(V
−1/2b)†, V −1/2b]〉H =
βω
V
, (4.21)
F3(V
−1/2b;V −1/2b) ≡ β2〈R†
1(V
−1/2b)R1(V
−1/2b) +R1(V
−1/2b)R†
1(V
−1/2b)〉H
= (βω)2
[
V −1〈b†b+ bb†〉H + V −1/2 λ
ω
〈b†A+ bA†〉H +
λ2
ω2
〈A†A+AA†〉H
]
,
(4.22)
F4(V
−1/2b;V −1/2b) = β3〈[R†
2(V
−1/2b), R1(V
−1/2b)]〉H
=
(βω)3
V
[
1 +
2λ2
ǫω2V
( ǫ
ω
− 2
)
〈T 〉H +
2λ3
ω3V 1/2
〈b†A〉H
]
. (4.23)
The right-hand side of these expressions can be evaluated using some relationships between
average values of different observables which follow from (4.15). Thus, by setting B = b and
X = b, we obtain,
(b; b)H =
1
βω
− V 1/2 λ
ω
(b;A)H . (4.24)
Next, from B = b and X = A it follows that
− (A; b)H = V 1/2 λ
ω
(A;A)H . (4.25)
Under the alternative choice B = b† and X = b† in (4.15), we derive
(b†; b†)H =
1
βω
− V 1/2 λ
ω
(b†;A†)H , (4.26)
which, due to the conjugate symmetry (A;B) = (B†;A†), is equivalent to
(b; b)H =
1
βω
− V 1/2 λ
ω
(A; b)H . (4.27)
By comparing this equality with (4.24) we conclude that (b;A)H = (A; b)H. Then, taking into
account (4.25) we derive the important relation
(b; b)H =
1
βω
+ V
λ2
ω2
(A;A)H , (4.28)
which was obtained in [32] using gage invariance arguments.
Proceeding further with the evaluation of (4.22) and (4.23), we note that since the right-hand
side of (4.23) is real, one must have
〈b†A〉H = 〈bA†〉H . (4.29)
Therefore, the application of the Schwarz inequality and the relation (4.28) yield the estimate
|〈V −1/2b†A〉H| = 〈V −1/2bA†〉H| 6 〈V −1b†b〉
1/2
H |〈AA†〉
1/2
H | 6 O(1). (4.30)
13003-14
Inequalities in the AHM
Finally, the application of the Harris inequality (2.30) in terms of b gives
(b; b)H 6 〈b†b〉H +
1
2
6 (b; b)H +
1
12
βω , (4.31)
which, in view of (4.28), implies 〈b†b〉H = O(V ), provided that (A;A)H = O(1). Taking into account
||A|| = O(1) and ||T || = O(V ), we conclude that F2(V
−1/2b;V −1/2b) and F4(V
−1/2b;V −1/2b) are
bounded from above by O(V −1), while |F3(V
−1/2b;V −1/2b)| 6 O(1).
Thus, we have proved that V −1/2b and Fn(V
−1/2b;V −1/2b) for n = 2, 3, 4 satisfy condi-
tions (3.24). Therefore, bounds of the form (3.25), (3.27), and (3.28) hold true at n = 1 even
for the unbounded operator JV = V −1/2b.
5. Discussion
The AHM makes it possible to rigorously obtain the exact thermodynamic properties of diverse
classes of model systems in quantum statistical mechanics. Each class of models requires a certain
structure of the interaction Hamiltonian and special properties of the operators in terms of which its
structure is defined. Initially, the mathematical technique for derivation of bounds which prove the
thermodynamic equivalence of the approximating and original Hamiltonians has been developed for
systems with interaction Hamiltonians constructed with bounded operators (see section 2). Later,
it was extended to the case of unbounded Bose operators (see section 4). This technique essentially
exploits the possibility of estimating the correlation functions in (2.14) or (4.9) from above by ex-
pressions containing the Bogoliubov-Duhamel inner product in combination with average values of
certain operators. As a rule, the latter are estimated by norm, which requires conditions (2.4)–(2.8).
It is well known that this procedure is not unique [7, 8]. The distinction comes from the different
inequalities used. At first, it was Bogoliubov’s Jr. inequality that was used in the case of bounded
operators. Latter it was realized that Ginibre’s inequality (2.33) is more convenient to be used, es-
pecially in the case of interactions involving Bose operators [7], see also section 4. Moreover, it made
it possible to derive a better estimate, namely O(|Λ|−1/2), in both cases of bounded and unbounded
operators [6–8]. However, in the latter case, two additional sufficient conditions have been imposed.
The interest that stimulates the search and use of different inequalities was prompted by the
wish to improve the upper bounds on the difference of the model and approximating free energy
densities and/or to enlarge the class of model systems rigorously solved by the AHM [21, 23].
In section 3, we have derived generalizations of the inequalities introduced in [21] and [23]. The
novel point is that a set of new functionals Fk(J ; J), k = 0, 1, 2, . . . , were defined by their spectral
representation (3.1). This made it possible to obtain different upper bounds in a unified fashion.
Beyond the use in the majorization step of the AHM, our new upper bounds (3.25)–(3.29) on
the difference between the Gibbs average values of a class of observables of a many-particle system
and the corresponding Bogoliubov-Duhamel inner products may find wider applications. Under
sufficiently mild conditions, these upper bounds have the same form and order of magnitude with
respect to the number of particles (or volume) for all the quantities derived by a finite number of
commutations Rn(J) of an original intensive observable J with the Hamiltonian of the system H.
In addition, we have obtained important relationship between average values of the different
observables in the framework of two types of exactly solved by the AHM model systems: one with
bounded separable attraction – the infinitely coordinated anisotropic Heisenberg model, and the
other, the Dicke superradiance model, describing interaction of a boson field with a subsystem of
matter.
Acknowledgements
N.S.T. was supported by the National Science Foundation of Bulgaria under grant TK–X–
1712/2007.
13003-15
J.G. Brankov, N.S. Tonchev
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13003-16
http://dx.doi.org/10.1103/PhysRev.108.1175
http://dx.doi.org/10.1070/RM1984v039n06ABEH003180
http://dx.doi.org/10.1103/PhysRevLett.17.1133
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http://dx.doi.org/10.1007/BF00739578
http://dx.doi.org/10.1007/s11232-009-0108-1
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http://dx.doi.org/10.1103/PhysRev.180.442
http://dx.doi.org/10.1103/PhysRev.93.99
http://dx.doi.org/10.1016/0003-4916(73)90039-0
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http://dx.doi.org/10.1007/BF01036490
http://dx.doi.org/10.1016/0378-4371(77)90037-1
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http://dx.doi.org/10.1016/0378-4371(75)90094-1
http://dx.doi.org/10.1007/BF01031033
Inequalities in the AHM
Узагальненi нерiвностi для внутрiшнього добутку
Боголюбова-Дюамеля iз застосуваннями в методi
апроксимуючого гамiльтонiану
Дж. Г. Бранков1,2, Н.С. Тончев3
1 Лабораторiя теоретичної фiзики iм. Боголюбова, Об’єднаний iнститут ядерних дослiджень,
141980 Дубна, Росiйська Федерацiя
2 Iнститут механiки, Болгарська академiя наук, Софiя, Болгарiя
3 Iнститут фiзики твердого тiла, Болгарська академiя наук, Софiя, Болгарiя
Отримано нескiнченнi набори нерiвностей, якi узагальнюють всi вiдомi нерiвностi, що можуть
бути використанi на етапi мажорування методу апроксимуючого гамiльтонiану Вони забезпечують
верхнi границi на рiзницю мiж квадратичними флуктуацiями iнтенсивних спостережуваних N -
частинкової системи i вiдповiдного внутрiшнього добутку Боголюбова-Дюамеля. Новою рисою є
те, що при достатньо м’яких умовах верхнi границi мають однакову форму i порядок величини
по вiдношенню до N для всiх величин, отриманих шляхом скiнченного числа перестановок
початкової iнтенсивої спостережуваної з гамiльтонiаном. Результати iлюструються на двох типах
точно розв’язуваних моделей: однiєї з обмеженим сепарабельним притяганням та iншої, що мiстить
взаємодiю бозонного поля з матерiєю.
Ключовi слова: кореляцiйнi функцiї, внутрiшнiй добуток Боголюбова-Дюамеля, нерiвностi
статистичної механiки, метод апроксимуючого гамiльтонiану, точно розв’язуванi моделi
13003-17
Introduction
Systems with bounded separable attraction
The Bogoluibov-Duhamel inner product
Main inequalities
Generalization of the Harris inequality
Generalization of the Plechko inequalities
Generalization of the Bogoliubov (Jr.)-Plechko-Repnikov inequalities
Alternative sets of inequalities
General features and comparison of upper bounds
Systems of matter interacting with Boson fields
Models and their treatment by the AHM
Application of the generalized inequalities to the Dicke model
Discussion
|