Asymptotic exactness of c-number substitution in Bogolyubov's theory of superfluidity
The Bogolyubov model of liquid helium is considered. The validity of substituting a c-number for the k=0 mode operator â0 is established rigorously. The domain of stability of the Bogolyubov's Hamiltonian is found. We derive sufficient conditions which ensure the appearance of the Bose condensa...
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Інститут фізики конденсованих систем НАН України
2010
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| Цитувати: | Asymptotic exactness of c-number substitution in Bogolyubov's theory of superfluidity / N.N. Bogolyubov, Jr., D.P. Sankovich // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23002: 1-6. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860018641655824384 |
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| author | Bogolyubov (jr), N.N. Sankovich, D.P. |
| author_facet | Bogolyubov (jr), N.N. Sankovich, D.P. |
| citation_txt | Asymptotic exactness of c-number substitution in Bogolyubov's theory of superfluidity / N.N. Bogolyubov, Jr., D.P. Sankovich // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23002: 1-6. — Бібліогр.: 14 назв. — англ. |
| collection | DSpace DC |
| container_title | Condensed Matter Physics |
| description | The Bogolyubov model of liquid helium is considered. The validity of substituting a c-number for the k=0 mode operator â0 is established rigorously. The domain of stability of the Bogolyubov's Hamiltonian is found. We derive sufficient conditions which ensure the appearance of the Bose condensate in the model. For some temperatures and some positive values of the chemical potential, there is a gapless Bogolyubov spectrum of elementary excitations, leading to a proper microscopic interpretation of superfluidity.
|
| first_indexed | 2025-12-07T16:46:43Z |
| format | Article |
| fulltext |
Condensed Matter Physics 2010, Vol. 13, No 2, 23002: 1–6
http://www.icmp.lviv.ua/journal
Asymptotic exactness of c-number substitution in
Bogolyubov’s theory of superfluidity
N.N. Bogolyubov, Jr., D.P. Sankovich
V.A. Steklov Institute of Mathematics, Gubkin Str. 8, Moscow, 119991, Russia
Received January 26, 2010, in final form March 16, 2010
The Bogolyubov model of liquid helium is considered. The validity of substituting a c-number for the k = 0
mode operator â0 is established rigorously. The domain of stability of the Bogolyubov’s Hamiltonian is found.
We derive sufficient conditions which ensure the appearance of the Bose condensate in the model. For some
temperatures and some positive values of the chemical potential, there is a gapless Bogolyubov spectrum of
elementary excitations, leading to a proper microscopic interpretation of superfluidity.
Key words: Bose-Einstein condensation, Bogolyubov’s Hamiltonian, superfluidity, c-number substitution
PACS: 05.30.Jp, 03.75.Fi, 67.40.-w
1. The model
Let us consider a system of N spinless identical nonrelativistic bosons of mass m enclosed in a
centered cubic box Λ ⊂ R3 of volume V = |Λ| = L3 with periodic boundary conditions for wave
functions. The Hamiltonian of the system can be written in the second quantized form as
ĤΛ(µ) ≡ ĤΛ − µN̂Λ =
∑
k∈Λ∗
(εk − µ)â†kâk +
1
2V
∑
p,q,k∈Λ∗
ν(k)â†pâ
†
qâp+kâq−k . (1)
Here â#
k = {â†k or âk} are the usual boson (creation or annihilation) operators for the one-particle
state ψk(x) = V −1/2 exp(ikx), k ∈ Λ∗, x ∈ Λ, acting on the Fock space FΛ = ⊕∞
n=0H
(n)
B , where
H(n)
B ≡ [L2(Λn)]symm is the symmetrized n-particle Hilbert space appropriate for bosons, and
H(0)
B = C. The sums in (1) run over the dual set
Λ∗ =
{
k ∈ R
3 : kα =
2π
L
nα, nα = 0,±1,±2, . . . , α = 1, 2, 3
}
,
εk = |k|2/(2m) is the one-particle energy spectrum of free bosons in the modes k ∈ Λ∗ (we propose
~ = 1), N̂Λ =
∑
k∈Λ∗ â
†
kâk is the total particle-number operator, µ is the chemical potential, ν(k)
is the Fourier transform of the interaction pair potential Φ(x). We suppose that Φ(x) = Φ(|x|) ∈
L1(R3) and ν(k) is a real function with a compact support such that 0 6 ν(k) = ν(−k) 6 ν(0) for
all k ∈ R3. Under these conditions the Hamiltonian (1) is superstable [1].
So long as the rigorous analysis of the Hamiltonian (1) is very knotty problem, Bogolyubov
introduced the model Hamiltonian of superfluidity theory [2, 3]. He proposed to disregard the
terms of the third and fourth order in operators â#
k , k 6= 0 in the Hamiltonian (1),
ĤB
Λ (µ) =
∑
k∈Λ∗
(εk − µ)â†kâk +
1
2V
∑
k 6=0
ν(k)(â†k â
†
−kâ0â0 + â†0â
†
0â−kâk)
+
1
V
â†0â0
∑
k 6=0
ν(k)â†kâk +
ν(0)
V
â†0â0
∑
k 6=0
â†kâk +
ν(0)
2V
â†0â
†
0â0â0 . (2)
c© N.N. Bogolyubov, Jr., D.P. Sankovich 23002-1
http://www.icmp.lviv.ua/journal
N.N. Bogolyubov, Jr., D.P. Sankovich
Then Bogolyubov takes advantage of the most relevant operators in the problem to replace the
corresponding creation and annihilation operators â#
0 by c-numbers,
â†0√
V
→ c̄,
â0√
V
→ c, (3)
where c ∈ C and the bar means complex conjugation. This idea has its roots in the work [4].
In §63 of this monograph Dirac analyses a many-body system within the framework of second
quantization. Bogolyubov developed the Dirac’s idea systematically to study Bose condensation
and superfluidity in the model (2).
Let ĤB
Λ (µ, c) be the Hamiltonian (2) after the Bogolyubov approximation (3). This Hamiltonian
is a bilinear form in boson operators â#
k (k 6= 0). So, one can diagonalize it by the Bogolyubov
canonical transformation. To determine the complex parameter c it is necessary to use some self-
consistent procedure.
In [2, 3] Bogolyubov considered the Hamiltonian (2) in the case of zero temperature θ. In the
main perturbation order he found that µ(θ = 0) = |c|2ν(0), where |c|2 = ρ0 is the density of Bose
condensate. In this case the structure of the collective excitation spectrum of the Hamiltonian
ĤB
Λ (µ, c) explains the superfluid properties of the system (2).
It should be noted that the main condition which makes it possible to replace the Hamiltonian
(1) by the model Hamiltonian (2) is
N −N0
N
� 1, (4)
where N0 is the number of condensate particles. Condition (4) means that the interaction is suffici-
ently weak and the case of very small temperatures must be considered. Thus in 1947 Bogolyubov
analysed the model (2) within the framework of (4). The validity of the Bogolyubov approximation
(3) has not been rigorously proved.
A rigorous justification for the c-number substitution in the case of the total, correct super-
stable pair Hamiltonian (1) was done in a classic paper of Ginibre [5]. Recently, this problem was
revisited in paper [6]. The authors of [5, 6] did not consider the truncated Bogolyubov’s Hamilto-
nian (2). Nevertheless, Lieb and others [6] mentioned that their device (based on the Berezin-Lieb
inequality) can be also used for the Hamiltonian (2). Earlier, in [7–9] the Bogolyubov prescription
concerning substitution of the zero-mode annihilation and creation operators by c-numbers and
concerning their choice was discussed. The authors of these articles arrived at a conclusion that the
Bogolyubov’s Hamiltonian (2) is unstable for µ > 0, which is the choice of the Bogolyubov theory.
So, from [7–9] it follows that the Bogolyubov’s model (2) does not explain the superfluidity. This
wrong conclusion is based on the erroneous Proposition 4.3 in [7] and Theorem 3.8 in [9].
Here we revert to the problem of justification of c-number substitution for the Bogolyubov’s
Hamiltonian (2). We show how to use the Ginibre approach for the non-superstable Hamiltonian
(2). To find the stability region of (2), i.e., the domain in a space of independent thermodynamical
variables (µ, θ), where the grand-canonical partition function associated for the Hamiltonian (2)
is finite, we use the Bogolyubov’s inequality [10], which efficiently gives the upper bound for
the pressure. Besides, we derive sufficient conditions which ensure the appearance of the Bose
condensate in the model.
2. Stability of the Hamiltonian
Let us first rewrite the Hamiltonian (2) in the following way:
ĤB
Λ (µ) = ĤB
Λ0(µ, c) + ĤB
Λ1(c), (5)
where
ĤB
Λ0(µ, c) ≡
∑
k∈Λ∗
(εk − µ− 1
2V
ν(k))â†k âk − Φ(0)
2
â†0â0 +
ν(0)
2V
â†0â
†
0â0â0 + ν(0)|c|2
∑
k 6=0
â†kâk , (6)
23002-2
Bogolyubov’s theory of superfluidity
ĤB
Λ1(c) ≡ ν(0)
V
(â†0â0 − V |c|2)
∑
k 6=0
â†kâk +
1
2V
∑
k 6=0
ν(k)(â†0âk + â0â
†
−k)†(â†0âk + â0â
†
−k). (7)
The complex parameter c in formulae (5)–(7) will be defined below. It is easy to see that the
Hamiltonian (6) is stable for µ 6 ν(0)|c|2 and any c ∈ C.
Denoting
δâ0 ≡ â0 − c
√
V , δâ†0 ≡ â†0 − c̄
√
V , Âk ≡ â†0âk + â0â
†
−k , k 6= 0,
we can write (7) in the form
ĤB
Λ1(c) =
ν(0)
V
∑
k 6=0
â†kâk
(
δâ†0δâ0 + c
√
V δâ†0 + c̄
√
V δâ0
)
+
1
2V
∑
k 6=0
ν(k)†
kÂk . (8)
Let us prove that
lim
V →∞
1
V
〈
ĤB
Λ1(c)
〉
ĤB
Λ
(µ)
> 0, (9)
where c is a solution of the equation
|c|2 =
1
V
〈â†0â0〉ĤB
Λ
(µ) .
From the Bogolyubov inequality for pressures
p
[
ĤB
Λ (µ)
]
6 p
[
ĤB
Λ0(µ, c)
]
− 1
V
〈ĤB
Λ1(c)〉ĤB
Λ
(µ)
we then obtain that the Hamiltonian (2) is stable for
µ 6 ν(0)|c|2. (10)
Let us introduce the Hamiltonian
ĤB
Λ (µ, ν) ≡ ĤB
Λ (µ) −
√
V (ν̄â0 + νâ†0)
with sources ν ∈ C breaking the symmetry of ĤB
Λ (µ). Using the Cauchy inequality, we get the
estimate
〈δâ†0N̂ ′〉ĤB
Λ
(µ,ν)| 6
[
〈δâ†0δâ0〉ĤB
Λ
(µ,ν)〈N̂ ′2〉ĤB
Λ
(µ,ν)
]1/2
6 ρV 〈δâ†0δâ0〉ĤB
Λ
(µ,ν) ,
where N̂ ′ ≡ ∑
k 6=0 â
†
kâk.
To obtain an upper bound for the average in the last inequality we can apply the usual procedure
of the Bogolyubov quasi-average method [11] and Bogolyubov, Jr. technique [12]. Define c by the
condition c = 〈â0〉ĤB
Λ
(µ,ν)/
√
V , |c| 6 M <∞. By the Harris inequality [13] one gets
1
2
〈
[δâ†0, δâ0]+
〉
ĤB
Λ
(µ,ν)
6 (δâ†0, δâ0)ĤB
Λ
(µ,ν) +
β
12
〈
[δâ0, [Ĥ
B
Λ (µ, ν), δâ†0]]
〉
ĤB
Λ
(µ,ν)
,
where [·, ·]+ is the anticommutator and (·, ·)Γ̂ denotes the Bogolyubov inner product (or the
Duhamel two-point function) with respect to the Hamiltonian Γ [10]. Literally reiterating the
standard for this method of calculations [5], we see that
1
V
〈δâ†0δâ0〉ĤB
Λ
(µ,ν) 6
η√
V
,
where η is some positive constant, independent of V . Therefore, it follows from the last inequality
that
|〈δâ#
0 N̂
′〉ĤB
Λ
(µ,ν)| 6 ρ
√
ηV 5/4.
23002-3
N.N. Bogolyubov, Jr., D.P. Sankovich
Thus, using the representation of the Hamiltonian ĤB
Λ1(c) in the form (8) one can see that the
condition (9) is actually justified. The parameter c should be chosen using the technique stereotyped
for the Bogolyubov–Ginibre technique. This parameter is connected with the Bose condensate
density as |c|2 = ρ0.
The above analysis confirms an assertion that if the system is stable after the c-number sub-
stitution (3), then so is the original one [6].
3. Self-consistency equation
In the manner similar to the work by Ginibre [5], one can prove that the model Hamiltonian
ĤB
Λ (µ) is thermodynamically equivalent to the approximating Hamiltonian
ĤB
Λ (µ, c) =
∑
k 6=0
[εk − µ+ |c|2(ν(0) + ν(k))]â†k âk
+
1
2
∑
k 6=0
ν(k)(c2â†kâ
†
−k + c̄2âkâ−k) +
1
2
ν(0)|c|4V − µ|c|2V. (11)
The self-consistency parameter c in the method is determined by the condition that the approximate
pressure p[ĤB
Λ (µ, c)] should be maximal. At the same time, the stability condition (10) must be
fulfilled.
A necessary condition for p[ĤB
Λ (µ, c)] to be maximum (self-consistency equation) in the case of
the Bogolyubov model is
〈
∂ĤB
Λ (µ, c)
∂c
〉
ĤB
Λ
(µ,c)
= 0. (12)
This equation always has a trivial solution c = 0 (no Bose condensation). By explicit calculations,
we get the following equation to obtain a nontrivial solution
µ− xν(0) =
1
2V
∑
k 6=0
[
(ν(0) + ν(k))
(
fk
Ek
coth
βEk
2
− 1
)
− ν(k)
hk
Ek
coth
βEk
2
]
, (13)
where
uk =
√
1
2
(
fk
Ek
+ 1
)
, vk = −
√
1
2
(
fk
Ek
− 1
)
,
fk = εk − µ+ x(ν(0) + ν(k)), hk = xν(k), Ek =
√
f2
k − h2
k
and we denote x ≡ |c|2.
Let us first consider the zero temperature case. In this case the right-hand side of (13) is
F (x) ≡ 1
2V
∑
k 6=0
[
(ν(0) + ν(k))
(
fk
Ek
− 1
)
− ν(k)
hk
Ek
]
.
One can see that F ′(x) < 0 and F ′′(x) > 0, i.e., the function F (x) is strictly monotonously
decreasing and convex on [0,∞). Furthermore, it is bounded from below. We are interested in the
case where µ > 0 (remind that always µ 6 xν(0)). For the unique nontrivial solution x∗ of the
equation (13) to exist, it is necessary that
∆(µ) ≡ F
(
x =
µ
ν(0)
)
6 0.
From now on, we shall use the function
ν(k) =
{
ν(0) for |k| 6 k0,
0 for |k| > k0
(14)
23002-4
Bogolyubov’s theory of superfluidity
as the Fourier transform of the pair potential. We suppose that ν(0) = 4πr0/m, k0r0 = 1, r0 =
2.56 Å,m = mHe4 . In the case of the potential (14) the nontrivial root µ∗ of the equation
∆(µ) = 0 is
µ∗ =
k2
0
36m
(4 −
√
1.6)2.
If
ν(0) >
1
2V
∑
k 6=0
ν2(k)
εk
, (15)
for θ = 0 and 0 < µ 6 µ∗, where µ∗ is the unique nontrivial solution of the equation ∆(µ) = 0, the
Bose condensation can be realized. For µ = µ∗, the Bose condensate density ρ∗0 has a maximum
and is defined by the equation µ∗ = ρ∗0ν(0). The stability condition µ 6 ν(0)ρ0 is fulfilled. Notice
that for all 0 < µ 6 µ∗, the curve ρ0(µ) very little differs from the straight line ρ0 = µ/ν(0) (for
the potential (14) the deflection is ∼ 0.1%). The condition of the existence of Bose condensate
at 0 < µ 6 µ∗ is in excellent agreement with the previous estimate of the Bogolyubov theory
correctness (see formula (29) in paper [2]) ν(0) � v/(2mr20). Here we must take into account that
v ∼ ρ−1 and suppose (as was conjectured by Bogolyubov) that µ = ρν(0).
Consider now the case θ > 0. Denote by Fβ(x) the right-hand side of the equation (13). As a
function of x, this function is strictly monotonously decreasing, convex and bounded from below
on [0,∞) for any β. As shown above, we reach a conclusion that if
∆β(µ) ≡ Fβ
(
x =
µ
ν(0)
)
6 0
then the equation (13) will have a nontrivial unique solution for µ > 0. The function ∆β(µ)
increases if β decreases. The “critical curve” θ = θ0(µ) is determined by the equation ∆β(µ) = 0.
One can verify that ∂2θ0/∂µ
2 < 0. On this curve, the Bose condensate density has a maximum
ρ0 = µ/ν(0). The curve θ = θ0(µ) is illustrated in figure 1.
µ[Κ]
θ[Κ]
0.5 1 1.5
0.01
0.02
0.03
0.04
0
Figure 1. Phase diagram.
Let µ 6 0 and the potential ν(k) satisfies (15). Then, the self-consistency equation for the
Bogolyubov model for any θ has a trivial solution only. In this case, the Bogolyubov model is
thermodynamically equivalent to the ideal Bose gas.
4. Conclusion
We have shown that if the potential ν(k) in the Bogolyubov model of superfluidity (2) satisfies
the condition (15), then there exists a domain of stability on the phase diagram {0 < µ 6 µ∗, 0 6
θ 6 θ0(µ)}, where the nontrivial solution of self-consistency equation takes place. In this domain,
there is a non-zero Bose condensate. At the boundary θ = θ0(µ) of this domain, the Bose con-
densate density equals ρ0 = µ/ν(0). In this case, the quasi-particles spectrum of the Bogolyubov
23002-5
N.N. Bogolyubov, Jr., D.P. Sankovich
Hamiltonian (2)
Ek =
√
εk(εk + 2ρ0ν(k))
is of a gapless type, and the famous criterion of superfluidity mink(Ek/|k|) > 0 holds.
As we have noted earlier, the Bogolyubov’s theory is a theory of a dilute weakly interacting Bose
gas at temperatures far below the λ-point. This is particularly evident from the phase diagram.
By contrast to the pair Hamiltonian (1), the Bogolyubov’s Hamiltonian (2) does not correspond
to some pair interaction and it is not superstable. Nevertheless, Bogolyubov’s approach forms the
basis for a systematic application of quantum theory to an interacting system of bosons (for a
review, see [14]).
References
1. Ruelle D., Helv. Phys. Acta, 1963, 36, 789–799.
2. Bogolyubov N.N., J. Phys. (USSR), 1947, 11, 23–32.
3. Bogolyubov N.N., Bull. Moscow State Univ., 1947, 7, 43–56.
4. Dirac P.A.M., The Principles of Quantum Mechanics, 2nd ed. Clarendon Press, Oxford, 1935.
5. Ginibre J., Commun. Math. Phys., 1968, 8, 26–51.
6. Lieb E.H., Seiringer R., Yngvason J., Phys. Rev. Lett., 2005, 94, 080401.
7. Angelescu N., Verbeure A., Zagrebnov V.A., J. Phys. A., 1992, 25, 3473–3491.
8. Bru J.B., Zagrebnov V.A., J. Phys. A., 1998, 31, 9377–9404.
9. Zagrebnov V.A., Bru J.B., Phys. Rep., 2001, 350, 291–434.
10. Bogolyubov N.N., Physikalishe Abhandlungen aus der Sowjetunion, 1962, 6, 229–284.
11. Bogolyubov N.N., Quasi-Averages in Problems of Statistical Mechanics. JINR, Dubna, 1961.
12. Bogolyubov, Jr. N. N., Phycica, 1966, 32, 933–944.
13. Harris B., J. Math. Phys., 1967, 8, 1044–1045.
14. Shi H., Griffin A., Phys. Rep., 1998, 304, 1–87.
Асимптотична точнiсть пiдстановки c-чисел в теорiї
надплинностi Боголюбова
Н.Н. Боголюбов (мол.), Д.П. Санковiч
Математичний iнститут iм. В.А.Стєклова Росiйської академiї наук, Москва, Росiйська Федерацiя
Розглядається модель Боголюбова для рiдкого гелiю. Обґрунтованiсть пiдстановки c-числа для
k = 0 модового оператора â0 строго встановлюється. Знайдено область стабiльностi гамiльтонiану
Боголюбова. Ми виводимо достатнi умови, що забезпечують появу бозе-конденсату в цiй моделi.
Для деяких температур та значень хiмiчного потенцiалу iснує безщiлинний боголюбiвський спектр
елементарних збуджень, що приводить до належної мiкроскопiчної iнтерпретацiї надплинностi.
Ключовi слова: бозе-ейнштейнiвська конденсацiя, гамiльтонiан Боголюбова, надплиннiсть,
пiдстановка c-числа
23002-6
The model
Stability of the Hamiltonian
Self-consistency equation
Conclusion
|
| id | nasplib_isofts_kiev_ua-123456789-32088 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T16:46:43Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Bogolyubov (jr), N.N. Sankovich, D.P. 2012-04-08T15:20:43Z 2012-04-08T15:20:43Z 2010 Asymptotic exactness of c-number substitution in Bogolyubov's theory of superfluidity / N.N. Bogolyubov, Jr., D.P. Sankovich // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23002: 1-6. — Бібліогр.: 14 назв. — англ. 1607-324X PACS: 05.30.Jp, 03.75.Fi, 67.40.-w https://nasplib.isofts.kiev.ua/handle/123456789/32088 The Bogolyubov model of liquid helium is considered. The validity of substituting a c-number for the k=0 mode operator â0 is established rigorously. The domain of stability of the Bogolyubov's Hamiltonian is found. We derive sufficient conditions which ensure the appearance of the Bose condensate in the model. For some temperatures and some positive values of the chemical potential, there is a gapless Bogolyubov spectrum of elementary excitations, leading to a proper microscopic interpretation of superfluidity. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Asymptotic exactness of c-number substitution in Bogolyubov's theory of superfluidity Article published earlier |
| spellingShingle | Asymptotic exactness of c-number substitution in Bogolyubov's theory of superfluidity Bogolyubov (jr), N.N. Sankovich, D.P. |
| title | Asymptotic exactness of c-number substitution in Bogolyubov's theory of superfluidity |
| title_full | Asymptotic exactness of c-number substitution in Bogolyubov's theory of superfluidity |
| title_fullStr | Asymptotic exactness of c-number substitution in Bogolyubov's theory of superfluidity |
| title_full_unstemmed | Asymptotic exactness of c-number substitution in Bogolyubov's theory of superfluidity |
| title_short | Asymptotic exactness of c-number substitution in Bogolyubov's theory of superfluidity |
| title_sort | asymptotic exactness of c-number substitution in bogolyubov's theory of superfluidity |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32088 |
| work_keys_str_mv | AT bogolyubovjrnn asymptoticexactnessofcnumbersubstitutioninbogolyubovstheoryofsuperfluidity AT sankovichdp asymptoticexactnessofcnumbersubstitutioninbogolyubovstheoryofsuperfluidity |