Bogolyubov approximation for diagonal model of an interacting Bose gas

0.13 MB·English

by  M. Corgini

Bogolyubov approximation for diagonal model
7 of an interacting Bose gas
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0
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n
a
J M. Corgini a, D.P. Sankovichb,
∗

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1 aDepartamento de Matema´ticas, Universidad de La Serena,

] Cisternas 1200, La Serena, Chile
h
c bV.A. Steklov Mathematical Institute, Gubkin str. 8, 119991 Moscow, Russia
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m
-
t
a
Abstract
t
s
.
t
a We study, using the Bogolyubov approximation, the thermodynamic behaviour of
m
a superstable Bose system whose energy operator in the second-quantized form
- contains a nonlinear expression in the occupation numbers operators. We prove
d
n that for all values of the chemical potential satisfying µ > λ(0), where λ(0) 0 is
≤
o the lowest energy value, the system undergoes Bose–Einstein condensation.
c
[

1 Key words: Bose–Einstein condensation, Bogolyubov approximation,
v
grand-canonical pressure
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PACS: 05.30.Jp, 03.75.Hh, 64.60.Cn
3
2
1
0
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0 1. Exactly solvable models of strongly interacting bosons could be helpful in
/ understanding the nature of Bose–Einstein condensation (BEC) and superflu-
t
a
idity in interacting Bose gases. In this paper we study the thermodynamic be-
m
havior of some theoretically relevant system of Bose particles. The considered
-
d
model has rather simplified character. Researches related to similar models
n
o are connected with the attempt to consider the effect of excitation-excitation
c coupling in Bogolyubov theory.
:
v
i
X The aim of this paper is to consider a model of Bose gas with diagonal interac-
r tion which shows an independent on temperature BEC. This kind of BEC was
a
theoretically discovered by Bru and Zagrebnov for some specific Bose system
with diagonalinteractions [1,2].Here we study a nonlinear modification of the
Bru–Zagrebnov model. Furthermore we want to clear up the influence of high-
Corresponding author.
∗
Email addresses: [email protected] (M. Corgini ), [email protected]
(D.P. Sankovich).
Preprint submitted to Physics Letters A 6 February 2008
order perturbations in the non-ideal Bose gas interaction on the magnitude of
the Bose condensate fraction.
The plan ofthe paper is asfollows. Firstly, we describe the basic mathematical
notions associated to this kind of systems. Secondly, we discuss the thermo-
dynamic behavior of a superstable Bose system whose Hamiltonian contains
a nonlinear term in the number operators. The nonlinearity can be physically
understood as the simultaneous creation of k-single one mode particles after
the disappearance of an equivalent amount of single particles associated to the
same mode. For k = 2 a variation of this kind of models has been extensively
studied in refs. [1,2,3,4] by using different mathematical techniques. (In con-
trast to the Bru–Zagrebnovmodel [2],the Hamiltonianwe consider contains a
specific nonlinear mean-field term (see also [5])). In our case use will be made
of the so-called Bogolyubov approximation [6]. In our proof a significative
role plays the fact that the involved Hamiltonian is written in terms of the
occupation numbers operators. It enables us to give a simplified proof of ther-
modynamic equivalence of the limit grand-canonical pressures corresponding
to the energy operator of the system and its respective approximative Hamil-
tonian.
We have to point out that although this modified approach leads to a straight-
forward proof of BEC based on the commutativity of the involved operators,
it does not apply for proving spontaneous breakdown of symmetry.
2. Let ∆ be the operator Laplacian. The one-particle free Hamiltonian corre-
sponds to the operator D = ∆/2 defined on a dense subset of the Hilbert
Λl −
space l = L2(Λ ), being Λ = [ l/2,l/2]d Rd a cubic box of boundary ∂Λ
l l l
H − ⊂
andvolumeV = ld.Weassume periodicboundaryconditionsunder whichD
l Λl
becomes a self-adjoint operator. In this case, as in ref. [3,7], we shall consider
a negative isolated lowest value of energy. This can be done by adding a term
to the original Hamiltonian which shifts the energy level of the zero mode of
the kinetic energy operator downward by a negative amount, creating in this
way a spectral gap.
We shall study model of interacting Bose particles whose Hamiltonian is given
as
γ γ Nˆ′
H = H0 + 0 (a†)kak + 1 (a†)kak +V g , (1)
l l Vlk−1 0 0 Vlk−1 p∈ΛX∗l\{0} p p l Vl !
where integer k 2 and g(x) is a suitable real valued function satisfying
≥
general conditions ensuring the superstability of the model. In particular, it
suffices to assume that g(x) is a continuously differentiable function on [0, )
∞
satisfying g(0) = 0 and g′(x) as x [8]. The sum in (1) runs over
→ ∞ → ∞
the set Λ∗ = p Rd : p = 2πn /l,n = 0, 1, 2,...,α = 1,2,...,d ,
l { ∈ α α α ± ± }
2
a†,a are the Bose operators of creation and annihilation of particles defined
p p
on the Fock space satisfying the usual Bose commutation rules: [a ,a†] =
FB p q
a a† a†a = δ , being δ the Kronecker’s delta. Denote by n = a†a
p q − q p p,q p,q p p p
the occupation-number operator in the mode p Λ∗. Let λ (p) = p2/2 for
∈ l l
p = 0 and λ (0) λ(0) 0. In this case H0 = λ (p)n , γ ,γ > 0.
6 l ≡ ≤ l p∈Λ∗l l p 0 1
N = n is the total number operator and N′ = n is the
p∈Λ∗ p P p∈Λ∗\{0} p
l l
number operator with exclusion of n . Note that the boson Fock space is
P 0 P FB
isomorphic to the tensor product B where B is the boson Fock space
⊗p∈Λ∗lFp Fp
constructed on the one-dimensional Hilbert space = γeipx .
p
H { }
Let
1
p (β,µ) = lnTr exp[ β(H µN)]
l βV ⊗p∈Λ∗FpB − l −
l l
be the grand-canonical pressure corresponding to H , where µ represents the
l
chemical potential and β = θ−1 is the inverse temperature.
If H (µ) H µN, the equilibrium Gibbs state is defined as
l ≡ l − h−iHl(µ)
−1
A Tr exp( βH (µ)) Tr Aexp( βH (µ)),
h iHl(µ) ≡ ⊗p∈Λ∗FpB − l ⊗p∈Λ∗FpB − l
(cid:20) l (cid:21) l
for any operator A acting on the symmetric bosonic Fock space. The density
of particles at infinite volume is defined as
N
ρ(µ) = lim ρ (µ) lim .
l
Vl→∞ ≡ Vl→∞(cid:28)Vl(cid:29)Hl(µ)
The density of particles of energy λ(0), corresponding to the mode p = 0, is
given as
n
0
ρ (β,µ) = lim ρ (β,µ) lim .
0 0,l
Vl→∞ ≡ Vl→∞(cid:28)Vl(cid:29)Hl(µ)
Inthispaperwewanttoprovetheoccurrenceofmacroscopicoccupationofthe
ground state, i.e. we attempt to find values of µ and β for which the inequality
ρ (β,µ) > 0 holds. Indeed, we prove for the studied model that macroscopic
0
occupation independent on temperature takes place, i.e. the density of the
condensate not depends on β.
3. In the framework of the Bogolyubov approximation we must replace the
operators a† and a in any operator A expressed in the normal form in the
0 0
Hamiltonian H (µ) with the complex numbers √V c¯ and √V c. We get the
l l l
following approximative Hamiltonian
3
γ N′
Happr(c,µ) = (λ (p) µ)n + 1 (a†)kak +V g
l p∈ΛX∗l\{0}" l − p Vlk−1 p p# l Vl !
+(λ(0) µ)V c 2 +γ V c 2k, c C. (2)
l 0 l
− | | | | ∈
Noting that (a†)k(a )k = n (n 1) (n k + 1) for all p Λ∗, it is not
p p p p − ··· p − ∈ l
hard to see that both Hamiltonians (1) and (2) are diagonal with respect to
the occupation-number operators.
In this case we have the following result [9].
Theorem 1 Tr⊗p∈Λ∗\{0}FpBe−βHlappr(c,µ) ≤ Tr⊗p∈Λ∗FpB e−βHl(µ).
l l
4. Let f (x) be the real valued function defined as
Vl
1 k 1
f (x) = (µ λ(0))x γ x x x − .
Vl − − 0 (cid:18) − Vl(cid:19)··· − Vl !
We define the functions
fL(x) = (µ λ(0))x γ xk
0
− −
and
k
k 1
fU(x) = (µ λ(0))x γ x − .
Vl − − 0 − Vl !
For x (k 1)/V the functions fL(x) and fU(x) are concave with unique
≥ − l Vl
maxima attained at xL and xU, respectively. Evidently,
Vl
1
(µ λ(0)) k−1
lim xU = xL = − + ,
Vl→∞ Vl " γ0k #
where (a) = max(0,a).
+
Lemma 2 For V sufficiently large such that xL > (k 1)/V , the following
l l
−
inequalities
fL(xL) sup f (x) fU(xU),
≤ Vl ≤ Vl Vl
[k−1,∞)
Vl
k 1 (2(k 1))!
sup f (x) (µ λ(0)) − +γ −
Vl ≤ − + V 0(k 2)!Vk
[0,kV−l1) l − l
4
hold.
PROOF. For V sufficiently large and x [(k 1)/V , ) we have the obvi-
l l
∈ − ∞
ous inequalities fL(x) f (x) fU(x), which for xL > (k 1)/V imply the
≤ Vl ≤ Vl − l
first statement of the proposition. On the other hand,
1 k 1
f (x) (µ λ(0)) x+γ x x+ x+ −
Vl ≤ − + 0 (cid:18) Vl(cid:19)··· Vl !
for x [0,(k 1)/V ). Then,
l
∈ −
k 1 k 1 k 2(k 1)
sup f (x) (µ λ(0)) − +γ − − .
Vl ≤ − + V 0 V V ··· V
[0,k−1) l l l l
Vl
2
Lemma 3 The following inequality takes place:
1
(µ λ(0)) k−1 k 1
lim sup f (x) fL(xL) = − + (µ λ(0)) − .
Vl→∞[0,∞) Vl ≤ " kγ0 # − + k
PROOF.
k 1 k 1
fU(xU) fL(xL) = fU xL + − fL(xL) = (µ λ(0)) − ,
| Vl Vl − | Vl Vl !− − + Vl
i.e. lim fU(xU) = fL(xL). Then, using this result and the first inequality
Vl→∞ Vl Vl
in lemma 2, we get
1
(µ λ(0)) k−1 k 1
lim sup f (x) = fL(xL) = − + (µ λ(0)) − .
Vl→∞[k−1,∞) Vl " kγ0 # − + k
Vl
The second inequality of lemma 2 yields
lim sup f (x) 0.
Vl→∞[0,k−1) Vl ≤
Vl
From these results, noting that
sup f (x) max sup f (x), sup f (x) ,
Vl ≤ { Vl Vl }
[0,∞) [0,k−1) [k−1,∞)
Vl Vl
5
and passing to the limit V , the proof of the lemma follows.
l
→ ∞
2
Letρˆ = n /V .ρˆ isadenselydefinedon B positiveself-adjointoperatorwith
0 0 l 0 F0
spectrum σ(ρˆ ) [0, ) and spectral family E (x) : x σ(ρˆ ) [10]. Let
0 ⊂ ∞ { ρˆ0 ∈ 0 }
h(x) : [0, ) R be a continuous function such that sup h(x) exists.
∞ → x∈[0,∞)
Assume also that h(x)exp(βV f ) is a bounded and continuous function in
l Vl
[0, ). Let A,B be self-adjoint bounded operators in the Hilbert space
∞ U
endowed with inner product denoted as ( , ). Then we say that A is smaller
· ·
than B and write A B if (φ,Aφ) (φ,Bφ) for every φ .
≤ ≤ ∈ U
Lemma 4 h(ρˆ ) sup h(x).
h 0 iHl(µ) ≤
x∈[0,∞)
PROOF. The spectral theorem for self-adjoint operators [10] enables us to
write
h(ρˆ )exp(βV f (ρˆ )) = h(x)exp(βV f (x))dE (x).
0 l Vl 0 l Vl ρˆ0
Z
σ(ρˆ0)
Although ρˆ is an unbounded operator, the above representation is valid for
0
any φ B since h(x)exp(βV f (x)) is a bounded continuous function. This
∈ F0 l Vl
implies that,
h(ρˆ )exp(βV f (ρˆ )) sup h(x) exp(βV f (x))dE (x),
0 l Vl 0 ≤ l Vl ρˆ0
x∈[0,∞) Z
σ(ρˆ0)
in the sense of operators for all φ B . Noting that
∈ F0
exp( βHˆ (µ)) = exp(βV f (ρˆ ))exp( βHˆ′(µ))
− l l Vl 0 − l
andsinceexp(βV f (ρˆ )),exp( βHˆ′(µ))arepositivelydefiniteoperatorscom-
l Vl 0 − l
muting with each other, it is not hard to see that
h(ρˆ )exp(βV f (ρˆ ))exp( βHˆ′(µ))
0 l Vl 0 − l
sup h(x)exp(βV f (ρˆ ))exp( βHˆ′(µ)).
≤ l Vl 0 − l
x∈[0,∞)
Then, taking into account that Hˆ (µ) defines a superstable system and since
l
is isomorphic B we get
FB ⊗p∈Λ∗lFp
6
Tr h(ρˆ )exp(βV f (ρˆ ))exp( βHˆ′(µ))
FB 0 l Vl 0 − l
ˆ′
sup h(x)Tr exp(βV f (ρˆ ))exp( βH (µ)).
≤ FB l Vl 0 − l
x∈[0,∞)
Hence, we conclude that
h(ρˆ ) sup h(x).
h 0 iHˆ(µ) ≤
x∈[0,∞)
2
Theorem 5 The systems with Hamiltonians H (µ) and Happr(c,µ) are ther-
l l
modynamically equivalent in the Wentzel sense [11], i.e.
lim sup pappr(β,c,µ) = p(β,µ),
l
Vl→∞|c|:c∈C
where
1
pappr(β,c,µ) lnTr exp[ β(Happr(c,µ))]
l ≡ βV ⊗p∈Λ∗\{0}FpB − l
l
and p(β,µ) lim p (β,µ).
≡ Vl→∞ l
PROOF. Notethatthisisanequivalence oftheWentzeltypebutconsidering
infimum for the real variable c instead for c C. Applying the Bogolyubov’s
| | ∈
convexity inequalities for pressures and theorem 1 we obtain that
1
0 p (β,µ) pappr(β,c,µ) Happr(c,µ) H (µ) ,
≤ l − l ≤ V h l − l iHl(µ)
l
where
1
Happr(c,µ) H (µ) = (λ(0) µ) c 2+γ c 2k
V h l − l iHl(µ) − | | 0| |
l
k 1
+ (µ λ(0))ρˆ γ ρˆ ρˆ − .
0 0 0 0
* − − ··· − Vl !+Hl(µ)
Hence, we obtain
0 p (β,µ) pappr(β,c,µ) (λ(0) µ) c 2 +γ c 2k + f (ρˆ ) . (3)
≤ l − l ≤ − | | 0| | h Vl 0 iHl(µ)
We have to distinguish two cases.
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(i) Case µ > λ(0). In this situation, inf|c|:c∈C (λ(0) µ) c 2 +γ0 c 2k is ob-
− | | | |
tained for c given by h i
0
| |
1
µ λ(0) k−1
c 2 = − .
0
| | kγ0 !
This leads to
1
µ λ(0) k−1 k 1
inf (λ(0) µ) c 2 +γ c 2k = − (µ λ(0)) −
0
|c|:c∈C − | | | | − kγ0 ! − k
h i
= fL(xL).
−
From the above result, using lemma 4 with h(x) = f (x) and considering the
Vl
infimum for c C in the right-hand side of inequality (3), it follows that
∈
1
µ λ(0) k−1 k 1
0 p (β,µ) sup pappr(β,c,µ) − (µ λ(0)) −
≤ l −|c|:c∈C l ≤ − kγ0 ! − k
+ sup f (x).
Vl
[0,∞)
Therefore, passing to the limit V and applying lemma 3, we get
l
→ ∞
p(β,µ) lim sup pappr(β,c,µ) = 0.
−Vl→∞|c|:c∈C l
(ii)Case µ λ(0). In this case inf|c|:c∈C (λ(0) µ) c 2 +γ0 c 2k is attained
≤ − | | | |
at c = 0. Moreover, since (a†)kak is ahpositive operator, for µi λ(0) we
| 0| 0 0 ≤
obtain
k 1
(µ λ(0))ρˆ γ ρˆ ρˆ −
0 0 0 0
* − − ··· − Vl !+Hl(µ)
(a†)kak
= (µ λ(0))ρˆ γ 0 0 0.
* − 0 − 0 Vlk +Hl(µ) ≤
Then, from inequality (3) we get,
p (β,µ) = sup pappr(β,c,µ)
l l
|c|:c∈C
for any finite V . Therefore, in both cases, passing to the limit V , we get
l l
→ ∞
p(β,µ) = lim sup pappr(β,c,µ).
l
Vl→∞|c|:c∈C
8
The proof is complete.
2
5. The following theorem holds.
Theorem 6 For all µ > λ(0) the system (1) displays independent on temper-
ature BEC and the amount of condensate is given by
1
µ λ(0) k−1
ρ (µ) = − .
0
γ0k !
PROOF. Using the convexity of p (β,µ) and pappr(β,c ,µ) with respect to
l l 0
λ(0), we get from a Griffiths theorem [12]
ρ (µ) = ∂ p (β,µ) = ∂ pappr(β,c ,µ) = c 2.
0,l − λ(0) l − λ(0) l 0 | 0|
Finally, passing to the thermodynamic limit, we obtain for all µ > λ(0),
1
µ λ(0) k−1
lim ρ (µ) = ρ (µ) = c 2 = − .
0,l 0 0
Vl→∞ | | γ0k !
2
6. For the class of Hamiltonians given in equation (1), in the framework of
the so-called Bogolyubov approximation [6,9], it has been given a simple
proof of thermodynamic equivalence of the limit grand-canonical pressures
corresponding to those systems and their respective approximating ones for
any integer k 2. Moreover, in contrast to the Bru–Zagrebnov models [1,2],
≥
we prove that independent on temperature BEC in the sense of macroscopic
occupation of the ground state holds for any integer k 2 and any µ > λ(0).
≥
A similar type of BEC is explained entirely by superstability of the model and
by absence of an interaction between the ground state occupation number
operators and the nonzero modes ones.
Acknowledgements
Partial financial support by Programa MECESUP PUC103 ( Pontificia Uni-
versidad Cat´olica de Chile), Project ECOS/Conicyt: Estudio Cualitativo de
SistemasDin´amicosCu´anticos(Chile),ProgramadeMag´ısterenMatem´aticas,
Universidad de La Serena, Chile andProgram ”Fundamental problems of non-
linear dynamics” of the Russian Academy of Sciences.
9
References
[1] J.B. Bru, V.A. Zagrebnov, Exactly soluble model with two kinds of Bose–
Einstein condensations, Physica A 268 (1999), 309–325.
[2] J.B. Bru, V.A. Zagrebnov, A model with coexistence of two kinds of Bose
condensation, J. Phys. A : Math. Gen. 33 (2000), 449-464.
[3] J.B. Bru,B. Nachtergaele, V.A. Zagrebnov, Theequilibriumstates for a model
with two kinds of Bose condensation. J. Stat. Phys. 109 (2002), 143-276.
[4] A. Bernal, M. Corgini, D.P. Sankovich, Nonideal Bose gases: correlation
inequalities and Bose condensation, Theor. Math. Phys. 139 (2004), 866–877.
[5] L. Vandevenne, A. Verbeure, Imperfect Bose gas with attractive boundary
conditions, Rep. Math. Phys. 56 (2005), 109–116.
[6] N.N. Bogolyubov, On the theory of superfluidity,J. Phys.(U.S.S.R.) 11 (1947),
23–32.
[7] L. Lauwers, A. Verbeure, V.A. Zagrebnov, Proof of Bose-Einstein condensation
for interacting gases with a one-particle spectral gap, J. Phys. A: Math. Gen.
36 (2002), 169-174.
[8] E.B. Davies, The thermodynamic limit for an imperfect boson gas, Commun.
Math. Phys. 28 (1972), 69–86.
[9] J. Ginibre, On the asymptotic exactness of the Bogolyubov approximation for
many bosons systems, Commun. Math. Phys. 8 (1968), 26–51.
[10] M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol.1.
(Academic Press, New York, 1979).
[11] G. Wentzel, Thermodynamically equivalent Hamiltonian for some many-body
problems, Phys. Rev. 120 (1960), 1572–1575.
[12] R. Griffiths, A proof that the free energy of a spin system is extensive, J. Math.
Phys. 5 (1964), 1215–1222.
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