Influence of the Finite Size Effect on Properties of a Weakly Interacting Bose G as in Improved Hatree-Fock Approximation
It is obviously that the effective mass is the same as that in one-loop approximation while the order
parameter is different. In one-loop approximation, the order parameter is constant and equals to unity.
Eq. (15) shows that in IHF approximation depends strongly on the distance between two plates,
especially in small- region and it turns out to be divergent when the distance approaches to zero.
Experimentally, consider for rubidi87 with parameters m a 1.44 10 kg, 5.05 10 m 25 9 s and
400 nm. Fig. 1 is the evolution of order parameter versus the distance between two plates.
When the distance increases, the order parameter decays fast and tends to constant at large .
Using the s-wave scattering length on can rewrite Eq. (15) in form
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VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 3 (2018) 43-47
43
Influence of the Finite Size Effect on Properties of a Weakly
Interacting Bose G as in Improved Hatree-fock Approximation
Nguyen Van Thu*, Luong Thi Theu
Department of Physics, Hanoi Pedagogical University 2, Nguyen Van Linh, Phuc Yen, Vinh Phuc, Vietnam
Received 28 July 2018
Revised 11 August 2018; Accepted 11 August 2018
Abstract: The finite size effect causes many interesting behaviors in properties of a weakly
interacting Bose gas. These behaviors were considered in one-loop approximation of quantum
field theory. In this paper the influence is investigated in improved Hatree-Fock approximation,
which gives more accurate results.
Keywords: Finite size effect, improved Hatree-Fock approximation, Bose-Einstein condensate.
1. Introduction
The finite size effect is one of the most interesting effects in quantum physics, which takes place in
all of real systems and has been considered thoroughly. It is a hot topic in magnetic material [1],
superconductivity [2], nuclear matter [3] and so on.
In Bose-Einstein condensate (BEC) field, the finite size effect causes the quantum fluctuation on
top of the ground state, which leads to Casimir effect [4]. For two-component Bose-Einstein
condensates, this effect was investigated in [5], in which two essential results are that the Casimir
force is not simple superposition of the one of two single component BEC due to the interaction
between two species and one of the most important result is that this force is vanishing in limit of
strong segregation. In a dilute BEC, using Euler–Maclaurin formula, author of Ref. [6] calculated the
Casimir force corresponding to Dirichlet and Robin boundary conditions. The result shows that the
Casimir force is attractive and divergent when distance between two slabs approaches to zero.
One common thing of these papers is that the finite size effect is studied in one-loop
approximation. In this respect, the effective mass and order parameter do not depend on the distance
between two slabs. In this paper, we consider the influence of finite size effect in a weakly interacting
Bose gas in improved Hatree-Fock (IHF) approximation.
_______
Corresponding author. Tel.: 84-912924226.
Email: nvthu@live.com
https//doi.org/ 10.25073/2588-1124/vnumap.4278
N.V. Thu, L.T. Theu / VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 3 (2018) 43-47
44
2. Research content
To begin, let us start from Lagrangianof a weakly interacting Bose gas [7],
2 2
2
2
* * ( * ) ,
2 2
g
L i
t m z
(1)
where m is atomic mass, is Plack’s constant, the coupling constant g is determined through
s-wave scattering length
sa as
24 / ;sg a m is the chemical potential and in case of dilute gas
one has 0gn with 0n being bulk density of the condensate; is field operator and its mean value
plays the role of order parameter. We limit our attention to order parameters that are translationally
invariant in the x and y directions.
In order to obtain the Hatree-Fock approximation, we shift the field operator as follows
0 1 2
1
,
2
i (2)
Substituting (2) into (1), among others, we obtain the interaction Lagrangian
2 2 2 2 2
int 0 1 1 2 1 2( ) ( ) .
82
g g
L (3)
At finite temperature in the Hatree-Fock approximation, the interaction Lagrangian (3) gives
Cornwall-Jackiw-Tomboulis (CJT) effective potential is defined as [8],
2 4 1 1
0 0 0 0 0
2 2
11 22 11 22
1
, Tr ln , 1
2 2
3 3
,
8 8 4
CJT gV D D k D k D k
g g g
D k D k D k D k
(4)
in which 0D and D are propagators at tree and Hatree-Fock approximation, respectively. Here we
use the symbol
3
3
( , ),
2
n
n
d k
f k T f k
where k is wave vector and n is Matsubara frequency at temperature T. We realized that
Goldstone's theorem is not satisfied in this approximation. To satisfy Goldstone's theorem, the
effective potential (4) need a quantity
2 2
11 22 11 222 ,
4
g
V P P P P
(5)
Combining (4) and (5) we get CJT effective potential, which restores Goldstone boson
N.V. Thu, L.T. Theu / VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 3 (2018) 43-47 45
2 4 1 10 0 0 0 0
2 2
11 22 11 22
1
, ln , 1
2 2
3
.
8 8 4
CJT gV D tr D k D k D k
g g g
P P P P
(6)
The dispersionrelationcan be obtained by request
1det 0D and
2 2
( ) ,
2 2
k k
E k M
m m
(7)
with M being the effective mass. Minimizing effective potential (6) one gets Schwinger–Dyson
(SD) and gap equations
2
0 13 ,M g (8)
2
0 2 0,g (9)
with
1 11 22
2 11 22
3
,
2 2
3
.
2 2
g g
P P
g g
P P
(10)
We consider the effect from the compactified space along z-direction for the time being. Our
system is confined between two parallel plates perpendicular to z-axis and separated by a distance .
Because of the confinement along z-axis, the wave vector is quantized as 2 2 2jk k k , in which the
wave vector component k is perpendicular to0z-axis and jk is parallel with0z-axis. For boson system
the periodic boundary condition is employed, which has the form after combining to Dirichlet
boundary condition at two plates
, 1,2,3,...j
j
k j
To seek the simplicity, we introduce dimensionless distance L
with
02mgn
being
healing length, 0n is density in bulk. By this way, the dimensionless wave vector k becomes
2 2 2 ,j (11)
with ,j
j j L
L
L L
.Using Euler–Maclaurin formula [9] one has
N.V. Thu, L.T. Theu / VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 3 (2018) 43-47
46
1/2
0
11 22 2
0
0, ,
12
.
gn mM
P P
M
M
gn
(12)
After scaling to bulk density n0the dimensionless order parameter is reduced to
0
0n
.
Combining (8), (9), (10) and (12), we obtain SD and gap equations in dimensionless form
1/2
2
2
1 3 ,
8
mgM
M
1/2
2
2
1 0.
24
mgM
(13)
It is easily to find the solution for (13), which is read as
2,M (14)
1/2
2
1 .
24
mgM
(15)
It is obviously that the effective mass is the same as that in one-loop approximation while the order
parameter is different. In one-loop approximation, the order parameter is constant and equals to unity.
Eq. (15) shows that in IHF approximation depends strongly on the distance between two plates,
especially in small- region and it turns out to be divergent when the distance approaches to zero.
Experimentally, consider for rubidi87 with parameters 25 91.44 10 kg, 5.05 10 msm a
and
400 nm. Fig. 1 is the evolution of order parameter versus the distance between two plates.
When the distance increases, the order parameter decays fast and tends to constant at large .
Using the s-wave scattering length on can rewrite Eq. (15) in form
3/2
1/2
3 1/2
0
3
1 .
2
s
n
L
a M
(16)
For a dilute Bose gas, 3
0 1,sn a expanding (16) in power series one arrives
1/2
3
0
1
3/2
,
3
s
L
n a
(17)
in which 1 1 is order parameter in one-loop approximation.
3. Conclusions
By mean of CJT effective action method, in IHF approximation we consider the finite size effect
on a weakly interacting Bose gas. Our main results are in order:
N.V. Thu, L.T. Theu / VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 3 (2018) 43-47 47
- The order parameter depends strongly on the distance between two plates, in which Bose gas is
confined. For a dilute Bose gas, this parameter equals to its value in one-loop approximation after
adding a term
1/2
3
0
3/2
.
3
s
n a
L
This term is significant in small region of distance .
- Because of independence of the effective mass M on distance between two plates, the finite
size effect have no extra contribution on Casimir force in comparing to the one in one -loop
approximation [6].
Acknowledgements
This work is funded by the Vietnam National Foundation for Science andTechnology
Development (NAFOSTED) under Grant No. 103.01-2018.02.
References
[1] C. Mocutaet. al. (2017), Scientific Reports 7, 16970.
[2] X. Y. Lang, Q. Jiang (2005), Solid State Communications134, 797.
[3] Tran Huu Phat and Nguyen van Thu (2014), Int. J. Mod. Phys. A 29, 1450078.
[4] J. Schiefele, C. Henkel (2009), J. Phys. A 42, 045401.
[5] Nguyen Van Thu, Luong Thi Theu (2017), J. Stat. Phys. 168, 1.
[6] Nguyen Van Thu (2018), Phys. Lett. A 382, 1078.
[7] L. Pitaevskii, S. Stringari (2003), Bose–Einstein Condensation, Oxford University Press.
[8] T. H. Phat, L. V. Hoa, D. T. M. Hue (2014), Comm. Phys 24, 343.
[9] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists,6
th
edn (San Diego: Academic, 2005).
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