Abstract. Quantum phase transition and dynamical stability in Bose-Einstein condensates
(BECs) are studied within the framework of the Cornwall-Jackiw-Tomboulis (CJT)
effective action approach. The effective potential is found in the double-bubble diagram
approximation. Numerical results show that in this approximation stability is strongly
influenced by the chemical potential and the quantum phase transition is of second order.
Keywords: Quantum phase transition, Bose-Einstein condensates, symmetry restoration,
inverse symmetry breaking.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2017-0030
Mathematical and Physical Sci., 2017, Vol. 62, Iss. 8, pp. 40-47
This paper is available online at
A QUANTUM PHASE TRANSITION
IN THE BOSE-EINSTEIN CONDENSATES
Le Viet Hoa1, Cao Luong Van Huong1 and Chu Thi Chung2
1Faculty of Physics, Hanoi National University of Education
2Tay Tuu High School, Dan Phuong, Hanoi
Abstract. Quantum phase transition and dynamical stability in Bose-Einstein condensates
(BECs) are studied within the framework of the Cornwall-Jackiw-Tomboulis (CJT)
effective action approach. The effective potential is found in the double-bubble diagram
approximation. Numerical results show that in this approximation stability is strongly
influenced by the chemical potential and the quantum phase transition is of second order.
Keywords: Quantum phase transition, Bose-Einstein condensates, symmetry restoration,
inverse symmetry breaking.
1. Introduction
In recent years, theoretical studies of Bose-Einstein Condensates (BECs) [1, 2] have
attracted special attention. Following this trend, one has focused on considerations of phase
transition in BECs, such as inverse symmetry breaking (ISB) and symmetry restoration (SR) [3-5].
In the present article, a theoretical formalism for studying BECs in one and two component
Bose gases is formulated by means of Cornwall-Jackiw-Tomboulis (CJT) effective action [6].
2. Content
2.1. Quantum Phase of BEC in one component Bose gase
2.1.1. Effective potential in double-bubble approximation
This section is devoted to the investigation of the quantum phase transition of one
component BEC. To this end, let us begin with the dilute Bose gases given by the Lagrangian:
£ = φ∗
(
−i ∂
∂t
− ∇
2
2m
)
φ− µφ∗φ+ λ
2
(φ∗φ)2 (2.1)
where µ represents the chemical potential of the field φ, m the mass of φ atom, and λ the coupling
constant.
Received July 23, 2017. Accepted September 13, 2017.
Contact Le Viet Hoa, e-mail: dr.viethoa@gmail.com.
40
A quantum phase transition in Bose-Einstein condensates
Now let us focus on the calculation of effective potential in double-bubble diagram
approximation. At first the field operator φ is decomposed
φ =
1√
2
(φ0 + φ1 + iφ2). (2.2)
Inserting (2.2) into (2.1) we get, among others, the interaction Lagrangian
£int =
λ
2
φ0φ1(φ
2
1 + φ
2
2) +
λ
8
(φ21 + φ
2
2)
2,
and the inverse propagator in the tree approximation
D−1
0
(k, φ0) =
(
~k2
2m
− µ+ 3λ
2
φ2
0
−ω
ω
~k2
2m
− µ+ λ
2
φ2
0
)
. (2.3)
Following closely [5] we arrive at the CJT effective potential in the double-bubble diagram
approximation at finite temperature, which preserves the Goldstone theorem:
V˜ CJTβ (φ0,D) = −
µ
2
φ20 +
λ
8
φ40 +
1
2
∫
β
tr
{
lnD−1(k) + [D−1
0
(k;φ0)D]− 1
}
+
λ
8
P 211 +
λ
8
P 222 +
3λ
4
P11P22, (2.4)
Here D is the propagator in the considered approximation and
∫
β
f(k) = T
∞∑
n=−∞
∫
d3k
(2π)3
f(ωn, ~k), Paa =
∫
β
Daa, (a = 1 or 2).
From (2.4) we get:
- The SD equation
D−1 = D−1
0
(k;φ0) + Σ, (2.5)
in which
Σ =
(
Σ1 0
0 Σ2
)
=
(
λ
2
P11 +
3λ
2
P22 0
0 3λ
2
P11 +
λ
2
P22
)
- The gap equation
−µ+ λ
2
φ20 +Σ2 = 0. (2.6)
Combining (2.3) and (2.5) we get the form for inverse propagator
D−1 =
(
~k2
2m
+M1 −ω
ω
~k2
2m
+M2,
)
.
41
Le Viet Hoa, Cao Luong Van Huong and Chu Thi Chung
in which
M1 = −µ+ 3λ
2
φ20 +Σ1, M2 = −µ+
λ
2
φ20 +Σ2. (2.7)
Owing to (2.6) M2 vanishes in broken phase and
D−1 =
(
~k2
2m
+M1 −ω
ω
~k2
2m
)
. (2.8)
It is obvious that the dispersion relation related to (2.8) reads
E =
√√√√ ~k2
2m
(
~k2
2m
+M1
)
−→
√
M1
2m
k as k → 0,
which express the Goldstone theorem. Due to the Landau criteria for superfluidity criteria [8] the
idealized Bose gas turns out to be superfluid in broken phase and speed of sound in condensate is
given by
C =
√
M1
2m
. (2.9)
Ultimately the one-particle-irreducible effective potential V˜β(φ0, T ) is read off from (2.4) with D
fulfilling (2.5),
V˜β(φ0, T ) = −µ
2
φ20 +
λ
8
φ40 +
1
2
∫
β
tr lnD−1(k) +
1
2
(
−M1 − µ+ 3λ
2
φ20
)
P11
+
1
2
(
− µ+ λ
2
φ20
)
P22 +
λ
8
P 211 +
λ
8
P 222 +
3λ
4
P11P22. (2.10)
2.1.2. Numerical study
In order to get some insight into the quantum phase transition of the Bose gases, let us
choose model parameters, which are close to the experimental settings, namely we chose λ =
10−11eV −2;m = 8.1010eV .
Next, let us study quantum phase transition which is generated by changing the chemical
potential µ at fixed temperature T . In Fig. 1 the µ dependence of M1 and φ0 are ploted at several
values of temperature. Based on Eq. (2.9), in Fig.1a it is shown that quantum condensation in
one component Bose gase appears only since M1 is positive, i.e. its sound speed is real. This
corresponds to the dynamic stability for condensation in a system. At the phase transition points,
in this case, M1 changes the sign. As is seen from Fig. 1b the inverse symmetry breaking (ISB)
takes place at critical chemical potential µ = µc which depends on T . On the other hand, Fig. 1b
also reveals that phase transition is of second order: as µ increases from µ = µc the condensate
density φ0 without interrupted increases from φ0 = 0 to φ0 6= 0. This statement is confirmed again
in Fig. 2, providing the evolution of effective potential (2.10) with respect to φ0: with the increase
in chemical potential, a minimum point of effective potential smoothly changes from φ0 = 0 to
φ0 6= 0.
42
A quantum phase transition in Bose-Einstein condensates
(a) (b)
Figure 1. µ-dependence of M1 and condensates φ0 at several values of temperature
Figure 2. The evolution of the V (φ0, µ)
as a function of the order parameter φ0 for several values of µ
2.2. Quantum phase of BEC in two component Bose gases
2.2.1. Effective potential indouble-bubble approximation
The theoretical studies of two immiscible BECs [3, 7] have allowed us to explore many
interesting physical properties of BECs. In this section we consider the dynamic stability and
43
Le Viet Hoa, Cao Luong Van Huong and Chu Thi Chung
quantum phase trasnition of two component BEC. To begin with, let us start with the Lagrangian:
£ = φ∗
(
−i ∂
∂t
− ∇
2
2mφ
)
φ+ ψ∗
(
−i ∂
∂t
− ∇
2
2mψ
)
ψ
− µ1φ∗φ+ λ1
2
(φ∗φ)2 − µ2ψ∗ψ + λ2
2
(ψ∗ψ)2 +
λ
2
(φ∗φ)(ψ∗ψ) (2.11)
where µ1 (µ2) represents the chemical potential of the field φ (ψ), m1 (m2) the mass of φ atom
(ψ atom), and λ1, λ2 and λ the coupling constants.
Following closely [3] we arrive at the CJT thermal effective potential in the double-bubble
diagram approximation, which preserves the Goldstone theorem:
V˜ CJTβ (φ0, ψ0,D,G) = −
µ1
2
φ20 +
λ1
8
φ40 −
µ2
2
ψ20 +
λ2
8
ψ40 +
λ
8
φ20ψ
2
0
+
1
2
∫
β
tr
{
lnD−1(k) + lnG−1(k) + [D−1
0
(k;φ0, ψ0)D] + [G
−1
0
(k;φ0, ψ0)G]− 21
}
+
λ1
8
P 211 +
λ1
8
P 222 +
3λ1
4
P11P22 +
λ2
8
Q211 +
λ2
8
Q222 +
3λ2
4
Q11Q22
+
λ
8
P11Q11 +
λ
8
P11Q22 +
λ
8
P22Q11 +
λ
8
P22Q22 (2.12)
Here D, G are the propagators and∫
β
f(k) = T
∞∑
n=−∞
∫
d3k
(2π)3
f(ωn, ~k), Paa =
∫
β
Daa, Qaa =
∫
β
Gaa, (a = 1 or 2).
From (2.12) we deduce the following equations:
- The Schwinger-Dyson (SD) equations:
D−1 =
(
~k2
2m1
+M1 −ω
ω
~k2
2m1
)
; M1 = −µ1 + 3λ1
2
φ20 +
λ
4
ψ20 +Σ
φ
1
,
G−1 =
(
~k2
2m2
+M2 −ω
ω
~k2
2m2
)
; M2 = −µ2 + 3λ2
2
ψ20 +
λ
4
φ20 +Σ
ψ
1
. (2.13)
- The gap equations
−µ1 + λ1
2
φ20 +
λ
4
ψ20 +Σ
φ
2
= 0
−µ2 + λ
4
φ20 +
λ2
2
ψ20 +Σ
ψ
2
= 0. (2.14)
Here
Σφ
1
=
λ1
2
P11(k) +
3λ1
2
P22(k) +
λ
4
Q11(k) +
λ
4
Q22(k)
Σφ
2
=
3λ1
2
P11(k) +
λ1
2
P22(k) +
λ
4
Q11(k) +
λ
4
Q22(k)
Σψ
1
=
λ2
2
Q11(k) +
3λ2
2
Q22(k) +
λ
4
P11(k) +
λ
4
P22(k)
Σψ
2
=
3λ2
2
Q11(k) +
λ2
2
Q22(k) +
λ
4
P11(k) +
λ
4
P22(k) (2.15)
44
A quantum phase transition in Bose-Einstein condensates
Eqs. (2.13) leads to the dispersion relations at small momentum
Eφ =
√√√√ ~k2
2m1
(
~k2
2m1
+M1
)
−→
√
M1
2m1
k,
Eψ =
√√√√ ~k2
2m2
(
~k2
2m2
+M2
)
−→
√
M2
2m2
k,
consequently, the speeds of sound in each condensate read
Cφ =
√
M1
2m1
, Cψ =
√
M2
2m2
.
The dynamic instability takes place for condensate j when its superfluidity is broken, that
is when the corresponding speed of sound and energy become complex, namely,
Mj < 0. (2.16)
All thermodynamical processes in the system are governed by Eqs.(2.12), (2.13) and (2.14).
2.2.2. Quantum phase transition
In order to get some insight to the hydrodynamic stability and quantum phase transition of
the two component system of Bose gas, we first choose a set of model parameters, which are close
to the experimental settings [3]:
λ1 = 5.10
−12eV −2;λ2 = 0, 4.10
−12eV −2;λ = 2.10−12eV −2;mφ = mψ =
80GeV ;µ1 = 5.10
−12eV .
(a) (b)
Fig. 3. µ2-dependence of condensates φ0, ψ0 (a) and M1,M2 (b) at temperature T = 5nK
45
Le Viet Hoa, Cao Luong Van Huong and Chu Thi Chung
(a) (b)
Figure 4. Evolutions of effective potential as a function of the order parameters φ0
around µ2φ = 2, 01.10
−12eV (a) and ψ0 around µ2ψ = 0, 93.10
−12eV (b)
Next, solving Eqs. (2.13) and (2.14) we get Fig. 3a which represents the condensation of
φ(ψ) components at constant temperature T = 5nK . As is seen from this figure the quantum
symmetry restoration in sector φ takes place at µ2φ = 2, 01.10
−12eV and phase transition is of
second order.
This statement is confirmed again in Fig. 4a, providing the evolution of effective potential
(2.12) with respect to φ0: with an increase of chemical potential µ2, a minimum point of effective
potential smoothly changes from φ0 6= 0 to φ0 = 0. Also from Fig. 3a quantum inverse symmetry
breaking in sector ψ takes place at µ2ψ = 0, 93.10
−12eV and phase transition is of second order
which is confirmed in Fig. 4b, providing the evolution of effective potential with respect to ψ0:
with increase of chemical potential µ2, a minimum point of effective potential smoothly changes
from ψ0 = 0 to ψ0 6= 0.
The corresponding graphs of M1,M2 plotted in Fig. 3b confirm the close connection
between dynamic stability and the formation of condensate as mentioned above for one component
Bose gase.
3. Conclusions
In the present paper we have considered the dynamic stability and quantum phase transition
in one and two component Bose gases within the framework of CJT formalism restricted to
double-bubble diagram approximation. Our main results are as follows:
(1) Dynamic stability is the main factor wich governs the formation of condensates and the
symmetry breaks in both one and two component Bose gases.
(2) Depending upon the coupling constant setting, there are two scenarios for the quantum
phase transition in the two component Bose gases: symmetry restoration (SR) and inverse
symmetry breaking (ISB).
(3) The quantum phase transition in both systems is of second order.
In order to better understand the physical properties of the BECs, a more detailed
study of phase structure could be carried out by means of numerical computation.
This is left for future study.
46
A quantum phase transition in Bose-Einstein condensates
Acknowledgment. The authors would like to thank the HNUE for financial support.
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