Abstract. The counting rule of Nambu-Goldstone (NG) modes in a system of
two-segregated Bose-Einstein condensates (BECs) is studied by means of the
hydrodynamic approach for the Gross - Pitaevskii equations. It is shown that although
the numbers of NG modes in system at rest and in moving system differ from each other,
but they obey the same modified counting rule. Moreover, depending on the velocities the
Landau instability may occur in moving system.

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HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0027
Natural Science, 2018, Volume 63, Issue 6, pp. 39-45
This paper is available online at
A NEW COUNTING RULE OF NAMBU-GOLDSTONE MODES IN SYSTEM
OF TWO SEGREGATED BOSE-EINSTEIN CONDENSATES
Le Viet Hoa
Faculty of Physics, Hanoi National University of Education
Abstract. The counting rule of Nambu-Goldstone (NG) modes in a system of
two-segregated Bose-Einstein condensates (BECs) is studied by means of the
hydrodynamic approach for the Gross - Pitaevskii equations. It is shown that although
the numbers of NG modes in system at rest and in moving system differ from each other,
but they obey the same modified counting rule. Moreover, depending on the velocities the
Landau instability may occur in moving system.
Keywords: Counting rule of Nambu-Goldstone (NG) modes, two-segregated Bose-Einstein
condensates, the Gross - Pitaevskii equations, the Landau instability.
1. Introduction
It is known that the low energy property of any system, relativistic or non-relativistic, is
entirely determined by its NG modes. In relativistic physics the existence and the number of
NG modes are determined by the Goldstone theorem [1, 2] which states that any relativistic
system for which a continuous, global symmetry is spontaneously broken, must contain in its
spectrum gapless modes called the NG modes and, moreover, the number of NG modes coincides
with the number of broken continuous symmetries. However, this counting rule is violated for
systems without Lorentz invariance or with spontaneously broken space-time symmetry [3, 4].
For example, in systems without Lorentz-invariance a new counting rule was formulated by
Nielsen and Chadha in Ref.[3]: The number of type I plus twice the number of type II of NG
modes are greater than or equal to the number of broken generators. The type I (type II) consists
of NG modes which possess the dispersion relations with odd (even) power momentum in the low
momentum limit. Recently, the Goldstone theorem has been re-examined in the non-relativistic
systems [5, 6]. In Ref. [7] Takeuchi and Kasamatsu indicated that in system of two segregated
Bose-Einstein condensates (BECs) there exist only two NG modes, a phonon and a ripplon, despite
the number of broken symmetries equals 3. The recent progress of experimental realization for
ultra-cold gases allows us to consider BECs as an important ground for re-considering the counting
rule of NG modes. In this regards, the present paper deals with the subject on how counting rule of
NG modes be modified in two segregated BECs. The hydrodynamic approach will be used since
it is very powerful to provide all expected dispersion relations of NG modes. To this end, let us
begin with the Lagrangian
Received April 12, 2018. Revised August 10, 2018. Accepted August 17, 2018.
Contact Le Viet Hoa, e-mail: hoalv@hnue.edu.vn
39
Le Viet Hoa
£ =
∫
d3x
(
P1 + P2 − g12|ψ1|
2|ψ2|
2
)
, (1.1)
where
Pj = i~ ψ
∗
j
∂ψj
∂t
+
~
2
2mj
|∇ψj|
2 + µj|ψj |
2 −
gjj
2
|ψj |
4 (1.2)
with mj and µj being the atomic mass and chemical potential of the j-th component, respectively.
The intra-and inter-component interaction parameters have the form gjk = 2π~
2ajk(m
−1
j +m
−1
k ).
We assume that ajk > 0 in what follows.
The Gross-Pitaevskii equations are derived from (1.1)
i~
∂ψ1
∂t
=
(
−
~
2
2m1
∇2 + g11|ψ1|
2 + g12|ψ2|
2
)
ψ1,
i~
∂ψ2
∂t
=
(
−
~
2
2m2
∇2 + g22|ψ2|
2 + g12|ψ1|
2
)
ψ2, (1.3)
which describe the immiscible condensates for g212 > g11g22. It is clear that (1.1) and (1.3)
generally allow the excitation of two independent phonons corresponding to the spontaneous
breaking of symmetry U(1)×U(1). For immiscible condensates the translational invariance is also
spontaneously broken by the presence of a domain wall-interface. This leads to the appearance of a
new gapless NG mode: the ripplon mode displaying the ripple waves propagating along interface.
2. Content
For two immiscible BECs we assume that the first (second) component occupies the space
left (right) to the z = 0 plane and and their interface is represented by a surface z = η(x, y, t).
Then the Lagrangian (1.1) is approximated by
£ =
∫
dx dy
(∫ η
−∞
dz P1 +
∫ +∞
η
dz P2
)
− α S, (2.1)
where α is the interface tension and the interface area S is given as
S =
∫
dx dy
[
1 +
(
∂η
∂x
)2
+
(
∂η
∂y
)2]1/2
. (2.2)
For interface with small deviation from a plane, i.e. |η| ≪ 1, from (2.1) and (2.2) we arrive at the
equation
P1(x, y, z = η, t)− P2(x, y, z = η, t) + α
(
∂2
∂x2
+
∂2
∂y2
)
η = 0, (2.3)
which corresponds to the Bernoulli equation in classic fluid mechanics [8].
To explore Eq. (2.3), let us first determine the wave functions in some approximation. To do
this, we follow closely [9] to write the wave function as
ψj(x, y, z, t) =
√
nj(x, y, x, t)e
i φj(x, y, z, t), (2.4)
40
A new counting rule of nambu-goldstone modes in system of two segregated Bose-Einstein condensates
in which nj and φj are real functions. Next the density and phase are separated as follows:
nj(x, y, z, t) = nj 0 + δnj(x, y, z, t), (2.5a)
φj(x, y, z, t) = −
gjjnj 0
~
t+ δφj(x, y, z, t). (2.5b)
Inserting (2.4), (2.5) into (1.3) and taking only the first order of δnj , δφj we get approximately
the equations
−→
∇−→vj = 0, (2.6a)
~
∂
∂t
(δφj) + gjj δnj = 0, (2.6b)
assuming that the relative density changes following a fluid particle are small compared to the
velocity gradients. The velocity ~vj in (2.6a) is defined as
−→vj =
~
mj
−→
∇δφj . (2.7)
Eq.(2.6a) says that in the approximation under consideration the condensates are subject to
incompressible flow.
Now we adopt the ansatz
δφj = ϕj(z)χj(σ), σ = k1x+ k2y − ωt, (2.8)
which together with (2.6a) and (2.7) provides
d2
dz2
ϕj(z)− k
2ϕj(z) = 0, (2.9a)(
∂2
∂x2
+
∂2
∂y2
+ k2
)
χj(σ) = 0, (2.9b)
where
k2 = k21 + k
2
2 .
Imposing the conditions
ϕ1(z)→ 0 as z →∞,
ϕ2(z)→ 0 as z → −∞,
we get from (2.9a)
ϕj(z) = exp[−(−1)
jkz], (2.10)
and Eq.(2.9b) yields the general solutions
χj(σ) = Aj cos σ +Bj sinσ (2.11)
41
Le Viet Hoa
with arbitrary small constants Aj , Bj .
The substitution of (2.10) and (2.11) into (2.8) gives
δφj = exp[−(−1)
jkz](Aj cos σ +Bj sinσ).
For the sake of simplicity in what follows we will restrict ourselves to the case
δφj = Aj exp[−(−1)
jkz] cos σ. (2.12)
Inserting (2.12) into (2.6b) provides
gjj δnj = −Aj~ ω exp[−(−1)
jkz] sin σ. (2.13)
Finally, taking into account (2.4), (2.5), (2.12) and (2.13) we are led to the analytical expression
for the wave functions in the chosen approximation
ψj =
√
nj0 −
Aj~ ω
gjj
exp[−(−1)jkz] sin σ.ei
(
−
gjjnjo
~
t + Aj exp[−(−1)
jkz]. cos σ
)
. (2.14)
Thus, we already established successfully the analytical expressions for the wave functions in the
linear approximation. Therefore from now we go over to determining the shape of the interface
and the dispersion relations for both phonon and ripplon. To do this, we first accept the boundary
condition
∂η
∂t
=
~
m1
(
∂δφ1
∂z
)
z=0
=
~
m2
(
∂δφ2
∂z
)
z=0
, (2.15)
which yields the differential equations for η = η(σ) for δφj given by (2.12)
ω
dη(σ)
dσ
= −A1
~ k
m1
cos σ = A2
~ k
m2
cosσ. (2.16)
The solution to Eq. (2.16) which represents the interface locating at z = 0 is derived immediately
η(σ) = η0 sinσ (2.17)
with |η0| ≪ 1 and ω satisfying the phonon dispersion relation
ω = −
~A1
η0m1
k =
~A2
η0m2
k. (2.18)
Note that the boundary condition (2.15) is justified in the low energy limit, |kη0| ≪ 1.
For simplicity we take η0 > 0, then the first (second) component of BECs is superfluid for
A1 0). In the opposite case, the Landau instability emerges. The speed of sound in
both condensate reads
c = −
~A1
η0m1
=
~A2
η0m2
. (2.19)
(2.18) and (2.19) prove the quantum character of phonon.
42
A new counting rule of nambu-goldstone modes in system of two segregated Bose-Einstein condensates
Next, substituting (2.12), (2.13), (2.17) and (2.18) into Eq. (2.3) and neglecting second and
higher orders of Aj and kη0 we find the ripplon dispersion relation
ω2
(
m1n10
k
+
m2n20
k
)
= αk2
or
ω =
√
α
ρ1 + ρ2
k3/2, (2.20)
where ρ1 = m1n10 = n1(−0); ρ2 = m2n20 = n2(+0).
In order to get a deeper insight into the problem, let us extend the investigation to the
case when the j-component flows with a velocity ~Vj parallel to the interface. Consequently, the
corresponding stationary state is also represented in the form (2.5) with
φj = −
gjjnj0
~
t+
mj
~
~Vj .~r⊥ + δφj ; ~r⊥ = (x, y), (2.21)
here δφj is given in (2.12). In this case the boundary condition (2.15) is modified as(
∂
∂t
+ ~Vj
∂
∂~r⊥
)
η(σ) =
~
mj
(
∂δφj
∂z
)
z=0
. (2.22)
The substitution of (2.12) and (2.17) into (2.22) yields the dispersion relations of two phonon
branches attaching to two components
ω1 = c1k,
ω2 = c2k, (2.23)
here c1, c2 are the speeds of sound in corresponding condensates
c1 = V1 cos(~V1, ~k)−
~A1
η0m1
,
c2 = V2 cos(~V2, ~k) +
~A2
η0m2
. (2.24)
It is evident from (2.24) that the Landau instability might occurs if ~Vj fulfill, at least, one of the
following inequalities
V1 cos(~V1, ~k) <
~A1
η0m1
,
V2 cos(~V2, ~k) < −
~A2
η0m2
. (2.25)
It is similar to what we did before, the Bernoulli equation (2.3) in combination with (2.12), (2.17)
and (2.22) leads to the ripplon spectrum
ω± = VC ±
√
α k3
ρ1 + ρ2
−
ρ1ρ2V 2Rk
2
(ρ1 + ρ2)2
, (2.26)
43
Le Viet Hoa
in which
~VC =
ρ1~V1 + ρ2~V2
ρ1 + ρ2
; ~VR = ~V1 − ~V2.
Eq.(2.26) indicates that the system of two-component BECs is dynamically unstable for
V 2R > 0, i.e
~V1 6= ~V2. This is the well-known Kelvin-Helmholtz instability in hydrodynamics.
Transforming to the laboratory frame ~VC = 0, Eq. (2.26) becomes
ω2 =
α k3
ρ1 + ρ2
−
ρ1V
2
1
ρ2
k2,
which describes the ripplon mode generalized to the moving system and the ripplon mode causes
dynamical instability of the system for
k <
ρ1(ρ1 + ρ2)
αρ2
V 21 .
3. Conclusion
Let us first resume what was presented in the preceding section. The NG modes of capillary
waves were investigated by means of the linear approximation applied to the Gross-Pitaevskii
equations. Two main ingredients of the approach are the boundary condition and the Bernoulli
equation. Our main results are in order as follows:
1- In the long-wave length limit the boundary condition and the Bernoulli equation provide
the dispersion relations of two NG modes for system at rest.
2- For moving system the modified boundary condition leads to the dispersion relations
of two distinct phonons attaching to two condensates. The Landau instability might take place
depending on the velocities of condensate flows. In addition, based on the Bernoulli equation we
arrive at the ripplon with fractional dispersion relation, whose frequency becomes complex when
the velocities of two components are different. This is the well-known Kelvin-Helmholtz instability
in hydrodynamics.
It is worth noting that the phonon mode always come out as a quantum object, whereas
ripplon remains classic in character.
What we found above evidently asserts that the number of NG modes in a system at rest and
in a moving differs from each other. For system at rest our result is in agreement with that of [7],
which, unfortunately, fails for moving system. However, both of them obey the new counting rule
replacing that of Nielsen and Chadha: The number of type I plus twice the number of type II of
NG modes are greater than or equal to the number of broken generators. Here the type I (type II)
is defined as a collection of NG modes with linear (fractional) dispersion relation.
As mentioned above, we made use of the appropriate approximation which leads to Eq.
(2.6a) characterizing the incompressible fluid. Now a question which arises is that how our
obtained results be modified when the compressibility is taken into account. Remember that the
NG modes appears as low-energy excitations in the spectrum of energies. Taking into consideration
the compressibility means that we must calculate additionally the correction of the orders of Mach
number
(
v/vs
)2
[10], here v is typical speed of the system and vs the speed of sound. In the low
44
A new counting rule of nambu-goldstone modes in system of two segregated Bose-Einstein condensates
energy limit this ratio can be treated as negligibly small, of course. Hence, the full low-energy
picture of our system remains unchanged with the involvement of compressibility.
Last but not least it is worth to point out that the numerical diagonalization of the
Bogoliubov-de Gennes equations is very powerful to provide all excited energies, in which the
lowest energies corresponding to NG modes [7] are derived from the hydrodynamic approach.
In view of the above discussion it is possible to conclude that the hydrodynamic approach
is sufficiently reliable and, furthermore, very insightful for studying NG modes in BEC systems,
including the BECs in confined geometries.
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[3] H.B.Nielsen and S.Chadha, 1976. Nucl.Phys. B105, 445.
[4] I.Low and A.V., 2002. Phys. Rev. Lett. 88, 101602.
[5] H.Watanabe and H.Murayama, 2012. Phys. Rev. Lett. 108, 251602.
[6] H.Watanabe and T. Brauner, 2011. Phys.Rev. D84, 125013.
[7] H.Takeuchi and K. Kasamatsu, 2013. Phys.Rev. A88, 043612.
[8] L.D.Landau and E.M.Lifshitz, 1987. Fluid Mechanics, 2nd ed. (Butterworth - Heinemann,
Oxford).
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[10] A.N.Malmi - Kakkada, O.T.Valls and C.Dasgupta, 2014. J.Phys. B47, 055301.
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