Density, spin and isospin correlations in low-density two-component Fermi superfluid
Finding the distinct features of the crossover from the regime of large overlapping Cooper pairs to the limit of non-overlapping pairs of fermions (Shafroth pairs) in multi-component Fermi systems remains a topical problem in a quantum many-body theory. Here this transition is studied by calculat...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1206162017-06-13T03:06:31Z Density, spin and isospin correlations in low-density two-component Fermi superfluid Isayev, A.A. Yang, J. Квантовые жидкости и квантовые кpисталлы Finding the distinct features of the crossover from the regime of large overlapping Cooper pairs to the limit of non-overlapping pairs of fermions (Shafroth pairs) in multi-component Fermi systems remains a topical problem in a quantum many-body theory. Here this transition is studied by calculating the two-body density, spin and isospin correlation functions in dilute two-component Fermi superfluid, taking as an example an infinite system of protons and neutrons (nuclear matter). It is shown that criterion of the crossover (Phys. Rev. Lett. 95, 090402 (2005)), formulated for ultracold fermionic atomic gases and consisting in the change of the sign of the density correlation function at low momentum transfer, fails to describe correctly the density-driven BEC–BCS transition at finite isospin asymmetry or finite temperature. As an unambiguous signature of the BEC–BCS transition, one can use the presence (BCS regime) or absence (BEC regime) of the singularity in the momentum distribution of the quasiparticle density of states. 2006 Article Density, spin and isospin correlations in low-density two-component Fermi superfluid / A.A. Isayev, J. Yang // Физика низких температур. — 2006. — Т. 32, № 10. — С. 1195–1202. — Бібліогр.: 22 назв. — англ. 0132-6414 PACS: 21.65.+f, 21.30.Fe, 71.10.Ay http://dspace.nbuv.gov.ua/handle/123456789/120616 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| topic |
Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы |
| spellingShingle |
Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы Isayev, A.A. Yang, J. Density, spin and isospin correlations in low-density two-component Fermi superfluid Физика низких температур |
| description |
Finding the distinct features of the crossover from the regime of large overlapping Cooper pairs
to the limit of non-overlapping pairs of fermions (Shafroth pairs) in multi-component Fermi systems
remains a topical problem in a quantum many-body theory. Here this transition is studied by
calculating the two-body density, spin and isospin correlation functions in dilute two-component
Fermi superfluid, taking as an example an infinite system of protons and neutrons (nuclear matter).
It is shown that criterion of the crossover (Phys. Rev. Lett. 95, 090402 (2005)), formulated
for ultracold fermionic atomic gases and consisting in the change of the sign of the density correlation
function at low momentum transfer, fails to describe correctly the density-driven BEC–BCS
transition at finite isospin asymmetry or finite temperature. As an unambiguous signature of the
BEC–BCS transition, one can use the presence (BCS regime) or absence (BEC regime) of the singularity
in the momentum distribution of the quasiparticle density of states. |
| format |
Article |
| author |
Isayev, A.A. Yang, J. |
| author_facet |
Isayev, A.A. Yang, J. |
| author_sort |
Isayev, A.A. |
| title |
Density, spin and isospin correlations in low-density two-component Fermi superfluid |
| title_short |
Density, spin and isospin correlations in low-density two-component Fermi superfluid |
| title_full |
Density, spin and isospin correlations in low-density two-component Fermi superfluid |
| title_fullStr |
Density, spin and isospin correlations in low-density two-component Fermi superfluid |
| title_full_unstemmed |
Density, spin and isospin correlations in low-density two-component Fermi superfluid |
| title_sort |
density, spin and isospin correlations in low-density two-component fermi superfluid |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| topic_facet |
Квантовые жидкости и квантовые кpисталлы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120616 |
| citation_txt |
Density, spin and isospin correlations in low-density
two-component Fermi superfluid / A.A. Isayev, J. Yang // Физика низких температур. — 2006. — Т. 32, № 10. — С. 1195–1202. — Бібліогр.: 22 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT isayevaa densityspinandisospincorrelationsinlowdensitytwocomponentfermisuperfluid AT yangj densityspinandisospincorrelationsinlowdensitytwocomponentfermisuperfluid |
| first_indexed |
2025-07-08T18:13:58Z |
| last_indexed |
2025-07-08T18:13:58Z |
| _version_ |
1837103516223537152 |
| fulltext |
Fizika Nizkikh Temperatur, 2006, v. 32, No. 10, p. 1195–1202
Density, spin and isospin correlations in low-density
two-component Fermi superfluid
A.A. Isayev
Kharkov Institute of Physics and Technology, 1 Academicheskaya Str., Kharkov 61108, Ukraine
J. Yang
Dept. of Physics and Center for Space Science and Technology, Ewha Womans University, Seoul 120-750, Korea
E-mail: jyang@ewha.ac.kr
Received December 6, 2005, revised February 23, 2006
Finding the distinct features of the crossover from the regime of large overlapping Cooper pairs
to the limit of non-overlapping pairs of fermions (Shafroth pairs) in multi-component Fermi sys-
tems remains a topical problem in a quantum many-body theory. Here this transition is studied by
calculating the two-body density, spin and isospin correlation functions in dilute two-component
Fermi superfluid, taking as an example an infinite system of protons and neutrons (nuclear mat-
ter). It is shown that criterion of the crossover (Phys. Rev. Lett. 95, 090402 (2005)), formulated
for ultracold fermionic atomic gases and consisting in the change of the sign of the density correla-
tion function at low momentum transfer, fails to describe correctly the density-driven BEC–BCS
transition at finite isospin asymmetry or finite temperature. As an unambiguous signature of the
BEC–BCS transition, one can use the presence (BCS regime) or absence (BEC regime) of the sin-
gularity in the momentum distribution of the quasiparticle density of states.
PACS: 21.65.+f, 21.30.Fe, 71.10.Ay
Keywords: BCS–BEC transition, bound state, correlation function, quasiparticle density of states.
Introduction
The transition from BCS superconductivity to Bo-
se—Einstein condensation (BEC) occurs in a Fermi
system if either the density is decreased or the attrac-
tive interaction between fermions is increased suffi-
ciently. The crossover from large overlapping Cooper
pairs to tightly bound pairs of fermions can be de-
scribed on the basis of BCS theory if the effects of
fluctuations are disregarded. This transition has been
studied in excitonic semiconductors [1], ordinary su-
perconductors [2], in an attractive Fermi gas [3], nu-
clear [4,5] and quark [6] matter. Although the BCS
and BEC limits are physically quite different, the
crossover between them was found to be smooth with-
in the BCS theory.
The recent upsurge of interest in the BEC–BCS
crossover was caused by finding BCS pairing in
ultracold quantum atomic gases [7–9]. Ultracold
atomic gases provide an experimental playground for
testing pairing phenomena due to the possibility of
controling the inter-atomic interactions via a magneti-
cally-tuned Feshbach resonance. In low-density
systems of two types of fermions, such as ultracold
gases of alkali atoms, the interaction can be character-
ized by a single parameter, the s-wave scattering
length as . If the scattering length is negative, one ex-
pects fermions in the two hyperfine levels to pair. On
the opposite side from the position of the Feshbach
resonance, when the scattering length is positive,
pairs of atoms can form weakly bound states in the
form of molecules. In this regime, one expects the sys-
tem to consist of a BEC of these molecules. In fact, as
one goes through the resonance starting from the nega-
tive scattering lengths, the Cooper pairs in the BCS
state shrink in size and go over continuously into a
Bose—Einstein condensate of bound states. An impor-
tant question arises what are the observable features
of the BCS–BEC transition.
Such signatures were attempted to be found in a re-
cent study [10]. The authors of that study argued that
© A.A. Isayev and J. Yang , 2006
the density correlation function of a two-component
ultracold fermionic gas of atoms changes sign at
low-momentum transfer and this represents an unam-
biguous signature of the BEC–BCS crossover. Here
we would like to extend their calculations taking into
account additional factors: finite isospin asymmetry
(nonequal densities of fermions of different species) or
finite temperature. Both factors are missing in the
study of Ref. 10. As an example of two-component
Fermi system, we choose nuclear matter, i.e., an infi-
nite Fermi system consisting of neutrons and protons.
In free space the strongest internucleon interaction is
the interaction between a neutron and a proton in the
state with the total spin of a pair S � 1, leading to the
formation of a real bound state — deuteron. At not
too low densities, a condensate of neutron—proton
(np) Cooper pairs is formed in nuclear matter [11,12].
Under decreasing density, np Cooper pairs go over to
BEC of deuterons. During this transition the chemical
potential changes sign at a certain critical density
(Mott transition), approaching half of the deuteron
binding energy at ultralow densities [13,14]. We will
study density, spin and isospin correlations in
low-density nuclear matter and will show that the cri-
terion of Ref. 10 fails to provide a correct description
of the density-driven BEC–BCS crossover and cannot
serve as the universal feature of transition between
two states of a Fermi system.
Basic equations
The states of two-component Fermi superfluid are
described by the normal f and anomalous g distribu-
tion functions of fermions
f a a g a a� � � � � � � �1 2 2 1 1 2 2 1
� ��Tr Tr� �, , (1)
where � � �� ( , , )k , k is the momentum, � �( ) is the
projection of spin (isospin) on the third axis, and � is
the density matrix of the system. For nuclear matter,
� � � 1 2/ corresponds to neutrons and � � � 1 2/ cor-
responds to protons. We shall study np pairing corre-
lations in the pairing channel with total spin S and
isospin T of a pair S � 1, T � 0 and the projections
S Tz z� � 0. In this case the distribution functions for
isospin asymmetric nuclear matter have the structure
f f f
g g
( ) ( ) ( ) ,
( ) ( ) ,
k k k
k k
� �
�
00 0 0 03 0 3
30 3 2 2
� � � �
� � �
(2)
where � i and �k are the Pauli matrices in spin and
isospin spaces, respectively. Using the minimum prin-
ciple of the thermodynamic potential and the proce-
dure of block diagonalization [12], one can obtain ex-
pressions for the distribution functions
f
E
E
T
E
T
f
k
k
k k
00
03
1
2 4 2 2
( ) ,
( )
k
k
� � �
�
�
�
�
� ��
tanh tanh
1
4 2 2
430
tanh tanh
tanh
E
T
E
T
g
E
E
k k
k
� �
�
�
�
�
� �
,
( )
( )
k
k� k k
T
E
T
� �
�
�
�
�
2 2
tanh .
(3)
(4)
(5)
Here
E E
mk k k k
� � � � � � � ��� � �� � �2 2
2
2
� ( ) , ,k
k
(6)
� being the energy gap in the quasiparticle excitation
spectrum, m being the effective nucleon mass, � and
�� being half of the sum and half of the difference of
the neutron and proton chemical potentials, respec-
tively.
Equations governing np pairing correlations in the
S T� �1 0, pairing channel can be obtained on the
base of the variational approach for the thermody-
namic potential [14,15] and have the form
�
�
( ) ( , )
( )
( ( ) ( )),k k k
k
k
� � �
�
� �
�
� �
�
�
�
��1
2
1
2
V
V
E
f E f E
k
k k
�
V E
f E f E
V
n
V
f
k
k
k k k1 1
2
2
� � �
�
�
�
�
�
� �� ��
�
[ ( ) ( )] ,
( (
k k
� E f Ek k
� �� �
k
) ( )) ,
(7)
(8)
(9)
where f E( ) is the Fermi distribution function. Equa-
tion (7) is the equation for the energy gap � and
Eqs. (8), (9) are equations for the total density
� � �� �p n and neutron excess � �� � � �� � �n p (�
being the asymmetry parameter). Note that since we
are considering a unitary superfluid state (��� � I),
Eqs. (7)–(9) formally coincide with the equations for
two-component isospin asymmetric superfluid with
singlet spin pairing between unlike fermions. Intro-
ducing the anomalous density
�( )
( )
[ ( ) ( )], ,k
k
� � � � � ��
�
� � �a a
E
f E f En k p k
k
k k
�
2
1
and using Eq. (8), one can represent Eq. (7) for the
energy gap in the form
k
m
n Vk
2
1 2� � ��( ) ( ) ( , ) ( ) ( ) .k k k k k
k
� � � � �
�
� (10)
In the limit of vanishing density, nk � 0, Eq. (10)
goes over into the Schr�dinger equation for the deu-
teron bound state [5,13]. The corresponding energy
1196 Fizika Nizkikh Temperatur, 2006, v. 32, No. 10
A.A. Isayev and J. Yang
eigenvalue is equal to 2�. The change in the sign
of the mean chemical potential � of neutrons and
protons under decreasing density of nuclear matter
signals the transition from the regime of large over-
lapping np Cooper pairs to the regime of non-overlap-
ping bound states (deuterons).
Let us consider the two-body density correlation
function
D( , ) �( ) �( ), �( ) �( ) � ,
�( )
x x x x x x
x
� � � � �
�
Tr �� � �n n n n n
n � ��
�
�
�
�
�
�
�
� ��
�
��
��
��
�� ��
�( ) ( )
( ) ,
�
x x
k k x
kk
k k
1
V
i a a
n
exp ( )
1
V
a ak
k
k��
��
��
�� .
(11)
Its general structure in the spatially uniform and iso-
tropic case reads [16]
D( , ) ( ) ( ),x x r r x x� � � � � �� �� D r . (12)
The function D r( ) is called the density correlation
function as well. We will be just interested in the be-
havior of the function D r( ). The trace in Eq. (10) can
be calculated, using definitions (1) and Wick rules.
Taking into account Eqs. (2) and going to the Fourier
representation
D q d i D r( ) ( )� � 3r qrexp ( ) ,
one can get
D q I q I q I qg f f( ) ( ) ( ) ( ),� � �30 00 03 (13)
where
I q dr r j rq dk k f k j rkf
00
3
0
2
0
0
2
00 0
4
( ) ( ) ( ) ( )�
�
�
�
�
�
�
� �
� �� �
�
�
�
� �
� �
2
03
3
0
2
0
0
2
03 0
4
,
( ) ( ) ( ) ( )I q dr r j rq dk k f k j rkf
� �
�
�
�
�
�
� �
� �
2
30
3
0
2
0
0
2
30 0
4
,
( ) ( ) ( ) (I q dr r j rq dk k g k j rg
� �
k) .
�
�
�
�
�
�
2
Here j0 is the spherical Bessel function of the first
kind and zeroth order. Functions I If f
00 03, , and Ig
30
represent the normal and anomalous contributions to
the density correlation function. Analogously, we can
consider the two-body spin correlation function
S
( , ) � ( ) � ( ) ,x x x x� � �Tr �� �s s
�� ( ) � ( ) � ,s s s x x� �
� ( ) ( )( ) ( )
(
s
V
i
��
�� �
�� � �� � �x x x
k k
� �
� ��
�
�
� ��1
2
1
2
exp ( ) ( ) ,x
kk
k k)
�� �
�� �� � ��
� �
�
� � �� a a
� ( ) ,s
V
a a
�� �
�� � �
��
�� �
�
� ��1
2 k
k
k
(14)
and the two-body isospin correlation function
T
( , ) � ( ) � ( ) ,
� ( ) � ( ) � ,
�
x x x x
x x
� � �
� �
Tr �� �
�
t t
t t t
t ��
�
���
�� ��� � �( ) ( )( ) ( )
( )
x x x
k k x
� �
� ��
�
� ��1
2
1
2V
iexp ( )
���
�� �� ��
��
���
�
�
� �
�
� � �
�
�
�
��
kk
k k
k
k
a a
t
V
a
( ) ,
� ( )
1
2 �� ��� �ak .
(15)
Their general structure for isospin asymmetric nuclear
matter without spin polarization is
S
T
� �
� �
�
( ) ( ) ( ),
( ) ( )
x, x r
x, x r
� � �
� � �
�
�
� �
4
4
S r
4 3i T r! �
( ) ( ).r � �
(16)
(17)
Then, calculating traces in Eqs. (14), (15), for the
Fourier transforms of the spin and isospin correlation
functions, one can get
S q I q I q
I
f f
g
�
� � �
( ) { ( ( ) ( ))
( )
� � � �
� �
1
4
2
00 03
3 3
30( )} ,
( ) { ( ( ) ( ))
(
q
T q I q I qf g
�
� � �
� � � �
� �
1
4
2
00 30
3 3 ) ( )} .I qf
03
(18)
(19)
Note that if one puts " �� � 3 in Eqs. (18), (19), one
gets the longitudinal spin Sl and isospin Tl correla-
tion functions, while setting � ", ,� 12 gives the trans-
verse spin and isospin correlation functions
S q I q I q I q S qt
f f g
t
�
�
�
( ) ( ( ) ( ) ( )) ( ),
,
� � � � �
4
00 03 30
"
�
�
�
� � � � �
1 2
4
00 03 30
, ,
( ) ( ( ) ( ) ( ))T q I q I q I q Tt
f f g
t ( ) .q
(20)
Density, spin and isospin correlations in low-density two-component Fermi superfluid
Fizika Nizkikh Temperatur, 2006, v. 32, No. 10 1197
The following relationships between the correla-
tion functions hold true:
S q
D q
S q T ql t l( )
( )
, ( ) ( ) .� �
4
(21)
At zero temperature and zero momentum transfer, the
correlation functions satisfy the sum rule
S q T q
dk k f k f k g
t l( ) ( )
( ( ) ( ) (
� � � �
� � � ��
0 0
1
2 2
2
00
2
03
2
30
2
� �
k)) ,� �
1
4
(22)
where the right-hand side is independent of density
and isospin asymmetry. Besides, the transverse
isospin correlation function satisfies the relationship
T q dk k f k f k g kt( ) ( ( ) ( ) ( ))� � � � � �
� �
�
�0
1
2
1
2
2
00
2
03
2
30
2
�
�
�
4
, (23)
where the right-hand side is independent of density.
Correlation functions in dilute two-component
Fermi superfluid
Further for numerical calculations we shall use the
effective zero range force developed in Ref. 17 to re-
produce the pairing gap in the S T� �1 0, pairing
channel with the Paris nucleon potential:
V v( , )r r
r r
1 2 0
1 2
0
1
2
� �
��
�
�
�
�
�
�
�
�
�
�
#
$
%
%
&
%
%
'
(
�
�
�
)
%
%
*
%
%
��( ) ,r r1 2 (24)
where �0
3016� �. fm is the nuclear saturation den-
sity, v m mG0
3530 0� � + � �MeV fm , ,( , mG being
the effective mass, corresponding to the Gogny force
D1S [18]. Besides, in the gap equation (7), Eq. (24)
must be supplemented with a cutoff parameter,
! c � 60 MeV.
To find the correlation functions, first, one should
solve the gap Eq. (7) self-consistently with Eqs. (8),
(9). Then the correlation functions can be determined
directly from Eqs. (13), (18), and (19). The results of
numerical determination of the energy gap as a func-
tion of density for different asymmetries at zero tem-
perature are shown in Fig. 1. As one can see, with in-
creasing asymmetry the magnitude of the energy gap is
decreased and the density interval, where a np conden-
sate exists, shrinks to lower density. In reality solu-
tions exist for any � � 1 (the phase curves for larger
values of � are not shown in Fig. 1) and correspond to
the formation of BEC of deuterons at very low densi-
ties of nuclear matter.
Correlation functions for isospin symmetric case and
zero temperature
Now we consider the correlation functions D q( )
and S qt ( ) for symmetric nuclear matter at zero tem-
perature, depicted in Fig. 2 (at � � 0, T q S qt t( ) ( )� ).
The density correlation function changes sign at low
momentum transfer when the system smoothly evolves
1198 Fizika Nizkikh Temperatur, 2006, v. 32, No. 10
A.A. Isayev and J. Yang
�
�
,
M
e
V
�,
–
Fig. 1. Energy gap as a function of density at zero temper-
ature and different asymmetries.
–0.4
–0.2
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20
–0.25
–0.20
–0.15
–0.10
–0.05
� = 0.0001 fm
–3
= 0.0006 fm
–3
= 0.003 fm
–3
= 0.01 fm
–3
= 0.02 fm
–3
= 0.06 fm
–3
� = 0
D
(q
)
S
t (q
)
q/k F
Fig. 2. Density and transverse spin correlation functions
as functions of momentum at zero temperature and differ-
ent densities for symmetric nuclear matter.
from the BEC regime to the BCS one. These two re-
gimes are distinguished by the negative and positive
values of the chemical potential �, respectively. In
view of Eq. (21), the longitudinal spin correlation
function S ql ( ) changes the sign through the
BEC–BCS crossover as well. The transverse spin cor-
relation function, and, according to Eq. (21), the lon-
gitudinal and transverse isospin correlation functions
change fluently between BEC and BCS limits. The be-
havior of the density correlation function in isospin
symmetric case at zero temperature agrees qualita-
tively with the behavior of the density correlation
function in ultracold fermionic atom gas with singlet
pairing of fermions [10]. In Ref. 10, the change in the
sign of the density correlation function at low momen-
tum transfer was considered as a signature of the
BEC–BCS crossover. We will extend their calcula-
tions with account of finite isospin asymmetry and fi-
nite temperature.
Figure 3 (top) shows the dependence of the density
correlation function D q( )� 0 at zero momentum
transfer from density for symmetric nuclear matter.
The density correlation function changes sign between
BEC and BCS limits, that is also true for the longitu-
dinal spin correlation function S ql ( )� 0 . Corre-
sponding normal I qf
00 0( )� and anomalous I qg
30 0( )�
contributions to D q( )� 0 are shown in Fig. 3 (bot-
tom). According to Eq. (20) and the sum rule for
S qt ( )� 0 , in isospin symmetric case they satisfy the
relationship I If g
00 300 0 1( ) ( )� � .
Correlation functions for isospin asymmetric case
and zero temperature
Now we will present the results of numerical deter-
mination of the correlation functions at finite isospin
asymmetry and zero temperature. The behavior of the
density correlation function D q( ) as a function of mo-
mentum at zero temperature for different asymmetries
and densities is shown in Fig. 4. In accordance with
the above criterion, the transition density from BEC
to BCS state shifts to lower densities with increasing
asymmetry (at � � 0 6. and � � �0 01 3. fm we have al-
ready the normal state, and, consequently, only three
curves are shown in the bottom right part of Fig. 4).
At strong enough isospin asymmetry (e.g., �� 0.6) we
would have only the BCS state. It is clearly seen from
Fig. 5, where the density correlation function
D q( )� 0 at zero momentum transfer is shown as a
function of density for a set of various isospin asymme-
try parameters and zero temperature. For large isospin
asymmetry the function D q( )� 0 is always negative,
that would correspond to the BCS state for all densi-
ties where a np condensate exists. Obviously, this con-
clusion is contradicted by the behavior of the mean
chemical potential �, being negative at very low densi-
ties for any � � 1, and, hence, giving evidence for the
formation of a BEC of bound states [14]. Thus, at
strong isospin asymmetry the criterion of the cross-
over, based on the change of the sign of the density
correlation function, fails to predict the transition to
Density, spin and isospin correlations in low-density two-component Fermi superfluid
Fizika Nizkikh Temperatur, 2006, v. 32, No. 10 1199
0.05 0.10 0.15 0.200
0.2
0.4
0.6
0.8
–1.0
–0.5
0
0.5
1.0
�, fm
–3
D
(q
=
0
)
I(
q
=
0
),
I
(q
=
0
)
f
g
Ig
30
If
00
Fig. 3. Zero-momentum transfer density correlation func-
tion D q( )� 0 , and its normal I qf
00 0( )� and anomalous
I qg
30 0( )� contributions as functions of density at zero
temperature for symmetric nuclear matter.
5 10 15 200 5 10 15 20
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.8
1.0
= 0.0001 fm–3
= 0.0006 fm–3
= 0.003 fm–3
� = 0.6
q/kF
� = 0.3
� = 0.1
D
(q
)
q/k F
= 0.0001 fm–3
= 0.0006 fm–3
= 0.003 fm–3
= 0.01 fm–3
� = 0
D
(q
)
Fig. 4. Density correlation function D q( ) as a function of
momentum at zero temperature for different asymmetries
and densities.
the BEC of bound states in a low-density two-compo-
nent Fermi system.
Figures 6 and 7 show the transverse spin and trans-
verse isospin correlation functions as functions of mo-
mentum at zero temperature for different asymmetries
and densities. These correlation functions at zero tem-
perature satisfy the sum rules (22), (23) and change
smoothly between their limiting values S qt ( )� 0 ,
T qt ( )� 0 , and zero value, respectively.
Density correlation function for isospin symmetric
case and nonzero temperature
Now we consider symmetric nuclear matter at finite
temperature. Figure 8 shows the dependence of the
density correlation function D q( )� 0 at zero momen-
tum transfer as a function of density for a set of vari-
ous temperatures. It is seen that for not too high
temperatures the density correlation function is
non-monotone and twice changes sign in the region of
low densities. Hence, in accordance with the above
criterion, we would have the density interval
� � �1 2� � with the BEC state, surrounded by the
density regions with the BCS state. However, this
conclusion is contradicted by the behavior of the mean
chemical potential � for these temperatures, being a
monotone function of density and giving indication of
the formation of the BEC state at low densities (� � 0)
and the BCS state at larger densities (� � 0). Thus, at
finite temperature the criterion of the crossover, for-
mulated in Ref. 10, fails to provide the correct de-
scription of the transition between two regimes.
Qualitatively the boundary between BEC and BCS
states corresponds to the point where the chemical po-
1200 Fizika Nizkikh Temperatur, 2006, v. 32, No. 10
A.A. Isayev and J. Yang
0 0.04 0.08 0.12 0.16 0.20
–1.0
–0.5
0
0.5
1.0
= 0�
= 0.1
= 0.3
= 0.6
D
(q
=
0
)
�, fm–3
Fig. 5. Density correlation function D q( )� 0 as a function
of density at zero temperature for different isospin asym-
metry parameters.
5 10 15 200 5 10 15 20
–0.25
–0.20
–0.15
–0.10
–0.05
0
–0.25
–0.20
–0.15
–0.10
–0.05
0
= 0.0001 fm –3
= 0.0006 fm –3
= 0.003 fm –3
S
(q
)
t
S
(q
)
t = 0.0001 fm
–3
= 0.0006 fm
–3
= 0.003 fm
–3
= 0.01 fm
–3
� = 0.6
q/kF
� = 0.3
� = 0.1
q/k F
� = 0
Fig. 6. Same as in Fig. 4, but for the transverse spin cor-
relation function S qt( ).
5 10 15 200 5 10 15 20
–0.25
–0.20
–0.15
–0.10
–0.05
0
–0.25
–0.20
–0.15
–0.10
–0.05
0
= 0.0001 fm–3
= 0.0006 fm –3
= 0.003 fm –3
T
(q
)
t
T
(q
)
t
= 0.0001 fm
–3
= 0.0006 fm
–3
= 0.003 fm
–3
= 0.01 fm
–3
� = 0.6
q/kF
� = 0.3
� = 0.1
q/k F
� = 0
�
Fig. 7. Same as in Fig. 4, but for the transverse isospin
correlation function T qt( ).
0.02 0.04 0.06 0.08 0.10 0.12
–1.0
–0.5
0
0.5
1.0
� = 0
T = 0 MeV
= 0.8 MeV
= 1.0 MeV
= 1.2 MeV
D
(q
=
0
)
�, fm–3
0
Fig. 8. Density correlation function D( )q � 0 as a function
of density at different temperatures for symmetric nuclear
matter.
tential changes sign. According to Eq. (6), at this
point there is a qualitative change in the quasiparticle
excitation spectrum: the momentum corresponding to
the minimum in the excitation spectrum shifts from a
finite value in the BCS state to the zero value in the
BEC state. The appearance of the minimum in the
quasiparticle excitation spectrum is clearly reflected
in the quasiparticle density of states
"
�
( )k
k
dE /dkk
�
1
2
2
.
The density of states "( )k is singular in the stationary
points of the excitation spectrum (dE /dkk � 0), ex-
cept the point k � 0, where "( )k vanishes, assuming
that dE /dkk goes to zero no faster than k2. Thus, the
appearance of the singularity in the momentum distri-
bution of the quasiparticle density of states represents
the universal signature of the BEC-BCS transition
(see also Ref. 19).
Discussion and Conclusions
The transition from large overlapping Cooper pairs
to tightly bound pairs of fermions can be described en-
tirely within the mean-field approximation (BCS the-
ory). Indeed, in free-space limit the gap equation re-
duces to the Schr�dinger equation for bound pairs. An
important question appears: to what extent can the
BCS theory be trusted and what is the role of the be-
yond-mean-field effects? In order to give a correct an-
swer, it is necessary to compare calculations based
on the BCS theory with the results of more exact
schemes. The value of � !/ F for ( � ��( )a ks F
1 0 is
usually referred to as the «unitary limit» [20]. In this
limit, when the density is held fixed, one expects that
all sensitivity to the detailed features of the interac-
tion is lost. As a result, this limit is particularly sensi-
tive to many-body correlation effects, and the BCS
calculations, based on the single-parameter potential,
predict this value as being equal to 0.59 in the isospin
symmetric case and at zero temperature [19]. Recent,
more-accurate calculations based on the Green’s func-
tion Monte Carlo approach [21] have lowered the up-
per limit on this ratio to (0.44 � 0.01), which shows
that beyond-mean-field effects account for at least a
25% improvement in the binding energy over the
mean-field result. Thus, the consideration based on
the BCS theory gives a qualitatively correct picture of
the BEC–BCS transition, but further work is needed
in order to include many-body effects in a quantitative
way and to study the changes in the features of the
BEC–BCS crossover.
In summary, we have calculated the density, spin
and isospin correlation functions in low-density
two-component Fermi superfluid, taking as an exam-
ple superfluid nuclear matter with a neutron—proton
condensate. It is shown that the transverse spin, and
longitudinal and transverse isospin correlation func-
tions satisfy the sum rule at zero momentum transfer
and zero temperature. In Ref. 10 it was learned that
the density correlation function in two-component
ultracold fermionic atom gas with singlet pairing of
fermions changes sign at low momentum transfer
across the BEC–BCS transition, driven by changing
the scattering length of the interaction at zero temper-
ature. We have shown that for spin triplet pairing the
longitudinal spin correlation function plays an analo-
gous role to the density correlation function and
changes the sign at low momentum transfer across the
crossover in symmetric nuclear matter at zero tempera-
ture. However, while giving the satisfactory descrip-
tion of the density-driven BEC–BCS crossover in di-
lute nuclear matter at zero temperature for isospin
symmetric case, this criterion fails to provide the cor-
rect description of the crossover at finite isospin asym-
metry (nonequal densities of fermions of different spe-
cies) or finite temperature. Hence, the criterion in
Ref. 10 cannot be considered as the universal indica-
tion of the BEC–BCS transition. During the Mott
transition, when the chemical potential changes sign,
there is a qualitative change in the quasiparticle en-
ergy spectrum: the minimum shifts from a finite (BCS
state) to zero-momentum value (BEC state). As such,
the presence (BCS) or absence (BEC) of the singular-
ity in the momentum distribution of the quasiparticle
density of states represents the universal signature of
the BEC–BCS transition.
From the general point of view, we have studied
the possible features of the BEC–BCS crossover in a
low-density two-component Fermi superfluid within
the mean-field approach. The conclusions of this study
hold true not only for nuclear matter, taken here as an
example, but, for instance, for a two-component
ultracold fermionic gas of atoms (e.g., 6Li, 40K). In
the last case, the presence or absence of the singularity
in the quasiparticle density of states can be deter-
mined spectroscopically [22].
J.Y. acknowledges support from the CHEP SRC
program of KOSEF(Korea).
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