A New Approach to the Ground State of Superfluid Fermi Gas near the Feshbach Resonance of d-Wave ()
1. Introduction
We study and investigate the Fermi-gases pairing near the d-wave Feshbach resonance. Using Nozieres and SchmittRink potential that produces the two-body energy scattering amplitude, we have obtained an analytic solution of the gap equation and ground states by minimizing of energy at T = 0. Currently, there are considerable evidences for the formation of fermion pairs near Feshbach resonance [1,2]. The Feshbach resonance is a region where these fermion pairs strongly interact. Generally, the interaction in atomic gases is defined by a parameter called s-wave scattering length (a0). This quantity can be adjusted near the Feshbach resonance. The sign and size of a0 can be determined using an external magnetic field [3]. With regards to the scattering length a0, the atomic interactions in Feshbach resonance are divided into two domains. The domain a0 < 0 possesses the interaction of cooper pairs (BCS), and the domain a0 > 0 possesses the molecular interaction of fermion pairs (BEC or Bose-Einstein condensation) [4,5].
2. The Two-Body Scattering Amplitude According to the Scattering Matrix
The partial scattering amplitude of the 
th wave (
) through scattering matrix 
relates to the potential V(r) as the form below [6-13]:
(1)
where 
fulfills the following integral equation:
(2)
In order to obtain the scattering matrix, the expansion 
is applied as follows [14]:
(3)
3. The Nozihres and Schmitt-Rink Potential
The Nozihres and Schmitt-Rink potential is used to avoid technical complexities [15]:
(4)
where 
is the cut-off wave vector.
To achieve the scattering amplitude, the function 
is defined as follows:
(5)
where 
is the volume. Now, if we input the parameter below to Equation (5):
(6)
we achieve Equation (2). Also, Equation (5) is solved for
. Through this method, we obtain
. By calculating the residues, the solution to Equation (5) is as the following form:
(7)
Finally, through employing Equations (1), (5), and (6), the amplitude of 
equals:
(8)
Therefore, the scattering amplitude for 
at low energies is achieved as the form below:
(9)
4. The Effective Range and Scattering Length for 
The partial two-body scattering amplitude of the 
th wave at low energies is as follows [16]:
(10)
where
, 
, and b are respectively the scattering amplitude, effective range, and potential range. The boundstate energy appears as a polar in the function 
when k is the analytic continuation of pure imaginary axis [17- 19]. For
, the effective range in Equation (10) dominates the imaginary domain. Then, the bound-state energy equals [20-26]:
(11)
Inputting 2 instead of 
to Equation (10), and comparing it with Equation (9), we will have:
(12)
(13)
Removing 
from the two above equations, 
is achieved as the form below:
(14)
On the other hand, in the Feshbach resonance, 
inclines toward infinity [27], then:
(15)
5. The System Energy and Energy Gap below the Transition Temperature
Through the BCS theory, the average energy and energy gap can be written as [28]:
(16)
(17)
where
, 
, 
, 
, Ω is the volume, and µ is the chemical potential. Applying the mentioned parameters, and inputting them to Equations (16) and (17), the system energy and energy gap are rewritten as follows:
(18)
where n, the number of particles per volume unit, equals:
(19)
(20)
in which the multipliers of 
fulfills Equation (21):
(21)
Through utilizing Equations (20) and (21), we obtain:
(22)
Under the transition temperature (
),
[29,30], and also considering that the gap is small, the system energy and n are written as follows:
(23)
(24)
(25)
On the other hand, the number of electrons under the Fermi level equals:
(26)
Applying Equations (20) and (25), n for 
can be rewritten:
(27)
Now, Equation (27) is input to Equation (15). Using Equation (26) and some calculations, we achieve:
(28)
Utilizing Equations (11), (12), and (28):
(29)
Through inserting Equation (29) into Equations (23) and (24):
(30)
(31)
where
, and the energy gap is transformed into the below form:
(32)
where
.
6. Results and Discussion
In this investigation and study, for the first time, we presented a novel work and obtained a new method and solution for calculating the ground state of superfluid Fermi gas near the Feshbach resonance of d-wave. The system energy must be minimized to obtain the ground state. If the achieved energies are considered, solely, 
or 
is variable; therefore, 
must be minimized to find the ground state. For this purpose, the Cartesian display of spherical harmonics is utilized, and 
can be written as follows:
(33)
for
:
(34)
Through computing the energy gap expansion for
a general equation for 
can also be obtained:
(35)
which a more general equation for 
1, 2, and 3, can be written as follows:
(36)
However, we investigate single-particle excitations at T = 0 in the BCS-BEC crossover regime of a superfluid Fermi gas. We solve the Bogoliubov-de Gennes equations in a trap, including a tunable pairing interaction associated with a Feshbach resonance. We show that the single-particle energy gap Eg is dominated by the lowest-energy Andreev bound state localized at the surface of the gas.
7. Conclusions
In this paper, for the first time, we proposed a novel analytical approach toward the ground state of superfluid Fermi gas near the Feshbach resonance of d-wave. Also, the system energy and 
for different 
s are achieved:
(37)
where 
for various 
s are also different. The minimization of the system energy and ground state of superfluid Fermi gas for 
will be provided in another article.
Since TrA = 0 and A is a symmetric matrix,
. Therefore, the ground state has a random degeneracy; it means that the two below states are minimized.
(38)
(39)
8. Acknowledgement
The work described in this paper was fully supported by grants from the Institute for Advanced Studies of Iran. The authors would like to express genuinely and sincerely thanks and appreciated and their gratitude to Institute for Advanced Studies of Iran.
NOTES