ABSTRACT

In this paper, the d-wave coupling near the Feshbach resonance is investigated. Furthermore, we investigate the analytic solution to the gap equation and the ground state at T = 0 through utilizing the Nozihres and Schmitt-Rink potential creating the two-body low-energy scattering amplitude, and minimizing energy. Necessarily, to obtain the ground state, the system energy must be minimized. With regards to the achieved energies, merely, μ₀ or γ are variable; therefore, γ must be minimized to find the ground state. Moreover, considering the Fermi gas’s superfluidity at the temperature of T = 0, and by employing the potential model providing the two-body scattering amplitude, we investigate the analytic solution to the BCS problem for the d-wave coupling.

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 fora 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 E_g 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.

REFERENCES

NOTES

^*Corresponding author.