Jia-Ying Yang(楊家營(yíng)) Xu Liu(劉旭) Ji-Hong Qin(秦吉紅) and Huai-Ming Guo(郭懷明)

1Department of Physics,University of Science and Technology Beijing,Beijing 100083,China

2Institute of Theoretical Physics,University of Science and Technology Beijing,Beijing 100083,China

3Department of Physics,Beihang University,Beijing 100191,China

Keywords: Fermi gas,entropy,specific heat,Land′e factor

1. Introduction

With the realization of many important results of ultracold atomic Fermi systems,[1–7]the study of Fermi systems has attracted a great deal of interest and becomes an active and rich field. Although the Fermi gases cannot occur Bose–Einstein condensation like Bose gases at low temperatures,the low-temperature behavior of the Fermi gas system is also worthy of note. Under appropriate conditions, fermions can form Cooper pairs and thus have the macroscopic quantum effects of Bose system, such as superconductivity and superfluidity.[8,9]The properties of the Fermi gas are influenced by a variety of factors, including the total number of particles in the system,the spatial dimension,the energy spectrum dispersion relation and the interaction between all Fermi gas particles. Previous studies have introduced a number of approximate methods to study the affected factors of the systems specifically. For example, the first order energy level of the Fermi system with hard-sphere interaction is calculated using pseudopotential method.[10]The thermodynamic properties of ideal Fermi gases in a harmonic potential are studied under the generalized uncertainty principle applying Thomas–Fermi approximation.[11]Furthermore,the free energy applied to various problems in statistical mechanics is also investigated based on variational principle of Bogoliubov.[12]

In the study of quantum systems, using external potential to restrict is also an important research method.[13,14]The statistical properties have been investigated for the nonrelativistic charged spinless bosons in constant magnetic field.[15]However,for further study of statistical properties of quantum gases,it is necessary to consider both orbital and spin degrees of freedom. The analytical result of thermodynamic potential of the weakly interaction Fermi gas has been calculated on the bases of considering the orbital motion and spin motion of particles. It has been found that using the magnetic field always causes energy and stability decrease and chemical potential increase of the system.[16]In our previous work,we have investigated the magnetic properties of charged spin Bose and Fermi gases with ferromagnetic coupling.[17,18]They are helpful to further exploration of macroscopic quantum phenomena in the system of cold atoms.

Since 2D quantum systems can exhibit behaviors which are not found in three dimensions,[19,20]low-dimensional configurations of Fermi gases have also become the subject of theoretical and experimental studies.[21,22]Researches on 2D Fermi systems contribute to the understanding of important frontier scientific problems such as unconventional superconductors,[23]graphene,[24]topological insulator materials[25]as well as Majorana mode.[26]On the theoretical side, the evolution from a superfluid state with large Cooper pairs to a superfluid state with compact molecules has been studied in the 2D attracting fermion system.[27]Recently, the thermodynamic quantities as functions of temperature, interaction strength and effective range,[28]and the role of a negative confinement-induced effective range on many-body pairing above the superfluid transition[29]in the strongly correlated 2D Fermi gases are also investigated. Experimentally, 2D Fermi gases have been solved their production,[30–32]pairing,[33,34]pseudogap,[35]pressure profiles,[36]the Berezinskii–Kosterlitz–Thouless transition,[37]etc. The study of 2D Fermi gases provides an important tool for understanding confined many-body systems in physics.

In this paper, we will focus on the entropy and specific heat of the 2D charged spin-1/2 Fermi gas. Under the meanfield approximation,the orbital motion and spin motion of particles are considered simultaneously. The Land′e factorgis introduced to measure the strength of paramagnetism. The effect of temperature,Land′e factor and magnetic field on the entropy and specific heat are investigated. The remainder of this paper is organized as follows. Section 2 introduces a model consisting of Landau diamagnetism and Pauli paramagnetism.The entropy, specific heat as well as the magnetization density are calculated respectively. Section 3 presents the result of detailed analysis and discussion. Section 4 concludes this article.

2. Theoretical model

We consider the orbital motion of 2D charged Fermi gas ofNparticles with massm*in a constant magnetic fieldBin thez-direction. The quantized Landau energy level is

whereμis the chemical potential, and the effective Hamiltonian of the system can be used to derive the grand thermodynamic potential

For simplicity of calculation, we introduce the following dimensionless parameters: ˉω= ˉhω/(kBT*),t=T/T*,ˉμ=μ/(kBT*), ˉS=S/(NkB), ˉMT/=0=m*cM/(nˉhq), ˉCV=CV/(NkB),thenx= ˉω/t,T*is the characteristic temperature given bykBT*=2πˉh2n/m*.

and ˉμis the dimensionless parameter of the chemical potential, which can be determined from the mean-field selfconsistent Eq.(16).

3. Results and discussion

The evolution of entropy and specific heat with the inverse of temperature of the 2D charged spin-1/2 Fermi gas in the external magnetic field is plotted in Fig. 1. It can be seen that both the entropy and specific heat decrease monotonically with the increases of the inverse of temperature, which means that there is no phase transition. At low temperature,both the entropy and specific heat tend to a constant for different magnetic fields and the specific heat approaches to zero.On the other hand, it can be seen that the curve of entropy as well as the specific heat at different magnetic fields tends to overlap when the temperature is high,and the specific heat reaches a constant value 1.It shows that the magnetic field has little effect on the entropy and specific heat under the hightemperature limit.

Fig. 1. The entropy ˉS and specific heat ˉCV as the function of 1/t for fixed Land′e factor g=0.1,the value of ˉω are in sequence as ˉω =0.01(solid line), 1 (dashed line), 2 (dotted line), 5 (dash dotted line), 10(dash dot dotted line).

Furthermore, to reconfirm our numerical results, we derive the analytical expressions in the limit case. Under the high-temperature limit, the Fermi–Dirac statistics reduce to Boltzmann statistics, the grand thermodynamic potential of the system can be rewritten as

To show the effect of paramagnetism, we plot the evolution of the entropy and specific heat with Land′e factorgunder different magnetic fields in Fig. 2. As can be seen from Fig. 2(a), the entropy decreases with the increase of Land′e factor, indicating that paramagnetism can reduce the entropy of the system. An intersection point is shown in the figure.On the left side of the intersection point,the entropy increases with the increase of the magnetic field,while on the right side of the intersection point, the entropy decreases with the increase of the magnetic field. The existence of intersection point indicates that there is a critical value of Land′e factorg.When the Land′e factor is less than this critical value,diamagnetism leads to the increase of entropy, but when the Land′e factor is greater than this critical value,the effect of paramagnetism on the decrease of entropy is greater than that of diamagnetism on the increase of entropy.It can also be found that the intersection point is basically unchanged under different magnetic fields in this range. As shown in Fig. 2(b), paramagnetism can cause the decrease of the specific heat. Furthermore, when the Land′e factorgis constant, the specific heat decreases with the increase of the magnetic field, indicating that the enhancement of the magnetic field can reduce the specific heat of the system. When the magnetic field is small,the variation of the entropy and specific heat as the function of Land′e factorgis very small,which suggests that the paramagnetism has little effect on the entropy and specific heat of the system under the weak magnetic field.

Fig.2. The entropy ˉS and specific heat ˉCV as a function of g at t =1.5 for different magnetic field ˉω =0.1 (solid line), 0.5 (dashed line), 1(dotted line),1.5(dash dotted line),2(dash dot dotted line).

In order to explain Fig. 2(a) more clearly, it is necessary to find the critical Land′e factorgcto describe the competition between the paramagnetism and diamagnetism whenˉMT/=0=0. Figure 3 plotsgcas the function of ˉω. Obviously,gcdecreases monotonically with decreasing magnetic field,andgc|ˉω→0≈0.58 for different temperatures. Although the exact value ofgcin the very high magnetic field can not obtained, we can estimate that the value ofgcranges from 0.58 to 1. The illustration in Fig.3 shows thatgcchanges little with the magnetic field in the low magnetic field,which can explain why the intersection point of Fig.2(a)is basically not affected by the magnetic field.

Fig. 3. The critical value of Land′e factor gc as the function of ˉω for fixed value of t, the temperature t is chosen as t =1 (solid line), 1.5(dashed line), 2 (dotted line). The inset depicts the relationship of gc versus magnetic field ˉω,and the temperature t=1.5.

Since the magnetic field is an important factor affecting entropy and specific heat,we plot Fig.4 to focus on the effect of magnetic field. As shown in Fig.4(a),when the Land′e factor is small,the entropy increases with the increase of the magnetic field. With the increase of the Land′e factor,the entropy first decreases with the increase of the magnetic field,reaches a minimum value,and then increases with the increase of the magnetic field.gc=0.58 is the critical value for this transition. Whengis less than the critical value,diamagnetism is stronger than paramagnetism,so the entropy increases with the increase of the magnetic field.However,with the increase ofg,paramagnetism gradually prevails. Whengis greater than the critical value, the paramagnetism exceeds the diamagnetism.When paramagnetism is dominant,the entropy first decreases with the increase of the magnetic field. With the increase of the magnetic field, diamagnetism overcomes paramagnetism after entropy reaches the lowest point,and the entropy begins to rise with the increase of the magnetic field. The illustration shows in detail the evolution of entropy with magnetic field near the critical value of Land′e factor.g=0.5 is below the critical value. It is obvious that the entropy increases monotonically with the magnetic field. Whileg=0.7 is above the critical value. The entropy initially shows a downward trend with the magnetic field at low magnetic field. Then we discuss the specific heat in Fig.4(b). The specific heat decreases with the increase of the magnetic field, which coincides with Fig.2(b). When the Land′e factor is larger than 0.58,the specific heat still decreases with the increase of the magnetic field,but the magnitude of the decrease becomes large.It shows that the reduction of specific heat is enhanced after paramagnetism is dominant. It also finds that the larger the Land′e factor, the smaller the values of entropy and specific heat for a constant magnetic field. This is consistent with the conclusion we obtained in Fig.2 that both the entropy and specific heat decrease with the growth of paramagnetism.

Fig.4.The entropy ˉS and specific heat ˉCV as a function of ˉω for t=1.5.The Land′e factor g is taken as g=0.1(solid line), 0.58(dashed line),1(dotted line), 1.5(dash dotted line), 2(dash dot dotted line), respectively. The inset depicts the relationship of ˉS versus magnetic field ˉω with the magnetic field region lies between 0 and 4.5, and the Land′e factor g is chosen as g=0.5(solid line),0.58(dashed line),0.7(dotted line).

4. Conclusion

This paper discusses the variation of the entropy and specific heat of the 2D charged spin-1/2 Fermi gas with respect to the inverse of temperature 1/t,Land′e factorg,and the magnetic field ˉω. Our results show that both the entropy and specific heat decrease with the increase of the Land′e factor. It is indicated that the increase of paramagnetism can decrease the entropy and specific heat. The effect of paramagnetism on the entropy and specific heat is relatively small when the magnetic field is weak. The specific heat decreases with the magnetic field and eventually approaches zero. There is a competition between the diamagnetism and paramagnetism in the system. The system always exhibits diamagnetism and entropy increases with magnetic field whengc＜0.58. However,whengc＞0.58, paramagnetism is gradually enhanced. The entropy first decreases with the increase of the magnetic field,and then increases after a minimum value. Furthermore, the specific heat increases monotonically with the temperature and ultimately reaches a constant value. Here no phase transition point appears. Our result shows that there is no spontaneous breaking of continuous symmetries in this system.

Acknowledgements

Project supported by the National Natural Science Foundation of China (Grant No. 11774019) and the Fundamental Research Funds for the Central Universities,China(Grant No.FRF-BR-16-014A).