Functional Integral Approach to Transition Temperature of a Homogeneous Imperfect Bose Gas

Commun. Theor. Phys.(Beijing, China)41(2004)pp. 895-898C International Academic PublishersVol.41,No.6,June15,2004Functional Integral Approach to Transition Temperature of a Homogeneous ImperfectHU Guang-Xi,DAI Xian-Xi, ,s DAI Ji-Xin, and William E. EvensonResearch group of Quantum Statistics and Methods of Theoretical Physics and State Key Laboratory of Surface Physics,Department of Physics, Fudan University, Shanghai 200433, ChinaSchool of Microelectronics, Fudan University, Shanghai 200433, China3Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602-4645, USA4 Department of Chemistry, New York University, Washington Place, New York, NY 10003, USA(Received October 27, 2003)Abstract A functional integral approach(FIA) is introduced to calculate the transition temperature of a uniformimperfect Bose gas. With this approach we find that the transition temperature is higher than that of the correspondingideal gas. We obtain the expression of the transition temperature shift as AT/To=2.492(na)"/6, where n is the densityof particle number and a is the scattering length. The result has never been reported in the literaturePACS numbers: 05. 30.Jp, 05. 30.-dKey words: Bose gas, functional integral, transition temperature1 Introductionwith volume V. The Hamiltonian of the system isIn 1995, Bose-Einstein condensation(BEC)was observed in different laboratories. [1-3 Since then there haveBn+Bm=∑(21V:)+2∑一),()been many discussions, see, for example Refs. 4 and 5and the references therein. More researches have been where M is the mass of the atompursuing the transition temperature of an interacting BoseOne takes the eigenfunction set of Ho as the basis ofgas 6-16 Unfortunately the issue still remains unsettled. the second quantizationLittle agreement is achieved among the authors. Espe-cially on the topic of how much the transition temperaturek(7)=_1will shift. For a brief review one can see Ref [11. Since itis a debated topic, we think it is worth while to deal with The eigenvalue of energy isthis problem with a different methodIn this paper we use functional integral approach(FIA)ek=2MV2/(4+k+k4)to deal with the problem. FIA was first developed by Hub- where kr, k y, and kz are all integersbard and Statonovich(17, 18 to calculate the grand partiWe use the pseudopotential form of the atom-atomtion function of statistical mechanics. It was developed interaction (26, 27] So we take the interaction betweenand applied to study the valence fluctuation, Kondo ef- atoms as S function and we havefect, dilute magnetic alloy by S.Q. Wang, W.E. Evensonand J. R. Schrieffer, [19] and Amit and Keiter, (20 and x.XH1nt=U(,7)=U06(7-r),Dai and his co-workers, see, for example Refs. 21-2and the references therein. Then it was developed as a4丌方third formulation of quantum statistical mechanics in ourMprevious paper recently [25called coupling constant, a is the S-wave scatteringthe Hamiltonian, the price one must pay is to introduce length. We start from the momentum representation inintegration. By doing so one can obtain the grand partsecond quantization. The matrix elements aretion function of the system analytically and the problemwill be solved, see also our previous paper 21,25U两≈U(①+,2 General Description of a Homogeneouswhere q= pi+ha auohnnge momentum. Itis well known中国煤化工 forward scatBose gastering is diverCN MHGgeneral cases, inConsider a system with n bosons confined in a 3d box a theoretical model, if the zero exchange momentumThe project supported in part by National Natural Science Foundation of China, under Grant Nos. 19975009, 10174016, and 10375012HU Guang-Xi, DAI Xian-Xi, DAI Ji-Xin, and William E. EvensonVol. 41dominant, then one can keep only those terms with q=0. grand partition function of the system isThen Hint can be expressed as==Tr e(uN-H)HA(k-)Bik-(U0/2V)∑This is a typical separable kernel model. In Eq.(7), where 6= 1/kg T, kB, and T are Boltzmann constantticle number ng,= N is the operator of occupation par- and temperature respectively. The energy is a non-linearSo the hamiltonian of the system isfunction of ik. It is complicated to calculate. By meansof modified Startonovich identity, 18)H2Mi+∑∑Bdye-7/=/2e-Vr Az+VT Be3 Grand Partition Function of the Systemwhere z=a+iy, z is the conjugate of z. LettingA=B=VUoB/2Vnk, one canlinearize the hamilFrom quantum statistical mechanics we know the tonianE=T/dx/dye-l∑(-a)a-21v∑√2V AR]Considering the trace is representation-independent, one can choose plane wave function set as the representation basisto obtain the trace]=∑∑∏[1By finishing the integration for a, we obtain the exact solution of the partition function Edye-myii[lThis is the exact solution. All the operator computations have been finished. DefineA-Ek6kB t2VkRTf(4,T,V,y)InInF()F(5)≡F(-2i96)Expanding F(S)as Taylor series we haveaEm 1e-0m(-2iy65)(18)obtain0mF(5)中国煤化工1CNMHG2m)川!F(0)=F(0)G,G≡No. 6Functional Integral Approach to Transition Temperature of a Homogeneous Imperfect Bose Gas897F"(0)=F(0)G2+F()e7k1)2F(0)G2+F(0)C05In appendix a we show that in the vicinity of transi- varies with z. When z approaches a critical value zc, antion temperature, the following inequality holds,reaches to its maximum, and at this point the tempera-22) ture corresponds to transition temperature. ic is also theBy taking the leading term we haveTo understand this more clearlly, one can compare thisF(2m)(0)≈F(0)C22m3) with the ideal case. For an ideal system, zc= 1, andAt last we obtain the grand partition functionwhen An reaches the maximum, the point is the transi-tion point. Next we will calculate zc to find the maximum02mF()point. Since a/A< 1, we have(2m)!!∑(-1)mF0a/2821m1(2m)!!The function gu(z) can be evaluated as follows. 28)When620<2<1, and l <0 or l# integer, the following equation(24)holdEo= F(O) is the grand partition function of an ideal sys9n(2)=r(1-1)(-1m)-14 Transition Temperature shiftWe know from quantum statistical mechanics that the It converges very fast to small(-In z), so we obtainaverage particle number is given bykrt/d93/2(20)≈(3/2)-2√ma1/22(xc)1/2With Eq(24)we haveWith Eqs. 30)(32)we have21/2Under thermodynamic limit, the summation in Eq (26) The maximum occurs when the derivative of the rightcan be replaced by integrationhand side of Eq. (33)with respect to o equals zero, andthis leads to14丌VPdpaclexp[B(E-u)-1With Eq (30)we have the following form393/2(2)where A= h/v2 Mker is the thermal wavelength,z√(3/2)(35)e/ is fugacity, h is Planck constant, P is the momentum From Eq(35 )we obtain the transition temperature of anof the particleinteracting Bose system by iteration(z)/3(3/2)We can also obtain the following equationwhere To is the transition temperature of an ideal system2丌h2k[(3/212/3With eqs.(5),(14),(26),(28),and(29) we haveSo the transits中国煤化工An=93/2(2CNMHGT=2.9270)(37)here n= N/V is the density of particle number. Itis noticed that the right-hand side of the above equation where AT=T-To898HU Guang-Xi, DAI Xian-Xi, DAI Ji-Xin, and William E. EvensonVol. 415 ConclusionSo far we have used FIA to study the transition tem- and 2< N/(N +1), where No is the number of particlesperature shift of an interacting Bose gas. For a system occupying the ground state. We know from Ref. [28]thatwith repulsive interactions, we have analytically obtained93/2(x-)≈(3/2)-2√o12≈((3/2)he transition temperature shift. The shift is positiveand similar to the result of K. huang but with differentN91/2(xe)≤9112\N+1N+1)1/2coefficient, 16We have developed the functional integral approachThe result is encouraging. This may make a promising2(2)≈N(3/2)future for the application of FIA in the study of BEC andin the field of quantum statistical mechanicsAppendix: Proof of Inequality G2》G′Here we show near the transition point the following491(2)≤92N+1inequality holds(N+1)2where(V/3)N<(3/2)1G"-(v/A3)√r(N+1)≈S(3/2)N2At the transition point we have(A2) So inequality(Al) holdsReferences[13 M. Holzmann and G. Baym, Phys. Rev. Lett. 87(2001)[1] M.H. Anderson, J. R Ensher, M.R. Matthews, C.E. Wie- [14] M. Wilkens, F. Illuminati, and M. Kramer, J Phys. A33man, and E.A. Cornell, Science 269(1995)1982 C.C. Bradley, C.A. Sackett, J.J. Tollett, and R.G. Hulet(2000)L77915 M. 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