Volume 101, Number 4, July–August 1996 Journal of Research of the National Institute of Standards and Technology
[J. Res. Natl. Inst. Stand. Technol. 101, 435 (1996)] Methods for a Nonuniform Bose Gas
Volume 101 Number 4 July–August 1996
Kerson Huang We review mathematical methods for the temperatures is pointed out. A Gaussian treatment of a system of Bose particles variational method is proposed. Center for Theoretical Physics, with nonuniform density. The use of the Laboratory for Nuclear Science, pseudopotential is explained, especially Key words: Bogoliubov transformation; and Department of Physics, with respect to negative scattering lengths. Bose-Einstein condensation; Bose gas; It is emphasized that the delta-function Gaussian wave function; hard-sphere inter- Massachusetts Institute of potential produces no scattering in three di- action; negative scattering length; pseu- Technology, mensions, and should not be used in the dopotential; self-consistent field; variational Cambridge, MA 02139 Bogoliubov self-consistent field method, method. which is variational in nature. A common and misuse of the Bogoliubov method at finite Accepted: May 15, 1996
Paolo Tommasini Institute for Theoretical Atomic and Molecular Physics, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138
1. Introduction
The observation of Bose-Einstein condensation in where a is the s-wave scattering length, the thermal atomic traps [1]-[3] has renewed theoretical interest in wavelength, L the size of the system, and n is the average studying a system of Bose particles with a nonuniform particle density. Thus, na 3 << 1, a /L << 1, a / << 1, and density. The experiments involve atoms of mass m con- we can treat the system as a low-temperature weakly-in- densed in a harmonic trap characterized by frequency teracting dilute gas using well-known methods [6]. . The relevant parameters, as discussed more fully in In this paper, we review the following theoretical Refs. [4] and [5], have the following orders of magni- methods, and point out some common errors and mis- tude: conceptions: (a) The pseudopotential method: One replaces the a ϳ 10Ϫ6 cm, potential by a ␦-function, and, with the help of a Bogoli- =(2ប2/mkT)1/2 ϳ 10Ϫ4 cm, ubov transformation, obtains a weak-coupling expan- L =(ប/m)1/2 ϳ 10Ϫ4 cm, na 3 ϳ 10Ϫ8, (1)
435 Volume 101, Number 4, July–August 1996 Journal of Research of the National Institute of Standards and Technology sion. This method seems adequate to describe current the singular nature of the potential at r = 0, one should experiments. be careful. By treating the ␦-function as the limit of a (b) The self-consistent field method: This is varia- square well, one can verify that in three dimensions the tional, and capable of treating strongly interacting cases s-waves are the same as those of a free particle except in principle. It reduces to the pseudopotential method in at r = 0, where they drop to zero discontinuously. This appropriate limits; but one should not use a ␦-function behavior, however, does not lead to any scattering, nor potential in a variational calculation, because such a level shift. potential in three spatial dimensions has no effect, if As a more careful analysis [9] shows, the ␦-function treated exactly. potential must be regarded as the first term in a pseudo- (c) The Gaussian variational method: One uses a potential, which correctly reproduces all scattering Gaussian trial wave function, and obtains results similar phase shifts, and depends on these phase shifts as input to the self-consistent field method. It has the advantage parameters. For the simple case of a hard-sphere poten- of easily adaptable to the description of time-dependent tial, for which there is only one parameter a, the hard- phenomena. sphere diameter, the pseudopotential has the form All three methods are closely related to one another.
Although we will not use Feynman graphs, it might be Vpseudo(r)= relevant to note that the Bogoliubov transformation cor- responds to summing ‘‘one-loop’’ graphs. The self-con- 4aប2 Ѩ a2 Ѩ ␦3(r) rϪ ␦3(r)᭞2 r+иии , (4) sistent field method consists of readjusting parameters m ͫ Ѩr 3 Ѩr ͬ in the one-loop sum in a variational sense. The Gaussian method includes all one-loop contribution, plus parts of The differential operator (Ѩ/Ѩr)r removes spurious sin- higher-loop contributions. Of these methods, the gularities in the wave function, which would otherwise straightforward one-loop summation yields an expan- make the ground-state energy diverge. In low-order per- sion in na 3. On the other hand, the accuracy of the other turbation theory, one can replace this operator by a sim- methods cannot be ascertained. ple subtraction procedure for the energy. We do not attempt a comprehensive review, and the Using the first term in the pseudopotential, and mak- references cited are by no means complete. In a subject ing a Bogoliubov transformation, one can calculate ex- with such a long history, it is difficult to be aware of all actly the first few terms of an expansion for the ground- relevant sources, and we apologize for any omissions [7]. state energy per particle of a uniform hard-sphere Bose gas [10, 11]:
2. Pseudopotential Method 2naប 128 na 3 1/2 E = 1+ 0 m ͫ 15 ͩ ͪ In low-temperature calculations, one can replace the interatomic potential with a ␦-function potential 4 +8 Ϫ͙3 na 3ln(na 3)+O(na 3) . (5) (4aប 2/m)␦(r), where a > 0 is the s-wave scattering ͩ3 ͪ ͬ length. Such a potential should be treated in low-order perturbation theory only, because the ␦-function poten- This shows that the energy is not analytic at a =0,and tial in three spatial dimensions has no effect if treated therefore cannot be continued to negative values of a. exactly [8]. Fetter [12] has given a systematic treatment of the Consider the Schro¨dinger equation nonuniform case based on the Bogoliubov transforma- tion. For application to the atomic-trap experiments, one Ϫ᭞2(r)+g␦3(r)(r)=E(r), (2) can also adapt the results of the uniform case using a Thomas-Fermi approximation [5]. where E is 2m /ប 2 times the energy. Taking the Fourier What happens when the scattering length is negative? transform of both sides leads to In this case, the two-body scattering is dominated by the attractive part of the potential. The attraction may or (E Ϫ k 2)(k)=g(0), (3) may not be strong enough to produce a two-particle bound state; but it will lead to an N-body bound state, where (k) is the Fourier transform of (r). This must for a sufficiently large N. The many-body problem is hold for all k , including k 2 = E. Therefore (0)=0. very different from that of the repulsive case, for it This condition is automatically satisfied for all partial depends on the details of the potential. Consider two waves except s-waves. For s-waves, this seems to re- potentials with the same negative scattering length, one quire that (r) be discontinuous at r = 0; but because of everywhere attractive, and the other having a repulsive
436 Volume 101, Number 4, July–August 1996 Journal of Research of the National Institute of Standards and Technology
core. The many-body system with the purely attractive where ⌿0 is the ground-state wave function. The excita- potential will collapse spatially, whereas the one with tion energy is hard core will yield extensive energies. When the scattering length is negative, we must there- ប 2k 2 Ek = , (7) fore use an actual potential, for example, one with a hard 2mSk core plus an attractive tail. We can replace the hard-core part by a pseudopotential, and do the Bogoliubov trans- where Sk is the liquid structure factor formation. In this manner, it has been shown [6] [13] that a dilute Bose system with such a potential exhibits 1 † Sk = ͗⌿0|k k |⌿0͘, (8) a first-order phase transition separating a gas phase n from a liquid phase. The transition line terminates in a critical point. The Bose-Einstein condensation occurs in where the liquid phase, and divides it into liquid I and liquid II. In short, one reproduces qualitatively the phase diagram = ⍀Ϫ1/2 a † a , (9) 4 k p+k p of He. Since the calculation is perturbative, one can p follow the collapse of the phase diagram to that of the ideal Bose gas when the potential is turned off. with ak the annihilation operator of a free particle of The results described above pertains to an infinite momentum បk,and⍀the volume. From general princi- Bose system in the thermodynamic limit, and may be ples [17,18], we expect modified for a finite system. The authors of Ref. [2] observed Bose condensation in an atomic trap filled → បk 7 Sk , (10) with in Li, which has a negative scattering length. In k→0 2mc this case, the smallness of the system may have rendered the first-order transition indistinct. It is also possible that where c is the sound velocity as computed from the the observed phenomenon is metastable. compressibility of the ground state. This leads to the phonon spectrum
3. Variational Methods and the Energy Ek = បck. (11) Gap Feynman’s argument can be sharpened by recasting it as A variational approach to the ground-state energy of the statement that the longitudinal sum-rules are satu- a Bose gas was introduced by Girardeau and Arnowitt rated by the phonon states [17] [18]. All the equations [14], who use (wisely) an actual potential instead of the above are verified in the dilute hard-sphere Bose gas ␦-function. Their results are for a uniform gas, but oth- [10] [18]. erwise very similar to what we obtain later in the self- Other types of states that can be built upon ⌿0 are consistent field method. The excitation spectrum in this those describing superfluid flow [10] [18]: approach, however, contains an energy gap, contrary to N i␣(rj) general expectations [15], and to the perturbation-the- ⌿superflow(r1,...,rN )=⌸e ⌿0(r1,...,rN ), (12) ory results in the hard-sphere case [10]. j A variational method is designed to yield the best where ␣(r) is the superfluid phase, which is related to ground-state energy, and may not yield the excitation the superfluid velocity vs through spectrum accurately. In this instance, we know that the gap should have been filled with phonons, which can be ប v (r)= ᭞␣(r). (13) described using general principles. We may therefore s m view the variational principle as a way to obtain an approximate ground-state wave function, and build the With this wave function one can describe quantized vor- phonon states upon this ground state. tices. Feynman [16] argues that the low-lying excited states of a Bose system are purely phonon states, owing to the statistics, and suggests the following one-phonon wave 4. Self-Consistent Field Method: Ground function of momentum k: State
N The self-consistent field method was first used by ⌿ (r ,...,r )= eikиrj⌿ (r ,...,r ), (6) k 1 N 0 1 N Bogoliubov in his treatment of superconductivity, and is j=1
437 Volume 101, Number 4, July–August 1996 Journal of Research of the National Institute of Standards and Technology therefore also known as the ‘‘Bogoliubov method.’’ We H (1) = d3x †(x)h(x)(x) shall follow a concise formulation due to De Gennes ͵ [19]. A recent review of similar methods is given by Griffin [20]. + d3xd3yV(x,y) †(x)[(y)(y,x)+(x)(y,y) We use a quantized-field representation with field op- ͵ erator ⌿(x), and canonical conjugate i⌿ †(x): +*(y)D(x,y)], (21) [⌿(x), ⌿ †(y)] = ␦ 3(x Ϫ y). (14) H (2) = d3x †(x)h(x)(x) The Hamiltonian is ͵
H = ͵ d3x⌿(x)†h(x)⌿(x) + ͵ d3xd3yV(x,y)[†(y)(x)R(x,y)
1 +†(x)(x)R(y,y)] + d3xd3y⌿(y)†⌿(x)†V(x,y)⌿(x)⌿(y), (15) 2 ͵ 1 + d3xd3yV(x,y)[†(x)†(y)D(x,y) where 2 ͵
ប2 +(x)(y)D*(x,y)], (22) h(x)=Ϫ ᭞2 +V (x ) Ϫ , (16) 2m ext where where Vext(x ) is the external potential, and is the chemical potential. We displace the field operator by its R(x,y) ≡ (x,y)+*(x)(y), ground-state expectation: D(x,y) ≡ ⌬(x,y)+(x)(y). (23)
⌿(x)=(x)+(x), (17) For arbitrary and ⌬, we can diagonalize the effec- tive Hamiltonian through a generalized Bogoliubov where (x) is the displaced field operator, and transformation:
(x)=͗⌿(x)͘. (18) * † (x)=[un(x)n Ϫvn(x)n]. (24) n The function (x) describes a nonuniform condensate, † and (x) annihilates particles not in the condensate. where n and n are annihilation and creation operators Variational trial states are taken to be the eigenstates satisfying of an effective Hamiltonian that is quadratic in the field † operators, chosen to be a mean-field version of the ac- [n ,m ]=␦nm . (25) tual Hamiltonian. There are two variational functions (x,y)and⌬(x,y), which will turn out to be To preserve the canonical commutators for (x), we (x,y)=͗†(x)(y)͘,and⌬(x,y)=͗(x)(y)͘. That require is, and ⌬ are, respectively, the direct and off-diagonal density correlation functions of the uncondensed parti- * * 3 [un (x)un (y) Ϫ vn (x)vn (y)] = ␦ (x Ϫ y). (26) cles. n The effective Hamiltonian is chosen to be
We now require Heff to have the form (0) (1) (1)† (2) Heff = H + H + H + H , (19) H = E + ⑀ † , (27) (0) (1) (2) eff 0 n n n where H is independent of ,andH and H terms n are respectively linear and quadratic in : or, equivalently, H (0) = d3x *(x)h(x)(x) ͵ Ϫ⑀ [Heff, n ]= n n, 1 [H , † ]=⑀†. (28) + d3xd3y *(y) *(x)V(x,y)(x)(y), (20) eff n n n 2 ͵
438 Volume 101, Number 4, July–August 1996 Journal of Research of the National Institute of Standards and Technology
These conditions determine the functions vn , un and . (31), we have a system of coupled nonlinear integro-dif- The condition for comes from the requirement that in ferential equations.
Heff the terms linear in (x) vanish. The resulting equa- tions are 5. The Uniform Case h(x)(x)+ d3yV(x,y)[(y)(y,x)+(x)(y,y) ͵ We put Vext = 0, assume V(x,y)=V(xϪy), and seek solutions with uniform density by putting +*(y)⌬(x,y)+|(y)|2(x)] = 0, (29) (r)=0, Ϫ1/2 ikиr uk(r)=⍀ e xk , 3 ⑀ u (x)=h(x)u (x)+ dyV(x,y)[u (x)R(y,y) Ϫ1/2 ikиr n n n ͵ n vk (r)=⍀ e yk , (37)
+u (y)R(y,x)Ϫv (y)D(x,y)], (30) where xk , yk and 0 can be taken to be real. The condi- n n 2 2 tion [Eq. (26)] becomes xk Ϫ yk = 1, which can be satis- fied by putting 3 Ϫ ⑀nvn (x)=h(x)vn(x)+͵dyV(x,y)[vn(x)R(y,y) xk = coshk , * yk = sinhk . (38) +vn(y)R(y,x)Ϫun(y)D (x,y)]. (31) This parametrization simplifies the calculations. For ex- To determine and ⌬, let |⌽͘ be the lowest eigenstate ample, the liquid-structure factor of the ground state can of H : eff be represented in the form ⌽ ⌽ Heff| ͘ = E0| ͘. (32) n S =1+ 0 (cosh 2 Ϫ sinh 2 Ϫ 1) k n k k We require that the ground-state energy be at a mini- mum with respect to variations in and ⌬: 1 + [sinh 2 sinh 2 4n⍀ p |k+p| ␦͗H͘ = 0, (33) p 2 2 + sinh p sinh |p+k|], (39) where the brackets denote expectation value with re- spect to |⌽͘. In calculating the expectation value above, where n0 is the condensate density given through we use Wick’s theorem: n = n + ⍀Ϫ1 sinh2 . (40) ͗ †(x) †(y)(x)(y)͘ = ͗ †(x)(y)͗͘ †(y)(x)͘ 0 p p
+ ͗ †(x)(x)͗͘ †(y)(y)͘ + ͗ †(x) †(y)͗͘(x)(y)͘. The Bogoliubov result is recovered by neglecting the p (34) sums, which is equivalent to putting n0 ഠ n:
1/2 The calculation is facilitated by noting that ␦͗Heff͘ =0, ⑀k S = , (41) ⌽ k ͫ⑀ ͬ since is an eigenstate. Thus we can use the equivalent k +2nVk condition 2 2 where ⑀k = ប k /2m,andVk is the Fourier transform of ␦͗H Ϫ Heff͘ = 0. (35) the interaction potential.
The results are as follows: 6. Reduction to ␦-Function Potential
† * (x,y)=͗ (x)(y)͘=vn(x)vn(y), The equations in the last section are rather complex, n and difficult to solve even numerically. To gain some * ⌬(x,y)=͗(x)(y)͘=Ϫun(x)vn(y). (36) insight, we reduce them to a simpler form by using a n ␦-function potential. We shall show that this potential When these are substituted into Eqs. (29), (30), and has no effect if self-consistency is enforced, at least in the case of a uniform system. We put
439 Volume 101, Number 4, July–August 1996 Journal of Research of the National Institute of Standards and Technology
V (x ,y )=g␦3(xϪy), (42) The first equation then gives, in the limit ⌳ → ϱ, where = 0. (51)
g =4ប2a/m. (43) That is, there is no depletion of the condensate. It is then straightforward to show that the system is an ideal Bose The functions and ⌬ now depend only on x: gas. This result is reassuring; it shows that the method is able to verify that the ␦-function potential has no
† * effect. (x)=͗ (x)(x)͘=vn(x)vn(x) n * ⌬(x)=͗(x)(x)͘=Ϫun(x)vn(x). (44) n 7. Self-Consistent Field Method: Finite Temperatures Note that is the depletion of the condensate due to interactions. Equations (29), (30), and (31) reduce to To extend the variational approach to finite tempera-
tures, we can use the eigenstates of Heff to calculate the [h +2g+g||2]+g⌬ * = 0, (45) partition function, and then minimize the free energy
* 2 (h +2g+2g )un Ϫg(⌬+ )vn =⑀un, F = ͗H͘ Ϫ TS , (52) * * *2 (h+2g+2g )vn Ϫg(⌬ + )un = Ϫ ⑀vn , (46) where the brackets now denote thermal average with where we have suppressed the x dependence of the func- respect to Heff,andSis the entropy. De Gennes [19] tions. The dilute uniform hard-sphere gas is recovered argues that the quantity by setting = ⌬ = 0, for they give higher-order contri- butions. Feff = ͗Heff͘ Ϫ TS (53) For = ⌬ = 0, Eq. (45) is the ‘‘nonlinear Schro¨dinger equation’’ [21] [22], which has been widely used in is stationary, and implements the variational principle by numerical studies of the condensate and its excitations minimizing F Ϫ Feff. But the assertion is correct only if [23]. One should be aware, however, that and ⌬ may be S were defined with respect to the eigenvalue spectrum important, especially for the excitations. of Heff. In fact, as we explain below, S is defined with In the uniform case, with Vext = 0, we obtain, in the respect to the energy levels ͗␣|H|␣͘, where |␣͘ are the notation of the last section, eigenstates of Heff. The finite-temperature results in Ref. [19] are therefore incorrect. g(n Ϫ + ⌬) A proper variational principle for the free energy [24] tanh 2k = , (47) ⑀k + g(n Ϫ Ϫ ⌬) is based on the inequality where ⑀ = ប 2k 2/2m, and the chemical potential has k ͗␣|eϪH|␣͘ Ն eϪ͗␣|H|␣͘. (54) been eliminated in terms of the density n. The self-con- ␣ ␣ sistency conditions [Eq. (44)] then lead to This shows that, to calculate the trial partition function, 1 ⌳ we must use the energy spectrum = dkk 2{f [⑀ + g(n Ϫ Ϫ ⌬)] Ϫ 1}, 2 ͵ k k 4 0 ⌳ g W␣ = ͗␣|H|␣͘. (55) ⌬ = Ϫ dkk 2f (n Ϫ + ⌬), (48) 2 ͵ k 4 0 In contrast, the other scheme leads to a free Bose gas where with single-particle energy ⑀n , the eigenvalues from Eqs. (30) and (31). Because of the simplicity, it is 2 2 Ϫ1/2 fk =[⑀k +2g⑀k(nϪϪ⌬)Ϫ4g (nϪ)⌬] . (49) widely used in the literature; but it does not follow from a variational principle. The integrals are cut off at ⌳, because otherwise ⌬ As an example of the difference between the two would be divergent. Solving for ⌬ from the second equa- schemes, the calculation of the partition of a hard- tion, we obtain in the limit ⌳ → ϱ sphere Bose gas in Ref. [25] corresponds to using Eq. (55), and yields the virial coefficients in the gas phase. ⌬ Ϫ + n = 0. (50) On the other hand, a calculation based on the other
440 Volume 101, Number 4, July–August 1996 Journal of Research of the National Institute of Standards and Technology scheme would give a free Bose gas above the transition ⌬(x,y)≡͗(x)(y)͘ temperature. 1 1 =Ϫ GϪ1(x,y)ϪG(x,y) . (61) 2ͫ4 ͬ 8. Gaussian Variational Method For further analysis, it is convenient to introduce We go to a representation in which the field operator ⌿(x) and its conjugate have the forms 1 X(x,y) ≡ GϪ1/2(x,y)+2G1/2(x,y) , 2͙2 ͫ ͬ 1 ␦ ⌿(x)= (x)+ , ͫ ␦(x)ͬ 1 ͙2 Y(x,y) ≡ GϪ1/2(x,y) Ϫ 2G 1/2(x,y) . (62) 2͙2 ͫ ͬ i ␦ i⌿ †(x)= (x)Ϫ , (56) ͙2ͫ ␦(x)ͬ it easy to show that (x,y)=͐d3zY(x,z)Y(z,y), and ⌬(x,y)=Ϫ͐d3zX(x,z)Y(z,y). Now expand X and Y in where (x)isac-number function. The more familiar terms of a basis {bn (x)}: representation, in which ⌿ is diagonal, would be awk- ward to use in a non-relativistic case, in which ⌿ and * i⌿ † are canoncial conjugates. (A similar situation hap- X(x,y)=Xnbn(x)bn(y), n pens in a relativistic fermion field obeying the Dirac * equation.) Y(x,y)=Ynbn(x)bn(y), (63) n The state of the system is described by a wave func- tional ⌽[]. We use a Gaussian form for a trial wave where Xn and Yn are real. We then find functional:
* 2 (x,y)= bn(x)bn(y)Yn, 1 3 3 ⌽[]=Cexp Ϫ d x d y [ (x ) n ͭ 4 ͵ * 2 ⌬(x,y)=Ϫbn(x)bn(y)Xn. (64) n Ϫ1 Ϫ 0(x )]G (x,y)[(y) Ϫ 0(y)]ͮ, (57) Comparison with Eq. (36) leads to the identification where C is a normalization constant and and G(x,y) 0 u (x)=b (x)X , are the variational parameters. Expectation values are n n n v (x)=b (x)Y . (65) given by functional integrals, for example n n n This shows the equivalence between this method and the ͗H͘ = ͵ (D)⌽ *[]H⌽[] (58) self-consistent field method. One of the advantages of the Gaussian method lies in the possibility of generalization to the ‘‘time-dependent where ⌽ is normalized to unity. It is easy to show that Hartree-Fock’’ method [26,27], which has been used successfully in nuclear physics. One of us (P.T.) plans to ͗⌿(x)͘ = 0(x), discuss this in a separate publication. ͗(x)(y)͘ = G(x,y)+0(x)0(y). (59) Acknowledgments The variational principle states that We thank R. Jackiw for suggesting the field represen- ␦͗H͘ ␦͗H͘ =0, = 0. (60) tation used in the last section. This work was supported ␦G(x,y) ␦ (x) 0 in part by funds provided by the U.S. Department of Energy under cooperative agreement #DE-FC02- The equation obtained for G and will not be given 0 94ER40818. P.T. was supported by Conselho Nacional directly. To make contact with the self-consistent field Desenvolvimento Cientifico e Tecnologico (CNPq), method, we quote the results Brazil. (x,y) ≡ ͗(x)†(y)͘
1 1 = GϪ1(x,y)+G(x,y)Ϫ␦3(xϪy) , 2 ͫ4 ͬ 441 Volume 101, Number 4, July–August 1996 Journal of Research of the National Institute of Standards and Technology
9. References
[1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 (1995). [2] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995). [3] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. D. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995). [4] G. Baym and C. Pethick, Phys. Rev. Lett. 76, 6 (1996). [5] T. T. Chou, C. N. Yang, and L. H. Yu, Phys. Rev. A 53, 4257 (1996). [6] For a review see K. Huang, Chap. 1 in Studies in Statistical Mechanics, Vol. II, J. De Boer and G. E. Uhlenbeck, eds., North- Holland, Amsterdam (1964). [7] For other references see Bose-Einstein Condensation, A. Griffin, D. W. Snoke, and S. Stringari, eds., Cambridge University Press, Cambridge, England (1995). [8] K. Huang, J. Mod. Phys. (Singapore) 4, 1037 (1989). [9] K. Huang and C. N. Yang, Phys. Rev. 105, 767 (1957); E. Lieb, Proc. Nat. Acad. Sci. 46, 1000 (1960) gives an exact pseudopo- tential for hard spheres. [10] T. D. Lee, K. Huang, and C. N. Yang, Phys. Rev. 106, 1135 (1957). [11] T. T. Wu, Phys. Rev. 115, 1390 (1959). [12] A. L. Fetter, Ann. Phys. (NY) 70, 67 (1972). [13] K. Huang, Phys. Rev. 115, 769 (1959); Phys. Rev. 119, 1129 (1960). [14] M. Girardeau and R. Arnowitt, Phys. Rev. 113, 755 (1959). [15] N. M. Hugenholtz and D. Pines, Phys. Rev. 116, 489 (1959). [16] R. P. Feynman, Phys. Rev. 94, 262 (1954). [17] K. Huang and A. Klein, Ann. Phys. (NY) 30, 203 (1964). [18] K. Huang, Chap. 13 in Statistical Mechanics, 2nd ed., Wiley, NewYork (1987). [19] P. G. De Gennes, Chap. 5 in Superconductivity of Metal and Alloys, Benjamin, New York(1966). Note that the finite-temper- ature treatment is incorrect. (See Sec.7 of this paper.) [20] A. Griffin, Phys. Rev. B 53, 9341 (1996). [21] E. P. Gross, N. Cimento 20, 454 (1961); J. Math. Phys. 4, 195 (1963). [22] V. L. Ginzburg and L. P. Pitaevskii, Sov. Phys. JETP 7, 858 (1958). [23] P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51, 4704 (1995). [24] K. Huang, Ref. [18], Sec. 10.4. [25] K. Huang, C. N. Yang, and J. M. Luttinger, Phys. Rev. 105, 776 (1957). [26] A. K. Kerman and S. E. Koonin, Phys. Rev. D 40, 504 (1989). [27] J. W. Negele, Rev. Mod. Phys. 54, 913 (1986).
About the authors: Kerson Huang is Professor of Physics at the Massachusetts Institute of Technology, Cambridge, MA. Paolo Tommasini is Visiting Fellow at the Institute for Theoretical Atomic and Molecular Physics, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA.
442