Rigorous Upper Bound on the Critical Temperature of Dilute Bose Gases

Rigorous Upp er Bound on the Critical T emp erature of Dilute Bose Gases Rob ert Seiringer Dep artment of Physics, Princeton University, Princ eton, NJ 08544, USA ∗ Daniel Ueltschi Dep artment of Mathematics, University of Warwick, Coventry, CV4 7AL, England † W e prov e exp onential deca y of the off-diagonal correlation function in the t wo-dimensional homo- geneous Bose gas when a 2 ρ is small and the temp erature T satisfies T > 4 π ρ ln | ln( a 2 ρ ) | . Here, a is the scattering length of the repulsive in teraction p otential and ρ is the density . T o leading order in a 2 ρ , this b ound agrees with the exp ected critical temp erature for sup erfluidity . In the three-dimensional Bose gas, exp onential deca y is prov ed when T − T (0) c T (0) c > 5 p aρ 1 / 3 , where T (0) c is the critical temp erature of the ideal gas. While this condition is not exp ected to b e sharp, it gives a rigorous upp er b ound on the critical temp erature for Bose-Einstein condensation. P ACS num b ers: 05.70.Fh, 03.75.Hh, 05.30.Jp Keywords: Dilute Bose gas, Bose-Einstein condensation, off-diagonal long-range order, scattering length I. INTR ODUCTION Quan tum man y-b ody effects due to particle in terac- tions and quantum statistics make the Bose gas a fasci- nating system and a c hallenge to theoretical ph ysics. It is increasingly relev ant to exp erimen tal ph ysics, esp ecially after the first realization of Bose-Einstein condensation in cold atomic gases. 1,2 It displays a stunning physical phenomenon: superfluidity . Several mec hanisms that are presen t in the Bose gas also play a rˆ ole in in teracting elec- tronic systems and in quantum optics. Both the tw o-dimensional and the three-dimensional gas ha ve physical relev ance, and they b ehav e rather dif- feren tly . W e consider them separately here. Throughout the pap er, we shall assume that units are chosen in such a wa y that ~ = 2 m = k B = 1, where m is the particle mass. A. The t wo-dimensional Bose gas There is no Bose-Einstein condensation in the tw o- dimensional Bose gas at p ositive temperature, as w as pro ved b y Hohenberg more than fort y years ago. 3 In con- trast to higher dimensions, the ideal Bose gas offers no in triguing features in tw o dimensions. But the interact- ing gas is exp ected to display a Kosterlitz-Thouless t yp e transition from a normal fluid to a sup erfluid, where the deca y of off-diagonal correlations go es from exp onential to p o wer law. The critical temp erature T c dep ends on the scattering length a of the in teraction potential, whic h w e consider to b e repulsive. F or dilute gases, i.e. when a 2 ρ  1, Popov 4 p erformed diagrammatic expansions in a functional integral approac h, finding that T c ≈ 4 π ρ ln | ln( a 2 ρ ) | . (1) This form ula was confirmed b y Fisher and Hohen b erg 5 using Bogoliubov’s theory , and b y Pilati et al. 6 using Mon te-Carlo sim ulations. No rigorous pro of is a v ailable to this date, ho wev er. In this article we prov e in a mathematically rigorous fashion that there is exponential deca y of the off-diagonal correlation function when the temp erature satisfies T > 4 π ρ ln | ln( a 2 ρ ) |  1 + O  ln ln | ln( a 2 ρ ) | ln | ln( a 2 ρ ) |  (2) for small a 2 ρ . Th us we prov e that T c cannot b e big- ger than the conjectured v alue (1), to leading order in a 2 ρ . The main nov el ingredient in our pro of is a rigorous b ound on the grand-canonical densit y of the in teracting Bose gas. This is explained in the next section. B. The three-dimensional Bose gas A three-dimensional Bose gas is interesting ev en in the absence of particle interactions. Bose-Einstein conden- sation takes place at the critical temp erature T (0) c = 4 π ( ρ/ζ ( 3 2 )) 2 / 3 (where ζ ( 3 2 ) ≈ 2 . 612, with ζ the Riemann zeta function). The effects of particle interactions on the critical temp erature hav e b een studied by many authors. A consensus has b een reached in recen t y ears but it is 2 ten uous; we give a survey of the main results, b oth for historical p ersp ectiv e and in order to gain a sense of the solidit y of the consensus. Let ∆ T c = T c − T (0) c denote the change of the critical temp erature. 1953 F eynman 7 argued that interactions increase the ef- fectiv e mass of the particles and hence decrease T c , i.e, ∆ T c < 0. 1958 Lee and Y ang 8 predict that the change of criti- cal temp erature is linear in the scattering length, namely ∆ T c /T (0) c ≈ c aρ 1 / 3 . No information on the constant c is provided, not ev en its sign. 1960 Glassgold, Kaufman, and W atson 9 find that the critical temp erature increases as ∆ T c /T (0) c ≈ C ( aρ 1 / 3 ) 1 / 2 with C > 0. 1964 Huang 10 giv es an argumen t suggesting that ∆ T c /T (0) c ≈ C ( aρ 1 / 3 ) 3 / 2 with C > 0. 1971 A Hartree-F o c k computation shows that ∆ T c < 0 (F etter and W aleck a 11 ). 1982 A lo op expansion of the quantum field represen ta- tion gives ∆ T c /T (0) c ≈ − 3 . 5( aρ 1 / 3 ) 1 / 2 (T o yoda 12 ). 1992 By studying the evolution of the interacting Bose gas, Stoof 13 finds that the change of critical tem- p erature is linear in the scattering length with c = 16 π/ 3 ζ (3 / 2) 4 / 3 = 4 . 66. 1996 A diagrammatic expansion in the renormalization group yields ∆ T c > 0 (Bijlsma and Sto of 14 ). 1997 A path integral Mon te-Carlo sim ulation yields c = 0 . 34 ± 0 . 06 (Gr ¨ uter, Cep erley , and Lalo¨ e 15 ). 1999 A virial expansion leads to c = 0 . 7 (Holzmann, Gr ¨ uter, and Lalo ¨ e 16 ). Another virial expan- sion leads Huang 17 to conclude that ∆ T c /T (0) c ≈ 3 . 5( aρ 1 / 3 ) 1 / 2 . In terchanging the limit a → 0 with the thermo dynamic limit, and using Monte- Carlo simulations, Holzmann and Krauth 18 find c = 2 . 3 ± 0 . 25. The dilute Bose gas can b e mapp ed on to a classical field lattice mo del (Baym et. al. 19 ); a self-consistent approach then yields c = 2 . 9. 2000 An exp erimental realization by Reppy et. al. 20 yields c = 5 . 1 ± 0 . 9. It was later p ointed out that the estimation of the scattering length b etw een par- ticles was not correct, how ev er. 21 2001 Arnold and Mo ore, 22 and Kashurnik ov, Prok of ’ev, and Svistunov 21 p erformed numerical sim ulations on the equiv alen t classical field model; 19 the former get c = 1 . 32 ± 0 . 02 and the latter get c = 1 . 29 ± 0 . 05. 2003 A v ariational perturbation theory p erformed by Kleinert 23 yields c = 1 . 14 ± 0 . 11. 2004 By studying the classical field mo del with v aria- tional p erturbations, Kastening 24 finds c = 1 . 27 ± 0 . 11. A path integral Mon te-Carlo simulation b y Nho and Landau 25 yields c = 1 . 32 ± 0 . 14. The last articles essentially agree with one another, and also with more recent articles. 6 The case for a linear cor- rection with constant c ≈ 1 . 3 is made rather convinc- ingly; it is not b ey ond reasonable doubt, though. Notice that the constant c is universal in the sense that it does not dep end on such sp ecial features as the mass of the particles or the details of the interactions. (The mass en ters the scattering length a , how ev er.) The question of the critical temp erature for interact- ing Bose gases is reviewed in Baym et. al. 26 and in Blaizot. 27 A comprehensive survey on many aspects of b osonic systems has b een written by Bloch, Dalibard, and Zwerger. 28 This question is also mentioned in addi- tional articles dealing with certain p erturbation metho ds. The v alue of c is assumed to be kno wn and its calculation serv es to test the metho d. Some of these references can b e found in Blaizot. 27 In this article we giv e a partial rigorous justification of the results in the literature b y proving that off-diagonal correlations decay exp onentially when T − T (0) c T (0) c > 5 . 09 p aρ 1 / 3  1 + O  p aρ 1 / 3   . (3) In particular, there is no Bos e-Einstein condensation when (3) is satisfied. This rigorous result is not sharp enough to disprov e any of the previous claims that ha ve b een just review ed, although it gets close to Huang’s 1999 result. As in the tw o-dimensional case, the pro of is based on b ounds of the grand-canonical density for the in ter- acting gas. C. Outline of this article In the next section, w e shall explain ho w the exp onen- tial decay of correlations can b e deduced from appropri- ate low er b ounds on the particle density in the grand- canonical ensemble. These bounds will b e prov ed in the remaining sections. In Section I I I, we shall state our main result, Theorem I I I.1, and we shall explain the pre- cise assumptions on the interparticle interactions under whic h it holds. Our main to ol is a path in tegral represen- tation which is explained in detail in Section IV. Finally , in Section V we in vestigate certain in tegrals of the differ- ence b etw een the heat k ernel of the Laplacian with and without p otential, and obtain b ounds that are needed to complete the pro of of Theorem I II.1. 3 I I. DECA Y OF CORRELA TIONS W e consider the grand-canonical ensemble at chemi- cal p oten tial µ and we denote the fugacit y b y z = e β µ . Let γ ( x, y ) = h a † ( x ) a ( y ) i denote the reduced one-particle densit y matrix of the in teracting system, and γ (0) the one of the ideal gas. An imp ortan t fact is that, when the in- teractions are repulsive, w e ha ve γ ( x, y ) 6 γ (0) ( x, y ) (4) for an y 0 < z < 1. See Bratteli-Robinson, 29 Theo- rem 6.3.17. In d spatial dimensions, γ (0) ( x, y ) = X n > 1 z n (4 π β n ) d/ 2 e − | x − y | 2 4 β n whic h b ehav es like exp( − p − β − 1 ln z | x − y | ) for large | x − y | . That is, off-diagonal correlations decay exp onen- tially fast when z < 1. In particular, the critical fugacity satisfies z c > 1. Next, let ρ ( z ) denote the grand-canonical densit y of the interacting system (it dep ends on β as well, although the notation do es not sho w it explicitly), and let ρ (0) ( z ) = (4 π β ) − d/ 2 g d/ 2 ( z ) (5) the density of the ideal system. Here, the function g d/ 2 is defined by g r ( z ) = X n > 1 z n n r . (6) The densit y ρ ( z ) is increasing in z . Then a sufficient condition for the exp onen tial deca y of correlations is that, for some z < 1, ρ < ρ ( z ) . (7) The obvious problem with this condition is that the den- sit y ρ ( z ) for the interacting system is not given by an explicit function. Our wa y out is to obtain bounds for ρ ( z ) (see Theorem I II.1 b elo w) and to use them with z < 1 suitably chosen. A. Tw o dimensions W e now explain the pro of of exp onen tial deca y of corre- lations under the condition (2) for d = 2. W e sho w b elow (see Theorem I I I.1 and the follo wing remarks) that the densit y satisfies the low er b ound ρ ( z ) > ρ (0) ( z ) − C 4 π β | ln(1 − z ) | 1 − z 1 | ln( a 2 /β ) | , (8) for some constant C > 0 and for aβ − 1 / 2 small enough. Here, a denotes the tw o-dimensional scattering length, whic h can b e defined similarly to the three-dimensional case via the solution of the zero-energy scattering equation. 30,31 In tw o dimensions, ρ (0) ( z ) = − (4 π β ) − 1 ln(1 − z ). F or the choice z = z 0 with z 0 = 1 − ln | ln( a 2 /β ) | | ln( a 2 /β ) | , the criterion (7) is fulfilled when ρ 6 ln | ln( a 2 /β ) | 4 π β  1 − O  ln ln | ln( a 2 /β ) | ln | ln( a 2 /β ) |  . Since β = 1 /T , one can chec k that this is equiv alen t to the condition (2). The situation is illustrated in Fig. 1 with qualitative graphs of ρ (0) ( z ) and ρ ( z ). The critical fugacity z c is kno wn to b e larger than 1. Our density b ound holds for z < 1, and this yields the low er b ound ρ 0 for the critical densit y . It turns out to b e equal to the conjectured crit- ical density (determined b y Eq. (1)) to leading order in the small parameter aβ − 1 / 2 . z c ρ (0) ρ 0 z 0 β −1 ~ 1 ρ 0 0 z ρ ρ c FIG. 1: Qualitativ e graphs of the grand-canonical densit y for d = 2. The shaded area represen ts our lo wer b ound for the in teracting density — the darker area is the function defined in Eq. (8) and it extends to the ligh ter area by monotonicity of the grand-canonical density . Our lo wer b ound ρ 0 for the critical density is obtained by c ho osing z 0 = 1 − ln | ln( a 2 /β ) | | ln( a 2 /β ) | . B. Three dimensions W e shall prov e exp onential decay under the condition (3), where the constan t 5 . 09 is really A = 2 7 / 2 π 1 / 2 3 ζ (3 / 2) 7 / 6 q 2 3 / 2 + ζ (3 / 2) ≈ 5 . 09 . 4 It is more con venien t to consider the c hange in the critical densit y rather than in the temp erature. Inequality (3) is equiv alent to ρ − ρ (0) c ρ (0) c 6 − A 0 p aβ − 1 / 2  1 + O  p aβ − 1 / 2   , (9) where ρ (0) c = ρ (0) (1) is the critical density of the ideal Bose gas at temp erature T , and where the constants A and A 0 are related by A 0 = 3 2 ζ (3 / 2) 1 / 6 (4 π ) 1 / 4 A ≈ 4 . 75 . W e show b elow that the low er b ound ρ ( z ) > ρ (0) ( z ) − a (2 π β ) 2 h  2 3 / 2 + ζ ( 3 2 )  r π − ln z + C i (10) holds for some p ositive constan t C and aβ − 1 / 2 small enough (see Theorem I II.1 and the follo wing remarks). W e use d g 3 / 2 / d z = z − 1 g 1 / 2 ( z ), as w ell as the b ound g 1 / 2 ( z ) 6 Z ∞ 0 z t √ t d t = r π − ln z (11) to obtain ρ (0) (1) − ρ (0) ( z ) 6 (4 π β ) − 3 2 Z 1 z r π − ln s d s s = (4 π ) − 1 β − 3 2 √ − ln z . The criterion (7) is thus fulfilled when ρ 6 ρ (0) (1) − (4 π ) − 1 β − 3 2 √ − ln z − a (2 π β ) 2 h  2 3 / 2 + ζ (3 / 2)  r π − ln z + C i for some z < 1. The righ t side of this expression dep ends on z only through w = √ − ln z . Since the minimum of Aw + B w o ver w > 0 is 2 √ AB , we get the condition (9). Notice that the optimal choice of z is z 0 = 1 − π − 1 / 2 (2 3 / 2 + ζ (3 / 2)) aβ − 1 / 2 to leading order in aβ − 1 / 2 . The three-dimensional situation is illustrated in Fig. 2. The critical fugacity z c is larger than 1 but our density b ound holds for z < 1. Our lo wer b ound for the criti- cal density , ρ 0 , is close to the conjectured expression for small aβ − 1 / 2 . I II. RIGOROUS DENSITY BOUNDS W e are left with proving the low er b ounds (8) and (10), resp ectiv ely . These will b e an immediate conse- quence of Theorem I I I.1 b elo w. In order to state our re- sults precisely , we shall first giv e a definition of the mo del and sp ecify the assumptions on the interaction potential. z 0 z c ρ c ρ (0) c ρ (0) β −2 a ~ a ~ 1/2 β −7/4 1 0 0 z ρ ρ ρ 0 FIG. 2: Qualitativ e graphs of the grand-canonical densit y for d = 3. The shaded area represen ts our lo wer b ound for the in teracting density — the darker area is the function defined in Eq. (10) and it extends to the ligh ter area by monotonicit y of the grand-canonical density . Our lo wer b ound ρ 0 for the critical density is obtained b y choosing z 0 = 1 − C aβ − 1 / 2 . The difference b et ween ρ (0) c and ρ c is exp ected to b e of the order aβ − 2 . This mak es it necessary to adopt a precise mathemati- cal tone from no w on. W e do so in order to make the results accessible also to readers with a more mathemat- ical background. Let Λ ⊂ R d b e an op en and b ounded domain. The state space for N bosons in Λ is the Hilb ert space L 2 sym (Λ N ) of square-in tegrable complex-v alued functions that are symmetric with resp ect to their argumen ts. The Hamiltonian is H Λ ,N = − N X i =1 ∆ i + X 1 6 i 0. W e assume that U is radial and has finite range, i.e., U ( x ) = 0 for | x | > R 0 . No regularity is as- sumed, how ev er; we only require that the Hamiltonian defines a self-adjoint operator on an appropriate domain, and that the F eynman-Kac formula for the heat kernel applies. In particular, U is allow ed to hav e a hard core. The scattering length of U is denoted by a . The grand-canonical partition function is Z ≡ Z ( β , Λ , z ) = X N > 0 z N T r e − β H Λ ,N . The thermo dynamic pressure is defined by p ( β , z ) = 1 β | Λ | ln Z ( β , Λ , z ) , 5 and the density is given by ρ ( z ) = β z ∂ ∂ z p ( β , z ) . (12) W e alwa ys work in finite v olume Λ. The existence of the thermo dynamic limit for the pressure, densit y and reduced density matrix is far from trivial. In particular, the limit for the latter has only b een pro ved when z is small enough. 29 This is of no relev ance to the presen t arti- cle, how ev er, since our b ounds apply to all finite domains uniformly in the volume. The one-particle reduced den- sit y matrix can b e written in terms of the in tegral k ernels of the op erators e − β H Λ ,N as γ ( x, y ) = 1 Z X N > 1 N z N Z Λ N − 1 d x 2 · · · d x N × e − β H Λ ,N ( x, x 2 , . . . , x N ; y , x 2 , . . . , x N ) . Relativ ely few rigorous results on interacting homo- geneous Bose gases are av ailable to this date. The only pro of of o ccurrence of Bose-Einstein condensation deals with the hard-core lattice mo del at half-filling. 32,33 Ro epstorff 34 used Bogoliubov’s inequality to get an up- p er b ound on the condensate density . Sev eral as- p ects of Bogoliub ov’s theory 31,35 ha ve b een rigorously justified. 36,37,38 A rigorous pro of of the leading order of the ground state energy p er particle in the lo w density limit w as given by Lieb and Yngv ason. 30,39 The next or- der correction term w as recen tly studied in a certain scal- ing limit. 40 Bounds of the free energy at p ositiv e temp er- ature were given in 41 . Cluster expansions give informa- tions on the phase without Bose-Einstein condensation, for repulsive or stable p oten tials. 42,43 Recen tly there has b een interest in F eynman cycles whic h should b e related to Bose-Einstein condensation. 44 The conditions (2) and (3) guarantee the absence of infinite cycles. This follows from the considerations here, and from the proof that all cycles are finite when the chemical potential is negative. 45 The following theorem giv es b ounds on the density ρ ( z ). Recall the function g r defined in (6). Let us define the following small parameter e a ( β ), which is asso ciated with the scattering length a : for d = 2, e a ( β ) =  | ln( a 2 /β ) | − 2 ln | ln( a/ p β ) |  − 1 + | ln( a 2 /β ) | − 2 ; and for d = 3, e a ( β ) = a   1 − ( a/ p β ) 1 / 2  − 1 + 1 3 ( a/ p β ) 1 / 2  . Theorem II I.1. L et us assume that √ β | ln( a/ √ β ) | − 1 > R 0 when d = 2 , or that a √ β > R 2 0 when d = 3 . Then we have, for 0 < z < 1 , ρ ( z ) > ρ (0) ( z ) − 4 z 2 (4 π β ) d − 1  h d ( z ) e a ( β ) + 2 d/ 2 e a ( β / 2)  , (13) wher e h d ( z ) =  2 d 2 + g d 2 ( z )  g d 2 − 1 ( z ) + 2 d 2 +1 g d 2 ( z ) + g d 2 ( z ) 2 . (14) Notice that ρ ( z ) 6 ρ (0) ( z ); this is an immediate consequence of (4). F or d = 2 w e b eliev e that for z close to 1 the low er b ound is optimal up to terms of higher order in ˜ a ( β ), while for d = 3 the prefactor is not optimal. This is based on the (yet unprov ed) as- sumption that the leading order correction to the pres- sure is equal to − 8 π e a ( β ) ρ (0) ( z ) 2 for z < 1. 30,41,46 . Us- ing (5) and (12), this suggests that ρ ( z ) ≈ ρ (0) ( z ) − 4 e a ( β )(4 π β ) 1 − d g d/ 2 ( z ) g d/ 2 − 1 ( z ). If this indeed holds as a low er b ound, one can replace the constan t 5 . 09 in (3) b y 3 . 52, yielding a b ound in agreement with Huang’s prediction. 17 F rom Theorem I II.1, we can easily deduce the b ounds (8) and (10), which we ha ve used in the previous section. Since g 0 ( z ) = z / (1 − z ) and g 1 ( z ) = − ln(1 − z ), we see that the function h 2 is b ounded by h 2 ( z ) 6 C (1 − z ) − 1 | ln(1 − z ) | for some constan t C < ∞ , which implies (8). T o obtain (10), note that the function g 3 / 2 ( z ) con verges to ζ (3 / 2) as z → 1. Using the b ound (11) we see that h 3 is less than h 3 ( z ) 6  2 3 / 2 + ζ ( 3 2 )  r π − ln z + 2 5 / 2 ζ ( 3 2 ) + ζ ( 3 2 ) 2 . W e are left with the pro of of Theorem I I I.1. In Sec- tion IV we use the F eynman-Kac representation of the Bose gas to obtain b ounds on the density . These b ounds are expressed in terms of in tegrals of the difference b e- t ween the heat k ernel of the Laplacian with and without p oten tial. Section V deals with b ounds of these in te- grals. It contains a no vel v ariational principle for inte- grals ov er heat k ernel differences (Lemma V.1), which allo ws to b ound these in terms of the scattering length of the interaction p oten tial. Theorem II I.1 then follows di- rectly from Prop osition IV.2 and from Lemmas V.2 and V.3. IV. FEYNMAN-KA C REPRESENT A TION OF THE INTERA CTING BOSE GAS F rom no w on we shall w ork in arbitrary dimension d > 1. Let W t x,y denote the Wiener measure for the Bro wnian bridge from x to y in time t ; the normalization is chosen so that Z d W t x,y ( ω ) = (2 π t ) − d/ 2 e −| x − y | 2 / 2 t ≡ π t ( x − y ) . The integral kernel of e 2 β ∆ − e β (2∆ − U ) will b e denoted b y K ( x, y ). By the F eynman-Kac formula, it can be ex- pressed as K ( x, y ) = Z  1 − e − 1 4 R 4 β 0 U ( ω ( s ))d s  d W 4 β x,y ( ω ) . (15) 6 Let us introduce the in teraction U ( ω , ω 0 ) b etw een tw o paths ω and ω 0 : [0 , 2 β ] → R d . Namely , U ( ω , ω 0 ) = 1 2 Z 2 β 0 U  ω ( s ) − ω 0 ( s )  d s. The following identit y , which will prov e useful in the se- quel, is obtained by changing to center-of-mass and rela- tiv e co ordinates. Lemma IV.1. F or any x, y , x 0 , y 0 ∈ R d , Z d W 2 β x,y ( ω ) Z d W 2 β x 0 ,y 0 ( ω 0 )  1 − e − U ( ω ,ω 0 )  = 2 d π 4 β ( x − y + x 0 − y 0 ) K ( x − x 0 , y − y 0 ) . Pr o of. The difference ω − ω 0 of tw o Bro wnian bridges is a Bro wnian bridge with double v ariance. Precisely , we ha ve Z d W 2 β x,y ( ω ) π 2 β ( x − y ) Z d W 2 β x 0 ,y 0 ( ω 0 ) π 2 β ( x 0 − y 0 )  1 − e − U ( ω ,ω 0 )  = Z d W 4 β x − x 0 ,y − y 0 ( ω ) π 4 β ( x − x 0 − y + y 0 )  1 − e − 1 2 R 2 β 0 U ( ω (2 s ))d s  . By the parallelogram iden tity , π 2 β ( x − y ) π 2 β ( x 0 − y 0 ) π 4 β ( x − x 0 − y + y 0 ) = 2 d π 4 β ( x − y + x 0 − y 0 ) . The result then follo ws from (15). W e also use the F eynman-Kac formula for the canoni- cal partition function. Namely , T r e − β H Λ ,N = 1 N ! X π ∈S N Z Λ N d x 1 · · · d x N × Z d W 2 β x 1 ,x π (1) ( ω 1 ) · · · Z d W 2 β x N ,x π ( N ) ( ω N ) ×  N Y i =1 χ Λ ( ω i )  exp  − X 1 6 i 1 Ω k ; the measure µ ab ov e nat- urally extends to a measure on Ω. The grand-canonical partition function can then b e written as 42 Z = X n > 0 1 n ! Z Ω n d µ ( ω 1 ) · · · d µ ( ω n ) × exp n − X 1 6 i 1 1 ( n − 1)! Z d µ ( ω 1 ) k 1 × Z d µ ( ω 2 ) · · · Z d µ ( ω n ) e − P i P 2 6 i 2 then yields Z , and hence ρ ( z ) 6 1 | Λ | Z d µ ( ω 1 ) k 1 6 ρ (0) ( z ) . The last inequalit y follo ws since the self-in teraction V ( ω ) is also p ositive. In the following prop osition we shall deriv e a lower b ound on ρ ( z ). W e use the function h d defined in (14), as well as the integral kernel K ( x, y ) in (15). Prop osition IV.2. F or d > 1 , β > 0 and 0 < z < 1 , we have the lower b ound ρ ( z ) > ρ (0) ( z ) − 2 z 2 (4 π β ) d  h d ( z ) Z K ( x, y )d x d y + 1 2 (8 π β ) d/ 2 Z [ K ( x, x ) + K ( x, − x )] d x  for any b ounde d (and me asur able) Λ ⊂ R d . 7 Pr o of. Isolating the interactions b etw een the first path and the others, we can b ound exp {− P 1 6 i h 1 − n X j =2 (1 − e − V 1 j ) i exp n − X 2 6 k 2 yields exactly Z . F or the remaining terms (the sum o ver j ), we also use the fact that the p otential is repulsive so as to drop the in teractions b etw een ω j and the other lo ops in the last term in (19) for a low er b ound. W e conclude that ρ ( z ) > 1 | Λ | Z d µ ( ω ) k (20) − 1 | Λ | Z d µ ( ω 1 ) k 1 Z d µ ( ω 2 )(1 − e − V 12 ) . In a similar fashion to (19), we hav e e − V ( ω ) > 1 − X 0 6 ` 1 − k 1 − 1 X ` 1 =0 k 2 − 1 X ` 2 =0  1 − e − U ( ω 1 ,` 1 ,ω 2 ,` 2 )  . Here, ω i,` denotes the ` -th leg of the path ω i . W e insert these inequalities into (20), and obtain ρ ( z ) > ρ (0) ( z ) − A − B , with A = 1 | Λ | X k > 2 z k Z Λ d x Z d W 2 β k x,x ( ω ) × X 0 6 ` 3 and tw o consecutive in teracting legs are A 2 = 1 | Λ | Z Λ 3 d x 1 d x 2 d x 3 Z d W 2 β x 1 ,x 2 ( ω 1 )d W 2 β x 2 ,x 3 ( ω 2 )  1 − e − U ( ω 1 , 1 ,ω 2 , 1 )  X k > 1 ( k + 2) z k +2 (4 π β k ) d/ 2 e − | x 1 − x 3 | 2 4 β k . Using Lemma IV.1 and b ounding the exp onen tials b y 1, w e get A 2 6 2 d/ 2 z 2 (4 π β ) d h g d/ 2 − 1 ( z ) + 2 g d/ 2 ( z ) i Z K ( x, y )d x d y . The terms where no consecutive legs in teract are A 3 = 1 2 | Λ | Z Λ 4 d x 1 d x 2 d x 3 d x 4 Z d W 2 β x 1 ,x 2 ( ω 1 ) Z d W 2 β x 3 ,x 4 ( ω 2 )  1 − e − U ( ω 1 , 1 ,ω 2 , 1 )  X k 1 ,k 2 > 1 ( k 1 + k 2 + 2) z k 1 + k 2 +2 (4 π β k 1 ) d/ 2 (4 π β k 2 ) d/ 2 e − | x 2 − x 3 | 2 4 β k 1 − | x 1 − x 4 | 2 4 β k 2 . Then A 3 6 2 d/ 2 − 1 (4 π β ) 3 d/ 2 Z R 3 d d x d y d z e − | x − y − 2 z | 2 8 β K ( x, y ) × X k 1 ,k 2 > 1 ( k 1 + k 2 + 2) z k 1 + k 2 +2 ( k 1 k 2 ) d/ 2 = z 2 (4 π β ) d g d 2 ( z )  g d 2 − 1 ( z ) + g d 2 ( z )  Z K ( x, y )d x d y . W e no w decompose the terms in B as B 1 + B 2 + B 3 according to the winding num b ers of ω 1 and ω 2 . The term B 1 in volv es tw o paths of winding n umbers 1, and with the aid of Lemma IV.1 we find B 1 6 z 2 (2 π β ) d/ 2 Z R d K ( x, x )d x. Next, B 2 in volv es a path of winding num ber 1 and an- other path of higher winding num b er. Dropping the self- in teraction terms yields the upper b ound B 2 6 2 d/ 2 (4 π β ) d Z R 2 d d x d y e − | x − y | 2 8 β K ( x, y ) × X k > 1 ( k + 2) z k +2 k d/ 2 6 2 d/ 2 z 2 (4 π β ) d [ g d/ 2 − 1 ( z ) + 2 g d/ 2 ( z )] Z K ( x, y )d x d y . 8 Finally , B 3 in volv es paths with winding num b ers higher than 2. W e hav e B 3 6 2 d/ 2 (4 π β ) 3 d 2 Z R 3 d d x d y d z e − | x − y − 2 z | 2 8 β K ( x, y ) × X k 1 ,k 2 > 1 ( k 1 + 1) z k 1 + k 2 +2 ( k 1 k 2 ) d/ 2 = z 2 (4 π β ) d g d 2 ( z )  g d 2 − 1 ( z ) + g d 2 ( z )  Z K ( x, y )d x d y . Collecting the bounds on A 1 , A 2 , A 3 , B 1 , B 2 , B 3 , w e get the low er b ound of Prop osition IV.2. V. SCA TTERING ESTIMA TES As b efore, let U ( x ) > 0 b e radial and supp orted on the set { x : | x | 6 R 0 } . Let a b e the scattering length of U . W e consider the Hilbert space L 2 ( R d ) and the inte- gral k ernel K ( x, y ) of the op erator e 2 β ∆ − e β (2∆ − U ) . It follows from the F eynman-Kac representation that K ( x, y ) > 0, see Eq. (15) in the previous section. W e introduce a ( β ) = 1 8 π β Z K ( x, y )d x d y . (21) W e shall see below that, for d = 3, a ( β ) is a goo d appro x- imation to the scattering length. In fact, a 6 a ( β ) 6 a 0 , with a 0 the first order Born approximation to a . In t wo dimensions a ( β ) is dimensionless and its relation to the scattering length is a ( β ) ≈ | ln( a 2 /β ) | − 1 for large β . F or t > 0, w e also introduce the function f ( t ) = t 1 − e − t t − 1 + e − t . Lemma V.1. We have a ( β ) = 1 8 π inf ψ ∈ H 1 ( R d ) E β ( ψ ) , (22) wher e E β ( ψ ) = Z R d  2 |∇ ψ ( x ) | 2 + U ( x ) | 1 − ψ ( x ) | 2  d x + 1 β h ψ | f ( β ( − 2∆ + U )) | ψ i . Note that f is monotone decreasing, with 1 6 f ( t ) 6 2 for all t > 0. F rom monotonicity it follo ws immediately that a ( β ) is monotone decreasing in β . Moreov er, for d = 3 it is not hard to see that lim β →∞ a ( β ) = a . F or an y d , lim β → 0 a ( β ) = (8 π ) − 1 R U ( x )d x . (This is also true when R U ( x )d x = ∞ .) Pr o of. W e first consider the cas e when U is b ounded. With the aid of the Duhamel formula we hav e e 2 β ∆ − e β (2∆ − U ) = Z β 0 e 2( β − t )∆ U e t (2∆ − U ) d t = Z β 0 e 2( β − t )∆ U e 2 t ∆ d t − Z β 0 Z t 0 e 2( β − t )∆ U e s (2∆ − U ) U e 2( t − s )∆ d s d t . Hence a ( β ) = 1 8 π Z U ( x ) (1 − ψ β ( x )) d x, (23) where ψ β ( x ) = ( L β U )( x ), with L β = Z β 0 (1 − s/β ) e s (2∆ − U ) d s. The functional E β ( ψ ) has a quadratic and a linear part in ψ , and it is not hard to see that the unique minimizer satisfies  − 2∆ + U + 1 β f ( β ( − 2∆ + U ))  ψ = U. (24) Since 1 t + f ( t ) = Z 1 0 (1 − s ) e − st d s it follows that ψ = ψ β , i.e. ψ β = L β U is the unique minimizer of E β . After multiplying (24) b y ψ β and inte- grating w e see that E β ( ψ β ) = R U ( x )(1 − ψ β ( x ))d x whic h, b ecause of (23), implies (22). Finally , the case of un b ounded U can b e dealt with using monotone con vergence. If we replace U by U s ( x ) = min { U ( x ) , s } then the k ernel K ( x, y ) corresp onding to U s is monotone increasing in s . W e can apply the argumen t ab o ve to U s and take the limit s → ∞ at the end. The con vergence of a ( β ) is guaranteed b y monotonicity . The v ariational principle of Lemma V.1 is conv enient for obtaining an upper b ound on a ( β ). Lemma V.2. F or d = 2 and √ β | ln( a/ √ β ) | − 1 > R 0 , a ( β ) 6 1 | ln( a 2 /β ) | − 2 ln | ln( a/ √ β ) | + 1 | ln( a 2 /β ) | 2 . (25) F or d > 3 and a √ β > R d − 1 0 , a ( β ) 6 π d/ 2 − 1 2 Γ( d/ 2) a h 1 −  aβ 1 − d/ 2  1 / ( d − 1) i − 1 + 1 d  aβ 1 − d/ 2  1 / ( d − 1)  . (26) Note that the prefactor in (26) is equal to 1 for d = 3. Lemma V.2 is the only place where the finiteness of the range R 0 of U is b eing used. Appropriate upp er b ounds on a ( β ) can also b e obtained without this assumption, 9 and hence our main results generalize to repulsive in ter- action potentials with infinite range (but finite scattering length). F or simplicity , w e shall not pursue this general- ization here. Pr o of. Let R > R 0 , and let ψ ∞ b e the minimizer of Z | x | 6 R  2 |∇ ψ | 2 + U | 1 − ψ | 2  d x (27) sub ject to the boundary condition ψ ( x ) = 0 for | x | = R . It can b e sho wn 30,31 that there exists a unique minimizer for this problem, whic h satisfies 0 6 ψ ∞ 6 1 and ψ ∞ ( x ) = ( 1 − ln( | x | /a ) ln( R/a ) for d = 2 1 − 1 − a | x | 2 − d 1 − aR 2 − d for d > 3 in the region R 0 6 | x | 6 R . Moreov er, the minim um of (27) is giv en b y E R = ( 4 π ln( R/a ) for d = 2 4 π d/ 2 a Γ( d/ 2)(1 − aR 2 − d ) for d > 3. T o obtain an upp er b ound on a ( β ), w e use the v aria- tional principle (22) with ψ ( x ) = ψ ∞ ( x ) for | x | 6 R , and ψ ( x ) = 0 for | x | > R . Using | ψ ∞ | 6 1 and f 6 2, w e obtain the b ound a ( β ) 6 E R 8 π + σ d R d 4 π β , where σ d = π d/ 2 / Γ(1 + d/ 2) denotes the v olume of the unit ball in R d . The choice R = √ β [ln( √ β /a )] − 1 for d = 2 and R = ( a √ β ) 1 / ( d − 1) for d > 3 yields (25) and (26). F or our low er b ound on the density in Prop osition IV.2, w e need a b ound on tw o more integrals of the kernel K . Since they app ear only in terms of higher order, a rough b ound will do. Lemma V.3. L et a 0 ( β ) = (8 π β ) d/ 2 − 1 Z K ( x, x )d x, a 00 ( β ) = (8 π β ) d/ 2 − 1 Z K ( x, − x )d x . (28) Then max { a 0 ( β ) , a 00 ( β ) } 6 2 d/ 2 a ( β / 2) . F or d = 3, it can b e shown that b oth a 0 ( β ) and a 00 ( β ) con verge to a as β → ∞ , but we do not need this here. Pr o of. Using the semi-group prop erty of the heat k ernel w e can write K ( x, z ) = Z R d  e β ∆ ( x, y ) e β ∆ ( y , z ) − e β (∆ − 1 2 U ) ( x, y ) e β (∆ − 1 2 U ) ( y , z )  d y . Since ab − cd 6 a ( b − d ) + b ( a − c ) for a > c and b > d , K ( x, z ) is b ounded ab o ve b y Z R d e β ∆ ( x, y )  e β ∆ ( y , z ) − e β (∆ − 1 2 U ) ( y , z )  d y + Z R d e β ∆ ( y , z )  e β ∆ ( x, y ) − e β (∆ − 1 2 U ) ( x, y )  d y . Using the b ound e β ∆ ( x, y ) 6 (4 π β ) − d/ 2 the claim fol- lo ws easily . VI. CONCLUSION W e hav e given rigorous upp er b ounds on the critical temp erature for tw o- and three-dimensional Bose gases with repulsiv e tw o-b o dy in teractions. In tw o dimensions, our b ound agrees to leading order in a 2 ρ with the ex- p ected critical temp erature for sup erfluidity . In three dimensions, our b ound shows that the critical temp era- ture is not greater than the one for the ideal gas plus a constan t times p aρ 1 / 3 . Our bounds are based on the observ ation that the one- particle reduced density matrix decays exp onen tially if the fugacity z satisfies z < 1. What is needed are lo wer b ounds on the particle densit y in the grand canonical en- sem ble. The F eynman-Kac path in tegral representation allo ws us to get b ounds in terms of certain in tegral ker- nels which, in turn, can be estimated by the scattering length of the interaction p otential using a suitable v ari- ational principle. Ac knowledgmen ts: It is a pleasure to thank Rup ert F rank and Elliott Lieb for many stimulating discussions, and Markus Holzmann for helpful comments on the physics literature. D.U. is grateful for the hospitalit y of ETH Z¨ urich, the Center of The- oretical Studies of Prague, and the Universit y of Arizona, where parts of this pro ject were carried forw ard. 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