58 I. Cabrera et al. renormalization group tehniques gives an universal expression for the ivergene as t ff, where ff<1=. In the two omponent salar fi

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1 Brazilian Journal of Physis, vol. 1, no. 4, Deember, Type Phase Transition for a Weakly Interating Bose Gas I. Cabrera a;b, D. Oliva a;b an H. Pérez Rojas a;b; a International Centre for Theoretial Physis, b P.O. Box Trieste, Italy Grupo e Fisia Teória, ICIMAF, Calle E No. 9, Veao, La Habana 4, Cuba Helsinki Institute of Physis, P.O. Box 9 (Siltavuorenpenger C), FIN-14, University of Helsinki, Finlan Reeive on November, 1999 It is suggeste a general mehanism through whih a -type behavior is proue in the speifi heat of a Bose gas near the ritial temperature T. It is essential that the quasipartile spetra have a gap proportional to the onensate. It works for a general lass of quasipartile spetra, an in partiular, for the weakly interating Bose gas. The introution of a hemial potential in the theory is briefly isusse. I Introution In his book on Statistial Mehanis, [1] Feynman refers to the point behavior an expresses his view that perhaps part of the explanation of the transition involves Bose onensation. One interesting moel of the behavior is the one presente by Ceperley [], but in it the boson system is simulate by means of Monte Carlo tehniques using path integrals. However, although the analyti behavior of the speifi heat in a neighborhoo of the ritial temperature is provie by renormalization group methos, whih gives an aurate esription of the behavior (see i.e. []), we are not aware of any moel exhibiting the mehanism through whih the onensate inues the ivergene of the speifi heat v at T. There are two phenomena whih are expete to be explaine by a satisfatory moel of the weakly interating bose gas: 1) the existene of a gapless moe, whih is more manifest at T =, an ) the behavior, whih woul appear at finite temperatures. In our opinion both phenomena must be manifest even at the one-loop level, as it is in the relativisti moels we mention below, but whih is not the ase in the usual non-relativisti moel, i.e., the behavior annot be foun by using the Bogoliubov's spetrum [4] for the weakly interating Bose gas near T =by taking it as vali at temperatures near the ritial point. Our aim is to show that if the bosoni quasipartile spetra has a gap proportional to the onensate, a ivergene in v appears at the transition temperature, an that this is the ase for the weakly interating Bose gas near the ritial temperature. First of all, we wish to remin the simple moel of the temperature self-interating salar one-omponent fiel ffi with Z symmetry breaking ο = h ffii in relativisti quantum statistis [5]. Here the symmetry breaking parameter plays the role of a fiel onensate. The effetive potential is V (ο;t) = 1 ο4 a ο + V 4 1 (T;ο), where 1 is the oupling onstant, a is the negative mass term an V 1 (T;ο) is the sum of T -epenent tapole iagrams. On the mass shell V (ο ;T)=Ωisthe thermoynamial potential, ο being the p extremum of V (ο;t), where the spetrum is ffl(p) = p + M, M = 1 ο. ßh p T T In the high temperature limit one has V 1 (T;ο) = 1T ο =8, the extremum of V (ο;t) leaing for T p< T to a temperature-epenent mass M(T ) = 1 T T, T = 4a = 1 being the ritial temperature for symmetry restoration. The infrare ontribution to the thermoynamial potential in the limit T fl M(T )is Ω ' M T 1ß~. One obtains then a speifi heat v = ' T 4 1 whih iverges as t 1=, t = j1 T=T j, for T! T,showing a -type behavior. This expression shows that the ivergene appears alreay at the one-loop approximation of v, although it is quantitatively satisfatory for t fl 1 [6], sine in the region loser to T, the ontribution from higher loops must be onsiere. In that region,

2 58 I. Cabrera et al. renormalization group tehniques gives an universal expression for the ivergene as t ff, where ff<1=. In the two omponent salar fiel ase, having U(1) symmetry, two moes appear after the symmetry breaking, the massive ffl 1 (p) = p p + M, M = 1 ο, an the massless ffl (p) =p. The ontribution to the thermoynami potential infrare term is the same as in the previous ase (the massless term oes not ontribute), an v iverges as t 1=. Atually, both in the relativisti an non-relativisti ases, with spetra ffl r = p p +,anffl n = p =m +, respetively. By taking some harateristi small momentum p an introuing x = p=t, alling = p =T, it is easy to see that there R is a ommon infrare ontribution to Ω ο AT 5= x4 x=[x + T 1 ] ο A T =, where the mass (gap) ο b[1 (T=T ) fl ], fl>1, an A; A ;bbeing onstants. Then one an state as a theorem that for T! T, the one-loop thermoynami potential obtaine from suh spetra leas to a ivergent v behavior as ο 1=. In the relativisti limit this omes from (A ontains a fator ~ ): Ω=AT The the infrare limit of the last expression is p p ln(1 e fflrfi )= A p 4 p ffl r (e fflrfi 1) (1) 5= AT ο This means that near T Z x 4 x x + T 1 ο AT 5= ( T 1 ) Ω ο AT 5= ( T 1 ) = Ar tan Z x x + T 1 + O( T 1 ) () = p T = AT ß : () 1 6 From V = ; we get that V iverges as 1= : In the non-relativisti ase we have a similar formula, by taking x = p= p mt, sine Ω=AT p p ln(1 e fflnfi )= A p 4 p )5= m(e fflnfi ο A(mT 1) m Z x 4 x : (4) x + T 1 from whih we get the same behavior near T : It is expete that any system having a spetrum of similar infrare properties than the previous ones, woul have also a speifi heat having an infinite behavior at the ritial point. We woul like to show that suh is the ase of the Bogoliubov moel of the weakly interating Bose gas near the ritial point. It ontains a gap ue to the onensate, whih leas to a ivergene in the speifi heat (in the region T=T < 1), alreay at the one-loop approximation. II Bogoliubov Hamiltonian near the ritial point We will start from the quantize Hamiltonian for a weakly interating Bose gas expane in terms of the elementary reation an annihilation operators for spinless boson fiels, we shall assume momentum onservation in the interations an all U(r) the repulsive potential of the two boy interation, U p 1 ;p p RR 1;p = re ip r ~ U(r). For p R = (zero momentum transfer in the ollision) U o = ru(r). In the temperature interval we are onsiering the momenta are very small an we an assume that the momentum transfer in eah ollision is almost zero, p '. For that reason it is possible to express approximately the matrix elements by using U o. The sattering length an be written then as a = m 4ß~ U, an a> sine the potential is repulsive. We shall assume the onitions a= fi 1 an ρa fi 1 where is the thermal e Broglie wavelength an ρ = N=V the partile ensity. Then, by starting from the fat that the oupation number of the groun state n is a large number, we will write a o = p n e i, a + o = p n e i. From this we woul get an infinite set of (non-equivalent) representations as a onsequene of breaking the first orer gauge symmetry

3 Brazilian Journal of Physis, vol. 1, no. 4, Deember, 1 59 [7] ψ(x)! e iff ψ(x) (ψ(x) =V 1= P e ikx a k ), an we take the physial representation as the one with = whih leas to, ^H = U X o p V (n o n o )+ m a+ p a p (5) p6= + U on o V + U o V X p6= X p i6=;p i6= fa + p a+ p + a p a p +4a + p a pg a + p 1 a+ p a p a p1 We assume onservation of the total number of partiles N an then P p a+ p a p + n = N. The usual proeure for T!, by following Bogoliubov [8] [9] (see also [1]), assuming that P a + p a p = P n p fi N, is to substitute in (5) n by [N P a + p a p] in the first an n by N in the thir term, whih leas to the anellation of a term U n P a + a p p in the last term in urly brakets of (5). V This means to neglet one term of seon orer in P n p, the number of exite partiles. But if P n p ß N, one annot neglet the term ( P n p ) = P a + p a pa + p a p. This is the ase for temperatures near an below the transition point, an even for very low temperatures; e.g., as in the ase n ' N=, P n p ' N=. Thus, we keep n (T ) expliit in our formulae as a quantity ereasing with inreasing temperature. We onlue that Bogoliubov' gapless spetrum is an approximation vali atually for T =,for almost all the system in the onensate. In our present approah the last term in (5) aounts for the energy ue to the interations of partiles with momenta p 6=. Sine we are onsiering an interval of temperatures very near (but below) the transition temperature, we assume that suh term is approximately onstant an take it as E N = O(U N =V ). Then the problem we are left with is the iagonalization of the sum of the seon an thir terms in (5). Our assumptions are vali for a wie range of values of n (if N ' 1 our approximation is goo up to, say, n ' 1 1 ). Thus, we keep our alulations in the one-loop approximation an in terms of n an when ompare with the stanar Bogoliubov's moel [8] it iffers in the oeffiient of the last term in urly brakets, whih is hange from to 4. The next step is to make the usual Bogoliubov's q anonial transformation a p = (b p ff p b + p)= 1 ff p an q a + p =(b + p ff p b p )= 1 ff p as a result of whih we obtain the iagonalize Hamiltonian, ^H = E o + E N + X p6= "(p)b + p b p ; (6) where E o = V Uo [n o n o ] Uono V V U on o [4 Uono V then P p6= ff p an ff p = + p Uon m "(p)], an if we enote by K = V, r "(p) = 1K +8K p m +(p m ) ; (7) is the spetrum of the new Bose quasipartiles representing the elementary exitations of the system, where b + p ;b p are their reation an annihilation operators. The limit T! oes not lea to Bogoliubov's spetrum, in whih the gap term p K is absent, sine both moels orrespon to ifferent approximations. Atually, having only one moe, the present moel bears in this respet more resemblane with the Z ase than with the U(1) one. In our moel the long wavelength limit is obviously not linear in p, as it is usually. We may argue that our limit is not T! an also that its small value probably makes iffiult to ientify it among the experimental errors for p! near T. Atually,asK fi kt is very small (for n ' 1 ;K ' 1 1 ev) it an be usually neglete, but it is able to proue the marosopi effet of a typial -type ivergent speifi heat near T. As lim p! "(p) = p K, the gap parameter K formally behaves as the analog of a rest energy in relativisti ynamis. This gap has the property that it ereases with temperature an goes to zero, as well as E ;ff p, for T! T : Below we will alulate the thermoynami potential by taking the hemial potential μ =by P assuming that the number of quasipartiles n q = p b+ b p p is not onserve (See, however, next setion). The term K is proportional to n, the eigenvalue of a with regar to the oherent groun state. In our present approximation, the quasi-partiles are atually massive sine their effetive mass is m Λ = [@ "(p)=@p ] 1 j p= = p m=4. An expliit omputation of v showing the ivergent behavior lose to the ritial temperature is easily one by oing first the temperature expansion of the thermoynami potential of the quasipartiles by taking the spetrum (7). Due to the interation term, the groun state energy E is ifferent from zero,

4 5 I. Cabrera et al. E o = ß~ an o mv where E, F are the usual ellipti integrals. ( r a n o 15 ßV " 7E ψ 1; r! ψ r!#) F 1; ; (8) III The speifi heat We will alulate the thermoynamial potential of the quasipartiles Ω. The total energy is then U = E + E N T (@Ω=@T )+Ω. We will o also the asymptoti expansion of Ω lose to Tq for T < T. By hanging to the variable x = fi mp it reas Ω= (m)= V R 1 xx fi ln[1 e p 1M +8Mx +x 4 ], where 5= ~ M = Kfi fi 1 an we obtain an expansion in terms of it, in the same way one in the ase of the effetive potential in the temperature salar moel [5]. After some alulations whih we outline in the appenix (these are given in etail in [11]), the perturbative expansion of Ω in powers of K is obtaine, an from it the speifi heat is given by as, v = [n + 8 (m) = qßa n (7E(1; r ) F (1; r + ~ (ß) f 5 (5 ) (5 )k5= T = ( ) ( )k= T 1= K (9) +4kT[ffßK 1= ( ) ( )k1= T 4kT [ ( ) ( )k1= T 1= ffßk 1= +ffßkt K 1= Here ff = ( p )(p 6 p ). The last term iverges at T. We shall write n o = Nf(T ) where f(t ) = (1 N e =N) an N e is the ensity of partiles not in the groun state. By starting from N p = n q (1 + ff p)=(1 ff p)+ff p =1 ff p, [1] the expression for N e = ~ R pn p is given in the Appenix as an expansion similar to the one for Ω. Uner the assumption Kfi fi 1, the leaing terms are N e = N e + CT; (1) where N e = AT = an A = (ßmk=h ) = (=), C = ß 1= ( p 6+ p )AK 1= =k 1= (=). This equation for N e is to be solve by iteration, by taking in M as a first approximation f(t )=[1 N e =N]. It must be observe that as T ereases, the rate of onensation in (1) is stronger than for the ieal gas. We will take N = AT =. By evaluating (9) with helium parameters an taking the sattering length as a =1 1 m, the urve for v (T ) efine in the region T» T is epite in Fig. 1. Then an infinite -type behavior of v (T ) similar to the salar fiel ase results. Suh a behavior annot be obtaine by using the usual Bogoliubov's spetrum [4]. The more exat part of the v urve liesinthe region near an below T sine we assume E N onstant an Kfi fi 1. But the last onition is vali even for very low T an E N ereases to small values as T!. Thus, the v urve rawn is approximately vali even in that region. The behavior of v for T >T is beyon the sope of the present letter. Suh problem must be investigate by starting from an aequate moel of imperfet gas in the region T T [1].

5 Brazilian Journal of Physis, vol. 1, no. 4, Deember, C v =Nk (T=T ) Figure 1: v =Nk-urve obtaine from our moel, where the shape is shown. For T >T the ieal gas urve has been rawn for omparison. IV The introution of a hemial potential The absene of a seon (gapless) Golstone moe oes not mean an inompleteness of our moel, sine although we are ealing with a two-imensional problem, the present representation behaves as similar to the one-imensional salar ase seen in the Introution, whose spetrum bears a gap. On the other han, an investigation of the Bose gas by using the Green's funtion metho mae by Beliaev [14] leas to the usual Bogoliubov spetrum in terms of n at stritly T =. However, this is mae by introuing a hemial potential μ, whih loses its usual meaning, sine the alulation of the Green funtions by starting from a ensity matrix, emans ommutation between the Hamiltonian H an the number N of exite partiles, whih is not the ase sine the quasipartile number is not onserve. If μ = in [14] our spetrum is reproue, an the gap annot be erase in higher loop alulations. The papers [14] were written before the formulation of Golstone theorem [15], whih lea to a better unerstaning of the theory of spontaneous symmetry breaking. Atually, Beliaev results are base in the same operator algebra representation use by Bogoliubov, an esribe the existene of only one moe, whih is gapless. It misses a seon moe having a gap, whih is expete to appear in usual U(1) gauge invariant moels with spontaneous symmetry breaking, as was pointe out in the Introution. If the hemial potential is unerstoo not in the usual sense, but as a parameter introue before the breaking of the symmetry, one fins that the representation use by Beliaev is not the unique one: there is a set of infinite unequivalent ones, whih is typial of the spontaneously broken symmetry ase [7]. One of these is being investigate by two of the present authors (D.O. an H.P.R.) for the temperature ase. In it, as in the U(1) salar relativisti moel onsiere in the introution, a two moe spetrum is foun. One of them is a gapless Golstone moe an the other one has a gap, the later prouing a lamba behavior in the speifi heat. Work in this problem is in progress. V Conlusions In relativisti moels of a salar bose gas with symmetry breaking, it is foun a ivergent behavior of the speifi heat at the one loop level of the thermoynami potential, whih omes from the temperature epenent mass (gap) in the spetrum, whih is onsistent qualitatively with renormalization group preitions. By analogy, we onsier the usual Bogoliubov moel near the ritial temperature for the phase transition, an a gap is shown to be present, leaing to a ivergene in the speifi heat. We isusse briefly the introution of a hemial potential in the moel, as one in Beliaev proeure [14], an the absene of a Golstone moe. We onlue that the moel we have isusse, although not being perhaps the final one, is an step towars it, sine it is able to esribe the property of lamba behavior in the non-ieal Bose gas.

6 5 I. Cabrera et al. Aknowlegments The authors woul like to thank Professor M. Virasoro, IAEA an UNESCO for hospitalityattheinternational Centre of Theoretial Physis. All authors thank A. Amezaga, A. Cabo, S. Fantoni, S. Giovannazi, F. Hussain, G. A. Mezinesu, C. Montonen, K. Narain A. Polls, A. Smierzi an R. Sorkin for valuable omments an suggestions. H.P.R. woul like to thank his former stuent R. Torres Rivero for having alle his attention to the possible relation of the -type spetrum with some temperature-epenent mass moel, an M. Chaihian for several illuminating isussions an hospitality in the High Energy Physis Division, Department of Physis, University of Helsinki. The partial support of the Aaemy of Finlan uner Projet No is greatly aknowlege. VI Appenix To make an asymptoti expansion of Ω in terms of M = Kfi, we write Ω = j M=M + R(M), where we stop our expansion in the first-orer term an R(M) is ertain funtion of M whih we are going to fin out by using R(M) j M= @Ω j M=. = (m)= V xx [4M +8x ] p fi 5= ~ (ß) 1M +8Mx + x 4 [exp( p 1M +8Mx + x 4 ) 1] ; j M= =4(m) = V (=) (=)=fi 5= ~ (ß) ) The R(M) funtion is obtaine by P 1 from these expressions, by using the Matsubara sum = 1 1 "[exp(") 1] " +4ßn 1 ", an performing the aequate regularizations, we get finally (see also [11]), 1 Ω = ρ 1 4 ß 1= (5 ) (5 )kt + 4 ( ) ( )K ff8ß (kt) 1= K = (kt) = 4p ß 15 K5= + ( )(kt) 1 O(K ) 1 p ß g (11) By a similar proeure,we get, N e = 1 f ( ) ß1= ( p 6+ p )M 1= 6 ß (=)M + O(M ) g (1) Referenes [1] R.P. Feynman, Statistial Mehanis Benjamin, Reaing, Mass. (198). [] D.M. Ceperley, Rev. Mo. Phys. 67, (1995). [] J.A. Lipa, D.R. Swanson, J.A. Nissen, T. P. Chuy an U.E. Israelsson, Phys. Rev. Lett. 76, 944 (1996). [4] K. Huang, in Bose-Einstein Conensation, Eite by A. Griffin, D.W. Snoke an S. Stringari, Cambrige University Press (1995). [5] L. Dolan an R. Jakiw, Phys. Rev. D9, (1974). [6] D.A. Kirzhnits an A.D. Line, Ann. Phys. 11, 195 (1976). [7] F. Strohi, Elements of Quantum Mehanis of Infinite Degrees of Freeom, Worl Sientifi, Singapore (1985). [8] R. K. Pathria, Statistial Mehanis, Pergamon Press (197). [9] N. N. Bogoliubov, J. Phys. USSR 11, (1947). [1] K.A. Bruekner an K. Sawaa, it Phys. Rev. 16, 1117 (1957) [11] I. Cabrera an H. Perez Rojas, Con-mat [1] L.D. Lanau an E. M. Lifshitz, Statistial Physis, Pergamon Press (1966). [1] A.L. Fetter an J.D. Waleka,Quantum Theory of Many Partile Systems, M Graw Hill Book Co, (197). [14] S.T. Beliaev, JETP 4, 89 (1958); JETP 4, 99 (1958). [15] J. Golstone, Nuovo Cimento 5, 154 (1961).

Chapter 2: One-dimensional Steady State Conduction

Chapter 2: One-dimensional Steady State Conduction 1 Chapter : One-imensional Steay State Conution.1 Eamples of One-imensional Conution Eample.1: Plate with Energy Generation an Variable Conutivity Sine k is variable it must remain insie the ifferentiation