Beyond the dilute Bose gas

ARTICLE IN PRESS

Physica A 359 (2006) 306–344 www.elsevier.com/locate/physa

Beyond the dilute Bose gas Jean-Bernard Bru Johannes-Gutenberg-Universita¨t Mainz, FB 08 - Institut f. Mathematik, Staudinger Weg 9, Geb. 2413, D-55099 Mainz, Germany Received 12 May 2005 Available online 27 June 2005

Abstract We discuss problems of three dimensional Bose gases in interaction but non-dilute. We then use the theory of a ‘‘weakly interacting’’ Bose gas recently analyzed as an attempt to obtain further insights into non-dilute systems. In particular, we develop the theory with additional remarks, discussions and a slight modification. The article concludes with a much more detailed analysis of the Bose condensate depletion, as well as a study of the two-fluid model of Tisza and Landau: the coexistence of normal and superfluid liquids at sufficiently low temperatures. In fact, even if it is based on one debatable hypothesis, this non-dilute gas qualitatively leads, up to Landau’s ‘‘T4 law’’, to a rigorous derivation of most of the previous theoretical studies on liquid helium. It also provides new observations, such as, for example, the coexistence of four subsystems of linked pair of particles: a thermal quasi-particle gas, a frozen jelly, a mixture between the previous ones, and a non-conventional Bose condensation of ‘‘Cooper-type pairs’’. Moreover, the entropic gas is shown to be different from Landau’s form associated with the normal liquid. r 2005 Elsevier B.V. All rights reserved. Keywords: Bogoliubov; Helium; Superfluidity; Landau spectrum; Cooper; Bose

Corresponding author. Tel.: +49 6131 39 25775; fax: +49 6131 39 20658.

E-mail address: [email protected]. 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.05.075

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

307

1. Introduction A challenging problem in quantum statistical mechanics is the understanding of Bose systems. Its history really started in 1925 with the first debatable description of a phenomenon of macroscopic accumulation of Bose particles in the ground state done by Einstein. Uhlenbeck criticized these arguments in 1927. But, through the concept of thermodynamic limit given by Kramers in 1937, he withdrew his objection and pointed out that the Einstein prediction is correct in the thermodynamic limit. After London, the physical mechanism of this Bose–Einstein condensation became so transparent that, at present, it is an essential part of any standard textbook on statistical physics. Rather more difficult is the study of more real Bose systems, i.e., a gas with interactions. Such questions pose enormous problems even for a few-body quantum system (e.g. atomic or molecular spectra beyond the hydrogen atom). Since the description of the Perfect Bose Gas, the theoretical researches were strongly motivated by liquid 4He. Indeed this Bose system, liquid (under normal pressure) even for T ! 0 K, has a transition from normal 4He (called He I) to superfluid phase He II at a temperature T l ¼ 2:17 K. From the thirties up to now, this extraordinary and peculiar liquid in sufficiently weak interaction has been the subject of many discussions, controversies and theories. For a recent historical overview on superfluidity, see [1]. A very fruitful procedure to derive properties of the Bose gas in interaction was performed by Bogoliubov in the forties. He was guided by Landau’s idea that (at least) the low energy part of the spectrum of atoms of 4He is defined by coherent collective movements of the system instead of individual ones. His starting point was to find a physical (or mathematical) mechanism which as in crystals with phonons, favors the collective motions of the ‘‘helium jelly’’, via some kind of ordering or coherence. A plausible conjecture already suggested in 1938 by Fritz London [2] relates to the phenomenon of Bose–Einstein condensation. To summarize, the Bose condensation together with interactions between bosons should transform individual excitations of the Perfect Bose Gas into collective excitations of the ‘‘helium jelly’’ with a Landau-type spectrum [3,4]. This first microscopic theory of superfluidity proposed in 1947 by Bogoliubov [5–9] for ‘‘weakly interacting’’ gases turns out to be unsuitable for a real superfluidity theory of 4He, i.e., for a non-dilute Bose gas. In many regards, his analysis was obviously crucial and has leaded to many fruitful developments or research directions, as the idea of quasi-particle or ‘‘pairing’’, concept ‘‘recaptured’’ within the framework of electrons to give the famous Cooper pairs in the BCS theory. However, his outstanding achievement, i.e., the derivation of the Landau-type excitation spectrum [3,4] from the full (non-dilute) interacting Hamiltonian, is based on inexact recipes or approximations [10–18]. In fact, the Bogoliubov procedure is coherent within the dilute approach in the canonical ensemble, see for example [19]. The term ‘‘dilute’’ means that the interparticle distance is much bigger than the range of the interaction potential strength. Its theoretical foundation was rigorously performed on the level of ground

ARTICLE IN PRESS 308

J.-B. Bru / Physica A 359 (2006) 306–344

states by Lieb and Yngvason [20,21] for the homogeneous gas with non-negative spherically symmetric potential,1 and in collaboration with Seiringer [22–25] in the inhomogeneous case. It was motivated by famous experiments for the derivation of Bose–Einstein condensation of a large dilute trapped system of n real particles ( 87Rb [26], 7Li [27], 23Na [28], but also more recently 85Rb, 41K, 133Cs, hydrogen, metastable triplet 4He, 174Yb, 85Rb2, and 6Li2). For dilute trapped gases with positive interaction potentials (see Footnote 1), the authors prove that the wellknown Gross-Pitaevskii formula2 adequately describes its ground state and its energy [22,23] in the dilute limit n ! 1, and that 100% Bose condensate occurs [24], corresponding also to 100% superfluid at zero-temperature [25]. For a recent overview on dilute gases, see Ref. [32]. The liquid helium is quite far from the Perfect Bose Gas, and also from any dilute (homogeneous or inhomogeneous) gases since 100% superfluid helium occurs, but corresponding only to 9% Bose condensate at zero-temperature [33–37]. The next step is therefore to go beyond this dilute limit. Actually, the papers [10,18] represent two starting points or steps to find a theory of a ‘‘weakly interacting’’ but non-dilute Bose gas. This new microscopic theory of superfluidity is properly defined and then rigorously performed at non-zero temperatures in Ref. [38]. In particular, it exhibits for low temperatures or high densities the Landau-type excitation spectrum in the presence of a non-conventional depleted3 Bose condensation. It is therefore natural to develop it, in order to find further insights on the gas in full interaction itself. Foremost, since Ref. [38] is based on a slightly inexact proof done in Ref. [39], we take here the opportunity to correct it and therefore to adapt the results of Ref. [38]. The main consequence of our corrections on theorems of Refs. [38,39] is the possible appearance, only at non-zero temperatures, of a range of particle densities (around the phase transition), where the theory can be thermodynamically derived only in the grand-canonical ensemble. Then, we analyze the ground state4 and mean energies of this purely quantum gas, and in much more details the depletion of the Bose condensate. In particular, outside the condensate, a thermal quasi-particle gas coexists with a system of linked pair of particles or ‘‘frozen quasi-particles’’. This frozen jelly is shown to be a quantum print of the effective attraction [12,38] induced on the Bose condensate: at zerotemperature two particles inside the condensate pair up to form interacting (virtual) pair of particles or ‘‘Cooper-type pairs’’ by going back and forth in the frozen jelly. In fact, this pairing leads to a concept, introduced in this paper, of spectrum of frozen, or quantum statistical, excitations. Moreover, the two subsystems, i.e., the thermal quasi-particle gas and the frozen jelly, are related with each other via an additional reservoir of particles on non-zero momenta, or maybe linked pair of particles, implied by interaction phenomena. In fact, this last physically intriguing 1

Decreasing sufficiently fast at infinity. This variational formula was first introduced in Refs. [29,30] and independently in Ref. [31] for again the study of superfluid 4He. 3 Even at zero-temperature. 4 The ground state energy is evaluated and it is strictly negative, but the ground state itself is not derived. 2

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

309

part corresponds to a specific mixture between the thermal quasi-particle subsystem and the frozen jelly. The depletion of the Bose condensate is then much more complex than usually considered, even in Ref. [38]. We also verify the existence of two fluids [3,4,33,34,40–44] suggested by Tisza, then by Landau, and experimentally verified for liquid helium by Adronikashvili in the forties.5 In fact, we could define two different superfluid fractions respectively based on its absence of viscosity (standard definition) or the zero-entropy criteria. The first definition leads to Landau’s form for the normal density. However, its asymptotics for extremely low temperatures is different from Landau’s ‘‘T4 law’’, which is experimentally found in liquid helium, see for example Refs. [43,44]. Indeed, the main divergence with Landau’s approach comes from the concept, introduced in this paper, of thermal excitation spectrum, which could be different from spectrum of elementary excitations. Using the second criteria, the entropic gas is different from Landau’s form and is the thermal quasi-particle subsystem. The appearance of the non-entropic gas, i.e., the Bose condensate, the ‘‘mixture’’ and the frozen jelly, is strongly correlated to the one of Bose condensation as in liquid helium [35–37]. At this point, we touch very lightly on one of the most fascinating problems of mathematical or quantum physics [1]—a proof of the existence of non-trivial superfluidity in a ‘‘weakly interacting’’, above all non-dilute, system. The structure of the paper is as follows. We first review and correct this ‘‘weakly interacting’’ non-dilute Bose gas in Section 2 by adding also a new critical discussion of our model. In Section 3, we develop its thermodynamics by explaining in much more details the depletion of the Bose condensate and its mean energies. Then, Section 4 concerns the analysis of normal and superfluid gases for this theory. Two appendices are devoted respectively to a proof correction on [38,39] and to some non-exhaustive discussions about the spectrum of elementary excitations. To fix the notations, b40 is here the inverse temperature, m the chemical potential, and r40 the fixed particle density. Then, T  ðkB bÞ1 X0 is the temperature where kB is the Boltzmann constant. Here hiH X ðb; rÞ and hiH X ðb; mÞ represent the (finite L volume) canonical and grand-canonical Gibbs statesL respectively for some X Hamiltonian H L . Also, d40 is always taken here as an arbitrary small parameter.

2. To a rigorous superfluidity theory for the non-dilute Bose gas Let an homogeneous gas of n spinless bosons with mass m be enclosed in a cubic box L R3 of volume V  jLj ¼ L3 : The one-particle energy spectrum is then ek  _2 k2 =2m and, using periodic boundary conditions, L  ðð2p=LÞZÞ3 R3 is the set of wave vectors k. The considered system is non-dilute and with interactions defined via a (real) two-body soft potential jðxÞ  jðkxkÞ such that: (A) jðxÞ 2 L1 ðR3 Þ (absolute integrability). 5

Below T l ¼ 2:17 K, a normal fluid and a superfluid liquid (no viscosity and zero-entropy) coexist in liquid helium.

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

310

(B) Its (real) Fourier transformation Z jðxÞeikx d3 x; k 2 R3 , lk  R3

satisfies: l0 40 and 0plk p lim lk for k 2 R3 . kkk!0þ

This homogeneous gas of bosons in full interaction is then described via a (second quantized) Hamiltonian explained in the first subsection. But, before going further, first note that the conditions may be relaxed in our weakly interacting non-dilute gas defined in the second subsection. In fact, this model turns out to be independent of the Fourier transformation l0 of jðxÞ for k ¼ 0: In particular, l0 may be zero or infinite for some interaction potentials, but the (effective coupling) constant Z 1 l2k 3 g00   d ko0 , (2.1) 4ð2pÞ3 R3 ek (cf. [12]) and jð0Þ have to exist. See discussions in Ref. [38]. Here, we have considered a soft potential jðxÞ by condition (A). In general, due to the strong divergence for x ! 0; this condition is not verified for a realistic (hard) interaction which should be of Lennard-Jones type [45]. A standard way to approximate this realistic interaction potential could be to cut it when kxkormin : However, one should choose rmin 5rmean ; where rmean r1=3 is the average length of the inter-particle distance at density r40. Actually, the problem linked to (A) is by far non-trivial. It is related with the concept of scattering length which should replace the L1 -norm of jðxÞ. This phenomenon has been theoretically predicted for general many-body problems [46, Ch. 14; 47]. Indeed, face to Bogoliubov’s interrogations, Landau combined a diagrammatic method (Born’s approximation of the scattering length) with Bogoliubov’s approximations to almost reconstruct, in the ground state, the scattering length from the L1 -norm of jðxÞ. Remark 2.1. Following Bogoliubov [5–8], we have taken the condition (B) to ensure stability of the system [45]. Note also that lk ¼ lkkk since jðxÞ  jðkxkÞ is real. 2.1. The Bose gas in full interaction Let a#k  fa k or ak g be the usual boson creation/annihilation operators in the oneparticle state ck ðxÞ  V 1=2 eikx ; k 2 L , x 2 L, acting on the boson Fock space þ1

FBL   HðnÞ B , n¼0

with

HðnÞ B

defined as the symmetrized n-particle Hilbert spaces

ð0Þ 2 n HðnÞ B  ðL ðL ÞÞsymm ; HB  C ,

see [45,48]. The full interacting Bose gas is then defined by the Hamiltonian H L;l0  T L þ U L þ U MF L

(2.2)

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

311

acting on FBL , with X TL  k a k ak , k2L

UL 

1 2V k

X 1 ;k2 ;qa02L



lq a k1 þq a k2 q ak1 ak2 ,

X

l0 l0 ðN 2  N L Þ, a a ak ak ¼ 2V k ;k 2L k1 k2 2 1 2V L 1 2 X a k ak . NL  U MF L 

ð2:3Þ

k2L

Under assumptions (A) and (B) on the interaction potential jðxÞ, the full Hamiltonian H L;l0 40 is superstable [45]. Considering the existence of a Bose condensation on the zero-kinetic energy state in liquid helium [2], a spontaneous description of the corresponding non-dilute gas at low temperatures should be deduced by carrying out the Bogoliubov approximation n pffiffiffiffi o pffiffiffiffi a0 = V ! c; a 0 = V ! c; c 2 C (2.4) directly in the full Hamiltonian H L;l0 , see for example Refs. [8,49,50]. We then obtain in the grand-canonical ensemble the model H L;l0 ðm; cÞ, which is well-defined on the boson Fock space F0B of symmetrized n-particle Hilbert spaces HðnÞ B;ka0 for non-zero momentum bosons. This heuristic approach is shown [51] to be rigorous in the grand-canonical thermodynamic limit as soon as the c-number is the solution e cðb; rÞ of the following variational problem: pðb; mb;r Þ  lim L

1 ln TrFB ebðH L;l0 mb;r N L Þ ¼ sup pðb; mb;r ; cÞ ¼ pðb; mb;r ; e cðb; rÞÞ , bV L c2C

(2.5) with pðb; m; cÞ  lim L

  1 NL ln TrF0 ebH L;l0 ðm;cÞ ; lim B L V H bV

ðb; mb;r Þ ¼ r40; mb;r 2 R . L;l0

In particular, the validity of the Bogoliubov approximation (2.4) has, a` priori, nothing to do with the existence, or not, of a Bose condensation [51]. See also Refs. [52,53], which are nice refinements of [51]. In fact, intuitively, if there is no Bose condensation on a specific mode or on an infinite number but non-macroscopic, then the system thermodynamically behaves exactly as if the corresponding modes are dropped out. It can easily be seen for the Perfect Bose Gas or many other simplified interacting models. It also means that the Bogoliubov approximation (2.4) is trivial on each corresponding mode, i.e., c ¼ 0. The papers [51,52] prove6 in fact this phenomenon for the full 6

The proofs in Ref. [51,52] are completely different from each other. In particular, the results of Ref. [52] are much more general.

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

312

interacting Bose gas. Nevertheless, even before solving the variational problem, one should then deal with a complicated system, i.e., the model H L;l0 ðm; cÞ in this case, still not thermodynamically solved at fixed densities and temperatures. 2.2. A ‘‘weakly interacting’’ but non-dilute Bose gas An alternative way to get interesting results is to follow the method mainly explained in Ref. [18]. In particular, this theory and the Bogoliubov one have behind them the fundamental hypothesis originally given by Fritz London [2] about existence in the system of a Bose condensation on the zero-mode. Indeed, intuitively, at high temperatures or low densities, the system is more or less a perfect, or dilute, Bose gas. This means that a Bose condensation could appear at low enough temperatures, if the interactions are still not too strong of course. This also implies that the interactions involving the Bose condensate in relation with the thermally excited bosons start to be macroscopically relevant, i.e., important in comparison with the rest of the interaction. The result of this phenomenon should then be a nontrivial quantum behavior at low temperatures or high densities. Therefore, in this scenario, we should caricature the full interaction in (2.2) by keeping only the terms in which at least two operators a 0 ; a0 appear, as Bogoliubov originally pointed it out. Indeed, if no Bose condensation exists then the system has no interaction, i.e., it is a perfect Bose gas. And, as soon as the Bose condensation appears, those interaction terms start to play a macroscopic roˆle: the gas should then become non-trivial. However, in contrast with the Bogoliubov theory, the truncation must be partial, in the sense that the Mean-Field interaction should not be taken into account since it is just a constant in the canonical ensemble. Within the framework of the canonical ensemble, this procedure then implies the new Hamiltonian: ND H BL;0  T L þ U D L þ UL ,

(2.6)

with UD L 

1 X lk a 0 a0 ða k ak þ a k ak Þ , 2V k2L nf0g

U ND L 

1 X 2 lk ða k a k a20 þ a 0 ak ak Þ . 2V k2L nf0g

ð2:7Þ

This Bose gas is quartic and non-diagonal. It is not a ‘‘mean-field’’ model but a purely quantum gas. It is coherently solved in the canonical ensemble [38] (up to the errata done in this paper). Within the framework of perturbation theory, this approach seems to ‘‘fall between two stools’’. Indeed, in the dilute gas where there is almost 100% of Bose condensate, i.e., X NL ¼ N0 þ N k N 0 ; with fN k  a k ak gk2L , k2L nf0g

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

313

one could, in a second step, truncate again H BL;0 in the same way, i.e., by keeping only the quadratic terms in operators fa k ; ak gka0 . For example, we could use on U D L the approximation X X X lk N 0 ðN k þ N k Þ ¼ lk ðN L  N k ÞðN k þ N k Þ k2L nf0g

k2L nf0g



X

k2L nf0g

lk N L ðN k þ N k Þ .

k2L nf0g

This coherent (dilute) approach [19] leads to the well-known Bogoliubov theory in the canonical ensemble with almost 100% of Bose condensate7 at zero-temperature. However, even if our procedure is applicable to the weakly interacting gas, it does not mean that we always reach this dilute limit. Actually, Bogoliubov (and Zubarev) early noticed the difficulty with his ansatz of 100% of Bose condensate. For example, at non-zero temperatures, the thermally induced depletion of condensate becomes also important even for dilute gases. By treating the condensate in some consistent fashion, Popov (see e.g. [54,55]) proposed in 1965 a generalization of the Bogoliubov theory for non-zero temperatures that gives the elementary excitation spectrum similar to that at zero-temperature, but now with temperature-dependent condensate. For further details of the Beliaev-Popov theory, based on Green’s functions applied on a dilute Bose-condensed gas, we address readers to the review by Shi and Griffin [56]. On the other hand, following a perturbation theory on the model H L;l0 ðm; cÞ with a condensate, the depletion of the condensate, i.e., N L  N 0 ; is zero in the first order, whereas in the second order we get the well-known depletion of the condensate [8,9,55–59]: see for example (2.108)–(2.109) in Section 2.4 of Ref. [17]. These studies are performed after considering the creation/annihilation operators as c-complex numbers. For the initial model H L;l0 ; they should be interpreted, at least for very low temperatures, by taking the solution e cðb; rÞ of the variational problem (2.5). Our crucial point is that, by keeping the creation/annihilation operators in the zero-mode, even on the ground state, the non-diagonal quartic interaction U ND L (2.7) implies a second order contribution on the interactions via the effective coupling constant g00 (2.1), see Figs. 1–2 and [12,38]. This contribution defined by g00 turns out to be amazingly crucial in our rigorous proofs [38,39]. Actually, for the Bose system H BL;0 ; the corresponding kinetic part only helps to turn on a Bose condensation phenomenon via the Bose distribution. But, as soon as the Bose condensate appears, the non-diagonal quartic interaction U ND L drastically changes all thermodynamic properties of the system by switching the usual perfect gas to a nontrivial system. It is coherent with the desired thermodynamic behavior explained in the beginning of this subsection. Actually, it implies a depletion of the condensate exactly equal, at zero-temperature, to (2.108)–(2.109) in Section 2.4 of Ref. [17], but for a c-number now solution of a variational problem coming from the 7

The assumption of 100% of Bose condensate was crucial in the (canonical) Bogoliubov theory in order to get the Landau-type excitation spectrum.

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

314

k

k’=0

k’=0

k

λk -k

λk k’=0

k’=0

-k

Fig. 1. Non-diagonal-interaction vertices corresponding to U ND L .

k

k=0

k=0

λk

λk -k

k=0

k=0

k=0

k=0

gΛ,00 < 0 k=0

k=0

Fig. 2. Effective attraction on the zero-mode induced by the non-diagonal interaction U ND L ; where g00 ¼ limL gL;00 o0; cf. (2.1).

thermodynamics of H BL;0 (see Theorem 2.3) and not solution of (2.5) (or an analogous variational problem in the canonical ensemble). To conclude, being a strong caricature of the Bose gas in full interaction, this theory cannot be a complete theory of ‘‘real superfluidity’’. But, at least, the thermodynamic behavior of the model H BL;0 qualitatively retains the main features of liquid helium as the Landau-type excitation spectrum for TpT c ðT c ]3:14 KÞ. Consequently, for low temperatures and for some ‘‘weakly interacting’’ gases, the model H BL;0 ; ‘‘being a second order thermodynamics’’, might capture the kernel of some thermodynamics properties of H L;l0 .

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

315

Remark 2.2. By combining in a different way the vertices associated with U ND L (Fig. 1), via a Fro¨hlich transformation (second-order term in lk ), one also gets another effective interaction [12,38]. It is a repulsion inside each quasi-particle, i.e., inside each couple of particles respectively with modes k and k (ka0). The corresponding positive coupling constant is proportional to the Bose condensate density. 2.3. Basic thermodynamic properties of the weakly interacting gas The analysis performed in Ref. [38] is based on a (maybe) controversial but unique truncation hypothesis. It involves the rigorous thermodynamics in the canonical ensemble of a quartic (non-diagonal and non-dilute) Bose system (i.e., H BL;0 ). (1) Let f BL;0 ðb; rÞ be the corresponding free-energy density8 defined for a fixed particle density r40 by f BL;0 ðb; rÞ  

1 bH B ðn¼½rV Þ L;0 g ln Tr ðnÞ ðfe Þ. H bV B

(2.8)

By the Bogoliubov approximation (2.4) applied on H BL;0 , the resulting model H BL;0 ðcÞ is well-defined on the boson Fock space F0B of symmetrized n-particle Hilbert spaces HðnÞ B;ka0 for non-zero momentum bosons. We then consider its (infinite volume) freeenergy density9 defined by f B0 ðb; r1 ; xÞ   lim L

1 bH B ðcÞ ðn ¼½r V Þ L;0 g 1 1 ln Tr ðn1 Þ ðfe Þ, H bV B;ka0

(2.9)

for any b40; r1 40 and x ¼ jcj2 X0. The (infinite volume) pressure of this gas is P ! bðH B ðcÞað a ak þVxÞÞ L;0 k 1 B k2L nf0g ln TrF0 e p0 ðb; a; xÞ  lim , B L bV for ap0: The Bogoliubov u-v transformation diagonalizes the quadratic operator X a k ak H BL;0 ðcÞ  a k2L nf0g

by using the new set of boson operators fbk ; b k gka0 : bk  uk ak þ vk a k ; b k  uk a k þ vk ak ,

(2.10)

for any ap0. The real even functions uka0 and vka0 have to satisfy: u2k  v2k ¼ 1; uk vk ¼

f k;0 xlk ; u2k þ v2k ¼ B , B 2E k;0 E k;0

with f k;0  ek  a þ xlk ; E Bk;0 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðek  aÞðek  a þ 2xlk Þ .

(2.11)

8 Here AðnÞ  A dHBðnÞ is the restriction of any operator A acting on FBL , whereas n ¼ ½rV is defined as the integer of rV : ðn1 Þ 9 Here Aðn;ka0Þ  A dHB;ka0 is the restriction of any operator A acting on F0B :

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

316

And, for ap0 we then get: 1 pB0 ðb; a; xÞ ¼ ax þ ð2pÞ3

Z ( R3

) 2 2 1 x l bE B 1 k lnð1  e k;0 Þ þ d3 k . b 2ðf k;0 þ E Bk;0 Þ

(2.12)

Now, we give the connection between f B0 ðb; r1 ; xÞ and the thermodynamic limit f B0 ðb; rÞ of f BL;0 ðb; rÞ (cf. Ref. [38] in relation with Appendix A). Theorem 2.3. There are two particle densities rc;inf ðbÞprc;sup ðbÞ or two inverse temperatures bc;inf ðrÞpbc;sup ðrÞ such that b; x bÞ; with x bor , f B0 ðb; rÞ ¼ inf ff B0 ðb; r  x; xÞg ¼ f B0 ðb; r  x x2½0;r

for re½rc;inf ðbÞ; rc;sup ðbÞ (b fixed) or be½bc;inf ðrÞ; bc;sup ðrÞ (r fixed). More explicitly, bÞg ¼ aðb bÞ . f B0 ðb; rÞ ¼ supfar  pB0 ðb; a; x xÞr  pB0 ðb; aðb xÞ; x ap0

Note that f B0 ðb; r1 ; xÞ (2.9) could have been directly defined as transformation of fpB0 ðb; a; xÞ  axg. Also, this theorem is valid for any

the Legendre densities r40

at zero-temperature since lim rc;inf ðbÞ ¼ lim rc;sup ðbÞ ¼ 0 ,

b!þ1

b!þ1

cf. (A.5). An illustration of this phase diagram is performed in Fig. 3. Remark 2.4. If rc;inf ðbÞorc;sup ðbÞ (bo þ 1), the canonical thermodynamics of H BL;0 may not exist almost surely or be, at least, really mysterious for r 2 ½rc;inf ðbÞ; rc;sup ðbÞ (cf. end of Appendix A). In any case, if it is not mentioned, the fixed particle density r or the fixed inverse temperature b is always taken such that re½rc;inf ðbÞ; rc;sup ðbÞ (b fixed) or be½bc;inf ðrÞ; bc;sup ðrÞ (r fixed). bðb; rÞ of the first variational problem of the theorem is x b¼0 Here, the solution x b 2 ð0; rÞ for r4rc;sup ðbÞ; with following asymptotics: for rorc;inf ðbÞ; and x bðb; rÞor; lim x

b!þ1

bðb; rÞ ¼ r . lim x

r!þ1

Therefore, at least on free-energy density level, the thermodynamics of H BL;0 is the one of the Perfect Bose Gas (elementary excitation spectrum ek ) for rorc;inf ðbÞ. Whereas for r4rc;sup ðbÞ; the Hamiltonian H BL;0 (2.6) seems to be thermodynamically equivalent, up to a constant term, to a perfect Bose gas of quasi-particles (2.10) for pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bðb; rÞ (elementary excitation spectrum ek ðek þ 2b ka0 with a density x ¼ x xlk Þ; cf. Appendix B). Moreover, the free-energy per particle is strictly negative at low enough temperatures (Appendix A). It is the solution of the second variational problem of the theorem, i.e., qr f B0 ðb; rÞ ¼ aðb xÞp0: A strictly negative ‘‘chemical potential’’ aðb xÞ is in fact the unique solution of the Bogoliubov density equation: bÞ  rB0 ðb; a; x bÞ for r40 . r ¼ qa pB0 ðb; a; x

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

317

β

^ x(β,ρ)>0

?

? βc,sup ? ^ x(β,ρ)=0

βc,inf

? ρc,inf

0

ρc,sup

ρ

Fig. 3. Illustration of the phase diagram, i.e., rc;inf ðbÞ and rc;sup ðbÞ (b fixed) or bc;inf ðrÞ and bc;sup ðrÞ (r fixed). The dotted line corresponds to the phase diagram of the Perfect Bose Gas.

Here 1 rB0 ðb; a; xÞ ¼ x þ ð2pÞ3

8 Z <

f k;0 B

R3 :E B ½ebE k;0 k;0

9 = x2 l2k d3 k . þ B B ; 2E ½f þ E  k;0 k;0 k;0  1

(2.13)

For more details, see Refs. [38,39] combined with Appendix A. (2) Now, we give the main results of Ref. [38] for the canonical thermodynamic behavior of H BL;0 combined with our corrections (Appendix A). In fact, there exist: (i) A non-conventional Bose condensation induced by the non-diagonal interaction U ND L (2.7) for high particle densities, or low temperatures (Fig. 4): (   ¼ 0 for rorc;inf ðbÞ or bobc;inf ðrÞ; a0 a0 bðb; rÞ ¼ ðb; rÞ ¼ x lim 40 for r4rc;sup ðbÞ or b4bc;sup ðrÞ ; L V HB L;0

(2.14) with no other Bose condensation (of any type I, II or III [60–62]) outside the zeromode.

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

318

x(β,ρ)

Bose Condensation ρ

ρc,inf ρc,sup

0

To zero-temperature bðb; rÞ as a function of r40. For Fig. 4. Illustration of the non-conventional Bose condensate density x bðb; rÞ ¼ 0. The dashed dotted line corresponds to a zero-temperature, i.e., for rorc;inf ðbÞ, recall that x b ! þ1. The straight line is x ¼ r.

(ii) A particle density outside the zero-mode equal to: 8 Z < f k;0 1 X 1 3 lim ha k ak iH B ðb; rÞ ¼ d k 3 : B bE Bk;0 L V L;0 ð2pÞ R3 k2L nf0g E k;0 ½e  1 9  = x2 l2k  þ B , 2E k;0 ½f k;0 þ E Bk;0  ; x¼b x;a¼aðb xÞ

ð2:15Þ

cf. Fig. 5. More precisely, for rorc;inf ðbÞ or bobc;inf ðrÞ; one has the Bose distribution: 8k 2 L : kkkXd40; limha k ak iH B ðb; rÞ ¼ L

L;0

1 ebðek að0ÞÞ

1

.

(2.16)

Whereas, for r4rc;sup ðbÞ or b4bc;sup ðrÞ; i.e., in the presence of the Bose condensation, we get another one, which we call the Bogoliubov distribution: 9 8 = < 2 2 f x l k;0 k  limhak ak iH B ðb; rÞ ¼ þ B , (2.17) B ; L : B bE Bk;0 L;0 2E ðf þ E Þ  k;0 k;0 k;0 E k;0 ðe  1Þ x¼b x;a¼aðb xÞ with again kkkXd40: Here f k;0 and E Bk;0 are defined by (2.11).

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

319

ρ-x(β,ρ)

0

ρc,inf ρc,sup

ρ

To zero-temperature

Fig. 5. Illustration of the depletion of the Bose condensate, i.e., the Bogoliubov system density ðr  bðb; rÞÞ as a function of r. The dashed dotted line corresponds to a zero-temperature, i.e., for b ! þ1. x

(3) The existence of particles outside the Bose condensate even at zero-temperature is well-known as the depletion of the Bose condensate, see for example Refs. [8,9,55–59]. But the paper [38] goes a step further in its interpretation. This behavior is considered as the coexistence of two systems10 for r4rc;sup ðbÞ or b4bc;sup ðrÞ: the Bose condensate (i) and the Bogoliubov system (ii). The ‘‘Bogoliubov system’’ is only a personnel terminology, which stands for the ‘‘out-of-condensate Bosons’’. It is roughly a gas of quasi-particles. It exists if and only if the non-conventional Bose condensate exists too. However, these two systems still remain in competition with each other since the Bose condensate implies an effective repulsion for particles of modes ka0, cf. Remark 2.2. For more details relating to this quantum interpretation, see Ref. [38]. The next step is now to deeply understand the Bogoliubov system (ii).

3. Depletion of the Bose condensate: new insights A deeper analysis of the depletion of the Bose condensate, i.e., the Bogoliubov system (ii), is performed here and will provide new informations. This study is not included in Ref. [38]. In particular, we would like first to analyze more closely the gas of quasi-particles defined by fbk ; b k gka0 (2.10), when the Bose condensate appears. Intuitively, the distribution of these quasi-particles should be a Bose distribution like b and a ¼ aðb (2.16) with ðek  að0ÞÞ replaced by E Bk;0 (2.11) for x ¼ x xÞp0. But then, what is the remaining density in (2.17)? Remark that hb k bk iH B ¼ L;0

f k;0 xlk x2 l2k . ha a i þ ha a þ a a i þ B B k k k H L;0 E Bk;0 k H L;0 2E Bk;0 k k 2E Bk;0 ½f k;0 þ E Bk;0  (3.1)

10

The two systems (i)–(ii) have nothing to do with normal and superfluid fractions, see Section 4.

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

320

In particular, the corresponding thermodynamic limit is not completely obvious for r4rc;sup ðbÞ or b4bc;sup ðrÞ. 3.1. The thermal quasi-particle subsystem, the frozen jelly and their mixture To be precise, the boson operators fbk ; b k g (2.10) really represent a quasi-particle for any k 2 L nf0g such that lk a0: To simplify our purpose, we consider that lk a0 for any non-zero wave vector k: Also, note that fbk ; b k g are well-defined only for a fixed Bose condensate density xX0; i.e., only after the Bogoliubov approximation in combination with a gauge transformation, and for the thermodynamic parameter ap0. In each finite volume, the parameter x in (2.10) should be taken as   a0 a0 b xL  ðb; rÞ , V HB L;0

xL Þ as the unique solution of the (finite and the thermodynamic parameter aL ðb volume) Bogoliubov density equation: bL Þ for r40 , r ¼ rBL;0 ðb; a; x with rBL;0 ðb; a; xÞ  x þ

8 X <

x2 l2k

9 =

f k;0 1 . þ V k2L nf0g:E B ½ebE Bk;0  1 2E Bk;0 ½f k;0 þ E Bk;0 ; k;0

Of course, in the appropriate domain of ðb; rÞ; b and lim aL ðb bL ¼ x lim x xL Þ ¼ aðb xÞ , L

L

see (2.13) and (2.14). Now, we study the distribution of fbk ; b k g for x ¼ xL and a ¼ aL ðxL Þ in the thermodynamic limit. In this subsection, we consider r4rc;sup ðbÞ or b4bc;sup ðrÞ to avoid triviality. (1) First, by (2.17), the thermodynamic limit of hb k bk iH B (3.1) is achieved by L;0

knowing the thermodynamic limit of the non-diagonal operator ha k a k þ ak ak iH B : L;0

To compute this limit, we first define EL  fk 2 L nf0g : kkk 2 Ig; with I Rnf0g any open set. Then, we use the auxiliary Hamiltonian g X H BL;0 ðgÞ  H BL;0  ða ak þ a k ak þ a k a k þ ak ak Þ , 2V k2E k L

for jgjod; d 2 ð0; 1Þ. We solve the thermodynamic properties of H BL;0 ðgÞ exactly like H BL;0 : application of the superstabilization method [63,64] as explained in Appendix A, in combination with the Griffiths lemma [65,66]. To simplify our purpose, we

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

321

omit the details. Using (2.17), we then get: (

lim L

Z 1 X 1 xlk 1 1 þ hak ak þ ak ak iH B ¼  2V k2E L;0 ð2pÞ3 R3 E Bk;0 ðebE Bk;0  1Þ 2 L

wI ðkkkÞ d3 k .

!)    

x¼b x;a¼aðb xÞ

ð3:2Þ

Here wI denotes the characteristic function of I. Note that ha k a k þ ak ak iH B ðb; rÞ L;0

is defined on the discrete set L nf0g R3 nf0g. Below we denote by 1 xb;r;L ðkÞ  ha k a k þ ak ak iH B ðb; rÞ 2 L;0

(3.3)

a continuous interpolation of these values from the set L nf0g to R3 nf0g and we define by xb;r its thermodynamic limit. This limit exists for any kkkXd since qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ha k a k þ ak ak iH B p2 ha k ak iH B hak a k iH B L;0

L;0

L;0

by Cauchy-Schwartz inequality, see (2.17). Consequently, Eqs. (3.2) and (3.3) imply (Z ) ! Z  xlk 1 1  3 3 xb;r ðkÞwI ðkkkÞ d k ¼  ðkkkÞ d k , w þ  B B  2 I R3 R3 E k;0 ðebE k;0  1Þ x¼b x;a¼aðb xÞ for any open set I Rnf0g. Therefore, ( 1 xlk lim hak ak þ ak ak iH B ðb; rÞ ¼  L 2 L;0 E Bk;0

1 ðe

bE B

k;0

1 þ  1Þ 2

!)    

, x¼b x;a¼aðb xÞ



for any k 2 L such that kkkXd40; and in the appropriate domain of ðb; rÞ: (2) Combining (2.17), (3.1) and the previous result, the quasi-particle distribution in the thermodynamic limit is ( )  1  , limhbk bk iH B ðb; rÞ ¼  bE B  L L;0 e k;0  1 x¼b x;a¼aðb xÞ with kkkXd40: The first part corresponds to a Bose distribution (2.16) with ðek  b40 and a ¼ aðb að0ÞÞ replaced by E Bk;0 (2.11) for x ¼ x xÞ, as expected. Actually, in the high density (or low temperature) regime, the k-density (2.17) of the Bose condensate depletion, i.e., the Bogoliubov system (ii), can be rewritten as a sum of three positive terms: limha k ak iH B ðb; rÞ ¼ bk ðb; rÞ þ rk ðb; rÞ þ jk;b ðrÞ , L

L;0

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

322

k-densities

(iia)

(iib)

(iic)

0

k

Fig. 6. Illustration, as functions of kkk; of the k-densities respectively of the thermal quasi-particle subsystem (iia), i.e., bk ðb; rÞ, the ‘‘mixture’’ (iib), i.e., rk ðb; rÞ, and the frozen jelly (iic), i.e., jk;b ðrÞ.

for kkkXd40, with

(

(iia) bk ðb; rÞ  limhb k bk iH B ðb; rÞ ¼ L

(

L;0

1 bE B

e )    

k;0

1

)    

,

x¼b x;a¼aðb xÞ

! f k;0 1 1 ¼ 2bk ðb; rÞjk;b ðrÞ , (iib) rk ðb; rÞ  bE B E Bk;0 e k;0  1 x¼b x;a¼aðb xÞ ( )  x2 l2k  , (iic) jk;b ðrÞ   B B 2E k;0 ðf k;0 þ E k;0 Þ  x¼b x;a¼aðb xÞ

ð3:4Þ

see Fig. 6. Now, we continue our investigation on the Bogoliubov system (ii) via energetic considerations in the next subsection, in order to understand its three parts, i.e., the thermal quasi-particle subsystem (iia), the ‘‘mixture’’ (iib), and the frozen jelly (iic). 3.2. Energetic considerations: ground state and mean energies We provide here an energetic study giving useful additional informations on the Hamiltonian H BL;0 ; but also on the subsystems (3.4). In particular, we give the ground state and mean energies, as well as the energy for each mode ka0 in relation with (iia), (iib), and (iic). To avoid triviality, we consider again r4rc;sup ðbÞ or b4bc;sup ðrÞ in our discussions, but the results below are also valid for rorc;inf ðbÞ or bobc;inf ðrÞ, b ¼ 0: where x

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

323

(1) Using Theorem 2.3, we can divide the free-energy density f B0 ðb; rÞ in three parts: f B0 ðb; rÞ ¼ f b ðb; rÞ þ Ej ðb; rÞ þ aðb xÞðr  xÞ .

(3.5)

The first part f b ðb; rÞ is the free-energy density produced by the gas (iia) of thermal quasi-particles: 9 8  > > = < 1 Z bE B  3 f b ðb; rÞ  lnð1  e k;0 Þ d k  p0 . (3.6) 3 > > ; :bð2pÞ R3 x¼b x;a¼aðb xÞ Whereas 8 > <

Z

x2 l2k

9  > =  3 dk  > ;

1 3 > f k;0 þ E Bk;0 2ð2pÞ : R3 x¼b x;a¼aðb xÞ Z  1 ¼ jk;b ðrÞfE Bk;0 gx¼b d3 kp0 x;a¼aðb xÞ ð2pÞ3

Ej ðb; rÞ  

ð3:7Þ

R3

is the free-energy density created by the frozen jelly (iic). It turns out to be also equal to the mean energy implied by the subsystem (iic), see below. This singular property already shows that the frozen jelly has zero-entropy, see Section 4.2. For the definition of subsystems (iia)–(iic), again see (3.4). Since qr f B0 ðb; rÞ ¼ aðb xÞ; the last part is related to the free-energy per particle. (2) Let 0 be the ground state energy of the Hamiltonian H BL;0 : 9 8  > > Z = < 1 x2 l2k  B 3 E0  lim f 0 ðb; rÞ ¼ aðr  xÞ  dk  B 3 > > b!þ1 f þ E 2ð2pÞ ; : k;0 k;0 R3 x¼b x;a¼aðb xÞ;b!þ1 Z  1 bÞ þ ¼ aðb xÞðr  x jk;1 ðrÞfE Bk;0 gx¼bx;a¼aðb d3 k . ð3:8Þ xÞ;b!þ1 ð2pÞ3 R3

bor for b ! þ1. xÞo0 and x In contrast with the Perfect Bose Gas, E0 o0 since aðb This strict negativity of the ground state energy E0 explains why the standard 100% Bose–Einstein condensation does not occur at zero-temperature. Note that the equality between E0 and ( ) hCjH BL;0 Ci lim inf L C hCjCi should be rigorously verified by large deviations techniques. The normalized ground b L;0 2 Hðn¼½rV Þ of the non-diagonal Hamiltonian H B is then also unknown state C L;0 B even in the thermodynamic limit. In fact, it should be a mixture between coherent and squeezed states. As a recent example, see Ref. [67].

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

324

(3) Since the (finite volume) free-energy density bf BL;0 ðb; rÞ (2.8) is a strict convex function of b, by using the Griffiths lemma [65,66] the mean energy Eðb; rÞ is equal in the thermodynamic limit to 1 hH BL;0 iH B ðb; rÞ ¼ qb ðbf B0 ðb; rÞÞ L V L;0 bÞ . xÞðr  x ¼ Eb ðb; rÞ þ Ej ðb; rÞ þ aðb

Eðb; rÞ  lim

Here

ð3:9Þ

9 8  > > B = < 1 Z E k;0  d3 k  Eb ðb; rÞ  B 3 bE > > ; :ð2pÞ e k;0  1 R3 x¼b x;a¼aðb xÞ Z  1 ¼ bk ðb; rÞfE Bk;0 gx¼b d3 kX0 , x;a¼aðb xÞ ð2pÞ3

ð3:10Þ

R3

and Ej ðb; rÞ (3.7) clearly represent the two mean energies generated by the thermal quasi-particle gas (iia) and the frozen jelly (iic) respectively, cf. (3.4). Formally, note that Z b2 l2k x b2 o0 ; if aðb Ej ðb; rÞ   d3 k  g00 x xÞ  0. (3.11) 3 e  að0Þ þ 4ð2pÞ k b b x!0 x!0þ R3

This asymptotics is coherent with results of the Fro¨hlich transformation done in Ref. [12], i.e., with the effective attraction induced by the non-diagonal term U ND L (2.7), cf. Figs. 1 and 2. (4) We now analyze in more details the energy of each mode ka0 in the thermodynamic limit. For non-zero wave vectors k; this energy is well-defined, at least almost surely, by hk ðb; rÞ ¼ hk ðb; rÞ  limhH fkg iH B ðb; rÞ , L

L;0

with H fkg ¼ H fkg 

ek a k ak

lk þ 2



  2 a 0 a0 a ðak ak þ ak ak Þ þ ak ak 0 þ V V

2

a 0 V

!

! ak ak

.

(3.12) Actually, for this study we consider again the subset EL  fk 2 L nf0g : kkk 2 Ig, with I Rnf0g any open set. We use the partial Hamiltonian X H fkg , H EL ¼ k2EL

and compute first ð2pÞ3 hH EL iH B ðb; rÞ ¼ lim L V L;0

Z R3

hk ðb; rÞwI ðkkkÞ d3 k ,

(3.13)

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

325

where we recall that wI denotes the characteristic function of I: As in the previous subsection, an auxiliary Hamiltonian H BL;0 ðgÞ now defined by H BL;0 ðgÞ  H BL;0  gH EL , for jgjod; d 2 ð0; 1Þ; is thermodynamically solved by applying the superstabilization method [63,64] (see Appendix A). Using in the last step the Griffiths lemma [65,66] combined with some direct computations, the limit (3.13) equals (Z ! Z B 2 2 E x l k;0 k  hk ðb; rÞwI ðkkkÞ d3 k ¼ wI ðkkkÞ B bE B 2ðE þ f k;0 Þ R3 R3 k;0 k;0 ðe  1Þ 0

þa@

f k;0 E Bk;0 ðe

bE B

k;0

9 1 = x2 l2k 3 A þ B d k  B ;  1Þ 2E k;0 ðE k;0 þ f k;0 Þ

,

x¼b x;a¼aðb xÞ

for any open set I Rnf0g. Therefore, almost surely, we have ("

1

hk ðb; rÞ ¼ ðe

bE B

k;0

) #  x2 l2k  B  B þ a lim ha a i ðb; rÞ E  B k k;0 k H B  L L;0 2E ðE þ f Þ k;0 k;0 k;0  1Þ

 ¼ ½bk ðb; rÞ  jk;b ðrÞfE Bk;0 gx¼b þ aðb xÞ limha k ak iH B ðb; rÞ , x;a¼aðb xÞ L L;0

x¼b x;a¼aðb xÞ

ð3:14Þ

with kkkXd40, see also (2.16)–(2.17) and (3.4). Note that the result (3.14) is coherent with ( ) " #  x2 l2k  B H fkg ðxÞ ¼ bk bk  B þ aa a , (3.15) E  k k;0 k B  2E k;0 ðE k;0 þ f k;0 Þ x¼b xL ;a¼aL ðb xL Þ where the operator H fkg ðxÞ is the result of a gauge transformation combined with the Bogoliubov approximation done on H fkg (3.12). In fact, we have limhH fkg  H fkg ðb xL ÞiH B ðb; rÞ ¼ 0 , L

L;0

which is not a trivial statement in the canonical ensemble (H fkg (3.12) is quartic and non-diagonal). An illustration of hk ðb; rÞ is given in Fig. 7. (5) It is clear that the non-diagonal interaction U ND (2.7) is the origin of the L strictly negative energy  jk;b ðrÞfE Bk;0 gx¼bx;a¼aðb o0 (3.16) xÞ in (3.14), since ek a k ak þ

  lk a 0 a0 ða k ak þ a k ak Þ40 . 2 V

Actually, without U ND L , the corresponding system would thermodynamically behave like the standard Perfect Bose Gas. Moreover, via (3.4) (iia) and from the same lines

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

326

mean energy

0

k zero-temperature

Fig. 7. Illustration of the mean energy hk ðb; rÞ as a function of kkk. The dashed dotted line corresponds to a zero-temperature, i.e., for b ! þ1.

of argument as for (3.14), note that * + !  2 2 lk a0 a 0 lim ak ak þ ak ak L 2 V V ( ¼

¼

x2 l2k E Bk;0

1 ðe

bE B

k;0

x2 l2  Bk limhb k bk 2E k;0 L

 1Þ

!)  1  þ   2

ðb; rÞ HB

L;0

x¼b x;a¼aðb xÞ

þ bk b k iH B ðb; rÞ ,

ð3:17Þ

L;0

with again kkkXd40 (almost surely). Therefore, from the point of view of the Bose condensate, the non-diagonal term U ND L really gives an effective attraction on the Bose condensate (cf. Figs. 1 and 2 and (3.11)), which, combined with the diagonal counterpart, implies (3.16) in (3.14). In particular, the term x2 in (3.17) is formally related with an attraction on the Bose condensate of the type 1 X l2k ge00 e a a a a ; g   lim lim hbk bk þ bk b k iH B ðb; rÞo0 . 0 0 00 B L;0 V2 0 0 d!0þ L 2V kkkXd40 E k;0 (3.18) From the point of view of the depletion of the condensate, we can also say that, in infinite volume, U ND L gives the negative energy (3.17) on the Bogoliubov system by destroying/creating each quasi-particle fbk ; b k g, since the condensate is reduced to a background in the thermodynamic limit.

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

327

3.3. Concluding remarks In more detailed investigations, the Bogoliubov system (ii) turns out to be more complex, even for this weakly interacting gas. In particular, in the thermodynamic limit at non-zero temperatures, inside the depletion (ii) of the condensate, via (3.4) (cf. Fig. 6), there exist: (iia) a thermal quasi-particle subsystem with k-density bk ðb; rÞ; this part is exactly a b40 and perfect Bose gas of quasi-particles with spectrum E Bk;0 for x ¼ x a ¼ aðb xÞ; (iib) but also a k-density rk ðb; rÞ; which seems to be a reservoir of particles or maybe linked pair of particles implied by interaction phenomena, or correlations/ transfers between the thermal quasi-particle gas (iia) and the ‘‘frozen jelly’’ (iic) described below; (iic) ‘‘frozen quasi-particles’’ creating in the thermodynamic limit a background characterized by jk;b ðrÞ, whose entropy turns out to be zero (as for (iib), see next Section 4). Note that lim bk ðb; rÞ ¼ 0 and jk;1 ðrÞ  lim jk;b ðrÞ40 .

b!1

b!1

The frozen jelly (iic) subsists at zero-temperature, whereas the thermal quasi-particle subsystem (iia) vanishes implying also the disappearance of the ‘‘mixture’’ (iib). In fact, the mean energy hk ðb; rÞ (3.14) justifies the use of bk ðb; rÞ (iia), rk ðb; rÞ (iib) and jk;b ðrÞ (iic) on the level of densities in (3.4). Indeed, for ka0, each linked pair of particles in (iia), (iib) and (iic) can be seen as having energy E Bk;0 ; 0 and E Bk;0 b and a chemical potential a ¼ aðb respectively, for a Bose condensate density x ¼ x xÞ, cf. Fig. 8. Each macroscopic group of particles on ka0 can then be in three possible energetic states: E Bk;0 ; 0 or E Bk;0 ; respectively with probability bk ðb; rÞ=r; rk ðb; rÞ=r and jk;b ðrÞ=r. In particular, since rk ðb; rÞ ¼ 2bk ðb; rÞjk;b ðrÞ , rk ðb; rÞ=r should represent a joint probability of each macroscopic group of indistinguishable particles to form maybe a specific pair with the first particle related to (iia) and the other to (iic). Within this framework, the k-energy of the ‘‘mixture’’ (iib) should be zero. Note that the k-energy E Bk;0 of (iic) appeals for attention, since it comes from the interaction U ND L (2.7). In fact, recall that this non-diagonal term implies an effective attraction for particles inside the Bose condensate (cf. Fig. 2, (3.11), (3.17)–(3.18)), which, counterbalanced by the other positive part of the Hamiltonian, directly imply the strictly negative energy (3.16) of the frozen jelly. Therefore, on the one hand, from the point of view of the Bose condensate (i), its particles then get an attraction via the Bogoliubov system (ii), and so the frozen jelly (iic), to form ‘‘Cooper-type

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

328

k-energies

(iia): E

B k,0

(iib): 0 0

k

(iic): -E

B k,0

Fig. 8. Illustration, as functions of kkk; of the k-energies E Bk;0 ; 0 and E Bk;0 respectively of the thermal quasi-particle subsystem (iia), the ‘‘mixture’’ (iib), and the frozen jelly (iic) for a Bose condensate density b and a chemical potential a ¼ aðb x¼x xÞo0.

B

Energy on each non-zero mode k

Ek,0

(iia)

0

(iib)

B

-E k,0

(iic)

Bogoliubov system

(i) mode k=0 attraction

Bose condensation

Fig. 9. Depletion of the Bose condensate, i.e., the Bogoliubov system (ii), versus Bose condensation (i). Recall that (iia), (iib) and (iic) represent respectively the thermal quasi-particle system, the ‘‘mixture’’, and the frozen jelly.

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

329

pairs’’. This is also why the depletion appears even at zero-temperature. On the other hand, from the point of view of the Bogoliubov system (ii), it is equivalent to say that each ‘‘frozen quasi-particles’’ in (iic) has an energy E Bk;0 , since the Bose condensate (i) is reduced to a background in the thermodynamic limit. To resume this phenomenon, the frozen jelly is a quantum print of this effective attraction induced on the Bose condensate: two particles inside the Bose condensate pair up to form interacting (virtual) pair of particles or ‘‘Cooper-type pairs’’ by going back and forth inside the Bogoliubov system (ii) (cf. (3.17)–(3.18)), and, in particular, inside the frozen jelly (iic). An illustration of the complexity of the depletion together with the Bose condensate is given in Fig. 9. To conclude, we can then define two different spectra of statistical excitations, namely the spectra of thermal and, frozen or quantum statistical, excitations, whereas the spectrum of elementary excitations (Appendix B) should correspond to individual or local excitations of the system. For example, let us consider the Perfect Bose Gas. The spectrum of its Hamiltonian is given by ek  _2 k2 =2m. Whereas, considering the chemical potential aPBG ðrÞ at a fixed particle density r40 in the grand-canonical ensemble, it equals fek  aPBG ðrÞg, i.e., this spectrum has a gap if aPBG ðrÞo0. The correct elementary excitation spectrum of the Perfect Bose Gas is of course ek : And, aPBG ðrÞ represents a macroscopic physical quantity, namely the free-energy per particle since aPBG ðrÞ ¼ qr f PBG ðb; rÞ: Here, the corresponding free-energy density f PBG ðb; rÞ is the Legendre transformation of its (grand-canonical) pressure for any density r40: f PBG ðb; rÞ ¼ f B0 ðb; r; 0Þ ¼ supfar  pB0 ðb; a; 0Þg ¼ aPBG ðrÞr  pB0 ðb; aPBG ðrÞ; 0Þ , ap0

cf. (2.9) and Theorem 2.3. In fact, the spectrum of thermal and frozen excitations are by definition equal to fek  aPBG ðrÞg and 0 respectively for the Perfect Bose Gas. For the Bose gas H BL;0 ; the corresponding free-energy per particle, namely the chemical potential, is qr f B0 ðb; rÞ ¼ aðb xÞ. It is then natural to define by E Bk;0 (2.11) for b and a ¼ aðb x¼x xÞ its spectrum of thermal excitations, cf. Fig. 8 (iia). Whereas E Bk;0 is the spectrum of frozen, or quantum statistical, excitations, cf. Fig. 8 (iic). This new second branch of excitations is quite interesting since it is the origin of the purely quantum aspect of the non-conventional Bose condensation, as we have already explained above. Also, the notion of thermal excitation spectrum is not useless since it plays a crucial roˆle in the statistical response of the model from macroscopic solicitations. A direct example is given in Section 4.

4. To a microscopic theory of the two-fluid model It is now natural to analyze the existence of two fluids suggested by Tisza, then Landau, and experimentally verified for liquid helium. Indeed, it is well-known [3,4,33,34,40–43] that below the critical temperature T l ¼ 2:17 K of the l-transition of 4He (He II phase), two fluids coexist: the normal and superfluid liquids, with

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

330

densities rnHe ðTÞ and rHe s ðTÞ respectively. More intriguing, a Bose condensate exists in He II as a small fraction of the superfluid even at zero-temperature, i.e., even in its ground state [33–37]. Moreover, the superfluid liquid is phenomenologically interpreted as having, of course, no viscosity but also zero-entropy. It has a heat conductivity much higher than any other substance. One of the most fascinating experiment on He II is the thermomechanical (fountain) effect: if two containers containing He II are connected by a narrow capillary and one of the two is heated, a flow of helium toward the heated vessel will occur. The opposite phenomenon is known as the mechanocaloric effect. The properties of these two fluids are summarized in the following table for ToT l :

Helium liquid Normal liquid

Density

Entropy

He r ¼ rHe n ðTÞ þ rs ðTÞ He r ðTÞ ¼ ð1  g Þr s

sHe  sHe

rsHe ðTÞ ¼ gs  r rHe ðTÞ ¼ g  r

5sHe 

n

Superfluid liquid Bose condensate

0

0

In particular, 0ogs o1 and 0og0 o0:09 for 0oToT l ; whereas for T ¼ 0; g0 ¼ 0:09  and gs ¼ 1, i.e., rHe n ð0Þ ¼ 0 [33,34]. For T ! T l , the apparitions of the Bose condensate and the superfluid liquid are strongly correlated to each other [35–37]: gs ðT l  TÞZ g0 ,

(4.1)

see Fig. 10. Therefore, within the framework of the ‘‘weakly interacting’’ non-dilute gas H BL;0 (2.6), we study the existence of normal and superfluid densities. In particular, we derive the fraction of normal fluid by a standard definition to get Landau’s form, but also the portion of entropic gas. We show that these two densities are different.

γ0(%)

γs(%)

100 9

0

Tλ

0

Tλ

Fig. 10. The fractions, gs of superfluid liquid and g0 of Bose condensate, as a function of temperature T for 4He.

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

331

4.1. Standard definition for the densities of normal and superfluid systems To distinguish the normal fluid from the superfluid, diverse criteria already exist, which may lead to different results [68]. In any case, we have to impose a twist on the system. For translation invariant models, the standard approach is to add _qL :PL =m to the corresponding Hamiltonian. Here PL is the total momentum operator, whereas qL 2 L nf0g. Therefore, the model H BL;0 (2.6) is modified to be equal to H BL;0;qL  H BL;0 

X _ qL : _ka k ak . m k2L nf0g

We then define the normal fluid density rn ðbÞ for the infinite system as in Ref. [69]:   m PL rn ðbÞ  lim j lim ðb; rÞj , q!0 _q L V HB L;0;qL

with qL ! q 2 R3 nf0g in the thermodynamic limit. Remark that the definition for the normal fluid density should formally be given by inverting the limits in the previous expression, but we do not want to enter in this mathematical problem. Actually, the aim of this subsection is not to provide a completely rigorous result but to explain the derivation of the normal fluid as in Ref. [69]. Let f BL;0 ðb; r; qL Þ be the corresponding free-energy density, see (2.8) with H BL;0;qL instead of H BL;0 : Then, the momentum flux is equal to   PL m ðb; rÞ ¼  rqL f BL;0 ðb; r; qL Þ , _ V HB L;0;qL

i.e., # rn ðbÞ ¼  lim

q!0

m2 lim rqL f BL;0 ðb; r; qL Þ _2 q L

$ .

(4.2)

The results [39,38] performed for the model H BL;0 can be generalized for the model all the more as jqjod can be taken arbitrary small. In particular, it is easy to see that the Bogoliubov u-v transformation (2.10) still works since X X PL ¼ _ka k ak ¼ _kb k bk . H BL;0;qL ;

k2L nf0g

k2L nf0g

As expected, the twist only influences the perfect Bose gas of thermal quasi-particles (iia), cf. (3.4), which should be the carrier of the total entropy of the system. Therefore, in the appropriate domain of ðb; rÞ and for jqL jod; the (infinite volume) free-energy density f B0 ðb; r; qÞ associated with H BL;0;qL exists and is equal to f B0 ðb; rÞ

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

332

(Theorem 2.3) but with 1 pB0 ðb; a; x; qÞ ¼ ax þ ð2pÞ3

Z ( R3

% & ) 2 b E B _m q:k 1 1 x2 l2k k;0 Þ þ d3 k , lnð1  e b 2ðf k;0 þ E Bk;0 Þ

instead of pB0 ðb; a; xÞ (2.12). Note that f BL;0 ðb; r; qL Þ is a convex function of each component of qL . Then, by using the Griffiths lemma [65,66], one gets   PL m ðb; rÞ ¼  lim rqL f BL;0 ðb; r; qL Þ lim L _ L V HB L;0;qL 9 8  > > = < 1 Z _k  3 ¼ . d k  2 3  > > bðE B _m q:kÞ ; :ð2pÞ k;0  e  1 R3 x¼b x;a¼aðb xÞ bðqÞ and aðq; x bðqÞÞ be the two even functions defined as solutions of Let x # $ B bðqÞÞr  pB0 ðb; aðq; x bðqÞÞ; x bðqÞÞ , inf supfar  p0 ðb; a; x; qÞg ¼ aðq; x x2½0;r

ap0

in the appropriate domain of ðb; rÞ. To simplify our purpose, let us consider that the bðqÞ and aðq; x bðqÞÞ are continuous as a function of q. partial derivatives of x Consequently, via (4.2) combined with the isotropy of the system and some direct computations, we finally obtain 9 8  > > Z bE B = < 2 k;0  m 1 2mbe e  k B 3 rn ðbÞ ¼  2 fDq f 0 ðb; r; qÞgq¼0 ¼ d k  . B 3 bE > > 2 3_ ; :ð2pÞ k;0  1Þ 3ðe R3 x¼b x;a¼aðb xÞ (4.3) b is This last expression agrees with Landau’s form, except that E Bk;0 (2.11) for x ¼ x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi not the gapless excitation spectrum ek ðek þ 2b xlk Þ if a ¼ aðb xÞo0, which is verified at low enough temperatures. In particular, since lk is spherically symmetric, i.e., lk ¼ lkkk ; one gets  fE Bk;0 gx¼b ¼ E 0 þ Aek þ oðek Þ , x;a¼aðb xÞ by assuming the absolute integrability of x2 jðxÞ 2 L1 ðR3 Þ, where      bl0 2b xl0 1=2 m 2b xl0 1=2 x b 2 l000 E 0 ¼ aðb þx xÞ 1  and A ¼ 1  , 1 aðb xÞ aðb xÞ aðb xÞ _ b and aðb with x xÞo0 taken for b ! þ1. Therefore, 9 8  > > = b!þ1 > k 0 ðe e Þ ; : 3ð2pÞ R3

. x¼b x;a¼aðb xÞ

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

333

More precisely, rn ðbÞ does not decay with Landau’s ‘‘T4 law’’ for the density of normal fluid in bulk 4He: rLandau ðbÞ  n

' ( 2p2 2p2 _ kB T 4 ¼ , 45c _c 45_3 c5 b4

with c ¼ 238 m:s1 defined as the speed of sound in bulk decays exponentially:

(4.4) 4

He. But, for b ! þ1; it

rn ðbÞoeC 1 b ; C 1 40 . Within this framework, note that the superfluid density is of course 9 8 > > Z bE B < 1 2mbek e k;0 3 = rs ðbÞ  r  rn ðbÞ ¼ r  dk  . bE B > > ð2pÞ3 ; : k;0  1Þ2 3ðe R3 x¼b x;a¼aðb xÞ 4.2. Entropy of the ‘‘weakly interacting’’ non-dilute gas (1) Let sB0 ðE; rÞ be the (infinite volume) entropy defined for a fixed energy EL ! EXE0 (3.8) and particle density r40 by sB0 ðE; rÞ  lim L

1 ln Tr ðnÞ ðP Þ, ðnÞ H fj2H :H B =V ¼EL g V B B L;0

where PA denotes the projection on a set A HðnÞ B . The mean energy Eðb; rÞ (3.9) as the derivative of the free-energy bf B0 ðb; rÞ is continuous (as a function of be½bc;inf ðrÞ; bc;sup ðrÞ). Therefore, via a Tauberian theorem proven in Ref. [70], the existence of bf B0 ðb; rÞ already implies the convexity of sB0 ðE; rÞ for EXE0 and the weak equivalence of the micro-canonical and canonical ensembles in the following sense: f B0 ðb; rÞ ¼ b1 sup fbE  sB0 ðE; rÞg ¼ Eðb; rÞ  b1 sB0 ðEðb; rÞ; rÞ , EXE0

sB0 ðE; rÞ

¼ supfbðE  f B0 ðb; rÞÞg ¼ bE;r ðE  f B0 ðbE;r ; rÞÞ ,

ð4:5Þ

b40

for the appropriate parameters ðE; b; rÞ (Remark 2.4). In fact, bE;r is the inverse function of Eðb; rÞ, i.e., bE;r is the unique solution of equation Eðb; rÞ ¼ EXE0 , see (3.9). Therefore, there are Ec;inf ðrÞpEc;sup ðrÞ such that Ee½Ec;inf ðrÞ; Ec;sup ðrÞ implies bE;r e½bc;inf ðrÞ; bc;sup ðrÞ and conversely. Also, we have sB0 ðE0 ; rÞ ¼ lim bðE0  f B0 ðb; rÞÞ ¼ 0 , b!þ1

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

334

i.e., by definition bE0 ;r  þ1: Consequently, by (3.5), (3.9), and (4.5), the entropy sB0 ðE; rÞ is equal to sB0 ðE; rÞ ¼ bE;r fEb ðbE;r ; rÞ  f b ðbE;r ; rÞgX0 , where f b ðb; rÞ and Eb ðb; rÞ are respectively defined by (3.6) and (3.10). (2) In other words, the entropy sB0 ðE; rÞ is completely produced by the thermal quasi-particle gas (iia) (3.4) with full density 9 8  > > Z Z = < 1 1 1  3 3 rb ðbÞ  d k  bk ðb; rÞ d k ¼ , (4.6) B 3 3 bE > > ð2pÞ ; :ð2pÞ k;0  1Þ ðe R3 R3 x¼b x;a¼aðb xÞ for an appropriate domain of ðb; rÞ. Therefore, the non-entropic gas, i.e., the Bose condensate (i), the ‘‘mixture’’ (iib) and the frozen jelly (iic), has a full density equal to Z 1 b r0;r;j ðbÞ  r  rb ðbÞ ¼ x þ ½rk ðb; rÞ þ jk;b ðrÞ d3 k 3 ð2pÞ R3 9 8 > > Z 2 B < 2 lk cothðbE k;0 =2Þ 3 = x . ¼ xþ d k  > > E Bk;0 ðE Bk;0 þ f k;0 Þ 2ð2pÞ3 ; : R3 x¼b x;a¼aðb xÞ For r4rc;sup ðbÞ; note that 0ob xðb; rÞor0;r;j ðbÞ (even for infinite b). And, using the same arguments as for rn ðbÞ (4.3), we have rb ðbÞoeC 2 b ; C 2 40 , for b large enough. Again, it is not Landau’s ‘‘T 4 law’’ (4.4). However, there is 100% of non-entropic gas at zero-temperature with only a small fraction of Bose b with condensate. Moreover, assuming (3.4) as exact for any fixed Bose condensate x aðb xÞo0; it would imply the relation b=r , gr;j  r0;r;j ðbÞ=r  g0  x bx!0þ which is similar to (4.1) for liquid helium, see Fig. 10. Note that the entropic and non-entropic gases are slightly different from the ones conjectured in Ref. [38] because of the ‘‘mixture’’ (iib). 4.3. Additional remarks We could have defined two different densities for the normal and superfluid gases respectively: (a) rn ðbÞ and rs ðbÞ based on the absence of viscosity, cf. Section 4.1, (b) rb ðbÞ and r0;r;j ðbÞ with the zero-entropy criteria, cf. Section 4.2.

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

335

In particular, it is important to remark in this model that the entropic density rb ðbÞ (4.6) is then different from the normal density rn ðbÞ (4.3) defined by standard definitions. Moreover, the first definition (a) gives Landau’s form (4.3) for the normal density, but the existence of a gap in E Bk;0 gives an asymptotics for very low temperatures different from Landau’s ‘‘T 4 law’’, which is experimentally found in liquid helium [43,44]. The difference with Landau’s approach is that the gapless elementary excitation spectrum E Bk ðb; rÞ (B.2) of the perfect Bose gas of quasiparticles could not be equal to the spectrum of thermal excitations E Bk;0 (2.11) for b, a ¼ aðb x¼x xÞ, r4rc;sup ðbÞ. This strictly negative free-energy per particle qr f B0 ðb; rÞ ¼ aðb xÞ at low enough temperatures and then this gap in the spectrum of thermal excitations is the direct consequence of the depletion of the condensate with a non-trivial ground state energy for this model. Note that the ‘‘T 4 law’’ (4.4) might be a good approximation for rn ðbÞ as soon as the temperature is not extremely low. b To check this, one has to evaluate the size of the possible gap of E Bk;0 (2.11) for x ¼ x and a ¼ aðb xÞp0 in comparison with its first local maximum. Or more likely, this ‘‘weakly interacting’’ gas fails in describing a such behavior of helium liquid, which is not surprising, since the important truncation performed on the gas in full interaction. Finally, we recall that several criteria already exist for (a) [68], which can lead to different results. As it is claimed in Ref. [25], it is not clear that there is a one-size-fits-all definition of superfluidity. A final answer, difficult to perform, might be the derivation of the macroscopic dynamics, as the Navier–Stokes equations, from quantum dynamics, as it is performed by B. Nachtergaele and H.–T. Yau for a Fermi gas [71]. Indeed, using an ‘‘Euler limit’’ the authors derived in Ref. [71] the Euler equations from the corresponding Schro¨dinger equation of a general fermionic system. In our (infinite volume) model, the density rb ðbÞ (4.6) of thermal quasi-particles, i.e., the entropic gas, should give the only non-trivial quantum dynamics, and a viscosity in a macroscopic scale.

Acknowledgements The author wants to express his gratitude to Y. Castin, J.-N. Fuchs, E. Presutti and V.A. Zagrebnov for their useful remarks. Special thanks go to S. Adams for his support.

Appendix A The aim of this section is to give the corrections of the proof of Theorem 2.3 in Ref. [39], which implies slightly different results for Theorem 2.5 in Ref. [39] or for Theorem 2.3 in Ref. [38].

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

336

A.1. Setup of the problem The thermodynamics of the Hamiltonian H BL;0 (2.6) in the canonical ensemble ðb; rÞ is performed by using, in the grand-canonical ensemble, the model B H SB L;l  H L;0 þ

l ðN 2  N L Þ , 2V L

cf. superstabilization method [63,64], with a sufficiently large parameter l40; i.e., for l4  2g00 ,

(A.1)

cf. (2.1). This model H SB L;l is just a technical tool. Actually, the canonical thermodynamic properties of H BL;0 correspond to the grand-canonical thermodynamic behavior of H SB L;l for a fixed particle density r outside the phase transition. For more details on this technique, see Refs. [38,63,64]. Now, we explain the thermodynamics of H SB L;l in the grand canonical ensemble ðb; mÞ in order to point out the (hopefully all and not imaginary) errors hidden in Ref. [39]. Note that (A.1) is the assumption (C1) in Ref. [39] with a strict inequality. A.2. Correction of the solution of the superstable Bogoliubov Hamiltonian In (1) and (2), we consider the chemical potential m as a fixed parameter, whereas in (3) the full particle density r is fixed in the grand-canonical and canonical ensembles. (1) The grand-canonical pressure and density associated with H SB L;l are denoted in the thermodynamic limit as pSB ðb; mÞ and   NL SB ðb; mÞ ¼ qm pSB ðb; mÞ , r ðb; mÞ  lim L V H SB L;l

respectively. They are defined and explicitly found in Ref. [39] for any ðb; mÞ 2 fb40g  fm 2 Rg. We have to solve two variational problems. The first one is characterized by aðxÞ  aðb; m; xÞp0; i.e., the unique solution of # $ # $ ðm  aÞ2 ðm  aÞ2  B B inf p0 ðb; a; xÞ þ (A.2) ¼ p0 ðb; a; xÞ þ  ap0 2l 2l a¼aðxÞ for any fixed xX0; where pB0 ðb; a; xÞ is defined by (2.12). Whereas the second variational problem directly related to pSB ðb; mÞ is # # $$ # $ ðm  aÞ2 ðm  aÞ2 SB B B bÞ þ p ðb; mÞ ¼ sup inf p0 ðb; a; xÞ þ ¼ inf p0 ðb; a; x , ap0 2l 2l xX0 ap0 (A.3)

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

337

b¼x bðb; mÞ is also unique outside the phase transition. In fact, for any which solution x b40; there is a unique mc ðbÞ such that the pressure pSB ðb; mÞ equals 8 2 > < pB0 ðb; að0Þ; 0Þ þ ðmað0ÞÞ ; for mpmc ðbÞ : 2l o pSB ðb; mÞ ¼ n B 2  ðmaðxÞÞ > ; for m4mc ðbÞ :  : p0 ðb; aðxÞ; xÞ þ 2l x¼b x40 The pressure pSB ðb; mÞ is continuous for m ¼ mc ðbÞ: b may not be (2) However, at the phase transition, i.e., for m ¼ mc ðbÞ, the solution x unique for non-zero temperatures: it could be zero or strictly positive. This is in fact the missing part of Ref. [39]. It comes from the proof done in Section 3.2 of Ref. [39]. A first error is localized at the analysis of Eq. (3.15) in Ref. [39], which integrals are in fact indeterminate in the limit ða; xÞ ! ð0; 0Þ at any finite b40 and if lk does not go sufficiently fast to zero when k ! 0: More crucial, Remark 3.2 in Ref. [39] is b¼x bðb; mÞ actually not obvious in the sense that, even if (C1) is verified, the solution x of (A.3) may not be continuous for m ¼ mc ðbÞ at, even low, non-zero temperatures. This possible phenomenon can be seen via a more detailed analysis for finite inverse temperatures of the function 9 8 ( !) > > Z = < 1 ðk  aÞ 2  3 qx F b ðaðxÞ; xÞ ¼ a þ lk 1  1þ , d k  B 3 bE B > > E 2ð2pÞ ; : k;0 e k;0  1 R3

a¼aðxÞ

where F b ða; xÞ  pB0 ðb; a; xÞ þ ðm  aÞ2 =2l. Therefore, one has rc;inf ðbÞ 

lim rSB ðb; mÞprc;sup ðbÞ 

m!m c ðbÞ

lim rSB ðb; mÞprB0 ðb; 0; 0Þ ,

m!mþ c ðbÞ

(A.4)

for non-zero temperatures, and lim rSB ð1; mÞ ¼ lim rSB ð1; mÞ ¼ rPBG ð1; 0Þ ¼ 0; with

m!0

m!0þ

lim mc ðbÞ ¼ 0 .

b!þ1

(A.5) Theorem 2.5 of Ref. [39] is then rigorously proven up to the continuity property for non-zero temperatures. As a consequence of (A.4), in Section 3.5 of Ref. [38], the following interpretation: ‘‘for the Bose system H BL;0 ; the corresponding kinetic part only turns on the Bose condensation phenomenon via the Bose distribution’’ is not rigorously exact at any finite b, but it is almost true at very low temperatures. Note b can be rigorously proven to be strictly also that the solution aðb xÞ of (A.2) for x ¼ x negative only at low enough temperatures. (3) By fixing the particle density r in the grand-canonical ensemble, we define a unique chemical potential mb;r satisfying mb;r  aðb xÞ ¼r. l Actually, at a fixed b40; the function mb;r is the inverse function of the mean particle density rSB ðb; mÞ. By bc;inf ðrÞ and bc;sup ðrÞ we can also denote the critical inverse temperatures defined for a fixed density r as the unique solutions, respectively, rSB ðb; mb;r Þ ¼

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

338

of equations r ¼ rc;inf ðbÞ and r ¼ rc;sup ðbÞ . Here bc;inf ðrÞobc;sup ðrÞ. For rorc;inf ðbÞ or bobc;inf ðrÞ note that mb;r omc ðbÞ, whereas mb;r 4mc ðbÞ for b4bc;sup ðrÞ or r4rc;sup ðbÞ. Moreover, the free-energy density f SB L ðb; rÞ  

1 bH SB ðn¼½rV Þ L;l g ln Tr ðnÞ ðfe Þ, H bV B

B associated with H SB L;l ; and f L;0 ðb; rÞ (2.8) are related to each other by l % 2 r & SB l 2 B B r  ðb; rÞ ¼ f ðb; rÞ þ f SB ; f ðb; rÞ  lim f SB L L;0 L ðb; rÞ ¼ f 0 ðb; rÞ þ r . L 2 V 2 (A.6)

The two models H BL;0 and H SB L;l are then equivalent in the canonical ensemble, in the sense that their (infinite volume) free-energy densities at fixed densities differ only by a constant. Their Gibbs states are equal to each other for all ðb; rÞ. In fact, coming back to the original model H BL;0 (2.6), we find that the canonical thermodynamic properties of H BL;0 corresponds to the grand-canonical thermodynamic behavior of H SB L;l for any fixed re½rc;inf ðbÞ; rc;sup ðbÞ. To conclude, via (A.6), the condition (A.1) is useful to partially solve the problem of convexity of f B0 ðb; rÞ by reducing the range of densities Drc ðbÞ  rc;sup ðbÞ  rc;inf ðbÞ , where f SB ðb; rÞ is not convex. Indeed, if lo  2g00 ; then Drc ðbÞ would have been much bigger. For example, ( 40 if lo  2g00 : lim Drc ðbÞ ¼ ¼ 0 if l4  2g00 ; cf. (A.1). b!þ1 But, if rc;inf ðbÞorc;sup ðbÞ; then the condition (A.1) would not be sufficient for fixed b, since for any large l; the gap still remains (Drc ðbÞ40). And, it does not depend anymore on l4  2g00 in this case. In spite of (A.6), l is never strong enough to make f SB ðb; rÞ strictly convex, at least almost surely, on the fixed interval r 2 ½rc;inf ðbÞ; rc;sup ðbÞ. This means that f B0 ðb; rÞ may not exist in this case, at least almost surely.

Appendix B. Elementary excitation spectrum The question of excitation spectrum is a difficult problem in general, since its structure could be rather complex. For example, the exact solution at zerotemperature of the one-dimensional gas of bosons interacting via repulsive deltafunction potential [72] already manifests two branches of excitation spectra. In spite of restrictions to the particular interaction and dimensionality, this example appeals for attention with classification of the hierarchy of different ‘‘excitations’’ (elementary, quasi-particle, collective), since all of them are components of a unique dynamical spectrum of the system Hamiltonian, see discussions in Ref. [72]. In this

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

339

appendix, our non-exhaustive discussions are only on the thermodynamic level, but also in relation with the property of non-locality (as it is already explained in Ref. [17]). Note that we do not plan here to go into details of the hierarchy of excitations in liquid 4He. For such questions, see Ref. [34]. B.1. Non-locality and spectrum of elementary excitations In the grand-canonical ensemble, the full Hamiltonian H L;l0 (2.2) can formally be rewritten in the x-space as  2   Z  Z _D 3 H L;l0 ¼ a ðxÞ  a ðxÞa ðyÞjðx  yÞaðyÞaðxÞ d3 x d3 y . aðxÞ d x þ 2m L L2 This Hamiltonian is invariant under the local gauge aðxÞ ! eijðxÞ aðxÞ; a ðxÞ ! eijðxÞ a ðxÞ: As a consequence, it verifies the 1=q2 -theorem ([50] Part I) and the Hugenholtz–Pines–Gavoret–Nozie`res study which guarantee a gapless spectrum of quasi-particle excitations even if one treats the k ¼ 0 mode operators using the Bogoliubov approximation (2.4), see an instructive discussion in Ref. [73,74] and the literature quoted there. In contrast to the full Hamiltonian, the model H BL;0 (2.6) violates this local gauge invariance, since (  2    Z  Z _ D a 0 a0 3 B H L;0 ¼ a ðxÞ  jðx  yÞ aðxÞ d x þ a ðxÞaðyÞþ 2m V L L2 ) !  2 2 a 0 a0 þ aðxÞaðyÞ þ a ðxÞa ðyÞ d3 x d3 y . 2V 2V Bogoliubov [75] and Hohenberg [76] in 1962, then Bell in 1963 [77] noted that nonlocal interactions break some exact relations known as the sum-rules for the particlenumber fluctuations [34]. In fact, these rules are related to the local particle-number conservation law, or to the equation of continuity. Then, these authors suggested that this breaking of the sum-rules should imply a ‘‘gap’’ in the spectrum of quasi-particles. However, an approximating Hamiltonian H BL;0 ðcÞ of H BL;0 equals #  2   Z  Z _D H BL;0 ðcÞ ¼ a ðxÞ  jðx  yÞ jcj2 a ðxÞaðyÞ aðxÞ d3 x þ 2m L L2 $ 2 2 c c þ aðxÞaðyÞ þ a ðxÞa ðyÞ d3 x d3 y . 2 2 This Hamiltonian can be diagonalized to obtain a perfect Bose gas of quasi-particles with two backgrounds defined by X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HBL;0 ðxÞ ¼ k ðk þ 2xlk Þb k bk þ ð k ðk þ 2xlk Þ  ðk þ xlk ÞÞ , 2 k2L nf0g (B.1) with x  jcj2 , see (2.10) for a ¼ 0. As Bogoliubov did in his Weakly Imperfect Bose Gas [5–9], the elementary excitation spectrum is the gapless spectrum

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

340

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ðk þ 2xlk Þ. This affirmation about gapless spectrum might be considered as a counter example to the previous conjecture. Actually, it arises the question of definition of spectrum of elementary excitations. As an example, the model HBL;0 ðxÞ for x40 has formally a gap in his exact spectrum coming from the constant term & 1 X %pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ðk þ 2xlk Þ  ðk þ xlk Þ . 2 k2L nf0g But our definition of spectrum of elementary excitation only involves the spectrum of the relevant, i.e., non-constant, or dynamical Hamiltonian related to the gas of quasi-particles defined by X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ðk þ 2xlk Þb k bk . k2L nf0g

At high densities r4rc;sup ðbÞ (or sufficiently low temperatures) the Bose gas H BL;0 seems to be equivalent for large volumes to a ‘‘gas of collective elementary excitations’’, but with two backgrounds: a Bose condensate on k ¼ 0 with density bðb; rÞ and a frozen jelly. In other words, the spectrum of elementary excitations x¼x for the Bose gas in the ‘‘infinite volume’’ equals the Bogoliubov gapless spectrum at inverse temperatures b40 and particle densities r40: 8 < k  _2 k2 =2m for rorc;inf ðbÞ or bobc;inf ðrÞ; E Bk ðb; rÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (B.2) : k ðk þ 2b xlk Þ for r4rc;sup ðbÞ or b4bc;sup ðrÞ ; b. An illustration of E Bk ðb; rÞ is given by Fig. 11. see (B.1) with x ¼ x B

E k (β,ρ)

h2k2 2m

0

krot

k

Fig. 11. Illustration, as a function of kkk, of the elementary excitation spectrum E Bk ðb; rÞ of the ‘‘weakly interacting’’ non-dilute Bose gas H BL;0 for r4rc;sup ðbÞ or b4bc;sup ðrÞ.

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

341

As it is claimed in Ref. [38], note that we do not rigorously know the exact spectrum of H BL;0 , since our analysis is only based on its thermodynamic properties. In infinite volume, this question also implies the problem of definition of limL H BL;0 ! In fact, the approximating Hamiltonian H BL;0 ðcÞ is sufficient to calculate exactly the free-energy density of H BL;0 but not dynamical properties such as the spectrum of the collective quantum fluctuations related to the broken symmetry and to the Goldstone mode [78]. To conclude, these discussions are in fact quite tricky and not satisfactory enough. And, we agree that one should avoid to perform various manipulations or interpretations with quantum system based on different kinds of ansa¨tze or definitions. As an interesting example, see the next subsection. B.2. An example: the Mean-Field Bose Gas The Mean-Field Bose Gas MF H MF L  TL þ UL

does not violate the local gauge invariance, since Z l0 MF UL ¼ a ðxÞa ðyÞaðyÞaðxÞ d3 x d3 y , 2V L2 see (2.3). From the Bogoliubov 1=q2 -theorem ([50] Part I), the spectrum of elementary excitations is gapless. In fact, it coincides with the free-gas spectrum k . However, from its thermodynamic properties, one can extract many misleading interpretations on its spectrum of elementary excitations, at least in the grandcanonical ensemble: (1) On the one hand, in the grand-canonical ensemble ðb; aÞ, i.e., MF H MF L ðaÞ  H L  aN L , it is exactly solved in Refs. [79–82]. And, in contrast to the gapless spectrum k ; the spectrum of the Hamiltonian X H MF ðrÞ ¼ ðk þ l0 rÞa k ak , (B.3) L k2L

in the (grand-canonical) thermodynamic limit, has always which approximates H MF L a gap. The corresponding chemical potential for a fixed particle density r is ( ol0 r for rorPBG ; c MF PBG a ðrÞ ¼ a ðrÞ þ l0 r ¼ PBG ¼ l0 r for rXrc ; in the thermodynamic limit.11 Therefore, taking now into account the chemical potential, the spectrum of the Hamiltonian X MF H MF ðrÞÞ ¼ ðk þ l0 r  aMF ðrÞÞa k ak L ðr; a k2L MF approximating H MF ðrÞÞ L ða MF MF spectrum of H L ða ðrÞÞ is 11 PBG

has a gap for rorPBG whereas it is k for rXrPBG . The c c PBG only gapless for rXrc .

rc and aPBG ðrÞ are respectively the critical density and the corresponding chemical potential for the Perfect Bose Gas.

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

342

(2) On the other hand, in the canonical ensemble ðb; rÞ, in terms of thermodynamic properties, the Mean-Field Bose Gas is completely equivalent to the Perfect Bose Gas with the correct gapless spectrum k , since the Mean-Field interaction U MF L (2.3) is simply a constant on the Hilbert space HBðn¼½rV Þ . In the same way, the spectrum of H MF L ðrÞ (B.3) is also k : However, the thermal excitation spectrum of the Mean-Field Bose Gas is in fact fk  aPBG ðrÞg: Note that an absence of gaps in the spectrum of thermal excitations is only a specific case. For example, it is only in the presence of the conventional Bose–Einstein condensation, i.e., for rXrPBG ; that this property holds for the c Perfect Bose Gas or the Mean-Field Bose Gas. This fact can not be generalized to any Bose system having a Bose condensation, i.e., a gap on the spectrum of thermal excitations may appear even if no gap exists in the spectrum of elementary excitations. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

A.J. Leggett, Rev. Mod. Phys. 71 (1999) S318. F. London, Nature 141 (1938) 643. L.D. Landau, J. Phys. (USSR) 5 (1941) 71. L.D. Landau, J. Phys. (USSR) 11 (1947) 91. N.N. Bogoliubov, J. Phys. (USSR) 11 (1947) 23. N.N. Bogoliubov, Izv. Akad. Nauk USSR 11 (1947) 77. N.N. Bogoliubov, Bull. Moscow State Univ. 7 (1947) 43. N.N. Bogoliubov, Lectures on Quantum Statistics, vol. 1: Quantum Statistics, Gordon and Breach Science Publishers, New York-London-Paris, 1970. N.N. Bogoliubov, in: Collection of papers, vol. 2, Naukova Dumka, Kiev, 1970, pp. 242–257. N. Angelescu, A. Verbeure, V.A. Zagrebnov, J. Phys. A: Math. Gen. 25 (1992) 3473. J.-B. Bru, V.A. Zagrebnov, Phys. Lett. A 244 (1998) 371. J.-B. Bru, V.A. Zagrebnov, Phys. Lett. A 247 (1998) 37. J.-B. Bru, V.A. Zagrebnov, J. Phys. A: Math. Gen. A 31 (1998) 9377. J.-B. Bru, V.A. Zagrebnov, in: S. Miracle-Sole, et al. (Eds.), Mathematical Results in Statistical Mechanics, World Scientific, Singapore, 1999, p. 313. J.-B. Bru, V.A. Zagrebnov, J. Stat. Phys. 99 (2000) 1297. V.A. Zagrebnov, Cond. Matter Phys. 3 (2000) 265. V.A. Zagrebnov, J.-B. Bru, Phys. Rep. 350 (2001) 291. S. Adams, J.-B. Bru, Physica A 332 (2004) 60. Y. Castin, Coherent Atomic Matter Waves, in: R. Kaiser, et al. (Eds.), Lecture Notes of Les Houches Summer School, EDP Sciences and Springer-Verlag, 2001, p. 1–136. E.H. Lieb, J. Yngvason, Phys. Rev. Lett. 80 (1998) 2504. E.H. Lieb, J. Yngvason, J. Stat. Phys. 103 (2001) 509. E.H. Lieb, R. Seiringer, J. Yngvason, Phys. Rev. A 61 (2000) 043602—1-13. E.H. Lieb, R. Seiringer, J. Yngvason, Commun. Math. Phys. 224 (2001) 17. E.H. Lieb, R. Seiringer, Phys. Rev. Lett. 88 (2002) 170409. E.H. Lieb, R. Seiringer, J. Yngvason, Phys. Rev. B 66 (2002) 134529. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Science 269 (1995) 198. C.C. Bradley, C.A. Sackett, J.J. Tollet, R.G. Hulet, Phys. Rev. Lett. 75 (1995) 1687. K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, W. Ketterle, Phys. Rev. Lett. 75 (1995) 3969. E.P. Gross, Nuovo Cimento, X. Ser. 20 (1961) 454. E.P. Gross, J. Math. Phys. 4 (2) (1963) 195.

ARTICLE IN PRESS J.-B. Bru / Physica A 359 (2006) 306–344

343

[31] P. Pitaevskii, Sov. Phys. JETP 13 (1961) 451. [32] E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason, The Quantum-Mechanical Many-Body Problem: The Bose Gas, arXiv:math-ph/0405004v1, 2004. [33] A. Griffin, D.W. Snoke, S. Stringari (Eds.), Bose–Einstein condensation, Cambridge University Press, Cambridge, 1996. [34] A. Griffin, Excitations in a Bose-Condensated Liquid, Cambridge University Press, Cambridge, 1993. [35] L. Aleksandrov, V.A. Zagrebnov, Zh.A. Kozlov, V.A. Parfenov, V.B. Priezzhev, Sov. Phys. JETP 41 (1975) 915. [36] E.V. Dokukin, Zh.K. Kozlov, V.A. Parfenov, A.V. Puchkev, Sov. Phys. JETP 48 (1978) 1146. [37] N.M. Blagoveshchenskii, I.V. Bogoyavlenskii, L.V. Karnatsevich, V.G. Kolobrodov, Zh.A. Kozlov, V.B. Priezzhev, A.V. Puchkov, A.N. Skomorokhov, V.S. Yarunin, Phys. Rev. B 50 (1994) 16550. [38] S. Adams, J.-B. Bru, Ann. Henri Poincare´ 5 (2004) 435. [39] S. Adams, J.-B. Bru, Ann. Henri Poincare´ 5 (2004) 405. [40] P. Kapitza, Nature 141 (1938) 74. [41] J.F. Allen, A.D. Misener, Nature 141 (1938) 75. [42] I.M. Khalatnikov, An Introduction to the Theory of Superfluidity, Benjamin-Reading, New York, 1965. [43] D.S. Betts, J. Wilks, An Introduction to Liquid Helium, 2nd ed., Clarendon Press, Oxford, 1987. [44] D. Pines, Ph. Nozie`res, The Theory of Quantum Liquids, vol. 2: Superfluid Bose Liquids, AddisonWesley Publishing Company Inc., Redwood City, 1989. [45] D. Ruelle, Statistical mechanics: Rigorous Results, Benjamin-Reading, New York, 1969. [46] A.L. Fetter, J.D. Walecka, Quantum Theory of Many Particle Systems, McGraw-Hill, New York, 1971. [47] V.N. Popov, Functional integrals in Quantum Field Theory and Statistical Physics, Riedel, Dordrecht, 1983. [48] O. Brattelli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, vol II, 2nd ed. Springer-Verlag, New York, 1996. [49] N.M. Hugenholtz, D. Pines, Phys. Rev. 116 (1959) 489. [50] N.N. Bogoliubov, Lectures on Quantum Statistics: Quasi-Averages, vol. 2, Gordon and BreachScience Publishers, New York-London-Paris, 1970. [51] J. Ginibre, Commun. Math. Phys. 8 (1968) 26. [52] E.H. Lieb, R. Seiringer, J. Yngvason, Phys. Rev. Lett. 94 (2005) 080401—1–4. [53] A. Su¨to¨, Phys. Rev. Lett. 94 (2005) 080402—1–4. [54] V.N. Popov, Functional Integrals in Quantum Field Theory and Statistical Physics, Dordrecht, Reidel, 1983. [55] V.N. Popov, Functional Integrals and Collective Excitations, University Press, Cambridge, 1987. [56] H. Shi, A. Griffin, Phys. Rep. 304 (1998) 1. [57] N.N. Bogoliubov, D.N. Zubarev, JETP 28 (1955) 129. [58] D.N. Zubarev, JETP 29 (1955) 881. [59] Yu.A. Tserkovnikov, Doklady Acad. Nauk USSR 143 (1962) 832. [60] M. van den Berg, J.T. Lewis, Physica A 110 (1982) 550. [61] M. van den Berg, J. Math. Phys. 23 (1982) 1159. [62] M. van den Berg, J.T. Lewis, J.V. Pule`, Helv. Phys. Acta 59 (1986) 1271. [63] J.-B. Bru, J. Phys. A: Math.Gen. 35 (2002) 8969. [64] J.-B. Bru, J. Phys. A: Math.Gen. 35 (2002) 8995. [65] R. Griffiths, J. Math. Phys. 5 (1964) 1215. [66] K. Hepp, E.H. Lieb, Phys. Rev. A 8 (1973) 2517. [67] J.P. Solovej, Upper Bounds to the Ground State Energies of the One- and Two-Component Charged Bose Gases, math-ph/0406014 2004. [68] N.V. Prokof’ev, B.V. Svistunov, Phys. Rev. B 61 (2000) 11282. [69] P.B. Weichman, Phys. Rev. B 38 (1988) 8739. [70] R.A. Minlos, A.Ja. Povzner, Trans. Moscow Math. Soc. 17 (1967) 269. [71] B. Nachtergaele, H.T. Yau, Commun. Math. Phys. 243 (2003) 485.

ARTICLE IN PRESS 344 [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82]

J.-B. Bru / Physica A 359 (2006) 306–344 E.H. Lieb, Phys. Rev. 130 (1963) 1616. N.M. Hugenholtz, Rep. Prog. Phys. 28 (1965) 201. E. Talbot, A. Griffin, Ann. Phys. (NY) 151 (1983) 71. N.N. Bogoliubov, in: Collection of Papers, vol. 3, pp. 174–243. P.C. Hohenberg, P.C. Martin, Ann. Phys. (NY) 34 (1965) 291. J.S. Bell, Phys. Rev. 129 (1963) 1896. H. Wagner, Z. Phys. 195 (1966) 273. E.B. Davies, Commun. Math. Phys. 28 (1972) 69. M. Fannes, A. Verbeure, J. Math. Phys. 21 (1980) 1809. M. van den Berg, J.T. Lewis, Ph. de Smedt, J. Stat. Phys. 37 (1984) 697. V.A. Zagrebnov, Vl.V. Papoyan, Theor. Math. Phys. 69 (1986) 1240.