Bound pair states beyond the condensate for Fermi systems below T : c the pseudogap as a necessary condition A. Yu. Cherny1 and A. A. Shanenko2 1Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research, 141980, Dubna, Moscow region, Russia 9 2Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980, Dubna, Moscow region, Russia 9 (November29, 1998) 9 As is known, the 1/q2 theorem of Bogoliubov asserts that the mean density of the fermion pair 1 stateswiththetotalmomentum¯hqobeystheinequalitynq ≥C/q2 (q→0)inthecaseoftheFermi n system taken at nonzero temperature and in the superconducting state provided the interaction a term of its Hamiltonian is locally gauge invariant. With the principle of correlation weakening it is J proved in this paper that the reason for the mentioned singular behaviour of nq is the presence of 1 theboundstatesofparticlepairswithnonzerototalmomenta. Thus,belowthetemperatureofthe 2 superconductingphasetransitiontherealwaysexisttheboundstatesofthefermioncouplesbeyond ] thepaircondensate. Ifthepseudogapobservedinthenormalphaseofthehigh–Tc superconductors n is stipulated by the presence of the electron bound pairs, then the derived result suggests, in a o model–independent manner,that thepseudogap survivesbelow Tc. c - r p PACS numbers: 74.20.-z, 05.30.Fk, 05.30.-d u s . t I. INTRODUCTION trix (see, e.g. Ref.8). In particular, the reduced density a m matrix of the second order is of use when a noncoher- ent superposition of the pure states of two particles is - At present the pseudogap is well established to be in d relevant rather than any wave function. If among these n the spectrum of the elementary excitations of undoped states there exist bound ones, then a part of particles o and optimally doped high–Tc superconductors (for ex- of the system involved form bound pair states9. In the c ample see the review1). The presence of the pseudogap superconducting phase a macroscopicalnumber of parti- [ implies that the electron subsystem in the normal phase 1 is not the Fermi liquid and, so, theoretical explanation cthleepcaoinrsdeNns0aoteccoufpypatihrseastamwehibcohutnhdesrtaattieo, Ni.e0./Vthe=reni0s v ofthe pseudogapisrecognizedasthekeypointofunder- is constant in the thermodynamic limit V . In the 4 standing the phenomenon of the high–T superconduc- → ∞ c space–uniformcasethecondensateisformedbythepairs 2 tivity2,3. Thereareagreatnumberofvarioustheoretical with the zero total momentum h¯q = 0, the binding en- 2 approaches of investigating this problem. Two of them 1 ergy εb of these pairs being just the value of the super- 0 consideredbelowareespeciallyinterestinginthecontext conducting gap10. The bound particle pairs beyond the of this paper. 9 condensate are characterizedby the continuousdistribu- 9 The pseudogap can be associatedwith the presence of tion over the total momentum of a couple11. The couple t/ the local pairing correlations without phase coherence. like these must also have the finite binding energy εb(q) a Theideaofthis approachassumingthesingletpairingof that, due to the continuity argument, should tend to ε m b fermions without the phase coherence, as applied to the when q 0. If these bound particle pairs are ’hard’ - high–Tc superconductivity, has been proposed in Ref.4. clusters,→likeinthe theoryofAlexandrovandMott, then d The more radical model of Alexandrov and Mott5 op- one may consider that the quantity ε (q) is practically n b o erates with, say, preformed bosons (bipolarons) existing independent of q. The binding energy εb(q) is just the c in the system above Tc, the pseudogap being treated as pseudogap, which manifests itself in the normal phase : coming from the binding energy of a bipolaron (of the when the bound couples survive at T >T . v c order of a few hundred K). This model dates back to Xi the Schafroth’s ideas according to which the supercon- In the BCS–theory there are no bound pair states be- yond the condensate absolutely7,12 (see below) which is r ductivityisaresultoftheBose–Einsteincondensationof a the bound pairs of electrons localized in the space and a consequence of the violation of the local gauge invari- appearing in the system before the condensation6. ance (see, for example, Ref.13). The concept of a bound state of two particles in In this paper we shall prove in a model–independent mediumcanconsistentlybeformulatedwiththereduced mannerthattheexistenceofthecondensateofthebound density matrix of the second order (2–matrix)7. Indeed, pairstates(BCS–pairs)impliesthepresenceofthebound the system of two particles is a subsystem of that of N couples beyond the condensate (Schafroth’s pairs). Em- particles. So, its state is not pure even in the situa- phasize, that we do not specify the size of the pairs. If tion when all the system of interest has a wave function. it is much more than the mean distance between par- In general a subsystem is specified by the density ma- ticles (the condensate pairs in the BCS–model), then, 1 following Bogoliubov7, one may call these pairs ’quasi– TheboundaryconditionsforF216followfromtheprin- molecules’. If the radius of the bound particle couples is ciple of the correlation weakening at macroscopical sep- of the order of the mean distance between particles or, arations7: even, less (the Schafroth–Alexandrov–Mott approach), then one may speak about an ordinary molecules. The ψ†(x1)ψ†(x2)ψ(x′2)ψ(x′1) h i→ proofisbasedonthewell–known1/q2theoremofBogoli- ψ†(x1)ψ†(x2) ψ(x′2)ψ(x′1) (5) ubov for the Fermi system7 which is valid in the space– h ih i uniform case and under the condition of the local gauge when invariance of the interaction term of the system Hamil- tonian. r1 r2 =const, r′1 r′2 =const, r′1 r1 ; (6) − − | − |→∞ Thepresentarticleisorganizedasfollows. InsectionII theconceptofin–mediumwavefunctionsoffermionpairs ψ†(x1)ψ†(x2)ψ(x′2)ψ(x′1) is considered. The properties of the pair condensate are h i→ discussedinthethirdsection. Atlast,theproofconcern- hψ†(x1)ψ(x′1)ihψ†(x2)ψ(x′2)i (7) ingthe noncondensedboundpairsoffermionsis givenin when section IV of the paper. r1 r′1 =const, r2 r′2 =const, r1 r2 . (8) − − | − |→∞ II. THE CONCEPT OF PAIR WAVE FUNCTIONS Asthekernel(4)isanon–negativeHermitianoperator FOR FERMIONS acting on the two–particle wave functions ψ(x1,x2), we canexpanditintheorthonormalsetofitseigenfunctions Thus, let us consider a homogeneous Fermi system of (EF): N particles with the spin s = 1/2 at nonzero tempera- tures. Suppose thatthe totalmomentumandspinofthe F2(x1,x2;x′1,x′2)= Nνψν∗(x1,x2)ψν(x′1,x′2), (9) system are conserved quantities. Let the forces exerted ν X by fermions on each other be described with the two– where particle interaction potential depending on the relative distance between them and, may be, on the spin vari- 2 ables like in the case of various effective Hamiltonians. dx1dx2|ψν(x1,x2)| =1. (10) A state of the whole system is specified by the density Z matrix corresponding to the canonical Gibbs ensemble: Eigenfunctions ψν(x1,x2), which at the same time are EF of 2–matrix (2), are called the pair wave functions, H H or PWF. ρ=exp Trexp , (1) (cid:18)−kBT(cid:19)(cid:30) (cid:18)−kBT(cid:19) With (4), (9) and (10) one can be convinced that b b where Hbis the system Hamiltonian14. In this case the dx1dx2F2(x1,x2;x1,x2)= N2 N 15 h − i 2–matrix is representedin the form (see, for example, ) Z b =Nb(N b1)= Nν. ρ2(x′1,x′2;x1,x2) − ν X 1 = ψ†(x1)ψ†(x2)ψ(x′2)ψ(x′1) , (2) Therefore,the non–negativequantityNν canbe inter- N(N 1)h i − pretedasthemeannumberofthepairsinthestateν,any pairbeingdoublytaken. Theratiow =N / N(N 1) where = Tr( ρ) stands for the average over the ν ν { − } h···i ··· is the probability of observinga particle pair in the pure state (1); x = (r,σ) represents the space coordinates r and spin z–projection σ = 1/2; ψ†(x), ψ(x) are the state with the wave function ψν(x1,x2). Here, as one field Fermi operators.bThe 2±–matrix obeys the normal- might expect, νwν =1. It follows from the definition (4) that ization condition P F2(x1,x2;x′1,x′2)= F2(x1,x2;x′2,x′1)= dx1dx2ρ2(x1,x2;x1,x2)=1, (3) =−F2(x2,x1;x′1,x′2). Z − here dx ··· = σ d3r ··· and integration is fulfilled So, ψν(x1,x2) = −ψν(x2,x1), i.e. PWF for fermions, as over the volume V. Therefore, the 2–matrix (2) has the usual,areantisymmetricwithrespecttopermutationsof asymRptotic behaPviouRr 1/V2 when V , n = N/V = particles. const. So, it is more convenient to →dea∞l with the pair Inanequilibriumstatethetotalpairmomentumh¯qis correlation function F2 differing by a norm from ρ2: agoodquantumnumberforPWFprovidedthatthetotal momentum of the whole system is a conserving quantity F2(x1,x2;x′1,x′2)= ψ†(x1)ψ†(x2)ψ(x′2)ψ(x′1) . (4) (see proof in Ref.17). The same is correct for the total h i 2 spinSofaparticlepairifthereisnomagneticordering18. Now, with the variables So,theindexν canberepresentedasν =(ω,q,S),where ω stands for other quantum numbers. As to the PWF, R=(r1+r2)/2, r=r1 r2 (18) − they can be written as and,respectively,R′andr′,theexpression(12)isrewrit- ψν(x1,x2) ten as =ψω,q,S(r1−r2,σ1,σ2)exp{iq(√r1V+r2)/2}. (11) F2(x1,x2;x′1,x′2)= NqV,S,iϕ∗q,S,i(r,σ1,σ2) q,S,i X Then expression (9) has the form ϕ (r′,σ′,σ′)exp iq(R′ R) q,S,i 1 2 × { − } N F2(x1,x2;x′1,x′2)=ω,q,S NωV,q,Sψω∗,q,S(r1−r2,σ1,σ2) +p,qX,S,mS p,Vq,S2,mSϕ∗p,q,S,mS(r,σ1,σ2) ψω,q,S(r1 r2,σ1,Xσ2)exp iq(r′1+r′2 r1 r2) . (12) ×ϕp,q,S,mS(r′,σ1′,σ2′)exp{iq(R′−R)}. (19) × − 2 − − Inthethermodynamiclimitallthesummationsovermo- n o For the wave function ψω,q,S(r,σ1,σ2) which can be menta can be replaced by the corresponding integrals: interpreted as the wave function of a particle pair in the center–of–mass system, from (10) and (11) we obtain F2(x1,x2;x′1,x′2)= d3qwS,i(q)ϕ∗q,S,i(r,σ1,σ2) S,i Z X d3r|ψω,q,S(r,σ1,σ2)|2 =1. (13) ×ϕq,S,i(r′,σ1′,σ2′)exp{iq(R′−R)} σX1,σ2VZ + d3pd3qwS,mS(p,q)ϕ∗p,q,S,mS(r,σ1,σ2) Itcanberelatedtoeitherdiscreteorcontinuousspectra. SX,mSZ In the former case ϕ (r′,σ′,σ′)exp iq(R′ R) . (20) × p,q,S,mS 1 2 { − } ψω,q,S(r,σ1,σ2) 0 (14) Thus, from equations (19) and (20) we can see that → V w (q)d3q is the number of the bound particle pairs when r and, so, we deal with the sector of bound S,i → ∞ with the spin S, in the state i and with the total couple states of particle pairs. The latter variant implies momentum ¯hq located in the infinitesimal volume d3q. ψω,q,S(r,σ1,σ2)→χS,mS(σ1,σ2)√2 csoins((pprr)) (15) Rbeerspoefctthivee’ldyi,sVso2ciwaSte,md’Sp(pa,rtqi)cdle3ppda3irqssintatnhdessftoarteth(eS,nmumS)- (cid:26) withtherelativemomentumh¯pandtotalmomentumh¯q for r . This is a ’dissociated’, or scattering, pair located in the infinitely small volumes d3p and d3q. state c→orr∞esponding to the relative motion with the mo- In the center–of–mass system the replacement p → mentum ¯hp. Here χS,mS(σ1,σ2) is the spin part of the −p, σ1 → σ2, σ2 → σ1 corresponds to the permutation pairwavefunction(spinor),m beingthez–projectionof of particles. So, the following symmetric relations take S the total pair spin S. When S = 0, m = 0 (the singlet place: S state)oneshouldtakecos(pr). ForS =1,m = 1,0,1 S (the triplet state) one should use sin(pr). Rema−rk that wS,mS(p,q)=wS,mS(−p,q), (21) in the situation when the fermion interaction does not dPeWpeFndcaonnbsepisnepvaarraiatbedlesf,rotmheosnpeinaannodthsepranceotpoanrtlsyowfhtehne ϕp,q,S,mS(r,σ1,σ2)= −ϕp,q,S,mS(−r,σ2,σ1) r but also for any r. = −ϕ−p,q,S,mS(r,σ2,σ1). (22) →∞ In the case of (14) ω = i, where i stands for the dis- As an example, let us consider the expansion of F2 in crete index enumerating the bound pair states. Let us terms of PWF for the BCS–model. Taken with an accu- denote ψω,q,S(r,σ1,σ2)=ϕq,S,i(r,σ1,σ2), so that racytotheasymptoticallysmallquantities,theHamilto- nianintheBCS–approachisrepresentedasthequadratic d3r ϕi,q,S(r,σ1,σ2)2 =1. (16) form of the Fermi operators19 that can be diagonalized | | σX1,σ2VZ with the Bogoliubov transformation. Therefore, one is able to use the theorem of Wick, Bloch and De Domini- Inthesituationof(15)ω =(p,mS). Hereitisconvenient cis20: Ftoroimntr(o1d3u)cietψfoωll,qo,wSs(rt,hσa1t,σ2)=ϕp,q,S,mS(r,σ1,σ2)/√V. F2(x1,x2;x′1,x′2)= ψ†(x1)ψ†(x2)ψ(x′2)ψ(x′1) h i V1 σX1,σ2VZ d3r|ϕp,q,S,mS(r,σ1,σ2)|2 =1. (17) ×=hhψψ††((xx21))ψψ(†x(′2x)2i)−ihhψψ(†x(′2x)1ψ)(ψx(′1x)′2i)+ihhψψ††((xx21))ψψ((xx′1′1))ii. (23) 3 Further, for the ’normal’ averages we have With (27), (29) and (30) one can easily be convinced that the normalization relations (16) and (17) are satis- hψ†(x1)ψ(x′1)i=hψ†(r1,σ1)ψ(r′1,σ1′)i fied. WithintheBCS–modelwS,i(q)=∆(S)∆(i)n0δ(q) d3k (δ(q)) is the δ-function), i.e. all the bound particle pairs = (2π3)n(k)exp{ik(r′1−r1)} ∆(σ1−σ1′), (24) are condensed. Z where n(k) = a† a gives the distribution of h k,σ k,σi fermions over momenta; and we introduced the function III. PROPERTIES OF THE CONDENSATE OF 0, σ =0, PAIRS ∆(σ)= 6 1, σ =0. (cid:26) ’Anomalous’ averagesare given by Letusdemonstrateinthemostgeneralcasethatifthe ’anomalous’average ψ(x1)ψ(x2) isnotequaltozero(off ψ(x1)ψ(x′1) = ψ(r1,σ1)ψ(r′1,σ1′) = diagonallong–rangehorder)thentihedistributionfunction h i h i d3k wS,i(q)acquirestheδ functionalsingularitycorrespond- = (2π)3hak,σ1a−k,−σ1iexp{ik(r′1−r1)}∆(σ1+σ1′). (25) ingtosomeindicesS0−andi0 or,inotherwords,theratio Z N /V inthe firstsumof(19)doesnotvanishinthe q,S0,i0 IntheBCS–model,thequantity a a canberep- thermodynamic limit: k,σ −k,−σ h i resented in the following form wS,i(q)=n0δ(q)∆(S S0)∆(i i0)+wS,i(q), (31) sign(σ) − − ak,σa−k,−σ =√n0 ϕ(k) , (26) h i √2 where wS,i(q) is the regular part of (3e1) giving the with ϕ(k) obeying the normalization condition bound–pair distribution over nonzero momenta. Todoethis,letustakethelimitrelation(5)andrewrite d3k 2 it with the variables (18) in the form ϕ(k) =1. (27) (2π)3 | | Z F2(x1,x2;x′1,x′2) ψ†(x1)ψ†(x2) Remarkthatonecanconsiderϕ(k)asarealquantitybe- →h i cause it can be made real with the corresponding phase ×hψ(x′2)ψ(x′1)i=n0ϕ∗(r,σ1,σ2)ϕ(r′,σ1′,σ2′), (32) transformation of the operators a and a†. Now, Eqs. (24),(25)and(26)allowus to rewkrite(23)kinthe follow- where the functions ϕ∗(r,σ1,σ2)and ϕ(r′,σ1′,σ2′) arein- ing form: troduced in sucha way that the normalizationcondition (16)shouldbefulfilled. Thiscanalwaysbedonebecause F2(x1,x2;x′1,x′2)=n0ϕ(r)χ0,0(σ1,σ2)ϕ(r′)χ0,0(σ1′,σ2′) according to the principle of correlation weakening7 d3pd3q q q + (2π)6 n 2 +p n 2 −p ϕp,S(r)χS,mS(σ1,σ2) hψ(x1)ψ(x2)i→hψ(x1)ihψ(x2)i=0 SX,mSZ (cid:16) (cid:17) (cid:16) (cid:17) ×ϕp,S(r′)χS,mS(σ1′,σ2′)exp{iq(R′−R)}. (28) wcohnetnribru→tio∞n o(fseteheRefifr.s2t1)s.inEgxuplarersstieornm(3o2f)(3is1)exinactotly(2t0h)e. Here ϕ(r) is the Fourier transform of ϕ(k), for ϕ (r) The contribution of the regular part of (31) and that of p,S we have the ’dissociated’ pair states into (20) are infinitely small in the situation of (6) according to the Riemann’s theo- ϕ (r)= √2cos(pr), S =0, (29) rem22 because p,S √2sin(pr), S =1. (cid:26) Respectively, the spinor χ stands for R′ R = r′1+r′2 r1+r2 . | − | 2 − 2 →∞ (cid:12) (cid:12) χS,mS(σ1,σ2) Remark that the p(cid:12)(cid:12)air distribution ov(cid:12)(cid:12)er the scattering ∆(σ1+σ2)sign(σ1)/√2, S =0, mS =0; states(21)doesnotcontainδ functionalterms. Indeed, − =Θ(−σ1)Θ(−σ2), S =1, mS =−1; (30) ionftthheeoonpep–opsairtteicclaessetattheesyliwkeouinldthleeasditutoattihone ocofnthdeenBsoastee ∆(σ1+σ2)/√2, S =1, mS =0; liquid17 which is impossible for the Fermi systems. Θ(σ1)Θ(σ2), S =1, mS =1. Eq. (32) allows us to treat the ’anomalous’ averages Here asthe wavefunctions ofthe condensedpairsoffermions, (of course, with an accuracy to the normalizing factor). 1, σ 0, For the density of the pairs like these Eq.(16) and (32) Θ(σ)= ≥ 0, σ <0. gives (cid:26) 4 n0 = Nq=V0,S0,i0 = d3r|hψ(r,σ1)ψ(0,σ2)i|2 = (36In)wthitehstphaecien–euqnuifaolirtmyocfaCseawucehcya–nScrhewadairlzy–fiBnodgorleiulabtoiovn7 σX1,σ2Z d3k 2 AB 2 AA† B†B . = (2π)3|hak,σa−k,−σi| , (33) |h i| ≤h ih i Xσ Z Indeed, assuming Abb=ak,σ abndbB =b ab−k,−σ we arriveat where it has been taken into account that the total mo- a a 2 a a† a† a mentum of the system and z–component of its spin are |h k,σ −k,−σi| ≤h k,σ k,σih −k,−σ −k,−σi conserved quantities. Keeping in mind these integrals of =(1 n(k)) n(k). − the motion, one could expect that in the most general Then, from (33) we derive case the wave function of the condensed pairs should be written as N0 1 2 n0 = = ak,σa−k,−σ 1 V V |h i| ϕ(r,σ1,σ2)= ψ(r1,σ1)ψ(r2,σ2) Xk,σ √n0h i 2 2 N 2 n(k) n (k) n(k)= =n. d3k ∆(σ1+σ2) ≤ V − ≤ V V = (2π)3hak,σ1a−k,−σ1i √n0 exp(ikr) Xk (cid:0) (cid:1) Xk Z It is interesting to note that n(k) n2(k) = = √1n0∆(σ√1+2 σ2)(ϕs(r)sign(σ1)+ϕt(r)), (34) ht(hae†km,σaeakn,σ)s2qiu−arhea†kd,eσvaika,tσiio2n=ofDthne(okc)cuispantoiot−hninngumelsbeerbuotf (cid:0) (cid:1) where ϕ (r) = ϕ ( r) and ϕ (r) = ϕ ( r). Accord- the (k,σ) one–particle state. So, the stronger inequality s s t t − − − ing to Eq. (30) the first term in (34) corresponds to the 2 singlet and the second, to the triplet components of the n0 D n(k) (37) ≤ V wave function of the condensed fermions. However, (34) k X (cid:0) (cid:1) is notquite correctbecause the totalpair spin shouldbe demonstrates that the number of the condensed pairs is an integral of the motion, even in the situation with the tightly connected with the ’wash–out’ of the Fermi sur- spin–dependentinteractionbetweenfermions. Therefore, face. In the BCS–model at zero temperature we are not able to obtain a superposition of the singlet and triplet states. Instead, in (34) one should select ei- n0 kBTc 1 ther ϕs(r) = 0, ϕt(r) = 0 or ϕs(r) = 0, ϕt(r) = 0. So, n ∝ EF ≪ 6 6 we have: because the bound pairs are formed by the particles lo- ϕs(r)χ0,0(σ1,σ2)/√n0, cated near the Fermi surface only. In general, n0 is the ϕ(r,σ1,σ2)= (35) most ’reliable’ order parameter of the superconducting ( ϕt(r)χ1,0(σ1,σ2)/√n0. phase transition. The phase coherence takes place for the condensed bound pairs due to the uncertainty relation∆ϕ∆N0 ≃1 IV. THE BOGOLIUBOV 1/Q2 THEOREM AND for the phase ϕ and number of the bound fermion pairs BOUND PAIR STATES BEYOND THE N0 = Nq=0,S0,i0 in the state (q = 0,S0,i0). In the CONDENSATE thermodynamic limit the macroscopical occupation of this state results in ∆N0 √N0 and, therefore, ∝ → ∞ Let us now prove with the principle of the correlation ∆ϕ 0. For the bound pair states beyond the conden- → weakeningthatthe distributionofthe particlepairsover sate N is limitedaboveevenforV . Thusthese states aq,rSe,inot correlated with respect t→o t∞he phase. the ’scattering’ states wS,mS(p,q) is expressed in terms of the occupation numbers of the one–particle states Remark that the total number of the bound particle n(k) = a† a . Indeed, on the one hand, in the lim- pairs (condensed and not) h k,σ k,σi iting situation of (8) we have the relation (7), which can N =V d3qw (q) be written as b S,i XS,i Z F2(x1,x2;x′1,x′2) → pisarptriocpuloarrt,iotnhaelretoistthheetionteaqlunaulimtybfeorrotfhepanrutmiclbeesrNof.thIne (d23πp)13n(p1)exp(ip1(r′1−r1))∆(σ1−σ1′) condensed bound pair states23 Z N0 ≤N. (36) × (d23πp)23n(p2)exp(ip2(r′2−r2))∆(σ2−σ2′) Z It should be emphasized that the inequality (36) is not n(q/2+p)n(q/2 p) trivial. Onecanconsider,forexample,adilutegasofm– = d3qd3p − exp(ip(r′ r)) (2π)6 − particle molecules. In this case we have Nb =(m 1)N, Z thus, one can obtain Nb >N provided m 3. − exp(iq(R′ R))∆(σ1 σ1′)∆(σ2 σ2′), (38) ≥ × − − − 5 where, passing to the last equality, we introduced the Inthiscasethe1/q2theoremofBogoliubovfortheFermi newvariablesq=p1+p2 andp=(p1 p2)/2andused systems7 isvalidwhichassertsthatinthepresenceofthe − notations (18). On the other hand, when (8) is true, we pair condensate we have the inequality for sufficiently have small q r =|r2−r1|→∞, r′ =|r′2−r′1|→∞, mω,aSxNω,q,S ≥ qC2, r+r′ , R′ R=const, r′ r=const. where Nω,q,S appears in Eq. (12) and ω is the set of the | |→∞ − − quantum numbers correspondingto both the continuous spectrum (ω = (p,m )) and the discrete one (ω = i)25. Therefore,it follows from (14), (15) and (20) that in the S However,Eq. (42) results in limiting case (8) we have N 1 F2(x1,x2;x′1,x′2) p,q,S,mS ≤ → d3qd3pwS,mS(p,q)ϕp,S(r)ϕp,S(r′) obneclyaupsoessnib(kil)ity≤a1twfohricfhertmheiosnins.guTlahreitryef1o/req,2wapepheaavresdtuhee SX,mSZ to the noncondensedbound pairs. It is reasonableto ex- ×χS,mS(σ1,σ2)χS,mS(σ1′,σ2′)exp(iq(R′−R)), (39) pect that these pairs have the quantum numbers of the condensate couples S0,i0: where we used notations (29) and (30). Further, the Riemann’s theorem22 used while integrating over p and C′ w (q) . (44) relation (21) allow us to rewrite (39) as S0,i0 ≥ q2 F2(x1,x2;x′1,x′2) The BCS–modeel is not locally gauge invariant which → results in absence of the noncondensed bound pairs: d3qd3p wS,mS(p,q)χS,mS(σ1,σ2) wS,i(q) = 0. It is important to note in this connection Z SX,mS that the bound pair states beyond the condensate may χ (σ′,σ′)exp(ip(r′ r))exp(iq(R′ R)). (40) pelay a noticeable role in calculating the gauge–invariant × S,mS 1 2 − − response of the system to the electromagnetic fields. We have proved that the noncondensed bound pairs The right–hand side of Eq. (38) is equal to that of (40) coexist with the condensed ones at T <T . So, any the- at all the values of the spin variables and space ones c ˜r = r′ r and R˜ = R′ R. Taking into account the ory ignoring the noncondensed bound pairs of fermions − − is not fully consistent. Remark that the distribution of completeness of the set of the spin functions (30) the bound fermion pairs over the center–of–mass mo- χS,mS(σ1,σ2)χS,mS(σ1′,σ2′)=∆(σ1−σ1′)∆(σ2−σ2′), mReefn.7ta).oTbehyesdtihsetriibnueqtiuoanlitoyf (t4h4e)pwairtthiclCes′ o∝vekrBmTonm0e(nsetae SX,mS in the Bose gas w(q) = n(q)/(2π)3 answers, at small q, we derive the following equality: the similar relation w(q) C′′/q2 with C′′ kBTn0 ≥ ∝ 7 (here n0 denotes the density of the condensed bosons) . n(q/2+p)n(q/2 p) Therefore, there are fundamental parallels between the wS,mS(p,q)= (2π)6 − . (41) Bose gas and the considered subsystem of the fermion boundpairs. Andthese parallelsarenotonlyreducedto Thus, in the thermodynamic limit one can write agreement between the fermion–pair statistics and the Bose one. Following this analogy, we can expect that Np,q,S,mS =n(q/2+p)n(q/2−p). (42) the bound fermion pairs exist even at T > Tc (appar- ently, in some temperature interval T < T < T∗, in c As it is seen, when there is no magnetic ordering (it is spiteofthedisappearanceofthe1/q2–singularity). Thus, obviously true for the superconducting phase), the func- it looks as if any superconducting phase transition is a tion of the pair distribution over the ’dissociated’ states particular case of the Bose–Einstein condensation. This is independent of the quantum numbers S,mS. conclusion can be of interest in the context of the dis- It is now easy to prove that the pair condensate must cussion concerning different approaches of investigating always be accompanied by the presence of the noncon- the high–T superconductivity (see Refs.26,27). Remark c densed fermion pairs: wS,i(q) =0 in (31) if n0 =0. Let that possible experimental consequencesof the existence 6 6 theinteractionenergyofthesystembeinvariantwithre- of fermion bound pairs beyond the condensate can be specttothelocalgaugeetransformationofthefieldFermi found in paper12 in the case of neutral Fermi systems. operators24 The space–uniformcharacterofthe Fermisystemis of use inthe proofgivenabove. Electronsin the crystalline ψ(r,σ) ψ(r,σ) exp(iχ(r)), field, of course, can not be treated on the same level. → ψ†(r,σ) ψ†(r,σ) exp( iχ(r)). (43) However, for q 0 (large wave lengths) a crystalline → − → 6 lattice can be considered as continuum. Therefore, the derived result remains correct in this case. Emphasize that the bound pair states can fully be a result of the collective effects. Indeed, as it was demon- strated by Cooper28, an arbitrary small attraction be- 1M. Randeria, E-print cond-mat/9710223 (unpublished). tween electrons leads to forming the condensate of the 2P. W.Anderson, Phys.World 8, 37 (1995). bound electron pairs. Hence, if we considered a suffi- 3N.F. Mott, Phys.World 9, 16 (1996). cientlyshallowwellasthetwo–fermioninteractionpoten- 4G.Baskaron,Z.ZouandP.W.Anderson,SolidSt.Comm. tial, we would observe formation of the condensed and, 63, 973 (1987). according to the obtained result, noncondensed pairs at 5A. S. Alexandrov and N. F. Mott, Rep. Progr. Phys. 57, low temperatures. However, the well can be chosen in 1197 (1994); A. S. Alexandrov, V. V. Kabanov and N. F. such a way as to prevent the bound states of two ’bare’ Mott, Phys. Rev.Lett. 77, 4796 (1996). fermionsfromappearingwithintheordinarytwo–particle 6J.M.Blatt,TheoryofSuperconductivity(NewYork–Lon- problem. don,Acad. Press, 1964). At last, it is important to make one more remark on 7N. N. Bogoliubov, Quasi–averages, preprint D–781, JINR, the connection between the 1/q2 theorem of Bogoliubov Dubna (1961) [English transl. N. N. Bogoliubov, Lectures and the Goldstone theorem29. As it has been demon- on Quantum Statistics, vol. 2 (New York, Gordon and strated in Ref.7, existence of the Goldstone mode in the Breach, 1970), p.1]. Bose system results from the Bogoliubov theorem pro- 8L. D. Landau and E. M. Lifshitz, Course of Theoreti- videdthe massoperatorΣ(ω,k)isregularinthe vicinity cal Physics, vol. 3, Quantum Mechanics – Non–relativistic of the point ω = 0, k = 0. Let us emphasize that there Theory (NewYork,Pergamon Press, 1977), §14. 9It is more correct to speak about the bound pair states aresituationswhentheBogoliubovtheoremisvalidwhile ratherthanaboutboundpairs.Indeed,aparticlecanform it is not the case for the Goldstone one. For example, in a bound state together with M(M ≥1) particles, while it the case of neutral weakly interacting Bose gas the con- forms ’dissociated’ states with theother N−M particles. dition mentioned above for the mass operator is correct, 10Strictly speaking, this is valid for the s–wave pairing. In and the Goldstone mode exists. On the contrary, for general case the gap in the single–particle spectrum be- the charged Bose gas the mass operator is not regular comesk–dependent,oneusuallysupposethat∆=∆(k)is at k = 0, and, thus, there is no Goldstone mode. The proportional to ’anomalous’ averages ha−k,αak,βi. We as- similar situation is realized for the Fermi systems (see, sociatethe’anomalous’ averageswiththewavefunctionof e.g. Ref.30). pairs in the condensate (see Eq.(34)). As to the exact re- lation between the binding energy εb and the gap ∆(k) in generalcase,itisrathercomplicatedquestion.However,at V. CONCLUSION any rate, thepresence of thegap implies that εb 6=0. 11The continuousdistribution can only be introduced in the thermodynamiclimit N/V =const, V →∞. 12A. J. Leggett, in Modern Trends in the Theory of Con- Concluding,letus take noticeofthe mainresultsonce densed Matter, ed. by A. Pekalski and J. Przystawa more. Thereduceddensitymatrixofthesecondorderisa (Springer–Verlag, Berlin, 1980). fundamentalcharacteristicofamany–particlesystem,its 13J. R. Schrieffer, Theory of Superconductivity (New York, eigenfunctions being the pure states of two particles se- Benjamin, 1964). lectedinanarbitraryway. Appearanceofthecondensate 14ToconsidertheGibbsgrandcanonical ensembleitis suffi- oftheboundpairstates(33)impliestheoccurrenceofthe cient to replace H byH−µN. δ functional term in the distribution of the bound pairs 15N. N. Bogoliubov, Lectures on Quantum Statistics, vol. 1 − over the momentum of the couple center of mass q (see (NewYork,Gordbon anbd Breabch, 1967) p. 39. Eq. (31)). Using the space homogeneity of the system 16Afterthe thermodynamiclimit V →∞. and the local gauge invariance (43) of the fermion inter- 17A. Yu. Cherny, E-print cond-mat/9807120, submitted to action, we have proved that there is the 1/q2-singularity Phys.Rev.A. inthedistributionfunctionwS,i(q)providedthatn0 =0. 18Of course, there is no magnetic ordering in the supercon- Thus,werefinedthe1/q2 theoremofBogoliubov,hav6 ing ductingstate. proved the singularity to appear in w (q). Therefore, 19N. N. Bogoliubov, preprint P–511, JINR, Dubna (1960) e S,i presence of the noncondensed bound pairs below T is [English transl. in Ref.7, p.76]. c the necessary condition of superconductivity. 20C. Bloch and C. De Dominicis, Nucl.Phys. 7, 459 (1958). e 21Eq. hψ(x)i=6 0 implies that there is the condensate of the AnewsimpleproofoftheYanginequalityfortheFermi one–particle states which is impossible due to the Pauli systems(36)andits strongervariant(37)havealsobeen principle. derived as results of secondary importance. 22The Riemann’s theorem asserts that for any regular func- This workwassupportedby the RFBRGrantNo. 97- 02-16705. Discussions with V. V. Kabanov and V. B. Priezzhev are gratefully acknowledged. 7 tion f(q) (i.e. f(q) does not contain δ–function) we have 25Bogoliubov proved the theorem in the particular case of the s–wave pairing, i.e. when in (35) ϕs(r) = ϕ(r), where lim d3qf(q)exp(iqr)=0, ϕ(r) is radially symmetric function. The proof can easily r→∞ beextended to themore general case. Z provided theintegral d3qf(q)exp(iqr) exists. 26B. K. Chakraverty, J. Ranninger, D. Feinberg, Phys. Rev. 23C. N. Yang,Rev. Mod. Phys. 34, 694 (1962). Lett. 81, 433 (1998). 24EmphasizethatsomeRotherfieldscanalsobeincludedinto 27A.S. Alexandrov,E-print cond-mat/9807185. 28L. N.Cooper, Phys. Rev. 104, 1189 (1956). the Hamiltonian, for example, the phonon one. Thus, the 29J. Goldstone, NuovoCim. 19, 154 (1961). Fr¨ohlichmodelisinvariantwithrespecttothetransforma- 30H.Wagner, Z. Phys. 195, 273 (1966). tion (43). 8