Ann. Physik l(1992) 15-23 Annalen der Physik 0 Johann Ambrosius Barth 1992 Crossover from BCS-superconductivity to Bose-condensation M. Drechsler and W. Zwerger Institut fur Theoretische Physik, Universitiit Gottingen, W-3400 Gottingen, Germany Received 28 October 1991, accepted 14 November 1991 Abstract. Starting from a model of free Fermions in two dimensions with an arbitrary strong effective interaction, we derive a Ginzburg-Landau theory describing the crossover from BCS-superconductivity to Bose-condensation. We find a smooth crossover from the standard BCS-limit to a Gross-Pitaevski type equation for the order parameter in a Bose superfluid. The mean field transition temperature exhibits a maximum at a coupling strength, where the behaviour crosses over from BCS to Bose like with corresponding values of 2A,/Tc = 5 which are characteristic for high Tc superconductors. Keywords: Fermion Systems; Quantum fluids; Theory of superconductivity. 1 Introduction The standard microscopic model for the explanation of superconductivity is certainly the BCS-model in which a weak attractive interaction in an ideal Fermi-gas leads to a manybody state built from pairs of Fermions which are all described by the same two particle wave function. In conventional superconductors the coherence length to which determines the characteristic spread of the pair wave function is large compared with the average distance k~ between two Fermions. Thus the superconducting state consists of highly overlapping pairs rather than a Bose condensate of tightly bound molecules built from two Fermions. The extraordinary success with which the BCS-model has been applied to conventional superconductors relies on the fact that in the weak coupling limit the details of the attractive interaction are irrelevant and indeed by a proper rescaling of variables all weak coupling superconductors behave identically. With the discovery of high T, superconductors, however, it has become clear that such a description no longer applies. In particular, since in these systems superconductivity is obtained by adding carriers to antiferromagnetic insulators, the normal state above T, can hardly be modelled as an ideal Fermi-gas and an adequate description is still a matter of debate. Nevertheless in many respects the superconducting state in high T, materials appears to be rather conventional except for the fact that tois very small, leading to k F t Ovalues of order 10 or smaller. Assuming that the pairing concept still applies to these systems, this suggests that the pairs are much more tightly bound than in conventional superconductors and thus at least a weak coupling model is inadequate. In this work we discuss the crossover from weak to strong coupling superconductivity using a functional integral representation of interacting Fermions. The crucial advantage of this formulation is that it deals directly with the central quantity describing a superconducting state in its most general form, namely the condensate wave function or order parameter y.Indeed a non- ' 16 Ann. Physik 1 (1992) vanishing expectation value of t y signals the breaking of gauge invariance as the basic phenomenon of superconductivity. Our main results are the following - With increasing strength of the interaction the Ginzburg-Landau functional for U,I smoothly crosses over from the well known BCS-limit to a Gross-Pitaevski type functional, describing the condensation of tightly bound bosonic molecules to a superfluid. - In two dimensions the mean field transition temperature as a function of the coupling strength exhibits a maximum where the behaviour crosses over from BCS to Bose like. The corresponding value for 2A0/T, is close to 5 and thus in the range which is typically observed in high T, superconductors. We emphasize, however, that our aim is not to give a theory which is directly applicable to high T, materials but rather to give a general discussion based on a Ginzburg-Landau (GL) formulation of the crossover from a BCS-superconductor with very weakly bound pairs and large coherence length to a Bose superfluid consisting of tightly bound pairs and short coherence length. In this respect this work is a continuation of previous treatments of this problem by Nozitres and Schmitt-Rink [ I ] in three dimensions and by Randeria, Duan and Shieh [2] in two dimensions at T = 0, both of which however, used different methods than our functional integral approach. 2 Ginzburg-Landau theory We start with the Gorkov model of Fermions with an attractive short range pair interaction of strength g > 0. Here B: = C cl+qlC'kt is a creation operator for a pair with total k momentum q and Q the system volume (or area in two dimensions). In order to derive the associated GLfunctional we use the Hubbard-Stratonovich trick to formally linearize the interaction term (2). This leads to a representation of the grand partition function Z = Tr exp(-PH) in the form [3, 4, 51 lA(q, 7)l2] L[A] (3) with a functional which describes a Fermi system in a 'time' and space dependent potential. Although the Fermi operators occur only quadratically, this problem can only be solved perturbatively in A. This is justified near the transition point, where the pair potential A which is proportional to the anomalous average M. Drechsler, W. Zwerger, Crossover from BCS-superconductivity to Bose-condensation 17 is small. Up to fourth order in A one finds Z = Zo j D A exp(-Y[A]) with Z , the grand partition function for free Fermions and a GL-functional + /3!2 b A(l)A*(2)A(3)A*(l - 2 - 2 + 3) 1 . ,3 where 1 is an abbreviation for q , , om, etc. Here om = 2nm//?h, rn = 0, f 1, . . . are the Matsubara frequencies associated with the Fourier transformation of the s-dependence in the interval [0, /3h] and the coefficients a@, p, q, urn)and b are determined by the free thermal Greensfunctions Go(k, on)= (ion - l k / h ) - ' where on = n(2n + l)/Ph, n = 0, + I , . . ., via and Here in evaluating b, we have assumed a slowly varying pair potential A(x, r) thus allowing to evaluate b at q = o = 0. By contrast, in the quadratic term in A it is necessary to keep the lowest nontrivial dependence on q and omwhich is obtained by expanding a@,p,q,o,) = a + c- h2q2 - diho, 2rn + ... (9) The explicit calculation of the coefficients a, b, c and d is greatly simplified in the case of two dimensions where the density of states per spin is constant, i.e. 1 Q F m = N(0) d 0 m &k with N(0) = 2nh2 . Moreover for a dilute gas of Fermions with kF R e I , R being the range of the interaction, only s-wave scattering is important [6]. Thus it is possible to represent an arbitrary interaction potential by a single parameter which for convenience we choose as the binding energy Ebof a two particle bound state in vacuum [7]. The relation between E , and g as defined in (2) may then be expressed by Here + 03 is a high energy cutoff which cancels in the explicit evaluation of a using (7), leaving Eb as the finite physical quantity which characterizes the interaction. In this way after some algebra we obtain Ann. Physik 1 (1992) 18 a = - [ N(O) 2 ~n 2 + ln- ' I P 2 and (F) o + tanh- PP 2 (,u) + sgn(,u) In PEb 4 m 1 ~ PIP - In' dx] cosh2x 2 which are valid for p =k 0. In the particular case of p = 0 we have The corresponding results for the coefficient c and d are and As we will see below in the BCS-limit E b / & F -, 0 the chemical potential at the transition point ,uc is equal to the Fermi energy eF and Pc p c -+ 03. In this limit the relation between the formal auxiliary field A and the order parameter t y is usually fixed by requiring that the gradient term has the form h2/4m I V WI as for a wave function describing particles of mass 2m. This is achieved by defining In this manner the functional Y takes the form, written in position space and M. Drechsler, W. Zwerger, Crossover from BCS-superconductivity to Bose-condensation 19 where we have taken p and p at the transition point and pu,BCs = cF (see (27) below). For a r-independent order parameter ~ ( x )this agrees precisely with the standard form of the GLtheory as derived microscopically by Gorkov [8] except for an additional factor 2/3 which arises due to the two-dimensionality of the system (instead of three). In the Bose-limit E b / E F S 1 the chemical potential at the transition approaches p,Bose = -Eb/2 (see (23) below) and thus Pc p c - 03. In this case our GLfunctional describes an interacting gas of Bosons with an effective 2 m and an effective chemical potential p*. To see this we choose the mass m* order parameter -+ + such that the coefficient of the term proportional to w becomes unity at the transition point. In this manner the partition function takes the standard form of the coherent state functional integral representation of an interacting Bose-gas with a free energy functional [9] Here p* = - a/d, rn* = m d/c and g* = b/d2. The effective chemical potential p* vanishes at the mean field transition determined by a = 0 and obeys dp*/d,u I pc = 2 quite generally. In the extreme Bose-limit Pc p c - 00 we find rn* + 2rn as -+ expected and g * 2/N(O). Neglecting the z-dependence, minimization of the functional (21) leads to the Gross-Pitaevski equation [ 101 describing weakly interacting Bosons with mass rn* 2m. The repulsive interaction between two composite Bosons due to the fourth order term 1/2 g* I lyI4 is independent of the original interaction between two Fermions and depends only on the density of states N(0). Thus it reflects the statistical interaction due to the Pauli principle. + -+ 3 The mean-field transition point In mean-field approximation the transition to superconductivity occurs at a = 0 which leads to the standard gap equation. In the BCS-limit p, p, 03 this determines the transition temperature -+ while in the Bose limit Pc p c + - 03 the temperature drops out and the condition a only fixes the value of the chemical potential at the transition = 0 Ann. Physik 1 (1992) 20 An additional equation follows from n = (1 /pa)(a InZ/dp) which relates the chemical potential p and the Fermion density n. Introducing nf = 5[ p + 2 In k c o s h $11 2EF as the number of free Fermions arising from the trivial part Zo of the full partition function and neglecting the quartic term in the GLfunctional the number equation takes the form with p* and m* as defined above. Equation (25) may be interpreted in the following way [I], [ l I]: The total density is the sum of the contribution n f of free unbound Fermions plus that of paired but not necessarily yet condensed particles. The number of pairs is identical with the result for a system of noninteracting Bosons with mass m* and chemical potential p*. The Bose system is ideal since we have neglected terms of higher order than quadratic in Y [ w ] .Inclusion of the quartic term in Y derived above will therefore lead to a replacement of the second term in (25)by the density of the corresponding repulsively interacting Bosons. Now, in the case of two dimensions considered here, such a system is known to exhibit a transition to a superfluid with quasi long range order and finite superfluid density [ 121 below a critical temperature with n B the Boson density. Here we have used the relation J = (hZ/m*)nB between the Boson density nB and the coupling constant J of a two-dimensional XY-model [ 131 and the numerical result T, = 0,89 J [14] for the critical temperature of the latter model. In a very simplified manner we may therefore incorporate the effect of Boson interactions and the finiteness of T, in two dimensions by replacing the ideal Boson density in (25) by (lr/0,89) (n/P, E ~ (m*/m) ) where we have used that m/lrh2 = n / c F . In the BCS-limit p, p, D 1 we have m*/m a @, p c ) p 2--t 0 and thus nBwc,p,) 4 n f @ , , p , ) . This implies that n = n f and thus, using (24), we find In the Bose-limit p, p, + - 00 the free Fermion density (24) becomes exponentially small n f a exp(- p I p I /2) and thus the total density n approaches 2n,. Similarly for the effective Boson mass we obtain m* --* 2m and therefore the transition temperature T, is determined by the Bose relation (26) with nB = n / 2 and m* = 2m. In between these two limits the values of T, and p , at the transition may be obtained by a very simple numerical calculation. Inserting (24) and (26) into (2'5) the number equation at the transition point takes the form p, E~ I 2 = - p, p, + In 2n m* M. Drechsler, W. Zwerger, Crossover from BCS-superconductivity to Bose-condensation 21 with m*/m = d/c, determined by equation (1 3 ) and (1 4). Since (1 I ) and (28) both contain in the form j?, p, we proceed as follows: A value for j?, p, is given which determines TJEF from (28) and T J E b from a = 0. Considering the Fermi energy eF 0: n and the two particle binding energy Eb as fixed microscopic input parameters this allows to determine the transition temperature T, and the free Fermion density n f as a function of the dimensionless coupling strength Eb/&F.The corresponding results are shown in Figs. 1 and 2. Obviously the BCS-prediction (22) that TCBcs 0: EL'* remains valid even for rather large values of the coupling up to In = -2 where T, has a maximum but still nf is close to n. Beyond this point the transition temperature decreases and approaches the Bose-limit where T p e 0: nB = n becomes independent of the coupling strength. At the same time nf very rapidly drops to zero and the original Fermi system starts to behave like a collection of Bosons. A similar behaviour has been found in three dimensions by Nozieres and Schmitt-Rink [ l ] with, however, a much less pronounced maximum in T,. In two dimensions recent numerical simulations of the negative-U-Hubbard model [15] also give a maximum for T, as a function of I U 1, however in this case T, decreases like 1 U 1 with increasing coupling. A quantity of considerable interest, at least for not too large coupling, is the ratio 2Ao/T, between the gap A. at zero temperature and T,. Since our GL theory is valid only close to T, it cannot be used to determine Ao. However it has been shown in ref. [2]that within a BCS-Ansatz for the many body ground state wave function one obtains pc only -' for arbitrary strength of the coupling Eb. The identical result is obtained by replacing the Gorkov interaction ( 2 ) by the corresponding BCS interaction, which amounts to neglecting all terms with q + 0 in (2). The simple relation (29)can then be derived [I61 from an exact functional integral evaluation of the associated partition function at all temperatures following the method given by Miihlschlegel [ 171. Since T , / E is~ known as a function of Eb/&F use of the relation (29) now allows to determine 2Ao/T, as a function of the dimensionless coupling Eb/cF as shown in Fig. 3. Evidently the ratio remains close to the BCS value 2A0/T, = 27r/eY = 3.52 (which follows from (22), (27) Ij -l 0 -2 - 3 . 2 L ) . -6 -5 T -4 . , -3 , ~. -2 , . -1 , . , . 0 1 , 2 In(Eb/EF) Fig. 1 Transition temperature in units of t Fas a function of the dimensionless coupling strength E b / t Fin double logarithmic scale 0 . -6 -5 -4 0 -3 -2 -1 0 0 1 2 In(Eb/&F) Fig. 2 Free Fermion density in units of the total Fermion density as a function of the dimensionless coupling strength E b / t I ;in logarithmic scale 22 Ann. Physik 1 (1992) 51 I -6 1 -5 -4 -3 -2 -1 0 1 2 In(Eb/"F) Fig. 3 Gap at zero temperature in units of T J 2 as a function of the dimensionless coupling strength E , / c F in double logarithmic scale Fig. 4 Transition temperature in units of cF as a function of the ratio 2 A 0 / T , in double logarithmic scale and (29)) up to coupling strengths of order In Eb/eF= - 3 and then starts to increase very strongly at values of the coupling where the behaviour crosses over from BCS to Bose like according to Figs. 1 and 2. In the crossover region typical values of 2A0/T, are around 5 . The rapid increase in 2A0/Tc in the Bose like regime also signifies the qualitative change in the nature of the phase transition between the two extreme limits. In the BCS-limit the dissociation energy 2A0 is small and T, is determined by the breaking of pairs due to thermal fluctuations which leads to T,oc Ao. In the Bose limit where A. % T, almost all Fermions are strongly bound into Boson pairs and the thermal energy near the transition point is much too small for breaking them up, instead the Bosons take up kinetic energy and T, is determined by the excitation of collective modes. Nevertheless the crossover between the two limits is smooth. Finally in Fig. 4 we show the relation between the transition temperature T, and the ratio 2Ao/Tc which exhibits a maximum with TC/eF= 0.22 at 2Ao/T, = 4.6. 4 Conclusion To summarize we have given a description of the crossover between BCS-superconductivity and Bose-condensation in terms of a Ginzburg-Landau theory. Starting from the Gorkov model in two dimensions we have calculated the coefficients of the corresponding functional for arbitrary interaction strength. In the Bose-limit the GL-functional takes the form used by Gross and Pitaevski to study superfluidity in an interacting Bose system. Quantitatively it turns out that the mean field transition temperature exhibits a maximum with increasing coupling which occurs at values of 2A0/Tc around 5 . To which extent these results are relevant to high T , superconductors remains to be seen. However irrespective of specific details and the microscopic origin of the interaction we believe that our approach captures some of the essential features which are expected in any GL type description of superconductivity with a short coherence length. In particular it is evident that the coefficients of the GLfunctional are quite different from those used in the standard weak coupling limit. M. Drechsler, W. Zwerger, Crossover from BCS-superconductivity to Bose-condensation 23 References P. Nozieres, S. Schmitt-Rink, J. Low Temp. Phys. 59 (1985) 195 M. Randeria, J. M. Duan, L. Y. Shieh, Phys. Rev. Lett. 62 (1989) 981 and Phys. Rev. B41(1990) 327 T. M. Rice, J. Math. Phys. 8 (1967) 1581 U. Everts, Z. Phys. 199 (1967) 211 W. B. Strickfaden, Physica 70 (1973) 320 see also A. J. Leggett, J. de Phys. 41 (1980) C7 Indeed in two dimensions the existence of a bound state in the two particle problem is a necessary condition for a pairing instability as has been shown by K. Miyake, Progr. Theor. Phys. 69 (1983) 1794 [8] see for instance A. L. Fetter, J. D. Walecka, ‘Quantum Theory of Many-Particle Systems’, chpt. 53, Mc Craw Hill, New York 1971 [9] see L. S. Schulman ‘Techniques and Applications of Path Integration’ chpt. 21, Wiley, New York 1981 [lo] see chpt. 55 in ref. [8] [ I l l S. Schmitt-Rink, C. M. Varma, A. E. Ruckenstein, Phys. Rev. Lett. 63 (1989) 445 [12] J. M. Kosterlitz, D. J. Thouless, J. Phys. C6, (1973) 1181 [13] P. Minnhagen, Rev. Mod. Phys. 59 (1987) 1001 [I41 R. Gupta et al., Phys. Rev. Lett. 61 (1988) 1996 [15] P. J. M. Denteneer, G. An, J. M. J. van Leeuven, Europhys. Lett. 16 (1991) 5 [16] M. Drechsler, Diplomarbeit Universitllt Gottingen (1991), unpublished; in this work it is also shown that the BCS model is unable to describe the crossover between weak and strong coupling since it neglects the existence of collective modes. As a result the order parameter has no gradient term and T, would scale like E , in the strong coupling limit describing the - then irrelevant - break up of strongly bound pairs. [17] B. Muhlschlegel, J. Math. Phys. 3 (1962) 522 [l] [2] [3] [4] [5] [6] [7]

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