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Crossover from BCS-superconductivity to Bose-condensation.

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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
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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
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