a r X i v : c o n d  m a t / 0 4 1 2 4 1 8 v 2 [ c o n d  m a t . s u p r  c o n ] 2 3 F e b 2 0 0 5
BEC, BCS AND BCSBOSE CROSSOVER THEORIESIN SUPERCONDUCTORS AND SUPERFLUIDS
M. de Llano,
a
F.J. Sevilla,
b,c
M.A. Sol´ıs
b
& J.J. Valencia
a,da
Instituto de Investigaciones en Materiales, UNAM, 04510 M´exico, DF, Mexico
b
Instituto de F´ısica, UNAM, 01000 M´exico, DF, 04510 Mexico
c
Consortium of the Americas for Interdisiplinary Science,University of New Mexico Albuquerque, NM 87131, USA
d
Universidad de la Ciudad de M´exico,San Lorenzo Tezonco, 09940 M´exico, DF, Mexico
1. INTRODUCTION
Though commonly unrecognized, a superconducting BCS condensate consists of equal numbers of twoelectron (2e) and twohole (2h) Cooper pairs (CPs). A
complete bosonfermion
(statistical)
model
(CBFM), however, is able to depart from thisperfect 2e/2hCP symmetry and yields [1] robustly higher
T
c
’s without abandoning electronphonon dynamics mimicked by the BCS/Cooper model interaction
V
k
,
k
′
which is a nonzero negative constant
−
V,
if and only if singleparticle energies
ǫ
k
,ǫ
k
′
are within an interval [max
{
0
,µ
−
¯
hω
D
}
,µ
+ ¯
hω
D
] where
µ
is the electron chemical potential and
ω
D
is the Debye frequency. The CBFM is “complete” only in thesense that 2hCPs are not ignored, and reduces to all the known statistical theories of superconductors (SCs), including the BCSBose “crossover” picture but goesconsiderably beyond it.Bosonfermion (BF) models of SCs as a BoseEinstein condensation (BEC) goback to the mid1950’s [25], predating even the BCSBogoliubov theory [68]. Although BCS theory only contemplates the presence of “Cooper correlations” of singleparticle states, BF models [25, 917] posit the existence of actual bosonic CPs. Indeed, CPs appear to be universally accepted as the single most important ingredientof SCs, whether conventional or “exotic” and whether of low or hightransitiontemperatures
T
c
. In spite of their centrality, however, they are poorly understood.The fundamental drawback of early [25] BF models, which took 2e bosons as analogous to diatomic molecules in a classical atommolecule gas mixture, is the notoriousabsence of an electron energy gap ∆(
T
). “Gapless” models cannot describe the superconducting state at all, although they are useful [16,17] in locating transition
temperatures if approached from above, i.e.,
T > T
c
. Even so, we are not aware of any calculations with the early BF models attempting to reproduce any empirical
T
c
values. The gap ﬁrst began to appear in later BF models [914]. With two [12, 13]exceptions, however, all BF models neglect the eﬀect of
hole
CPs accounted for on anequal footing with electron CPs, except the CBFM which consists of
both
bosonic CPspecies coexisting with unpaired electrons, in a
ternary
gas mixture. Unfortunately,no experiment has yet been performed, to our knowledge, that distinguishes betweenelectron and hole CPs.The “ordinary” CP problem [18] for two distinct interfermion interactions (the
δ
well [19, 20] or the Cooper/ BCS model [6, 18] interactions) neglects the eﬀect of 2hCPs treated on an equal footing with 2e [or, in general, twoparticle (2p)] CPs. Onthe other hand, Green’s functions [21] can naturally deal with hole propagation andthus treat both 2e and 2hCPs [22, 23]. In addition to the generalized CP problem, acrucial result [12, 13] is that the BCS condensate consists of
equal numbers of 2p and 2h CPs
. This was already evident, though widely ignored, from the perfect symmetryabout
ǫ
=
µ
of the wellknown Bogoliubov [24]
v
2
(
ǫ
) and
u
2
(
ǫ
) coeﬃcients, where
ǫ
is the electron energy.Here we show: a) how the crossover picture
T
c
s, deﬁned selfconsistently by
both
the gap and fermionnumber equations, requires unphysically large couplings(at least for the Cooper/BCS model interaction in SCs) to diﬀer signiﬁcantly fromthe
T
c
from ordinary BCS theory deﬁned
without
the number equation since herethe chemical potential is assumed equal to the Fermi energy; how although ignoringeither 2h
or
2eCPs in the CBFM b) one obtains the precise BCS gap equation forall temperatures
T
, but c) only
half
the
T
= 0 BCS condensation energy emerges.The gap equation gives ∆(
T
) as a function of coupling, from which
T
c
is found as thesolution of ∆(
T
c
) = 0. The condensation energy is simply related to the groundstateenergy of the manyfermion system, which in the case of BCS is a rigorous upperbound to the exact manybody value for the given Hamiltonian. Results (b) and (c)are also expected to hold for neutralfermion superﬂuids (SFs)—such as liquid
3
He[25, 26], neutron matter and trapped ultracold fermion atomic gases [2738]—wherethe pairforming twofermion interaction of course diﬀers from the Cooper/BCS onefor SCs.
2. THE COMPLETE BOSONFERMION MODEL
The CBFM [12, 13] is described in
d
dimensions by the Hamiltonian
H
=
H
0
+
H
int
. The unperturbed Hamiltonian
H
0
corresponds to a nonFermiliquid “normal”state, being an
ideal
(i.e., noninteracting) ternary gas mixture of unpaired fermionsand both types of CPs namely, 2e and 2h. It is
H
0
=
k
1
,
s
1
ǫ
k
1
a
+
k
1
,
s
1
a
k
1
,
s
1
+
K
E
+
(
K
)
b
+
K
b
K
−
K
E
−
(
K
)
c
+
K
c
K
where as before
K
≡
k
1
+
k
2
is the CP centerofmass momentum (CMM) wavevectorwhile
ǫ
k
1
≡
¯
h
2
k
21
/
2
m
are the singleelectron, and
E
±
(
K
) the 2e/2hCP
phenomenological,
energies. Here
a
+
k
1
,
s
1
(
a
k
1
,
s
1
) are creation (annihilation) operators for fermions
and similarly
b
+
K
(
b
K
) and
c
+
K
(
c
K
) for 2e and 2hCP bosons, respectively. TwoholeCPs are considered
distinct
and
kinematically independent
from 2eCPs.The interaction Hamiltonian
H
int
(simpliﬁed by dropping all
K
=
0
terms, asis done in BCS theory in the
full
Hamiltonian but kept in the CBFM in
H
0
) consistsof four distinct BF interaction vertices each with twofermion/oneboson creationand/or annihilation operators. The vertices depict how unpaired electrons (subindex+) [or holes (subindex
−
)] combine to form the 2e (and 2h) CPs assumed in the
d
dimensional system of size
L
, namely
H
int
=
L
−
d/
2
k
f
+
(
k
)
{
a
+
k
,
↑
a
+
−
k
,
↓
b
0
+
a
−
k
,
↓
a
k
,
↑
b
+
0
}
+
L
−
d/
2
k
f
−
(
k
)
{
a
+
k
,
↑
a
+
−
k
,
↓
c
+
0
+
a
−
k
,
↓
a
k
,
↑
c
0
}
(1)where
k
≡
12
(
k
1
−
k
2
) is the relative wavevector of a CP. The interaction vertexform factors
f
±
(
k
) in (1) are essentially the Fourier transforms of the 2e and 2hCPintrinsic wavefunctions, respectively, in the relative coordinate of the two fermions.In Refs. [12, 13] they are taken as
f
±
(
ǫ
) =
f
if 12[
E
±
(0)
−
δε
]
< ǫ <
12[
E
±
(0) +
δε
]0 otherwise
.
(2)One then introduces the quantities
E
f
and
δε
as
new
phenomenological dynamicalenergy parameters (in addition to the positive BF vertex coupling parameter
f
) thatreplace the previous
E
±
(0) parameters, through the deﬁnitions
E
f
≡
14[
E
+
(0) +
E
−
(0)] and
δε
≡
12[
E
+
(0)
−
E
−
(0)] (3)where
E
±
(0) are the (empirically
un
known) zeroCMM energies of the 2e and 2hCPs, respectively. Alternately, one has the two relations
E
±
(0) = 2
E
f
±
δε.
(4)The quantity
E
f
serves as a convenient energy scale; it is not to be confused with theFermi energy
E
F
=
12
mv
2
F
≡
k
B
T
F
where
T
F
is the Fermi temperature. The Fermienergy
E
F
equals
π
¯
h
2
n/m
in 2D and (¯
h
2
/
2
m
)(3
π
2
n
)
2
/
3
in 3D, with
n
the totalnumberdensity of chargecarrier electrons, while
E
f
is the same with
n
replaced by,say,
n
f
. The quantities
E
f
and
E
F
coincide
only
when perfect 2e/2hCP symmetryholds, i.e. when
n
=
n
f
.The grand potential Ω for the full
H
=
H
0
+
H
int
is then constructed viaΩ(
T,L
d
,µ,N
0
,M
0
) =
−
k
B
T
ln
Tr
e
−
β
(
H
−
µ
ˆ
N
)
(5)where “Tr” stands for “trace.” Following the Bogoliubov recipe [39], one sets
b
+
0
, b
0
equal to
√
N
0
and
c
+
0
,
c
0
equal to
√
M
0
in (1), where
N
0
is the
T
dependent number
of zeroCMM 2eCPs and
M
0
the same for 2hCPs. This allows
exact
diagonalization,through a Bogoliubov transformation, giving [40]Ω
L
d
=
∞
0
dǫN
(
ǫ
)[
ǫ
−
µ
−
E
(
ǫ
)]
−
2
k
B
T
∞
0
dǫN
(
ǫ
)ln
{
1 + exp[
−
βE
(
ǫ
)]
}
+[
E
+
(0)
−
2
µ
]
n
0
+
k
B
T
∞
0
+
dεM
(
ε
)ln
{
1
−
exp[
−
β
{
E
+
(0) +
ε
−
2
µ
}
]
}
+[2
µ
−
E
−
(0)]
m
0
+
k
B
T
∞
0
+
dεM
(
ε
)ln
{
1
−
exp[
−
β
{
2
µ
−
E
−
(0) +
ε
}
]
}
(6)where
N
(
ǫ
) and
M
(
ε
) are respectively the electronic and bosonic density of states,
E
(
ǫ
) =
(
ǫ
−
µ
)
2
+ ∆
2
(
ǫ
) where ∆(
ǫ
)
≡ √
n
0
f
+
(
ǫ
) +
√
m
0
f
−
(
ǫ
), with
n
0
(
T
)
≡
N
0
(
T
)
/L
d
and
m
0
(
T
)
≡
M
0
(
T
)
/L
d
being the 2eCP and 2hCP number densities,respectively, of BEcondensed bosons. Minimizing (6) with respect to
N
0
and
M
0
,while simultaneously ﬁxing the total number
N
of electrons by introducing the electron chemical potential
µ
, namely
∂
Ω
∂N
0
= 0
, ∂
Ω
∂M
0
= 0
,
and
∂
Ω
∂µ
=
−
N
(7)speciﬁes an
equilibrium state
of the system with volume
L
d
and temperature
T
. Here
N
evidently includes both paired and unpaired CP electrons. The diagonalizationof the CBFM
H
is
exact
, unlike with the BCS
H
, so that the CBFM goes beyondmeanﬁeld theory. Some algebra then leads [40] to the three coupled integral Eqs. (7)(9) of Ref. [12]. Selfconsistent (at worst, numerical) solution of these
three coupled equations
then yields the three thermodynamic variables of the CBFM
n
0
(
T,n,µ
)
, m
0
(
T,n,µ
)
,
and
µ
(
T,n
)
.
(8)Fig.1 displays the three BE condensed phases—labeled
s
+,
s
−
and
ss
—along withthe normal phase
n
, that emerge [13] from the CBFM.Vastly more general, the CBFM contains [1] the key equations of all
ﬁve
distinctstatistical theories as special cases; these range from BCS to BEC theories, which arethereby uniﬁed by the CBFM. Perfect 2e/2h CP symmetry signiﬁes equal numbers of 2e and 2hCPs, more speciﬁcally,
n
B
(
T
) =
m
B
(
T
)
as well as
n
0
(
T
) =
m
0
(
T
)
.
With(4) this implies that
E
f
coincides with
µ
, and the CBFM then reduces to the gap andnumber equations [viz., (11) and (12) below] of the
BCSBose crossover picture
withthe Cooper/BCS model interaction—if its parameters
V
and ¯
hω
D
are identiﬁed withthe BF interaction Hamiltonian
H
int
parameters
f
2
/
2
δε
and
δε
, respectively. Thecrossover picture for unknowns ∆(
T
) and
µ
(
T
) is now supplemented by the centralrelation∆(
T
) =
f
n
0
(
T
) =
f
m
0
(
T
)
.
(9)Both ∆(
T
) and
n
0
(
T
) and
m
0
(
T
) are the familiar “halfbellshaped” orderparametercurves. These are zero above a certain critical temperature
T
c
, rising monotonicallyupon cooling (lowering
T
) to maximum values ∆(0)
, n
0
(0) and
m
0
(0) at
T
= 0
.
The energy gap ∆(
T
) is the order parameter describing the superconducting (or
S

SS
n
0
SS
S
+
n
m
0
Figure 1.
Illustration in the
n
0

m
0
plane of three CBFM condensed phases (thepure 2eCP
s
+ and 2hCP
s
−
BE condensate phases and a mixed phase
ss
) alongwith the normal (ternary BF nonFermiliquid) phase
n
.superﬂuid) condensed state, while
n
0
(
T
) and
m
0
(
T
) are the BEC order parametersdepicting the macroscopic occupation that arises below
T
c
in a BE condensate. This∆(
T
) is precisely the BCS energy gap if the bosonfermion coupling
f
is made tocorrespond to
√
2
V
¯
hω
D
. Note that the BCS and BE
T
c
s are the same. Writing (9)for
T
= 0, and dividing this into (9) gives the much simpler
f
independent relationinvolving order parameters
normalized
in the interaval [0
,
1]∆(
T
)
/
∆(0) =
n
0
(
T
)
/n
0
(0) =
m
0
(
T
)
/m
0
(0)
−−→
T
→
0
1
.
(10)The ﬁrst equality, apparently ﬁrst obtained in Ref. [9], simply relates the two heretofore unrelated “halfbellshaped” order parameters of the BCS and the BEC theories.The second equality [12, 13] implies that a BCS condensate is precisely a BE condensate of equal numbers of 2e and 2hCPs. Since (10) is
independent
of the particulartwofermion dynamics of the problem, it can be expected to hold for either SCs andSFs.
3. BCSBOSE CROSSOVER THEORY
The crossover theory (deﬁned by two simultaneous equations, the gap and number equations) was introduced by many authors beginning in 1967 with Friedel and