PHYSICAL REVIEW B 74, 144517 共2006兲
Two-band superfluidity from the BCS to the BEC limit
M. Iskin and C. A. R. Sá de Melo
School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
共Received 22 March 2006; revised manuscript received 18 July 2006; published 25 October 2006兲
We analyze the evolution of two-band superfluidity from the Bardeen-Cooper-Schrieffer 共BCS兲 to the BoseEinstein condensation 共BEC兲 limit. When the interband interaction is tuned from negative to positive values, a
quantum phase transition occurs from a 0-phase to a ␲-phase state, depending on the relative phase of the two
order parameters. Furthermore, population imbalances between the two bands can be created by tuning the
intraband or interband interactions. We also find two undamped low-energy collective excitations corresponding to in-phase and out-of-phase modes. Lastly, we derive the coupled Ginzburg-Landau equations, and show
that they reduce to coupled Gross-Pitaevskii equations for two types of bosons in the BEC limit.
DOI: 10.1103/PhysRevB.74.144517
PACS number共s兲: 74.20.Fg, 05.30.Fk, 03.75.Hh, 03.75.Ss
I. INTRODUCTION
The evolution from the Bardeen-Cooper-Schrieffer 共BCS兲
state to Bose-Einstein condensation 共BEC兲 is a very important topic of current research in the condensed matter,
nuclear, atomic, and molecular physics communities.1–7 In
atomic physics, the BCS to BEC evolution is studied via
Feshbach resonances8–17 in engineered one-band fermion
systems involving two hyperfine states of alkali-metal atoms
such as 6Li or 40K. However, two-band fermion systems may
also be produced experimentally with ultracold atomic Fermi
gases in optical lattices17 or in single traps of several hyperfine states. In this case, 共intraband and interband兲 interactions
may be tuned using Feshbach resonances which allow for the
study of BCS to BEC evolution of two-band superfluidity.
This evolution in the two-band problem is much richer than
the one-band case 共where a smooth crossover of physical
properties has been experimentally found8–13兲, since additional interaction parameters may be controlled externally.
An early two-band theory of superconductivity was introduced to allow for multiple band crossings at the Fermi
surface.18 This theory and its extensions are strictly limited
to the BCS regime, and they have been applied to MgB2,
where experimental properties can be well described by a
two-band BCS theory.19–22 Unfortunately, interband or intraband interactions cannot be tuned in MgB2, and presently its
physical properties cannot be studied away from the BCS
limit. However, new experimental techniques utilizing the
field effect23 may allow some tuning of the particle density
and thus indirectly the tuning of interactions. In contrast,
ultracold fermions with several hyperfine states may already
provide a unique opportunity to explore new phenomena in
two-band superfluids during the BCS to BEC evolution.
Thus, due to recent developments and advances in atomic
physics described above, and in anticipation of future experiments, we describe here the BCS to BEC evolution of twoband superfluids at zero and finite temperatures.
The main results of our paper are as follows. We show
that a quantum phase transition occurs from a 0-phase to a
␲-phase state depending on the relative phase of the two
order parameters, when the interband interaction is tuned
from negative to positive values. We found that population
imbalances between the two bands can be created by tuning
1098-0121/2006/74共14兲/144517共8兲
intraband or interband interactions. In addition, we describe
the evolution of two undamped low-energy collective excitations corresponding to in-phase phonon 共Goldstone兲 and
out-of-phase exciton 共finite frequency兲 modes. Near the critical temperature, we derive the coupled Ginzburg-Landau
共GL兲 equations, and show that they reduce to coupled GrossPitaevskii 共GP兲 equations for two types of bosons in the BEC
limit.
II. TWO-BAND MODEL
To obtain the results mentioned above, we start from a
two-band Hamiltonian describing continuum superfluids
with singlet pairing,
H=
兺 ␰n共k兲an,† ␴共k兲an,␴共k兲 − n,m,q
兺 Vnmb†n共q兲bm共q兲,
n,k,␴
共1兲
where 兵n , m其 = 兵1 , 2其 label different bands, and ␴ labels spins
†
共k兲 and b†n共q兲
共or pseudospins兲. The operators an,↑
*
†
†
= 兺k⌫n共k兲an,↑共k + q / 2兲an,↓共−k + q / 2兲 create a single and a
pair of fermions, respectively. The symmetry factor ⌫n共k兲
characterizes the chosen angular momentum channel, where
2
+ k2 is for the s-wave interaction in three
⌫n共k兲 = kn,0 / 冑kn,0
−1
dimensions. Here, kn,0 ⬃ Rn,0
sets the scale at small and large
momenta, where Rn,0 plays the role of the interaction range.
In addition ␰n共k兲 = ⑀n共k兲 − ␮n, where ⑀n共k兲 = ␧n,0 + k2 / 共2M n兲 is
the kinetic energy 共ប = 1兲 and M n is the effective band mass
of the fermions.
From now on, we focus on a two-band system with distinct intraband 共V11 , V22 ⬎ 0兲 and interband 共V12 = V21 = J兲 interactions. Notice that we allow pairing between states 兩1, ↑典
and 兩1, ↓典 or 兩2, ↑典 and 兩2, ↓典, and that pairing between 兩1, ↑典
and 兩2, ↓典 or 兩1, ↓典 and 兩2, ↑典 is not included. Here, 兩n , ␴典
represents a spin component with band index n and pseudospin ␴. Furthermore, notice that J plays the role of the
Josephson interaction which couples the two energy bands.
In addition, we assume that the total number of fermions N is
fixed such that ␮n = ␮, and that the reference energies are
such that ␧1,0 = 0 and ␧2,0 = ⑀D 艌 0. Here, ⑀D Ⰶ ⑀2,F where
2
⑀n,F = kn,F
/ 共2M n兲 is the Fermi energy 共⑀1,F 艌 ⑀2,F兲, and kn,F is
the Fermi momenta 关see Fig. 1共a兲兴. In addition, notice that
kn,0 Ⰷ kn,F in dilute systems.
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©2006 The American Physical Society
PHYSICAL REVIEW B 74, 144517 共2006兲
M. ISKIN AND C. A. R. SÁ DE MELO
particles and ⌬n共k兲 = ⌬n,0⌫n共k兲 is the order parameter. Here,
the matrix elements of g are associated with the inverse
interaction matrix V, and are given by g11 = −V22 / det V, g22
= −V11 / det V, and g12 = g21 = J / det V with det V = V11V22
− J2 ⬎ 0.
The action leads to the thermodynamic potential ⍀Gauss
= ⍀0 + ⍀fluct, where ⍀0 = S0 / ␤ is the saddle point and ⍀fluct
= 共1 / ␤兲兺qln det关F−1共q兲 / 共2␤兲兴 is the fluctuation contribution
to ⍀Gauss. Expressing ⌬n,0 in terms of its amplitude and phase
⌬n,0 = 兩⌬n,0兩exp共i␸n兲
共3兲
shows explicitly the Josephson coupling energy
− g11兩⌬1,0兩2 − g22兩⌬2,0兩2 − 2g12兩⌬1,0⌬2,0兩cos共␸2 − ␸1兲
of ⍀0. When J ⬎ 0, only the 0-phase 共or in-phase兲 ␸2 = ␸1
solution is stable. However, when J ⬍ 0, only the ␲-phase 共or
out-of-phase兲 ␸2 = ␸1 + ␲ solution is stable. Thus, a phase
transition occurs from the 0 phase to the ␲ phase when the
sign of J is tuned from negative to positive values as shown
in Fig. 1共b兲.
FIG. 1. 共Color online兲 Schematic 共a兲 figure of two bands with
reference energies ␧1,0 = 0 and ␧2,0 = ⑀D, and 共b兲 phase diagram of
0-phase and ␲-phase states.
We would like to emphasize that the model Hamiltonian
described in Eq. 共1兲 may be applicable to atomic and condensed matter systems. The case of identical bands 共M 1
= M 2 = M兲 corresponds physically to a situation that may be
encountered in atomic systems when four hyperfine states of
a given type of atom 共labeled as 兩1, ↑典; 兩1, ↓典; 兩2, ↑典; 兩2, ↓典兲
are trapped. In this case, the Josephson terms are responsible
for transferring pairs of atoms between states 兩1, ↑; 1, ↓典 and
兩2, ↑; 2, ↓典, and thus mixing different bands. Notice that the
case of nonidentical bands 共M 1 ⫽ M 2兲 is not applicable to
mixtures of two atomic species 共e.g., 6Li and 40K兲, because
the Josephson term would convert Li pairs into K pairs,
which is not physically allowed. However, the case of nonidentical bands may be applicable to standard condensed
matter systems such as MgB2, since electrons may have different effective masses in different bands, and the Josephson
term is physically allowed. In Sec. V, we discuss a generalized two-band model that includes the case of nonidentical
masses for atomic systems.
B. Self-consistency equations
From the stationary condition ⳵S0 / ⳵⌬*n共q兲 = 0, we obtain
the order parameter equation
冉
S0 = 兺 兵␤关␰n共k兲 − En共k兲兴 − 2 ln关1 + e−␤En共k兲兴其
n,k
*
⌬m,0 ,
− ␤ 兺 gnm⌬n,0
共2兲
n,m
where En共k兲 = 关␰2n共k兲 + 兩⌬n共k兲兩2兴1/2 is the energy of the quasi-
冊冉 冊
⌬1,0
= 0,
⌬2,0
共4兲
where the matrix elements are given by Onm = −gnm
− ␦nm兺k兩⌫m共k兲兩2 tanh关␤Em共k兲 / 2兴 / 关2Em共k兲兴. Here, ␦nm is the
Kronecker delta. The order parameter equations can also be
written in a more familiar form as
⌬n,0 = 兺
m,k
Vnm⌬m,0兩⌫m共k兲兩2
␤Em共k兲
tanh
.
2
2Em共k兲
共5兲
Notice that the order parameter amplitudes are the same for
both the 0 and ␲ phases as can be shown directly from Eq.
共4兲, but their relative phases are either 0 or ␲. In what follows, we analyze only the 0-phase state, keeping in mind that
analogous results 共with appropriate relative phase changes兲
apply to the ␲-phase state. We can eliminate Vnn in favor of
scattering length ann via the relation
M nV
兩⌫n共k兲兩2
1
,
=−
+兺
Vnn
4␲ann k 2⑀n共k兲
A. Effective action
The Gaussian action for the Hamiltonian H 共in units of
kB = 1 , ␤ = 1 / T兲 is SGauss = S0 + 共␤ / 2兲兺q⌳†共−q兲F−1共q兲⌳共q兲,
where q = 共q , ivᐉ兲 denotes both momentum and bosonic Matsubara frequency vᐉ = 2ᐉ␲ / ␤. Here, the vector ⌳†共−q兲 is the
order parameter fluctuation field, and the matrix F−1共q兲 is the
inverse fluctuation propagator. The saddle point action is
O11 O12
O21 O22
共6兲
where V is the volume. This relation can be rewritten as
2
=␲
␭nn
冑
⑀n,0
␲
−
,
⑀n,F kn,Fann
共7兲
2
where ⑀n,0 = kn,0
/ 共2M n兲, and ␭nm = VnmDm are the dimensionless interaction parameters with Dm = M mVkm,F / 共2␲2兲 being
the density of states per spin 共pseudospin兲 of noninteracting
fermions at the Fermi energy.
The order parameter equation needs to be solved selfconsistently with the number equation N = −⳵⍀ / ⳵␮ leading
to NGauss = N0 + Nfluct, and is given by
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TWO-BAND SUPERFLUIDITY FROM THE BCS TO THE…
NGauss =
兺
k,␴,n
N0,n共k兲 −
⳵关det F−1共q兲兴/⳵␮
1
兺 det F−1共q兲 .
␤ q
共8兲
Here, the first term is the saddle point 共N1 + N2兲 and the second term is the fluctuation 共Nfluct兲 contribution, where N0,n
= 1 / 2 − ␰n共k兲tanh关␤En共k兲 / 2兴 / 关2En共k兲兴 is the momentum distribution. The inclusion of Nfluct is very important near the
critical temperature, however, N0 may be sufficient at low
temperatures.4,5
In the remainder of the paper, numerical results are given
only for identical bands 共M 1 = M 2 = M兲 with zero offset 共⑀D
= 0兲, which correspond physically to situations that may be
encountered in atomic physics when four hyperfine states of
a given atomic system are trapped. However, we also present
analytical results for the case of unequal band masses, which
may be applicable to standard condensed matter systems
such as MgB2. In this case, the particle density may be tuned
using field effect techniques,23 and it may be possible to
study two-band systems away from the higher-density
共weakly interacting兲 BCS regime toward the lower-density
共strongly interacting兲 BEC regime.
On the other hand, the BEC limit is characterized by
negative chemical potential with respect to the bottom of the
lowest band ␮ ⬍ 0 and max兵兩⌬1,0 兩 , 兩⌬2,0 兩 其 Ⰶ 兩␮ 兩 Ⰶ ⑀0, where
pairing occurs between all fermions. In this limit, the solution of the order parameter equations is
␮± = − ⑀0关␲冑⑀0/共4⑀F兲/␾± − 1兴2 ,
while the number equation leads to
兩⌬n,0兩2 = 共8␲Nn/M n兲冑兩␮兩/共2M n兲.
Here, Nn = Nn / V is the density of the fermions in band
n. Notice that the total density of fermions is N = N1
3
3
+ N2 = 共k1,F
+ k2,F
兲 / 共3␲2兲. The familiar one-band results
are again recovered when J → 0 leading to ␮n
= −⑀n,0关␲␭nn冑⑀n,0 / 共4⑀n,F兲 − 1兴2 for the chemical potentials,3
which can be written in terms of the scattering lengths5 as
2
␮n = −1 / 共2M nann
兲, using Eq. 共7兲, and the condition kn,0ann
Ⰷ 1. The one-band order parameter amplitudes are also given
2 冑
兩␮n 兩 / ⑀n,F / 共3␲兲, where we used Nn
by 兩⌬n,0兩2 = 16⑀n,F
3
2
= kn,F / 共3␲ 兲.
B. Saddle point: BCS to BEC evolution
III. ZERO TEMPERATURE
In this section, we analyze the saddle point order parameter amplitudes 兩⌬n,0兩 and the chemical potential ␮ at T = 0.
We first obtain analytical results in the strictly BCS and BEC
limits, and then perform numerical calculations in the evolution from BCS to BEC, which are discussed next.
A. Saddle point: BCS and BEC limits
In the strictly BCS and BEC limits, the self-consistent
共order parameter and number兲 equations are decoupled, and
play reversed roles. In the BCS 共BEC兲 limit, the order parameter equations determine 兩⌬n,0兩 共␮兲, and the number equation determines ␮ 共兩⌬n,0兩兲.
The BCS limit is characterized by a positive chemical
potential with respect to the bottom of the fermion band ␮
⬎ 0 and max兵兩⌬1,0兩 , 兩⌬2,0兩其 Ⰶ ⑀2,F, where pairing occurs between the fermions whose energy are close to ⑀1,F. In this
limit, the solutions of the order parameter equation are
max兵兩⌬1,0兩,兩⌬2,0兩其 ⬃ 8⑀F exp关− 2 + ␲冑⑀0/共4⑀F兲 − ␾−兴
for the larger of the order parameter amplitudes and
min兵兩⌬1,0兩,兩⌬2,0兩其 ⬃ 8⑀F exp关− 2 + ␲冑⑀0/共4⑀F兲 − ␾+兴
for the smaller of the order parameter amplitudes, while the
number equation leads to ␮ ⬇ ⑀F. Here, ␾± = ␭+ ± 关␭+2
− 1 / det ␭兴1/2 where ␭± = 共␭11 ± ␭22兲 / 共2 det ␭兲 , det ␭ = ␭11␭22
− ␭12␭21, and we assumed ⑀1,F ⬃ ⑀2,F = ⑀F and ⑀1,0 ⬃ ⑀2,0 = ⑀0.
The familiar one-band results are again recovered when J
→ 0 leading to 兩⌬n,0 兩 ⬃ 8⑀n,F exp关−2 + ␲冑⑀n,0 / 共4⑀n,F兲
− 1 / ␭nn兴 for the order parameters and ␮ = ⑀n,F for the chemical potentials. The previous expression can be simplified by
using Eq. 共7兲, leading to 兩⌬n,0兩 = 8⑀n,F exp关−2 + ␲ / 共2kn,Fann兲兴,
which is the standard one-band result.5
Next, we analyze numerically the evolution of the order
parameter amplitudes and the chemical potential from the
BCS to the BEC limit for identical bands 共M 1 = M 2 = M兲 with
zero offset 共⑀D = 0兲 at zero temperature. For this purpose, we
solve the coupled self-consistency equations Eqs. 共5兲 and 共8兲,
for parameters k1,0 = k2,0 = k0 ⬇ 256kF and V22 = 0.001 in units
of 5.78/ 共MkFV兲 关or 1 / 共kFa22兲 ⬇ −3.38兴, and analyze two
cases.
In the first case, we solve for ␮, 兩⌬1,0兩 and 兩⌬2,0兩 as a
function of V11 关or 1 / 共kFa11兲兴, and show ⌬n,0 in Fig. 2共a兲 for
fixed values of J. The unitarity limit is reached at V11
⬇ 1.0132V22 关or 1 / 共kF 兩 a11 兩 兲 ⬇ 0兴 while ␮ changes sign at
V11 ⬇ 1.0159V22 关or 1 / 共kFa11兲 ⬇ 0.69兴. The evolution of 兩⌬1,0兩
is similar to the one-band result4 where it grows monotonically with increasing V11. However, the evolution of 兩⌬2,0兩 is
nonmonotonic having a hump at approximately V11
⬇ 1.0155V22 关or 1 / 共kFa11兲 ⬇ 0.58兴, and it decreases for stronger interactions until it vanishes 共not shown兲. In the case of
ultracold atoms, the order parameter may be measured using
similar techniques as for one-band systems14 involving only
6
Li. In Fig. 2共b兲, we show that both bands have similar populations for V11 ⬃ V22. However, as V11 / V22 increases, fermions from the second band are transferred to the first, where
bound states are easily formed and reduce the free energy. In
coupled two-band systems, spontaneous population imbalances are induced by the increasing scattering parameter 共interaction兲 in contrast with the one-band case, where population imbalances are prepared externally.15,16
In the second case, we solve for ␮, 兩⌬1,0兩, and 兩⌬2,0兩 as
functions of J, and show ␮ in Fig. 3共a兲 and band populations
Nn in Fig. 3共b兲 for fixed values of V11 / V22. The order parameters 兩⌬1,0兩 and 兩⌬2,0兩 grow with increasing J 共not shown兲.
Notice the population imbalance and the presence of maxima
共minima兲 in N1 共N2兲 for finite J, which is associated with the
sign change of ␮ shown in Fig. 3共a兲. When V11 ⬎ V22, there is
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M. ISKIN AND C. A. R. SÁ DE MELO
FIG. 2. Plots of 共a兲 order parameter amplitude 兩⌬n,0兩 共in units of
⑀F兲, and 共b兲 fraction of fermions Nn / N versus V11 关in units of
5.78/ 共MkFV兲兴 and versus 1 / 共kFa11兲 for J = 0.001V22 共hollow
squares兲 and J = 0.0001V22 共solid squares兲.
population imbalance even for J = 0, because atom pairs can
be easily transferred from the second band to the first until an
optimal J0 is reached. Further increase in J produces also
transfer from the first band to the second, leading to similar
populations for J Ⰷ J0. Therefore, the Josephson coupling parameter J acts as a knob to tune the populations of bands 1
and 2.
FIG. 3. Plots of 共a兲 chemical potential ␮ 共in units of ⑀F兲, and 共b兲
fraction of fermions Nn / N versus J 关in units of 5.78/ 共MkFV兲兴 for
V11 = ␥V22 where ␥ = 1 共dotted lines兲, 1.010 共solid squares兲, 1.014
共hollow squares兲, 1.016 共crossed lines兲, and 1.030 共solid lines兲; or
1 / 共kFa11兲 ⬇ −3.38, −0.81, 0.20, 0.70, and 4.17, respectively.
The phase-only collective excitations in the BCS and
BEC limits lead to a Goldstone mode w2共q兲 = v2兩q兩2 characterized by the speed of sound,
v2 = 共D1v21 + D2v22兲/共D1 + D2兲,
and a finite-frequency mode w2共q兲 = w20 + u2兩q兩2 characterized
by the finite frequency
w20 = 4␣兩g12⌬1,0⌬2,0兩冑⑀F/兩␮兩共D1 + D2兲/共D1D2兲
and the speed
C. Collective Excitations
Next, we discuss the low-energy collective excitations at
T = 0, which are determined by the poles of the propagator
matrix F共q兲 for the pair fluctuation fields ⌳n共q兲, which describe the Gaussian deviations about the saddle point order
parameter. The poles of F共q兲 are determined by the condition
det F−1共q兲 = 0, which leads to collective 共amplitude and
phase兲 modes, when the usual analytic continuation ivᐉ
→ w + i0+ is performed. The easiest way to get the phase
collective mode is to integrate out the amplitude field to
obtain a phase-only effective action. Notice that a welldefined low-frequency expansion is possible only at T = 0 for
s-wave systems, due to Landau damping present at finite
temperatures, causing the collective modes to decay into the
two-quasiparticle continuum.
u2 = 共D1v22 + D2v21兲/共D1 + D2兲.
In the BCS limit, ␣ = 1 and vn = vn,F / 冑3, while ␣ = 1 / ␲ and
vn = 兩⌬n,0 兩 / 冑8M n 兩 ␮兩 in the BEC limit. Here, vn,F is the Fermi
velocity. Here, we assumed ⑀1,F ⬃ ⑀2,F = ⑀F and kn,0 / kn,F → ⬁.
The familiar one-band results are again recovered when J
→ 0 leading to v = v1, and w0 = 0 and u = v2.
It is also illustrative to analyze the eigenvectors associated
with these solutions in the BCS and BEC limits. In the limit
of q → 0, we obtain
共␪1, ␪2兲 ⬀ 共兩⌬1,0兩,兩⌬2,0兩兲
for the Goldstone mode corresponding to an in-phase solution, while we obtain
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共␪1, ␪2兲 ⬀ 共D2兩⌬1,0兩,− D1兩⌬2,0兩兲
for the finite-frequency 共exciton兲 mode corresponding to an
out-of-phase solution. Notice that, in the case of identical
bands with zero offset and identical order parameter amplitudes, the collective mode simplifies to be perfectly in phase
where 共␪1 , ␪2兲 ⬀ 共1 , 1兲, and perfectly out of phase where
共␪1 , ␪2兲 ⬀ 共1 , −1兲. These in-phase and out-of-phase collective
modes are associated with the in-phase and out-of-phase
fluctuations of the order parameters around their saddle point
values, respectively. Our findings generalize Leggett’s BCS
results.24,25 Notice that measurements of collective modes
are already possible in one-band atomic systems,26,27 but the
two-band problem provides an additional richness which is
the existence of the finite-frequency 共Leggett兲 mode in addition to the low-frequency 共Goldstone兲 mode.
IV. FINITE TEMPERATURES
Next, we discuss two-band superfluidity near the critical
temperature Tc, where 兩⌬1,0兩 ⬃ 兩⌬2,0兩 → 0. Our basic motivation here is to investigate the low-frequency and longwavelength behavior of the order parameter near Tc. For T
= Tc, the order parameter equation reduces to
detO = O11O22 − O12O21 = 0,
共9兲
and the saddle point number equation N0 = 兺k,␴,nnF关␰n共k兲兴
corresponds to the number of unbound fermions, where
nF共x兲 = 1 / 关exp共␤x兲 + 1兴 is the Fermi distribution. While N0 is
sufficient in the BCS limit, the inclusion of Nfluct is crucial in
the BEC limit to produce the qualitatively correct physics,
and can be obtained as follows.
A. Fluctuations near Tc
Near T = Tc, the fluctuation action Sfluct reduces to Sfluct
−1
= ␤兺q,n,mLnm
共q兲⌳*n共q兲⌳m共q兲 where
−1
= − gnn − 兺
Lnn
k
1 − nF共␰n+兲 − nF共␰n−兲
兩⌫n共k兲兩2
␰n+ + ␰n− − ivᐉ
共10兲
corresponds to the fluctuation propagator of band n,
−1
Ln⫽m
共q兲 = gnm, and ␰n± = ␰n共k ± q / 2兲. Thus, the resulting action leads to ⍀fluct = 共1 / ␤兲兺q ln关det L−1共q兲 / ␤2兴, where
−1
−1
det L−1共q兲 = L11
共q兲L22
共q兲 − g12g21. Notice that det L−1共0兲 = 0
also produces Eq. 共9兲, which is the Thouless condition. After
−1
共q兲 to
the analytic continuation ivᐉ → w + i0+, we expand Lnn
first order in w and second order in q such that
−1
共q兲 = an + 兺
Lnn
i,j
cijn
qiq j − dnw.
2M n
共11兲
−1
The zeroth-order coefficient Lnn
共0 , 0兲 is given by
an = − gnn − 兺
k
Xn
兩⌫n共k兲兩2 ,
2␰n共k兲
共12兲
where Xn = tanh关␤␰n共k兲 / 2兴. The second-order coefficient cijn
−1
= M n⳵2Lnn
共q , 0兲 / 共⳵qi⳵q j兲 evaluated at q = 0 is given by
cijn = 兺
k
冋冉
册
冊
Xn
␤ 2X nY n
␤Y n
−
␦
+
kik j 兩⌫n共k兲兩2 ,
ij
16M n␰n共k兲
8␰2n共k兲 16␰n共k兲
共13兲
where Y n = sech2关␤␰n共k兲 / 2兴 and ␦ij is the Kronecker delta.
Notice that cijn = cn␦ij is isotropic for the s wave considered
here. The coefficient dn has real and imaginary parts, and for
the s-wave case is given by
dn = 兺
k
Xn
␤␲Dn⑀n,0
兩⌫n共k兲兩2 + i
8共⑀n,0 + ␮兲
4␰2n共k兲
冑
␮
⌰共␮兲,
⑀n,F
共14兲
where ⌰共x兲 is the Heaviside function. Its imaginary part reflects the decay of Cooper pairs into the two-particle continuum for ␮ ⬎ 0. However, for ␮ ⬍ 0, the imaginary part of
dn vanishes and the behavior of the order parameter is propagating, reflecting the presence of stable bound states. In addition, the coefficient of the fourth-order term in ⌳n共q兲 is
bn = 兺
k
冉
冊
Xn
␤Y n
− 2
兩⌫n共k兲兩4 ,
3
4␰n共k兲 8␰n共k兲
共15兲
which is necessary to derive the time-dependent GL 共TDGL兲
equations.
For completeness, we present the asymptotic forms
of an, bn, cn, and dn. The BCS limit is characterized by
positive chemical potential with respect to the bottom
of the lowest band ␮ ⬎ 0 and ␮ ⬇ ⑀1,F. In this limit, we
find an = −gnn + Dn关ln共T / Tc兲 + ␾−兴, bn = 7Dn␨共3兲 / 共8T2c ␲2兲, cn
= 7⑀n,FDn␨共3兲 / 共12T2c ␲2兲, and dn = Dn关1 / 共4⑀n,F兲 + i / 共8Tc兲兴,
where ␨共x兲 is the zeta function, and Tc is the physical
critical temperature. On the other hand, the BEC limit
is characterized by negative chemical potential with
respect to the particle band ␮ ⬍ 0 and ⑀n,0 Ⰷ 兩␮兩 Ⰷ Tc. In this
limit, we find an = −gnn − ␲Dn⑀n,0 / 关2冑⑀n,F共冑兩␮兩 + 冑⑀n,0兲兴,
bn = ␲Dn / 共4兩␮兩冑2兩␮兩⑀n,F兲, cn = ␲Dn / 共16冑兩␮兩⑀n,F兲, and dn
= ␲Dn / 共8冑兩␮兩⑀n,F兲.
In order to obtain ⍀fluct, there are two contributions, one
from the scattering states and the other from poles of L共q兲.
The pole contribution dominates in the BEC limit. In this
case, we evaluate det L−1共q兲 = 0 and find the poles
w±共q兲 = A+ + B+兩q兩2 ±
冑
共A− + B−兩q兩2兲2 +
g12g21
,
d 1d 2
and
B±
where
A± = 共a1d2 ± a2d1兲 / 共2d1d2兲
= 共M 2d2c1 ± M 1d1c2兲 / 共4M 1M 2d1d2兲. Notice that, when J → 0,
we recover the limit of uncoupled bands with wn共q兲 = an / dn
+ 兩q兩2cn / 共2M ndn兲. It is also illustrative to analyze the eigenvectors associated with these poles. In the q → 0 limit, the
eigenvectors 关⌳†1共0兲 , ⌳†2共0兲兴 = 关g12 , a1 − d1w±共0兲兴 correspond
to an in-phase mode for w+共q兲 and an out-of-phase mode for
w−共q兲 when J ⬎ 0; however, they correspond to an out-ofphase mode for w+共q兲 and an in-phase mode for w−共q兲 when
J ⬍ 0. Thus, we obtain ⍀fluct = 共1 / ␤兲兺±,q ln兵␤关ivᐉ − w共q兲兴其,
which leads to
144517-5
PHYSICAL REVIEW B 74, 144517 共2006兲
M. ISKIN AND C. A. R. SÁ DE MELO
Nfluct = 兺
±,q
⳵w共q兲
nB关w共q兲兴.
⳵␮
共16兲
In the BEC limit, ⳵w±共q兲 / ⳵␮ = 2, and the poles can also be
written as w±共q兲 = −␮B,± + 兩q兩2 / 共2M B,±兲, where ␮B,± is the
chemical potential and M B,± is the mass of the corresponding
bosons. In the case of identical bands 共M 1 = M 2 = M兲 with
zero offset 共⑀D = 0兲, c1 = c2 = c and d1 = d2 = d, ␮B,± = −关a1
+ a2 ± 冑共a1 − a2兲2 + 4g12g21兴 / 共2d兲 and M B,± = 2M in the BEC
limit. Notice that the ⫹ bosons always condense first for any
J 共independent of its sign兲 since ␮B,+ → 0 first. The familiar
one-band results4 are again recovered when J → 0, leading to
2
␮B,n = −an / dn = 2␮n − 1 / 共2M nann
兲 and M B,n = M ndn / cn = 2M n.
Next, we analyze Tc in the strict BCS and BEC limits, where
self-consistency equations are uncoupled, allowing analytical
results.
B. Critical temperature
In the BCS limit, solutions to the order parameter equation Eq. 共9兲 are
functional near Tc, we need to expand the effective action Seff
to fourth order in ⌳n共x兲 around the saddle point order parameter 兩⌬n兩 → 0 leading to
冋
冋
Tc,+ = 共2␲/M B,+兲关NB,+/␨共3/2兲兴2/3 ,
since the ⫹ bosons condense first. Notice also that the physical critical temperature is Tc = max兵Tc,+ , Tc,−其 = Tc,+ for any J.
Therefore, Tc grows continuously from an exponential dependence on interaction to a constant BEC temperature.
The familiar one-band results are again recovered when
J → 0, leading to ␮n = −⑀n,0关␲␭nn冑⑀n,0 / 共4⑀n,F兲 − 1兴2 for the
chemical potentials,3 and Tc,n = 共2␲ / M B,n兲关NB,n / ␨共3 / 2兲兴2/3
for the critical temperatures, where NB,n = Nn / 2 is the number and M B,n = 2M n is the mass of the corresponding bosons.
Using Eq. 共7兲 and the condition kn,0ann Ⰷ 1, ␮n can be written
2
兲. In
in terms of the scattering lengths as ␮n = −1 / 共2M nann
addition, Tc,n can be expressed in terms of ⑀n,F as Tc,n
= 0.218⑀n,F, as expected from the one-band calculations.4
C. TDGL equations
Next, we obtain the time-dependent Ginzburg-Landau
equations for T ⬇ Tc. To study the evolution of the TDGL
T2 + b2兩⌳2共x兲兩2
册冋 册
⌳1共x兲
= 0 共17兲
⌳2共x兲
T̃1R11 + g12R21
0
0
g21R12 + T̃2R22
册冋 册
⌽+共x兲
= 0, 共18兲
⌽−共x兲
where
T̃1 = T1 + b1兩R11⌽+共x兲 + R12⌽−共x兲兩2
and
T̃2 = T2
+ b2兩R21⌽+共x兲 + R22⌽−共x兲兩2. The matrix elements Rnm are determined from the relations 关T̃1R12 + g12R22兴⌽−共x兲 = 0 and
关T̃2R21 + g21R11兴⌽+共x兲 = 0, and the unitarity condition R†R
= 1. Notice that Eq. 共18兲 shows explicitly terms coming from
density-density interactions such as U±±兩⌽±共x兲兩2⌽±共x兲 or
U±⫿兩⌽±共x兲兩2⌽⫿共x兲.
The familiar one-band results are again recovered when
J → 0 共g12 = g21 → 0兲 leading to
冋
␮± = − ⑀0关␲冑⑀0/共4⑀F兲/␾± − 1兴2 ,
while the number equation N / 2 = NB,+ + NB,− with NB,+
Ⰷ NB,− leads to
g12
g21
in the real space x = 共x , t兲 representation, where Tn = an
− cnⵜ2 / 共2M n兲 − idn⳵ / ⳵t The coefficient bn of the nonlinear
term given in Eq. 共15兲 is always positive and guarantees the
stability of the theory. In the BCS limit, Eq. 共17兲 reduces to
the coupled GL equations of two BCS-type superconductors.
However, in the BEC limit, it is more illustrative to derive
TDGL equations in the rotated basis of ⫹ and ⫺ bosons such
that ⌳1共x兲 = R11⌽+共x兲 + R12⌽−共x兲 and ⌳2共x兲 = R21⌽+共x兲
+ R22⌽−共x兲 where Rnm are the matrix elements of the inverse
transformation matrix R. Notice that the matrix R−1 is the
unitary matrix that diagonalizes the linear part of the TDGL
equations. In this basis, Eq. 共17兲 reduces to generalized GP
equations of ⌽+共x兲 and ⌽−共x兲 bosons
Tc,⫿ = 共8⑀F/␲兲exp关␥ − 2 + ␲冑⑀0/共4⑀F兲 − ␾±兴,
while the number equation Eq. 共8兲 leads to ␮ ⬇ ⑀F. Here,
we assumed ⑀1,F ⬃ ⑀2,F = ⑀F and ⑀1,0 ⬃ ⑀2,0 = ⑀0. Notice that
the physical critical temperature is Tc = max兵Tc,+ , Tc,−其 = Tc,+
for any J. The familiar one-band results are again recovered when J → 0, leading to Tc,n = 共8⑀n,F / ␲兲exp关␥ − 2
+ ␲冑⑀n,0 / 共4⑀n,F兲 − 1 / ␭nn兴 for the order parameters and
␮ = ⑀n,F for the chemical potentials. The previous expression can be simplified using Eq. 共7兲, leading to Tc,n
= 共8⑀n,F / ␲兲exp关␥ − 2 + ␲ / 共2kn,Fann兲兴.
On the other hand, in the BEC limit, the solution of the
order parameter equation is
T1 + b1兩⌳1共x兲兩2
an + bn兩⌳n共x兲兩2 −
册
⳵
cn 2
ⵜ − idn
⌳n共x兲 = 0.
⳵t
2M n
共19兲
This expression reduces to the conventional TDGL equation
upon rescaling of the pairing field as ⌿n共x兲 = 冑bn / Dn⌳n共x兲 in
the BCS limit, while it reduces to the conventional GP equation upon rescaling of the pairing field as ⌿n共x兲 = 冑dn⌳n共x兲 in
the BEC limit.4
V. ULTRACOLD FERMI GASES
We would like to emphasize that our results for identical
bands 共M 1 = M 2 = M兲 may be applicable to ultracold Fermi
gases when four hyperfine states of a given type of atom are
trapped. However, in atomic systems involving mixtures of
two different alkali-metal atoms 共e.g., 6Li and 40K兲, a more
general two-band theory may be necessary to model future
experiments. These more general two-band theories should
allow pairing interactions between different Fermi atoms as
well as the same Fermi atoms.
The phase diagram and the BCS to BEC evolution of
superfluid mixtures involving only one hyperfine state
共single pseudospin component兲 from each type of atom 共A or
B兲 and pairing only between different types of atoms 共A-B
144517-6
PHYSICAL REVIEW B 74, 144517 共2006兲
TWO-BAND SUPERFLUIDITY FROM THE BCS TO THE…
pairing兲 has been recently addressed.28–30 These models rely
on the assumption that the only important Feshbach resonance is that between A and B atoms. However, generalized
models to describe Fermi mixtures should allow for the inclusion of pairing between the same species, as well as more
than one hyperfine state 共pseudospin component兲 of each
species. For example, a model describing the mixture of one
hyperfine state of Fermi atom A and one hyperfine state of
Fermi atom B might need to allow for A-A, A-B, and B-B
interactions, if A-A, A-B, and B-B Feshbach resonances turn
out to be close to each other. In addition, it may be possible
to trap two hyperfine states 共pseudospin components兲 of
Fermi atom A 共兩A , 1典 and 兩A , 2典兲 and two hyperfine states of
Fermi atom B 共兩B , 1典 and 兩B , 2典兲. Thus, a model to describe
this case may require interactions between fermions in all
four states considered if Feshbach resonances between any
two pairs of states are close to each other. Therefore, superfluidity in Fermi mixtures may be studied using a generalized
multicomponent 共“multiband”兲 Hamiltonian
H=
␰n␣共k兲an†␣共k兲an␣共k兲
兺
n,␣,k
−
兺
n,m,k,k⬘q
†
␣␤␥␦
Vnm
共k,k⬘兲bm
␥;n␦共k⬘,q兲bn␣;m␤共k,q兲,
where n and m label different types of Fermi atoms,
␣ , ␤ , ␥ , ␦ label the different hyperfine states, and Einstein’s
notation is used to indicate summation over greek indices.
Here, ␰n␣共k兲 = ⑀n␣共k兲 − ␮n␣, where ⑀n␣共k兲 = ␧n␣,0 + k2 / 共2M n兲 is
the kinetic energy 共ប = 1兲, M n is the corresponding mass of
fermion type n, ␧n␣,0 is the reference energy for hyperfine
state ␣, and ␮n␣ is the chemical potential that fixes the number of fermions of type n in each hyperfine state ␣. Further-
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†
†
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␥;n␦共k , q兲 = am␥共k
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VI. CONCLUSIONS
In conclusion, we analyzed the evolution of two-band superfluidity from the BCS to the BEC limit as a function of
interaction strength at zero and finite temperatures. At zero
temperature, we showed that a quantum phase transition occurs from a 0-phase to a ␲-phase state depending on the
relative phase of the two order parameters, when the interband interaction is tuned from negative to positive values.
We found that population imbalances between the two bands
can be created by tuning intraband or interband interactions.
In addition, we described the evolution of two undamped
low-energy collective excitations corresponding to in-phase
phonon 共Goldstone兲 and out-of-phase exciton 共finitefrequency兲 modes. Near the critical temperature, we derived
the coupled time-dependent Ginzburg-Landau functional
near Tc. We recovered the usual TDGL equations for BCS
superfluids in the BCS limit, whereas in the BEC limit we
recovered the coupled Gross-Pitaevskii equations for two
types of weakly interacting bosons 共tightly bound fermions兲.
ACKNOWLEDGMENTS
We thank the NSF 共Grant No. DMR-0304380兲 for support.
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