PHYSICAL REVIEW A 74, 013608 共2006兲
Nonzero orbital angular momentum superfluidity in ultracold Fermi gases
M. Iskin and C. A. R. Sá de Melo
School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
共Received 6 February 2006; revised manuscript received 18 April 2006; published 13 July 2006兲
We analyze the evolution of superfluidity for nonzero orbital angular momentum channels from the BardeenCooper-Schrieffer 共BCS兲 to the Bose-Einstein condensation 共BEC兲 limit in three dimensions. First, we analyze
the low-energy scattering properties of finite range interactions for all possible angular momentum channels.
Second, we discuss ground-state 共T = 0兲 superfluid properties including the order parameter, chemical potential,
quasiparticle excitation spectrum, momentum distribution, atomic compressibility, ground-state energy, and
low-energy collective excitations. We show that a quantum phase transition occurs for nonzero angular momentum pairing, unlike the s-wave case where the BCS to BEC evolution is just a crossover. Third, we present
a Gaussian fluctuation theory near the critical temperature 共T = Tc兲, and we analyze the number of bound,
scattering, and unbound fermions as well as the chemical potential. Finally, we derive the time-dependent
Ginzburg-Landau functional near Tc, and compare the Ginzburg-Landau coherence length with the zerotemperature average Cooper pair size.
DOI: 10.1103/PhysRevA.74.013608
PACS number共s兲: 03.75.Ss, 03.75.Hh, 74.25.Bt, 74.25.Dw
I. INTRODUCTION
Experimental advances involving atomic Fermi gases enabled the control of interactions between atoms in different
hyperfine states by using Feshbach resonances 关1–7兴. These
resonances can be tuned via an external magnetic field and
allow the study of dilute many-body systems with fixed density, but varying interaction strength characterized by the
scattering parameter aᐉ. This technique allows for the study
of new phases of strongly interacting fermions. For instance,
the recent experiments from the MIT group 关8兴 marked the
first observation of vortices in atomic Fermi gases corresponding to a strong signature of superfluidity in the s-wave
共ᐉ = 0兲 channel. These studies combined 关1–8兴 correspond to
the experimental realization of the theoretically proposed
Bardeen-Cooper-Schrieffer 共BCS兲 to Bose-Einstein condensation 共BEC兲 crossover 关9–13兴 in three-dimensional 共3D兲
s-wave superfluids. Recent extensions of these ideas include
trapped fermions 关14,15兴 and fermion-boson models
关16–18兴.
Arguably one of the next frontiers of exploration in ultracold Fermi systems is the search for superfluidity in higher
angular momentum states 共ᐉ ⫽ 0兲. Substantial experimental
progress has been made recently 关19–23兴 in connection to
p-wave 共ᐉ = 1兲 cold Fermi gases, making them ideal candidates for the observation of novel triplet superfluid phases.
These phases may be present not only in atomic, but also in
nuclear 共pairing in nuclei兲, astrophysics 共neutron stars兲, and
condensed-matter 共organic superconductors兲 systems.
The tuning of p-wave interactions in ultracold Fermi
gases was initially explored via p-wave Feshbach resonances
in trap geometries for 40K in Refs. 关19,20兴 and 6Li in Refs.
关21,22兴. Finding and sweeping through these resonances is
difficult since they are much narrower than the s-wave case,
because atoms interacting via higher angular momentum
channels have to tunnel through a centrifugal barrier to
couple to the bound state 关20兴. Furthermore, while losses due
to two body dipolar 关21,24兴 or three-body 关19,20兴 processes
challenged earlier p-wave experiments, these losses were
1050-2947/2006/74共1兲/013608共22兲
still present but were less dramatic in the very recent optical
lattice experiment involving 40K and p-wave Feshbach resonances 关23兴.
Furthermore, due to the magnetic dipole-dipole interaction between valence electrons of alkali atoms, the nonzero
angular momentum Feshbach resonances corresponding to
projections of angular momentum ᐉ 关mᐉ = ± ᐉ , ± 共ᐉ
− 1兲 , . . . , 0兴 are nondegenerate 共separated from each other兲
with total number of ᐉ + 1 resonances 关20兴. Therefore, in
principle, these resonances can be tuned and studied independently if the separation between them is larger than the
experimental resolution. Since the ground state is highly dependent on the separation and detuning of these resonances,
it is possible that p-wave superfluid phases can be studied
from the BCS to the BEC regime. For sufficiently large splittings, it has been proposed 关25,26兴 that pairing occurs only in
mᐉ = 0 and does not occur in the mᐉ = ± 1 states. However, for
small splittings, pairing occurs via a linear combination of
the mᐉ = 0 and mᐉ = ± 1 states. Thus the mᐉ = 0 or mᐉ = ± 1
resonances may be tuned and studied independently if the
splitting is large enough in comparison to the experimental
resolution.
The BCS to BEC evolution of d-wave 共ᐉ = 2兲 superfluidity
was discussed previously in the literature using continuum
关27–29兴 and lattice 关30,31兴 descriptions in connection to
high-Tc superconductivity. More recently, p-wave superfluidity was analyzed at T = 0 for two hyperfine state 共THS兲 systems in three dimensions 关32兴, and for single hyperfine state
共SHS兲 systems in two dimensions 关33–35兴, using fermiononly models. Furthermore, fermion-boson models were proposed to describe p-wave superfluidity at zero 关25,26兴 and
finite temperature 关36兴 in three dimensions. In this manuscript, we present a generalization of the zero and finite temperature analysis of both THS pseudospin singlet and SHS
pseudospin triplet 关37兴 superfluidity in three dimensions
within a fermion-only description.
The rest of the paper is organized as follows. In Sec. II,
we analyze the interaction potential in both real and momentum space for nonzero orbital momentum channels. We in-
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©2006 The American Physical Society
PHYSICAL REVIEW A 74, 013608 共2006兲
M. ISKIN AND C. A. R. SÁ DE MELO
troduce the imaginary-time functional integration formalism
in Sec. III, and obtain the self-consistency 共order parameter
and number兲 equations. There we also discuss the lowenergy scattering amplitude of a finite range interaction for
all possible angular momentum channels, and relate the selfconsistency equations to scattering parameters. In Sec. IV,
we discuss the evolution from BCS to BEC superfluidity at
zero temperature. There we analyze the order parameter,
chemical potential, quasiparticle excitation spectrum, momentum distribution, atomic compressibility, and groundstate energy as a function of scattering parameters. We also
discuss Gaussian fluctuations and low-energy collective excitations at zero temperature in Sec. V. In Sec VI, we present
the evolution of superfluidity from the BCS to the BEC regimes near the critical temperature. There we discuss the
importance of Gaussian fluctuations, and analyze the number
of unbound, scattering, and bound fermions, critical temperature, and chemical potential as a function of scattering parameters. In Sec. VII, we derive time-dependent GinzburgLandau 共TDGL兲 equation and extract the Ginzburg-Landau
共GL兲 coherence length and time. There, we recover the GL
equation in the BCS and the GP equation in the BEC limit. A
short summary of our conclusions is given in Sec. VIII. Finally, we present in Appendixes A and B the coefficients for
the low-frequency and long-wavelength expansion of the action at zero and finite temperatures, respectively.
II. GENERALIZED HAMILTONIAN
The Hamiltonian for a dilute Fermi gas is given by
H = 兺 ␰共k兲ak,s†1ak,s1 +
k,s1
⫻
兺 s ,s兺,s ,s
k,k⬘,q 1 2 3 4
The two fermion interaction can be expanded as
*
V共k,k⬘兲 = 4␲ 兺 Vᐉ共k,k⬘兲Y ᐉ,mᐉ共k̂兲Y ᐉ,m
共k̂⬘兲,
ᐉ
ᐉ,mᐉ
where Y ᐉ,mᐉ共k̂兲 is the spherical harmonic of order 共ᐉ , mᐉ兲,
⬁
ᐉ
兺ᐉ,mᐉ = 兺ᐉ=0
兺m
, and k̂ denotes the angular dependence
ᐉ=−ᐉ
共␪k , ␾k兲. The interaction should have the necessary symmetry under the Parity operation, where the transformation k
→ −k or k⬘ → −k⬘ leads to V共k , k⬘兲 for singlet, and
−V共k , k⬘兲 for triplet pairing. Furthermore, V共k , k⬘兲 is invariant under the transformation 共k , k⬘兲 → 共−k , −k⬘兲, and
V共k , k⬘兲 reflects the Pauli exclusion principle. The 共k , k⬘兲
dependent coefficients Vᐉ共k , k⬘兲 are related to the real-space
potential
V共r兲
through
the
relation
Vᐉ共k , k⬘兲
= 4␲兰⬁0 drr2 jᐉ共kr兲jᐉ共k⬘r兲V共r兲, where jᐉ共kr兲 is the spherical
Bessel function of order ᐉ. The index ᐉ labels angular momentum states in three dimensions, with ᐉ = 0 , 1 , 2 , . . . corresponding to s , p , d , . . . channels, respectively.
Under these circumstances, we choose to study a model
potential that contains most of the features described above.
One possibility is to retain only one of the ᐉ terms in Eq. 共2兲,
by assuming that the dominant contribution to the scattering
process between Fermionic atoms occurs in the ᐉth angular
momentum channel. This assumption may be experimentally
relevant since atom-atom dipole interactions split different
angular momentum channels such that they may be tuned
independently. Using the properties discussed above, we
write
Vᐉ共k,k⬘兲 = − ␭ᐉ⌫ᐉ共k兲⌫ᐉ共k⬘兲,
1
2V
共2兲
共3兲
where ␭ᐉ ⬎ 0 is the interaction strength, and the function
Vss3,s,s4共k,k⬘兲bs† ,s 共k,q兲bs3,s4共k⬘,q兲,
1 2
1 2
⌫ᐉ共k兲 =
共1兲
where sn labels the pseudospins corresponding to trapped
hyperfine states and V is the volume. These states are repre†
, and bs† ,s 共k , q兲
sented by the creation operator ak,s
1
1 2
†
†
= ak+q/2,s a−k+q/2,s . Here, ␰共k兲 = ⑀共k兲 − ␮ where ⑀共k兲
1
2
= k2 / 共2M兲 is the energy and ␮ is the chemical potential of
fermions.
The interaction term can be written in a separable form
Vss3,s,s4共k , k⬘兲 = ⌫ss3,s,s4V共k , k⬘兲, where ⌫ss3,s,s4 is the spin and
1 2
1 2
1 2
V共k , k⬘兲 is the spatial part, respectively. In the case of THS
case, where sn ⬅ 共↑ , ↓ 兲, both pseudospin singlet and pseudospin triplet pairings are allowed. However, we concentrate
on the pseudospin singlet THS state with ⌫ss3,s,s4
1 2
= ⌫ss3,s,s4␦s1,−s2␦s2,s3␦s3,−s4. In addition, we discuss the SHS case
1 2
共sn ⬅ ↑ 兲, where only pseudospin triplet pairing is allowed,
and
the
interaction
is
given
by
⌫ss3,s,s4
1 2
s3,s4
= ⌫s ,s ␦s1,s2␦s2,s3␦s3,s4␦s4,↑. In this manuscript, we analyze
1 2
THS singlet and SHS triplet cases for all allowable angular
momentum channels. THS triplet pairing is more involved
due to the more complex nature of the vector order parameters, and therefore we postpone this discussion for a future
paper.
共k/k0兲ᐉ
共1 + k2/k20兲共ᐉ+1兲/2
共4兲
describes the momentum dependence. Here, k0 ⬃ R−1
0 plays
the role of the interaction range in real space and sets the
scale at small and large momenta. In addition, the diluteness
condition 共nᐉR30 Ⰶ 1兲 requires 共k0 / kF兲3 Ⰷ 1, where nᐉ is the
density of atoms and kF is the Fermi momentum. This function reduces to ⌫ᐉ共k兲 ⬃ kᐉ for small k, and behaves as ⌫ᐉ共k兲
⬃ 1 / k for large k, which guarantees the correct qualitative
behavior expected for Vᐉ共k , k⬘兲 according to the analysis
above.
III. FUNCTIONAL INTEGRAL FORMALISM
In this section, we describe in detail the THS singlet case
for even angular momentum states. A similar approach for
the SHS triplet case for odd angular momentum states can be
found in Ref. 关34兴, and therefore we do not repeat the same
analysis here. However, we point out the main differences
between the two cases whenever it is necessary.
A. THS singlet effective action
In the imaginary-time functional integration formalism
共ប = kB = 1 and ␤ = 1 / T兲, the partition function for the THS
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PHYSICAL REVIEW A 74, 013608 共2006兲
NONZERO ORBITAL ANGULAR MOMENTUM¼
singlet case can be written as Zᐉ = 兰D共a† , a兲e−Sᐉ with action
†
Sᐉ = 兰0␤d␶关兺k,sak,s
共␶兲共⳵␶兲ak,s共␶兲 + Hᐉ共␶兲兴 where the Hamiltonian for the ᐉth angular momentum channel is Hᐉ共␶兲
4␲␭
†
†
共␶兲ak,s共␶兲 − V ᐉ 兺q,mᐉbᐉ,m
共q , ␶兲bᐉ,mᐉ共q , ␶兲.
= 兺k,s␰ᐉ共k兲ak,s
ᐉ
−1
Here, bᐉ,mᐉ共q , ␶兲 = 兺k⌫ᐉ共k兲Y ᐉ,mᐉ共k̂兲ak+q/2,↑ak−q/2,↓ and ␰ᐉ共k兲
= ⑀共k兲 − ␮ᐉ. We first introduce the Nambu spinor ␺†共p兲
= 共a†p,↑ , a−p,↓兲, where p = 共k , iw j兲 denotes both momentum and
Fermionic Matsubara frequency w j = 共2j + 1兲␲ / ␤, and use a
Hubbard-Stratonovich transformation to decouple Fermionic
and Bosonic degrees of freedom. Integration over the Fermionic part 关D共␺† , ␺兲兴 leads to the action
Seff
ᐉ =␤兺
q,mᐉ
兩⌽ᐉ,mᐉ共q兲兩2
4␲V−1␭ᐉ
+ 兺 关␤␰ᐉ共k兲␦q,0 − Tr ln共Gᐉ/␤兲−1兴,
p,q
where q = 共q , iv j兲, with Bosonic Matsubara frequency v j
*
= 2␲ j / ␤.
Here,
G−1
ᐉ = ⌽ᐉ共q兲⌫ᐉ共p兲␴− + ⌽ᐉ共−q兲⌫ᐉ共p兲␴+
+ 关iw j␴0 − ␰ᐉ共k兲␴3兴␦q,0 is the inverse Nambu propagator,
⌽ᐉ共q兲 = 兺mᐉ⌽ᐉ,mᐉ共q兲Y ᐉ,mᐉ共k̂兲 is the Bosonic field, and ␴±
= 共␴1 ± ␴2兲 / 2 and ␴i is the Pauli spin matrix. The Bosonic
field ⌽ᐉ,mᐉ共q兲 = ⌬ᐉ,mᐉ␦q,0 + ⌳ᐉ,mᐉ共q兲 has ␶-independent ⌬ᐉ,mᐉ
and ␶-dependent ⌳ᐉ,mᐉ共q兲 parts.
Performing an expansion in Seff
ᐉ to quadratic order in
⌳ᐉ,mᐉ共q兲 leads to
␤
−1
†
˜
共q兲Fᐉ,m ,m⬘共q兲⌳
兺 ⌳˜ ᐉ,m
ᐉ,mᐉ⬘共q兲,
ᐉ
ᐉ ᐉ
2
q,m ,m
Ssp
ᐉ +
ᐉ
共6兲
ᐉ⬘
˜ † 共q兲 is such that ⌳
˜ † 共q兲
where the vector ⌳
ᐉ,mᐉ
ᐉ,mᐉ
−1
†
= 关⌳ᐉ,m 共q兲 , ⌳ᐉ,mᐉ共−q兲兴, and Fᐉ,m ,m⬘共q兲 are the matrix eleᐉ
ᐉ
⍀sp
ᐉ =
ᐉ
ments of the inverse fluctuation propagator matrix F−1
ᐉ 共q兲.
Furthermore, Ssp
is
the
saddle-point
action
given
by
Ssp
ᐉ
ᐉ
−1
= ␤兺mᐉ 4␲V−1␭ + 兺 p关␤␰ᐉ共k兲 − Tr ln共Gsp
ᐉ / ␤兲 兴, and the saddleᐉ
−1
point inverse Nambu propagator is 共Gsp
ᐉ 兲 = iw j␴0 − ␰ᐉ共k兲␴3
*
+ ⌬ᐉ共k兲␴− + ⌬ᐉ共k兲␴+, with saddle-point order parameter
⌬ᐉ共k兲 = ⌫ᐉ共k兲兺mᐉ⌬ᐉ,mᐉY ᐉ,mᐉ共k̂兲. Notice that ⌬ᐉ共k兲 may involve several different mᐉ for a given angular momentum
channel ᐉ.
The matrix elements of the inverse fluctuation matrix
F−1
ᐉ 共q兲 are given by
−1
ᐉ,mᐉ⬘
兲11 = −
冉 冊
*
ᐉ
−1
共Fᐉ,m
兲12 =
ᐉ,mᐉ⬘
冉 冊
1
q
q
兺 共Gspᐉ 兲11 2 + p 共Gspᐉ 兲11 2 − p
␤ p
⫻⌫2ᐉ共p兲Y ᐉ,mᐉ共k̂兲Y ᐉ,m⬘共k̂兲 +
冉 冊
␦mᐉ,mᐉ⬘V
4␲␭ᐉ
,
−
*
ᐉ
−1
−1
共8兲
Notice that while 共Fᐉ,m ,m⬘兲12共q兲 = 共Fᐉ,m ,m⬘兲21共q兲 are even unᐉ ᐉ
ᐉ ᐉ
der the transformations q → −q and iv j → −iv j;
␰ᐉ共k兲 − Eᐉ共k兲
冊
共9兲
1
兺 ln det关F−1ᐉ 共q兲/共2␤兲兴
␤ q
共10兲
are the saddle-point and fluctuation contributions, respectively. Here, Eᐉ共k兲 = 关␰2ᐉ共k兲 + 兩⌬ᐉ共k兲兩2兴1/2 is the quasiparticle
energy spectrum. Having completed the presentation of the
functional integral formalism, we discuss next the selfconsistency equations for the order parameter and the chemical potential.
B. Self-consistency equations
*
The saddle-point condition ␦Ssp
ᐉ / ␦⌬ᐉ,mᐉ = 0 leads to the
order-parameter equation
*
⌬ᐉ共k兲⌫ᐉ共k兲Y ᐉ,m
共k̂兲
1
␤Eᐉ共k兲
ᐉ
tanh
= 兺
,
2
2Eᐉ共k兲
4␲␭ᐉ V k
⌬ᐉ,mᐉ
共11兲
which can be expressed in terms of experimentally relevant
parameters via the T-matrix approach 关32兴.
The low-energy two-body scattering amplitude between a
pair of fermions in the ᐉth angular momentum channel is
given by 关38兴
f ᐉ共k兲 = −
k2ᐉ
,
1/aᐉ − rᐉk2 + ik2ᐉ+1
共12兲
where rᐉ ⬍ 0 and aᐉ are the effective range and scattering
parameter, respectively. Here rᐉ has dimensions of L2ᐉ−1 and
aᐉ has dimensions of L2ᐉ+1, where L is the size of the system.
The energy of the two-body bound state is determined from
the poles of f ᐉ共k → i␬ᐉ兲, and is given by Eb,ᐉ = −␬2ᐉ / 共2M兲.
Bound states occur when a0 ⬎ 0 for ᐉ = 0, and aᐉ⫽0rᐉ⫽0 ⬍ 0
for ᐉ ⫽ 0. Since rᐉ ⬍ 0, bound states occur only when aᐉ
⬎ 0 for all ᐉ, in which case the binding energies are given by
共7兲
冉 冊
冉
2
ln兵1 + exp关− ␤Eᐉ共k兲兴其 ,
␤
Eb,0 = −
1
q
q
+ p 共Gsp
−p
共Gsp
兺
ᐉ 兲12
ᐉ 兲12
␤ p
2
2
⫻⌫2ᐉ共p兲Y ᐉ,mᐉ共k̂兲Y ᐉ,m⬘共k̂兲.
+兺
兺
−1
k
m 4␲V ␭ᐉ
ᐉ
兩⌬ᐉ,mᐉ兩2
共Fᐉ,m
兩⌬ᐉ,mᐉ兩2
⍀fluct
=
ᐉ
共5兲
=
Sgauss
ᐉ
−1
共Fᐉ,m ,m⬘兲11共q兲 = 共Fᐉ,m ,m⬘兲22共−q兲 are even only under q → −q,
ᐉ ᐉ
ᐉ ᐉ
having no defined parity in iv j.
The Gaussian action Eq. 共6兲 leads to the thermodynamic
fluct
= ⍀sp
potential ⍀gauss
ᐉ
ᐉ + ⍀ᐉ , where
Eb,ᐉ⫽0 =
1
Ma20
,
1
.
Maᐉrᐉ
共13兲
共14兲
Notice that only a single parameter 共a0兲 is sufficient to describe the low-energy two-body problem for ᐉ = 0, while two
parameters 共aᐉ , rᐉ兲 are necessary to describe the same problem for ᐉ ⫽ 0. The point at which 1 / 共kF2ᐉ+1aᐉ兲 = 0 corresponds
to the threshold for the formation of a two-body bound state
in vacuum. Beyond this threshold, a0 for ᐉ = 0 and 兩aᐉ⫽0rᐉ⫽0兩
for ᐉ ⫽ 0 are the size of the bound states.
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PHYSICAL REVIEW A 74, 013608 共2006兲
M. ISKIN AND C. A. R. SÁ DE MELO
For any ᐉ, the two-body scattering amplitude is related to
the T matrix via f ᐉ共k兲 = −M / 共4␲兲Tᐉ关k , k ; 2⑀共k兲 + i0+兴, where
the T matrix is given by
T共k,k⬘,E兲 = V共k,k⬘兲 +
1
V共k,k 兲T共k ,k⬘,E兲
⬙ ⬙
.
兺
Vk
E − 2⑀共k 兲 + i0+
⬙
⬙
tant difference between pseudospin singlet and pseudospin
triplet states. For pseudospin singlet states, the order parameter is a scalar function of k, while it is a vector function for
pseudospin triplet states discussed next.
In general, the triplet order parameter can be written in
the standard form 关39兴
Using the spherical harmonics expansion for both V共k , k⬘兲
and T共k , k⬘ , E兲 leads to two coupled equations,
⌫2ᐉ共k兲
1
1
M
,
=−
+ 兺
2ᐉ
␭ᐉ
4␲k0 aᐉ V k 2⑀共k兲
共15兲
⌫2ᐉ共k兲 ᐉ + 1
␲k2ᐉ
0
−
,
兺
M 2V k ⑀2共k兲 k20aᐉ
共16兲
rᐉ⫽0 = −
relating ␭ᐉ and k0 to aᐉ and rᐉ. Except for notational differences, notice that these relations are identical to previous
results 关32兴. After performing momentum integrations we
obtain
aᐉ =
k2ᐉ+1
0
−
Mk0␭ᐉ冑␲
,
˜ ᐉ − 4␲冑␲
Mk0␭ᐉ␾
共17兲
2k20冑␲
1
= 2ᐉ+1
,
aᐉ⫽0rᐉ⫽0 k0 aᐉ␾ᐉ + 2共ᐉ + 1兲冑␲
共18兲
˜ ᐉ = ⌫共ᐉ + 1 / 2兲 / ⌫共ᐉ + 1兲 and ␾ᐉ = ⌫共ᐉ − 1 / 2兲 / ⌫共ᐉ + 1兲.
where ␾
Here ⌫共x兲 is the gamma function. Notice that k2ᐉ+1
aᐉ di0
˜ ᐉ = 4␲冑␲, which corverges and changes sign when Mk0␭ᐉ␾
responds to the critical coupling for Feshbach resonances
共the unitarity limit兲.
In addition, the scattering parameter has a maximum
value in the zero 共␭ᐉ → 0兲 and a minimum value in the infinite 共␭ᐉ → ⬁兲 coupling limits given, respectively, by
max
冑 ˜
aᐉ⫽0
= −2共ᐉ + 1兲冑␲ / ␾ᐉ 共aᐉ ⬍ 0兲 and k2ᐉ+1
amin
k2ᐉ+1
0
0
ᐉ = ␲ / ␾ᐉ
共aᐉ ⬎ 0兲. The first condition 共when ␭ᐉ → 0兲 follows from Eq.
共18兲 where rᐉ⫽0 ⬍ 0 has to be satisfied for all possible aᐉ⫽0.
However, there is no condition on r0 for ᐉ = 0, and k0amax
0
= 0 in the BCS limit. The second condition 共when ␭ᐉ → ⬁兲
follows from Eq. 共17兲, which is valid for all possible ᐉ. The
minimum aᐉ for a finite range interaction is associated with
the Pauli principle, which prevents two identical fermions to
occupy the same state. Thus while the scattering parameter
cannot be arbitrarily small for a finite range potential, it may
go to zero as k0 → ⬁. Furthermore, the binding energy is
aᐉ␾ᐉ兲, when k2ᐉ+1
a ᐉ␾ ᐉ
given by Eb,ᐉ⫽0 = −2冑␲ / 共Mk2ᐉ−1
0
0
Ⰷ 2共ᐉ + 1兲冑␲.
Thus the order-parameter equation in terms of the scattering parameter is rewritten as
MV⌬ᐉ,mᐉ
16␲2k2ᐉ
0 aᐉ
=
兺
k,mᐉ⬘
冋
tanh关␤Eᐉ共k兲/2兴
1
−
2⑀共k兲
2Eᐉ共k兲
*
⫻⌬ᐉ,m⬘⌫2ᐉ共k兲Y ᐉ,m
共k̂兲Y ᐉ,m⬘共k̂兲.
ᐉ
ᐉ
ᐉ
册
共19兲
This equation is valid for both THS pseudospin singlet and
SHS pseudospin triplet states. However, there is one impor-
Oᐉ共k兲 =
冉
− dxᐉ共k兲 + idᐉy 共k兲
dzᐉ共k兲
dzᐉ共k兲
dxᐉ共k兲 + idᐉy 共k兲
冊
,
共20兲
where the vector dᐉ共k兲 = 关dxᐉ共k兲 , dᐉy 共k兲 , dzᐉ共k兲兴 is an odd function of k. Therefore all up-up, down-down, and up-down
components may exist for a THS pseudospin triplet interaction. However, in the SHS pseudospin triplet case only the
up-up or down-down component may exist leading to
⌬ᐉ共k兲 ⬀ 共Oᐉ兲s1s1共k兲. Thus for the up-up case dzᐉ共k兲 = 0 and
dxᐉ共k兲 = −idᐉy 共k兲, leading to dᐉ共k兲 = dxᐉ共k兲共1 , i , 0兲, which breaks
time-reversal symmetry, as expected from a fully spin polarized state. The corresponding down-down state has dᐉ共k兲
= dxᐉ共k兲共1 , −i , 0兲. Furthermore, the simplified form of the
SHS triplet order parameter allows a treatment similar to that
of THS singlet states. However, it is important to mention
that the THS triplet case can be investigated using our approach, but the treatment is more complicated.
The order-parameter equation has to be solved selfconsistently with the number equation Nᐉ = −⳵⍀ᐉ / ⳵␮ᐉ where
⍀ᐉ is the full thermodynamic potential. In the approximafluct
= Nsp
has two contributions. The
tions used, Nᐉ ⬇ Ngauss
ᐉ
ᐉ + Nᐉ
saddle-point contribution to the number equation is
Nsp
ᐉ = 兺 nᐉ共k兲,
共21兲
k,s
where nᐉ共k兲 is the momentum distribution given by
nᐉ共k兲 =
冋
册
1
␰ᐉ共k兲
␤Eᐉ共k兲
1−
.
tanh
2
2
Eᐉ共k兲
共22兲
For the SHS triplet case, the summation over s is not present
in Nsp
ᐉ . The fluctuation contribution to the number equation is
Nfluct
=−
ᐉ
⳵关det F−1
1
ᐉ 共q兲兴/⳵␮ᐉ
,
兺
␤ q
det F−1
ᐉ 共q兲
共23兲
where F−1
ᐉ 共q兲 is the inverse fluctuation matrix defined in Eqs.
共7兲 and 共8兲.
In the rest of the paper, we analyze analytically the superfluid properties at zero temperature 共ground state兲 and near
the critical temperatures for the THS singlet 共only even ᐉ兲
and the SHS triplet 共only odd ᐉ兲 cases. In addition, we analyze numerically the s-wave 共ᐉ = 0兲 channel of the THS singlet and the p-wave 共ᐉ = 1兲 channel of the SHS triplet cases,
which are currently of intense theoretical and experimental
interest in ultracold Fermi atoms.
IV. BCS TO BEC EVOLUTION AT T = 0
At low temperatures, the saddle-point self-consistent 共order parameter and number兲 equations are sufficient to describe ground-state properties in the weak-coupling BCS and
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PHYSICAL REVIEW A 74, 013608 共2006兲
NONZERO ORBITAL ANGULAR MOMENTUM¼
strong-coupling BEC limits 关10兴. However, fluctuation corrections to the number equation may be important in the
intermediate regime 关40兴.
Ground-state properties 共T = 0兲 are investigated by solving
saddle-point self-consistency 共order parameter and number兲
equations to obtain ⌬ᐉ,mᐉ and ␮ᐉ, which are discussed next.
A. Order parameter and chemical potential
We discuss in this section ⌬ᐉ,mᐉ and ␮ᐉ. In weak coupling,
we first introduce a shell about the Fermi energy 兩␰ᐉ共k兲兩
艋 wD such that ⑀F Ⰷ wD Ⰷ ⌬ᐉ共kF兲, inside of which one may
ignore the 3D density-of-states factor 共冑⑀ / ⑀F兲 and outside of
which one may ignore ⌬ᐉ共k兲. While in sufficiently strong
coupling, we use ␰ᐉ共k兲 Ⰷ 兩⌬ᐉ共k兲兩 to derive the analytic results
discussed below. It is important to notice that, in strictly
weak and strong coupling, the self-consistency equations
共21兲 and 共19兲 are decoupled, and play reversed roles. In weak
共strong兲 coupling the order-parameter equation determines
⌬ᐉ,mᐉ 共␮ᐉ兲 and the number equation determine ␮ᐉ 共⌬ᐉ,mᐉ兲.
In weak coupling, the number equation Eq. 共21兲 leads to
␮ᐉ = ⑀F
共24兲
⑀F = kF2 / 共2M兲
for any ᐉ where
is the Fermi energy. In strong
coupling, the order-parameter equation 共19兲 leads to
␮0 = −
␮ᐉ⫽0 = −
1
2Ma20
共25兲
,
冑␲
共26兲
,
Mk2ᐉ−1
a ᐉ␾ ᐉ
0
where ␾ᐉ = ⌫共ᐉ − 1 / 2兲 / ⌫共ᐉ + 1兲 and ⌫共x兲 is the Gamma function. This calculation requires that a0k0 ⬎ 1 for ᐉ = 0 and that
aᐉ␾ᐉ ⬎ 共ᐉ + 1兲冑␲ for ᐉ ⫽ 0 for the order-parameter equak2ᐉ+1
0
tion to have a solution with ␮ᐉ ⬍ 0 in the strong-coupling
limit. In the BEC limit ␮0 = −k20 / 关2M共k0a0 − 1兲2兴 for ᐉ = 0.
Notice that ␮0 = −1 / 共2Ma20兲 when k0a0 Ⰷ 1 关or 兩␮0兩 Ⰶ ⑀0
= k20 / 共2M兲兴, and thus we recover the contact potential 共k0
→ ⬁兲 result. In the same spirit, to obtain the expressions in
Eqs. 共25兲 and 共26兲, we assumed 兩␮ᐉ兩 Ⰶ ⑀0. Notice that ␮ᐉ
= Eb,ᐉ / 2 in this limit for any ᐉ.
On the other hand, the solution of the order-parameter
equation in the weak-coupling limit is
冋
冉 冊 冋冉 冊
兩⌬0,0兩 = 16冑␲⑀F exp − 2 +
兩⌬ᐉ⫽0,mᐉ兩 ⬃
k0
kF
ᐉ
⑀F exp tᐉ
册
␲ kF
␲
−
,
2 k0 2kF兩a0兩
k0
kF
2ᐉ−1
−
␲
2kF2ᐉ+1兩aᐉ兩
共27兲
册
FIG. 1. Plot of reduced order parameter ⌬r = 兩⌬0,0兩 / ⑀F vs interaction strength 1 / 共kFa0兲 for k0 ⬇ 200kF.
兩⌬ᐉ⫽0,m 兩2 =
兺
m
ᐉ
ᐉ
64冑␲
⑀F共⑀F⑀0兲1/2
3␾ᐉ
共30兲
to order ␮ᐉ / ⑀0, where we assumed that ␰ᐉ共k兲 Ⰷ 兩⌬ᐉ共k兲兩 for
sufficiently strong couplings with 兩␮ᐉ兩 Ⰶ ⑀0.
Next, we present numerical results for two particular
states. First, we analyze the THS s-wave 共ᐉ = 0, mᐉ = 0兲 case,
where ⌬0共k兲 = ⌬0,0⌫0共k兲Y 0,0共k̂兲 with Y 0,0共k̂兲 = 1 / 冑4␲. Second, we discuss the SHS p-wave 共ᐉ = 1, mᐉ = 0兲 case, where
⌬1共k兲 = ⌬1,0⌫1共k兲Y 1,0共k̂兲 with Y 1,0共k̂兲 = 冑3 / 共4␲兲cos共␪k兲. In
all numerical calculations, we choose k0 ⬇ 200kF to compare
s-wave and p-wave cases.
In Figs. 1 and 2, we show 兩⌬0,0兩 and ␮0 at T = 0 for the
s-wave case. Notice that the BCS to BEC evolution range in
1 / 共kFa0兲 is of order 1. Furthermore, 兩⌬0,0兩 grows continuously without saturation with increasing coupling, while ␮0
changes from ⑀F to Eb,0 / 2 continuously and decreases as
−1 / 共2Ma20兲 for strong couplings. Thus the evolution of 兩⌬0,0兩
and ␮0 as a function of 1 / 共kFa0兲 is smooth. For completeness, it is also possible to obtain analytical values of a0 and
⌬0,0 when the chemical potential vanishes. When ␮0 = 0, we
at
obtain
for
兩⌬0,0兩 = 8⑀F关␲2冑␲ / ⌫4共1 / 4兲兴1/3 ⬇ 3.73⑀F
3冑
which
1 / 共kFa0兲 = 共2␲ ␲⑀F / 兩⌬0,0兩兲1/2 / 关2⌫2共3 / 4兲兴 ⬇ 0.554,
also agrees with the numerical results. Here ⌫共x兲 is the
gamma function.
In Figs. 3 and 4, we show 兩⌬1,0兩 and ␮1 at T = 0 for the
p-wave case. Notice that the BCS to BEC evolution range in
1 / 共kF3a1兲 is of order k0 / kF. Furthermore, 兩⌬1,0兩 grows with
increasing coupling but saturates for large 1 / 共kF3a1兲, while ␮1
changes from ⑀F to Eb,1 / 2 continuously and decreases as
−1 / 共Mk0a1兲 for strong couplings. For completeness, we
,
共28兲
ᐉ+1
where t1 = ␲ / 4 and tᐉ⬎1 = ␲2 共2ᐉ − 3兲!! / ᐉ!. These expressions are valid only when the exponential terms are small.
The solution of the number equation in the strong-coupling
limit is
兩⌬0,0兩 = 8⑀F
冉 冊
␮0
9⑀F
1/4
,
共29兲
FIG. 2. Plot of reduced chemical potential ␮r = ␮0 / ⑀F 共inset兲 vs
interaction strength 1 / 共kFa0兲 for k0 ⬇ 200kF.
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M. ISKIN AND C. A. R. SÁ DE MELO
FIG. 3. Plots of reduced order parameter ⌬r = 兩⌬1,0兩 / ⑀F vs interaction strength 1 / 共kF3a1兲 for k0 ⬇ 200kF.
present the limiting expressions 兩⌬1,0兩 = 24 kF0 ⑀F exp关− 38 + 4kF0
k
␲k
− 2k3␲兩a 兩 兴 and 兩⌬1,0兩 = 8⑀F共 9⑀0F 兲 , in the weak- and strongF 1
coupling limits, respectively.
The evolution of 兩⌬1,0兩 and ␮1 are qualitatively similar to
recent T = 0 results for THS fermion 关32兴 and SHS fermionboson 关26兴 models. Due to the angular dependence of ⌬1共k兲,
the quasiparticle excitation spectrum E1共k兲 is gapless for
␮1 ⬎ 0, and fully gapped for ␮1 ⬍ 0. Furthermore, both ⌬1,0
and ␮1 are nonanalytic exactly when ␮1 crosses the bottom
of the fermion energy band ␮1 = 0 at 1 / 共kF3a1兲 ⬇ 0.48. The
nonanalyticity does not occur in the first derivative of ⌬1,0 or
␮1 as it is the case in two dimensions 关35兴, but occurs in the
second and higher derivatives. Thus, in the p-wave case, the
BCS to BEC evolution is not a crossover, but a quantum
phase transition occurs, as can be seen in the quasiparticle
excitation spectrum to be discussed next.
⑀
1/4
B. Quasiparticle excitations
The quasiparticle excitation spectrum Eᐉ共k兲 = 关␰2ᐉ共k兲
+ 兩⌬ᐉ共k兲兩2兴1/2 is gapless at k-space regions where the conditions ⌬ᐉ共k兲 = 0 and ⑀共k兲 = ␮ᐉ are both satisfied. Notice that
the second condition is only satisfied in the BCS side ␮ᐉ
艌 0, and therefore the excitation spectrum is always gapped
in the BEC side 共␮ᐉ ⬍ 0兲.
For ᐉ = 0, the order parameter is isotropic in k space without zeros 共nodes兲 since it does not have any angular dependence. Therefore the quasiparticle excitation spectrum is
fully gapped in both BCS 共␮0 ⬎ 0兲 and BEC 共␮0 ⬍ 0兲 sides,
since
min兵E0共k兲其 = 兩⌬0共k␮0兲兩
共␮0 ⬎ 0兲,
共31兲
FIG. 5. 共Color online兲 Plots of quasiparticle excitation spectrum
E0共kx = 0 , ky , kz兲 when 共a兲 ␮0 ⬎ 0 共BCS side兲 for 1 / 共kFa0兲 = −1 and
共b兲 ␮0 ⬍ 0 共BEC side兲 for 1 / 共kFa0兲 = 1 vs momentum ky / kF and
k z / k F.
min兵E0共k兲其 = 冑兩⌬0共0兲兩2 + ␮20
共32兲
Here, k␮ᐉ = 冑2M ␮ᐉ. This implies that the evolution of the
quasiparticle excitation spectrum from weak-coupling BCS
to strong-coupling BEC regime is smooth when ␮0 = 0 for
ᐉ = 0 pairing.
In Fig. 5, we show E0共kx = 0 , ky , kz兲 for an s-wave 共ᐉ = 0,
mᐉ = 0兲 superfluid when 共a兲 ␮0 ⬎ 0 共BCS side兲 for 1 / 共kFa0兲
= −1 and 共b兲 ␮0 ⬍ 0 共BEC side兲 for 1 / 共kFa0兲 = 1. Notice that
the quasiparticle excitation spectrum is gapped for both
cases. However, the situation for ᐉ ⫽ 0 is very different as
discussed next.
For ᐉ ⫽ 0, the order parameter is anisotropic in k space
with zeros 共nodes兲 since it has an angular dependence.
Therefore while the quasiparticle excitation spectrum is gapless in the BCS 共␮ᐉ⫽0 ⬎ 0兲 side, it is fully gapped in the BEC
共␮ᐉ⫽0 ⬍ 0兲 side, since
min兵Eᐉ⫽0共k兲其 = 0
min兵Eᐉ⫽0共k兲其 = 兩␮ᐉ兩
FIG. 4. Plots of reduced chemical potential ␮r = ␮1 / ⑀F 共inset兲 vs
interaction strength 1 / 共kF3a1兲 for k0 ⬇ 200kF.
共␮0 ⬍ 0兲.
共␮ᐉ ⬎ 0兲,
共␮ᐉ ⬍ 0兲.
共33兲
共34兲
This implies that the evolution of quasiparticle excitation
spectrum from weak-coupling BCS to strong-coupling BEC
regime is not smooth for ᐉ ⫽ 0 pairing having a nonanalytic
behavior when ␮ᐉ⫽0 = 0. This signals a quantum phase transition from a gapless to a fully gapped state exactly when
␮ᐉ⫽0 drops below the bottom of the energy band ␮ᐉ⫽0 = 0.
In Fig. 6, we show E1共kx = 0 , ky , kz兲 for a p-wave 共ᐉ = 1,
mᐉ = 0兲 superfluid when 共a兲 ␮1 ⬎ 0 共BCS side兲 for 1 / 共kF3a1兲
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FIG. 6. 共Color online兲 Plots of quasiparticle excitation spectrum
E1共kx = 0 , ky , kz兲 in 共a兲 ␮1 ⬎ 0 共BCS side兲 for 1 / 共kF3a1兲 = −1 and 共b兲
␮1 ⬍ 0 共BEC side兲 for 1 / 共kF3a1兲 = 1 vs momentum ky / kF and kz / kF.
FIG. 7. 共Color online兲 Contour plots of momentum distribution
n0共kx = 0 , ky , kz兲 when 共a兲 ␮0 ⬎ 0 共BCS side兲 for 1 / 共kFa0兲 = −1 and
共b兲 ␮0 ⬍ 0 共BEC side兲 for 1 / 共kFa0兲 = 1 vs momentum ky / kF and
k z / k F.
= −1 and 共b兲 ␮1 ⬍ 0 共BEC side兲 for 1 / 共kF3a1兲 = 1. The quasiparticle excitation spectrum is gapless when ⌬1共k兲 ⬀ kz / kF
= 0 and k2x + k2y + kz2 = 2M ␮1 are both satisfied in certain regions
of k space. For kx = 0, these conditions are met only when
kz = 0 and ky = ± 冑2M ␮1 for a given ␮1. Notice that these
points come closer as the interaction 共␮1兲 increases 共decreases兲, and when ␮1 = 0 they become degenerate at k = 0.
For ␮1 ⬍ 0, the second condition cannot be satisfied, and thus
a gap opens in the excitation spectrum of quasiparticles as
shown in Fig. 6共b兲.
The spectrum of quasiparticles plays an important role in
the thermodynamic properties of the evolution from BCS to
BEC regime at low temperatures. For ᐉ = 0, thermodynamic
quantities depend exponentially on T throughout the evolution. Thus a smooth crossover occurs at ␮0 = 0. However, for
ᐉ ⫽ 0, thermodynamic quantities depend exponentially on T
only in the BEC side, while they have a power-law dependence on T in the BCS side. Thus a nonanalytic evolution
occurs at ␮ᐉ⫽0 = 0. This can be seen best in the momentum
distribution which is discussed next.
the interaction increases the Fermi sea with locus ␰0共k兲 = 0 is
suppressed, and pairs of atoms with opposite momenta become more tightly bound. As a result, n0共k兲 broadens in the
BEC side since fermions with larger momentum participate
in the formation of bound states. Notice that the evolution is
a crossover without any qualitative change. Furthermore,
n0共kx , ky = 0 , kz兲 and n0共kx , ky , kz = 0兲 can be trivially obtained
from n0共kx = 0 , ky , kz兲, since n0共kx , ky , kz兲 is symmetric in kx,
ky, and kz.
In Fig. 8, we show n1共kx = 0 , ky , kz兲 for a p-wave 共ᐉ = 1,
mᐉ = 0兲 superfluid when 共a兲 ␮1 ⬎ 0 共BCS side兲 for 1 / 共kF3a1兲
= −1 and 共b兲 ␮1 ⬍ 0 共BEC side兲 for 1 / 共kF3a1兲 = 1. Notice that
n1共kx = 0 , ky , kz兲 is largest in the BCS side when kz / kF = 0, but
it vanishes along kz / kF = 0 for any ky / kF in the BEC side. As
the interaction increases the Fermi sea with locus ␰1共k兲 = 0 is
suppressed, and pairs of atoms with opposite momenta become more tightly bound. As a result, the large momentum
distribution in the vicinity of k = 0 splits into two peaks
around finite k reflecting the p-wave symmetry of these
tightly bound states. Furthermore, n1共kx , ky , kz = 0兲 = 兵1
− sgn关␰1共k兲兴其 / 2 for any ␮1, and n1共kx , ky = 0 , kz兲 is trivially
obtained from n1共kx = 0 , ky , kz兲, since n1共k兲 is symmetric in
kx,ky. Here, sgn is the sign function.
Thus n1共k兲 for the p-wave case has a major rearrangement
in k space with increasing interaction, in sharp contrast to the
s-wave case. This qualitative difference between p-wave and
s-wave symmetries around k = 0 explicitly shows a direct
measurable consequence of the gapless to gapped quantum
phase transition when ␮1 = 0, since n1共k兲 depends explicitly
on E1共k兲. These quantum phase transitions are present in all
C. Momentum distribution
In this section, we analyze the momentum distribution
nᐉ共k兲 = 关1 − ␰ᐉ共k兲 / Eᐉ共k兲兴 / 2 in the BCS 共␮ᐉ ⬎ 0兲 and BEC
sides 共␮ᐉ ⬍ 0兲, which reflect the gapless to gapped phase
transition for nonzero angular momentum superfluids.
In Fig. 7, we show n0共kx = 0 , ky , kz兲 for an s-wave 共ᐉ
= 0 , mᐉ = 0兲 superfluid when 共a兲 ␮0 ⬎ 0 共BCS side兲 for
1 / 共kFa0兲 = −1 and 共b兲 ␮0 ⬍ 0 共BEC side兲 for 1 / 共kFa0兲 = 1. As
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M. ISKIN AND C. A. R. SÁ DE MELO
FIG. 9. Plot of reduced isothermal atomic compressibility ␬r
= ␬T0 共0兲 / ˜␬0 vs interaction strength 1 / 共kFa0兲 for k0 ⬇ 200kF. The inset
shows the numerical derivative of d␬r / d关共kFa0兲−1兴 vs 1 / 共kFa0兲.
Here, ˜␬0 is the weak-coupling compressibility.
FIG. 8. 共Color online兲 Contour plots of momentum distribution
n1共kx = 0 , ky , kz兲 in 共a兲 ␮1 ⬎ 0 共BCS side兲 for 1 / 共kF3a1兲 = −1 and 共b兲
␮1 ⬍ 0 共BEC side兲 for 1 / 共kF3a1兲 = 1 vs momentum ky / kF and kz / kF.
nonzero angular momentum states, and can be further characterized through the atomic compressibility as discussed in
the next section.
D. Atomic compressibility
At finite temperatures, the isothermal atomic compressibility is defined by ␬Tᐉ 共T兲 = −共⳵V / ⳵P兲T,Nᐉ / V where V is the
volume and P is the pressure of the gas. This can be rewritten as
␬Tᐉ 共T兲 = −
1
N2ᐉ
冉 冊
⳵ 2⍀ ᐉ
⳵␮2ᐉ
=
T,V
冉 冊
1 ⳵Nᐉ
N2ᐉ ⳵␮ᐉ
,
共35兲
T,V
where the partial derivative ⳵Nᐉ / ⳵␮ᐉ at T ⬇ 0 is given by
兩⌬ᐉ共k兲兩
⳵Nᐉ
⳵Nsp
ᐉ
⳵␮ᐉ ⬇ ⳵␮ᐉ = 兺k,s 2E3ᐉ共k兲
2
.
The expression above leads to ␬T0 共0兲 = 2N共⑀F兲 / N20 in weakcoupling BCS and ␬T0 共0兲 = 2N共⑀F兲⑀F / 共3 兩 ␮0 兩 N20兲 in strongcoupling BEC limit for ᐉ = 0, where N共⑀F兲 = MVkF / 共2␲2兲 is
the density of states per spin at the Fermi energy. Notice that
␬T0 共0兲 decreases as a20 in strong coupling since 兩␮0兩
= 1 / 共2Ma20兲. However, we only present the strong-coupling
results for higher angular momentum states since they exhibit an interesting dependence on aᐉ and k0. In the case of
T
␬ᐉ⬎1
共0兲
THS
pseudospin
singlet,
we
obtain
2
¯
= 4N共⑀F兲⑀F␾ᐉ / 共⑀0␾ᐉNᐉ兲 for ᐉ ⬎ 1, while in the case of SHS
states we obtain ␬T1 共0兲 = N共⑀F兲⑀F / 共冑⑀0兩␮1兩N2ᐉ兲 for ᐉ = 1 and
T
¯ ᐉ / 共⑀0␾ᐉN2兲 for ᐉ ⬎ 1. Here ␾ᐉ = ⌫共ᐉ
␬ᐉ⬎1
共0兲 = 2N共⑀F兲⑀F␾
ᐉ
¯ ᐉ = ⌫共ᐉ − 3 / 2兲 / ⌫共ᐉ + 1兲, where ⌫共x兲 is
− 1 / 2兲 / ⌫共ᐉ + 1兲 and ␾
the gamma function. Notice that ␬T1 共0兲 decreases as 冑a1 for
T
ᐉ = 1 since 兩␮1兩 = 1共Mk0a1兲 and ␬ᐉ⬎1
共0兲 is a constant for ᐉ
⬎ 1 in strong coupling.
In Fig. 9, we show the evolution of ␬T0 共0兲 for a s-wave
共ᐉ = 0 , mᐉ = 0兲 superfluid from the BCS to the BEC regime.
␬T0 共0兲 decreases continuously, and thus the evolution is a
crossover 共smooth兲 as can be seen in the inset where the
numerical derivative of ␬T0 共0兲 with respect to 1 / 共kFa0兲 is
shown 兵d␬T0 共0兲 / d关共kFa0兲−1兴其. This decrease is associated with
the increase of the gap of the excitation spectrum as a function of 1 / 共kFa0兲. In this approximation, the gas is incompressible 关␬T0 共0兲 → 0兴 in the extreme BEC limit.
In Fig. 10, we show the evolution of ␬T1 共0兲 for a p-wave
共ᐉ = 1, mᐉ = 0兲 superfluid from the BCS to the BEC regime.
Notice that there is a change in qualitative behavior when
␮1 = 0 at 1 / 共kF3a1兲 ⬇ 0.48 as can be seen in the inset where the
numerical derivative of ␬T1 共0兲 with respect to 1 / 共kF3a1兲 is
shown 兵d␬T1 共0兲 / d关共kF3 a1兲−1兴其. Thus the evolution from BCS
to BEC is not a crossover, but a quantum phase transition
occurs when ␮1 = 0 关25,33–35兴.
The nonanalytic behavior occurring when ␮ᐉ⫽0 = 0 can be
understood from higher derivatives of ␬ᐉ with respect to ␮ᐉ
given by 关 ⳵␮ᐉ 兴 T , V = −2Nᐉ关␬Tᐉ 共T兲兴2 + N12 共 ⳵␮2ᐉ 兲 . For inᐉ
ᐉ T,V
2
stance,
the
second
derivative
⳵2Nsp
ᐉ / ⳵␮ᐉ
5
2
= 3兺k , s 兩 ⌬ᐉ共k兲兩 ␰ᐉ共k兲 / 关2Eᐉ共k兲兴 tends to zero in the weak共␮ᐉ ⬇ ⑀F ⬎ 0兲 and strong- 共␮ᐉ ⬇ Eb,ᐉ / 2 ⬍ 0兲 coupling limits.
2
On the other hand, when ␮ᐉ = 0, ⳵2Nsp
ᐉ / ⳵␮ᐉ is finite only for
⳵␬Tᐉ 共T兲
⳵ 2N
FIG. 10. Plot of reduced isothermal atomic compressibility ␬r
= ␬T1 共0兲 / ˜␬1 vs interaction strength 1 / 共kF3a1兲 for k0 ⬇ 200kF. The inset
shows the numerical derivative of d␬r / d关共kF3a1兲−1兴 vs 1 / 共kF3a1兲.
Here ˜␬1 is the weak-coupling compressibility.
013608-8
PHYSICAL REVIEW A 74, 013608 共2006兲
NONZERO ORBITAL ANGULAR MOMENTUM¼
ᐉ = 0, and it diverges for ᐉ ⫽ 0. This divergence is logarithmic
for ᐉ = 1, and of higher order for ᐉ ⬎ 1. Thus we conclude
again that higher derivatives of Nsp
ᐉ are nonanalytic when
␮ᐉ⫽0 = 0, and that a quantum phase transition occurs for ᐉ
⫽ 0.
Theoretically, the calculation of the isothermal atomic
compressibility ␬Tᐉ 共T兲 is easier than the isentropic atomic
compressibility ␬Sᐉ 共T兲. However, performing measurements
of ␬Sᐉ 共T兲 may be simpler in cold Fermi gases, since the gas
expansion upon release from the trap is expected to be nearly
isentropic. Fortunately, ␬Sᐉ 共T兲 is related to ␬Tᐉ 共T兲 via the therCVᐉ 共T兲
␬Sᐉ 共T兲 = CPᐉ 共T兲 ␬Tᐉ 共T兲, where ␬Tᐉ 共T兲
heat capacities CPᐉ 共T兲 ⬎ CVᐉ 共T兲. Fur-
modynamic relation
⬎ ␬Sᐉ 共T兲 since specific
thermore, at low temperatures 共T ⬇ 0兲 the ratio
CPᐉ 共T兲 / CVᐉ 共T兲 ⬇ const, and therefore ␬Sᐉ共T ⬇ 0兲 ⬀ ␬Tᐉ 共T ⬇ 0兲.
Thus we expect qualitatively similar behavior in both the
isentropic and isothermal compressibilities at low temperatures 共T ⬇ 0兲.
The measurement of the atomic compressibility could
also be performed via an analysis of particle density fluctuations 关41,42兴. As it is well known from thermodynamics
关43兴, ␬Tᐉ 共T兲 is connected to density fluctuations via the relation 具n2ᐉ典 − 具nᐉ典2 = T具nᐉ典2␬Tᐉ 共T兲, where 具nᐉ典 is the average density of atoms. From the measurement of density fluctuations
␬Tᐉ 共T兲 can be extracted at any temperature T.
It is important to emphasize that in this quantum phase
transition at ␮ᐉ⫽0 = 0, the symmetry of the order parameter
does not change as is typical in the Landau classification of
phase transitions. However, a clear thermodynamic signature
occurs in derivatives of the compressibility suggesting that
the phase transition is higher than second order according to
Ehrenfest’s classification. Next, we discuss the phase diagram at zero temperature.
FIG. 11. The phase diagram of s-wave superfluids as a function
of 1 / 共kFa0兲.
in Fig. 11, where k␮ᐉ = 冑2M ␮ᐉ. Notice that there is no qualitative change across ␮0 = 0 at small but finite temperatures.
This indicates the absence of a QCR and confirms there is
only a crossover for s-wave 共ᐉ = 0兲 superfluids at T = 0.
For ᐉ ⫽ 0, quasiparticle excitations are gapless in the BCS
side and are only gapped in the BEC side, and therefore
while thermodynamic quantities such as atomic compressibility, specific heat, and spin susceptibility have power-law
dependences on the temperature ⬃T␤ᐉ⫽0 in the BCS side,
they have exponential dependences on the temperature and
the minimum energy of quasiparticle excitations
⬃exp关−min兵Eᐉ⫽0共k兲其 / T兴 in the BEC side. Here, ␤ᐉ⫽0 is a
real number which depends on particular ᐉ state. For ᐉ = 1,
using Eqs. 共33兲 and 共34兲 leads to ⬃T␤1 in the BCS side
共␮1 ⬎ 0兲 and ⬃exp共−兩␮1 兩 / T兲 in the BEC side 共␮1 ⬍ 0兲 as
shown in Fig. 12. Notice the change in qualitative behavior
across ␮1 = 0 共as well as other ᐉ ⫽ 0 states兲 at small but finite
temperatures. This change occurs within the QCR and signals the existence of a quantum phase transition 共T = 0兲 for
ᐉ ⫽ 0 superfluids.
V. GAUSSIAN FLUCTUATIONS
E. Phase diagram
To have a full picture of the evolution from the BCS to
the BEC limit at T = 0, it is important to analyze thermodynamic quantities at low temperatures. In particular, it is important to determine the quantum critical region 共QCR兲
where a qualitative change occurs in quantities such as the
specific heat, compressibility, and spin susceptibility. Here,
we do not discuss in detail the QCR, but we analyze the
contributions from quasiparticle excitations to thermodynamic properties. However, the discussion can be extended
to include collective excitations 关28兴 共see Sec. V兲.
Next, we point out a major difference between ᐉ = 0 and
ᐉ ⫽ 0 states in connection with the spectrum of the quasiparticle excitations 共see Sec. IV B兲 and their contribution to
low-temperature thermodynamics.
For ᐉ = 0, quasiparticle excitations are gapped for all couplings, and therefore thermodynamic quantities such as
atomic compressibility, specific heat, and spin susceptibility
have an exponential dependence on the temperature
and the minimum energy of quasiparticle excitations
⬃exp关−min兵E0共k兲其 / T兴. Using Eqs. 共31兲 and 共32兲 leads to
⬃exp关−兩⌬0共k␮0兲 兩 / T兴 in the BCS side 共␮0 ⬎ 0兲 and
⬃exp关−冑兩⌬0共0兲兩2 + ␮20 / T兴 in the BEC side 共␮0 ⬍ 0兲 as shown
Next, we discuss the fluctuation effects at zero temperature. The pole structure of Fᐉ共q , iv j兲 determines the twoparticle excitation spectrum of the superconducting state
with iv j → w + i0+, and has to be taken into account to derive
⍀fluct
ᐉ . The matrix elements of Fᐉ共q , iv j兲 are Fᐉ,mᐉ,mᐉ⬘共q , iv j兲
for a given ᐉ. We focus here only on the zero-temperature
limit and analyze the collective phase modes. In this limit,
−1
we separate the diagonal matrix elements of Fᐉ,m ,m⬘共q兲 into
ᐉ ᐉ
even and odd contributions with respect to iv j,
FIG. 12. The phase diagram of p-wave superfluids as a function
of 1 / 共kF3a1兲.
013608-9
PHYSICAL REVIEW A 74, 013608 共2006兲
M. ISKIN AND C. A. R. SÁ DE MELO
−1
共Fᐉ,m
E
兲11
=
ᐉ,mᐉ⬘
共␰ ␰ + E E 兲共E + E 兲
兺k 2E++E−−关共iv j+兲2 −− 共E++ + E−−兲2兴 ⌫2ᐉ共k兲Y ᐉ,m 共k̂兲
ᐉ
*
⫻Y ᐉ,m⬘共k̂兲 −
ᐉ
−1
共Fᐉ,m
O
兲11
=
ᐉ,mᐉ⬘
−兺
k
␦mᐉ,mᐉ⬘V
4␲␭ᐉ
共36兲
,
共␰+E− + ␰−E+兲共iv j兲
⌫2共k兲
2E+E−关共iv j兲2 − 共E+ + E−兲2兴 ᐉ
*
⫻Y ᐉ,mᐉ共k̂兲Y ᐉ,m⬘共k̂兲.
ᐉ
decay into the two quasiparticle continuum. A well defined
expansion 关collective mode dispersion w兴 must satisfy the
following condition: w Ⰶ min兵E+ + E−其. Thus a zerotemperature expansion is always possible when Landau
damping is subdominant 共underdamped regime兲. To obtain
the long-wavelength dispersions for the collective modes at
−1
to second
T = 0, we expand the matrix elements of F̃ᐉ,m
ᐉ,mᐉ
order in 兩q兩 and w to get
−1
共F̃ᐉ,m
共37兲
ᐉ,mᐉ⬘
兲11 = Aᐉ,mᐉ,m⬘ + 兺 Cᐉ,m
i,j
ᐉ
i,j
ᐉ,mᐉ⬘
qiq j − Dᐉ,mᐉ,m⬘w2 ,
ᐉ
共39兲
The off-diagonal term is even in iv j, and it reduces to
−1
共Fᐉ,m
ᐉ
兲 =−兺
,m⬘ 12
ᐉ
k
⌬+⌬−共E+ + E−兲
2E+E−关共iv j兲2 − 共E+ + E−兲2兴
*
⫻⌫2ᐉ共k兲Y ᐉ,mᐉ共k̂兲Y ᐉ,m⬘共k̂兲.
−1
共F̃ᐉ,m
Here the labels ⫾ denote that the corresponding variables are
functions of k ± q / 2.
In order to obtain the collective mode spectrum, we express
⌳ᐉ,mᐉ共q兲 = ␶ᐉ,mᐉ共q兲ei␽ᐉ,mᐉ共q兲 = 关␳ᐉ,mᐉ共q兲 + i␪ᐉ,mᐉ共q兲兴 / 冑2
where ␶ᐉ,mᐉ共q兲, ␽ᐉ,mᐉ共q兲, ␳ᐉ,mᐉ共q兲, and ␪ᐉ,mᐉ共q兲 are all real.
Notice that the new fields ␳ᐉ,mᐉ共q兲 = ␶ᐉ,mᐉ共q兲cos关␽ᐉ,mᐉ共q兲兴 and
␪ᐉ,mᐉ共q兲 = ␶ᐉ,mᐉ共q兲sin关␽ᐉ,mᐉ共q兲兴 can be regarded essentially as
the amplitude and phase fields, respectively, when ␽ᐉ,mᐉ共q兲 is
small. This change of basis can be described by the following unitary transformation:
冉 冊冋
1 1 i
冑2 1 − i
␳ᐉ,mᐉ共q兲
␪ᐉ,mᐉ共q兲
册
−1
ᐉ
−1
ᐉ
兲 = 共Fᐉ,m
,m⬘ 22
ᐉ
ᐉ
−1
ᐉ
ᐉ
ᐉ
−1
兲E − 共Fᐉ,m
,m⬘ 11
ᐉ
−1
ᐉ
兲 ; and the off-diagonal
,m⬘ 12
ᐉ
i,j
ᐉ,mᐉ⬘
qiq j − Rᐉ,mᐉ,m⬘w2 ,
ᐉ
共40兲
−1
共F̃ᐉ,m
−1
*
O
= i共Fᐉ,m ,m⬘兲11
with the
elements are 共F̃ᐉ,m ,m⬘兲12 = 共F̃ᐉ,m ,m⬘兲21
ᐉ ᐉ
ᐉ ᐉ
ᐉ ᐉ
q dependence being implicit.
A. Collective (Goldstone) modes
The collective modes are determined by the poles of the
propagator matrix Fᐉ共q兲 for the pair fluctuation fields
⌳ᐉ,mᐉ共q兲, which describe the Gaussian deviations about the
saddle-point order parameter. The poles of Fᐉ共q兲 are determined by the condition det F−1
ᐉ 共q兲 = 0, which leads to 2共2ᐉ
+ 1兲 collective 共amplitude and phase兲 modes, when the usual
analytic continuation iv j → w + i0+ is performed. Among
them, there are 2ᐉ + 1 amplitude modes which we do not
discuss here.
The easiest way to get the phase collective modes is to
integrate out the amplitude fields to obtain a phase-only effective action. Notice that for ᐉ ⫽ 0 channels at any temperature, and for ᐉ = 0 channel at finite temperature, a well defined low-frequency expansion is not possible for ␮ᐉ ⬎ 0 due
to Landau damping which causes the collective modes to
ᐉ,mᐉ⬘
兲12 = iBᐉ,mᐉ,m⬘w.
共41兲
ᐉ
The expressions for the expansion coefficients are given in
Appendix A.
i,j
i,j
= C0,0,0␦i,j and Q0,0,0
For ᐉ = 0, the coefficients C0,0,0
= Q0,0,0␦i,j are diagonal and isotropic in 共i , j兲, and P0,0,0 = 0
vanishes. Here, ␦i,j is the Kronecker delta. Thus the collective mode is the isotropic Goldstone mode with dispersion
W0,0共q兲 = C0,0兩q兩,
.
ᐉ
ᐉ
i,j
ᐉ
C0,0 =
From now on, we take ⌬ᐉ,mᐉ as real without loss of generality. The diagonal elements of the fluctuation matrix in the
−1
−1
−1
E
+ 共Fᐉ,m ,m⬘兲12, and
rotated basis are 共F̃ᐉ,m ,m⬘兲11 = 共Fᐉ,m ,m⬘兲11
共F̃ᐉ,m
兲22 = Pᐉ,mᐉ,m⬘ + 兺 Qᐉ,m
共38兲
ᐉ
⌳ᐉ,mᐉ共q兲 =
ᐉ,mᐉ⬘
冉
A0,0,0Q0,0,0
2
A0,0,0R0,0,0 + B0,0,0
共42兲
冊
1/2
共43兲
,
where C0,0 is the speed of sound. Notice that the quasiparticle
excitations are always fully gapped from weak to strong coupling, and thus the Goldstone mode is not damped at T = 0 for
all couplings.
For ᐉ ⫽ 0, the dispersion for collective modes is not easy
to extract in general, and therefore we consider the case
when only one of the spherical harmonics Y ᐉ,mᐉ共k̂兲 is dominant and characterizes the order parameter. In this case,
Pᐉ,mᐉ,mᐉ = 0 due to the order-parameter equation, and the collective mode is the anisotropic Goldstone mode with dispersion
Wᐉ⫽0,mᐉ共q兲 =
i,j
Cᐉ⫽0,m
ᐉ
=
冉
冋兺
i,j
i,j
共Cᐉ,m
兲 2q iq j
ᐉ
i,j
Aᐉ,mᐉ,mᐉQᐉ,m
册
1/2
ᐉ,mᐉ
2
Aᐉ,mᐉ,mᐉRᐉ,mᐉ,mᐉ + Bᐉ,m
ᐉ,mᐉ
共44兲
,
冊
1/2
.
共45兲
Notice that the speed of sound has a tensor structure and is
anisotropic. Furthermore, the quasiparticle excitations are
gapless when ␮ᐉ⫽0 ⬎ 0, and thus the Goldstone mode is
damped even at T = 0. However, Landau damping is subdominant and the real part of the pole dominates for small
momenta. In addition, quasiparticle excitations are fully
gapped when ␮ᐉ⫽0 ⬍ 0, and thus the Goldstone mode is not
fluct
comes from
damped. Therefore the pole contribution to ⍀ᐉ⫽0
the Goldstone mode for all couplings. In addition, there is
013608-10
PHYSICAL REVIEW A 74, 013608 共2006兲
NONZERO ORBITAL ANGULAR MOMENTUM¼
also a branch cut representing the continuum of two particle
scattering states, but the contribution from the Goldstone
mode dominates at sufficiently low temperatures.
It is also illustrative to analyze the eigenvectors of F̃−1
ᐉ 共q兲
in the amplitude-phase representation corresponding to small
Wᐉ,mᐉ共q兲 mode
冋 册冤
␳ᐉ,mᐉ共q兲
␪ᐉ,mᐉ共q兲
=
−i
Bᐉ,mᐉ,mᐉ
Aᐉ,mᐉ,mᐉ
Wᐉ,mᐉ共q兲
1
冥
.
Notice that, when Bᐉ,mᐉ,mᐉ → 0 the amplitude and phase
modes are not mixed.
Next, we discuss the dispersion of collective modes in the
weak- and strong-coupling limits, where the expansion coefficients are analytically tractable for a fixed 共ᐉ , mᐉ兲 state.
B. Weak-coupling (BCS) regime
The s-wave 共ᐉ = 0 , mᐉ = 0兲 weak-coupling limit is characterized by the criteria ␮0 ⬎ 0 and ␮0 ⬇ ⑀F Ⰷ 兩⌬0,0兩. The expansion of the matrix elements to order 兩q兩2 and w2 is performed
under the condition 关w , 兩q兩2 / 共2M兲兴 Ⰶ 兩⌬0,0兩. Analytic calculations are particularly simple in this case since all integrals for
the coefficients needed to calculate the collective-mode dispersions are peaked near the Fermi surface. We first introduce a shell about the Fermi energy 兩␰0共k兲 兩 艋 wD such that
⑀F Ⰷ wD Ⰷ ⌬0共kF兲, inside of which one may ignore the 3D
density-of-states factor 冑⑀ / ⑀F and outside of which one may
ignore ⌬0共k兲. In addition, we make use of the nearly perfect
particle-hole symmetry, which forces integrals to vanish
when their integrands are odd under the transformation
␰0共k兲 → −␰0共k兲. For instance, the coefficient that couple
phase and amplitude modes vanish 共B0,0,0 = 0兲 in this limit.
Thus there is no mixing between phase and amplitude fields
in weak coupling, as can be seen by inspection of the fluctuation matrix F̃0共q兲.
For ᐉ = 0, the zeroth-order coefficient is
A0,0,0 =
N共⑀F兲
,
4␲
共46兲
and the second-order coefficients are
N共⑀F兲vF2
Qi,j
i,j
= 0,0,0 =
␦i,j ,
C0,0,0
3
36兩⌬0,0兩2
D0,0,0 =
N共⑀F兲
R0,0,0
=
.
3
12兩⌬0,0兩2
C0,0 =
Here, vF = kF / M is the Fermi velocity and N共⑀F兲
= MVkF / 共2␲2兲 is the density of states per spin at the Fermi
energy.
2
Ⰶ Aᐉ,mᐉ,mᐉRᐉ,mᐉ,mᐉ, the
In weak coupling, since Bᐉ,m
ᐉ,mᐉ
i,j
i,j
sound velocity is simplified to Cᐉ,m ⬇ 关Qᐉ,m
/ Aᐉ,mᐉ,mᐉ兴 for
ᐉ
ᐉ,mᐉ
any ᐉ. Using the coefficients above in Eq. 共43兲, for ᐉ = 0, we
obtain
共49兲
冑3
which is the well-known Anderson-Bogoliubov relation. For
ᐉ ⫽ 0, the expansion coefficients require more detailed and
lengthy analysis, and therefore we do not discuss them here.
On the other hand, the expansion coefficients can be calculated for any ᐉ in the strong-coupling BEC regime, which is
discussed next.
C. Strong-coupling (BEC) regime
The strong-coupling limit is characterized by the criteria
␮ᐉ ⬍ 0, 兩␮ᐉ 兩 Ⰶ ⑀0 = k20 / 共2M兲, and 兩␰ᐉ共k兲 兩 Ⰷ 兩⌬ᐉ共k兲兩. The expansion of the matrix elements to order 兩q兩2 and w2 is performed under the condition 关w , 兩q兩2 / 共2M兲兴 Ⰶ 兩␮ᐉ兩. The situation encountered here is very different from the weakcoupling limit, because one can no longer invoke particlehole symmetry to simplify the calculation of many of the
coefficients appearing in the fluctuation matrix F̃ᐉ共q兲. In particular, the coefficient Bᐉ,mᐉ,mᐉ ⫽ 0 indicates that the amplitude and phase fields are mixed. Furthermore, Pᐉ,mᐉ,mᐉ = 0,
since this coefficient reduces to the order-parameter equation
in this limit.
For ᐉ = 0, the zeroth-order coefficient is
A0,0,0 =
␬兩⌬0,0兩2
,
8 ␲ 兩 ␮ 0兩
共50兲
the first-order coefficient is
B0,0,0 = ␬ ,
共51兲
and the second-order coefficients are
i,j
i,j
= Q0,0,0
=
C0,0,0
D0,0,0 = R0,0,0 =
␬
␦i,j ,
4M
共52兲
␬
,
8兩␮0兩
共53兲
where ␬ = N共⑀F兲 / 共32冑兩␮0 兩 ⑀F兲.
Using the expressions above in Eq. 共43兲, we obtain the
sound velocity
C0,0 =
共47兲
共48兲
vF
冉
兩⌬0,0兩2
32M兩␮0兩␲
冊 冑
1/2
= vF
k Fa 0
.
3␲
共54兲
Notice that the sound velocity is very small and its smallness
is controlled by the scattering length a0. Furthermore, in the
theory of weakly interacting dilute Bose gas, the sound ve2
. Making the identilocity is given by CB,0 = 4␲aB,0nB,0 / M B,0
fication that the density of pairs is nB,0 = n0 / 2, the mass of the
pairs is M B,0 = 2M, and that the Bose scattering length is
aB,0 = 2a0, it follows that Eq. 共54兲 is identical to the Bogoliubov result CB,0. Therefore our result for the Fermionic system
represents in fact a weakly interacting Bose gas in the strongcoupling limit. A better estimate for aB,0 ⬇ 0.6a0 can be
found in the literature 关44–47兴. This is also the case when we
construct the TDGL equation in Sec. VII B.
013608-11
PHYSICAL REVIEW A 74, 013608 共2006兲
M. ISKIN AND C. A. R. SÁ DE MELO
For ᐉ ⫽ 0, the zeroth-order coefficient is
Aᐉ⫽0,mᐉ,mᐉ =
ˆ ᐉ˜␬
15␾
2 ⑀ 0 冑␲
兩⌬ᐉ,mᐉ兩2␥ᐉ,兵mᐉ其 ,
共55兲
the first-order coefficient is
Bᐉ⫽0,mᐉ,m⬘ =
ᐉ
␾ᐉ˜␬
冑␲ ␦mᐉ,mᐉ⬘ ,
共56兲
and the second-order coefficients are
i,i
Cᐉ⫽0,m ,m⬘
ᐉ ᐉ
=
i,i
Qᐉ⫽0,m ,m⬘
ᐉ ᐉ
D1,mᐉ,m⬘ = R1,mᐉ,m⬘ =
ᐉ
ᐉ
=
4M 冑␲
3˜␬
8冑⑀0兩␮1兩
Dᐉ⬎1,mᐉ,m⬘ = Rᐉ⬎1,mᐉ,m⬘ =
ᐉ
␾ᐉ˜␬
ᐉ
␦mᐉ,mᐉ⬘ ,
␦mᐉ,mᐉ⬘ ,
¯ ᐉ˜␬
3␾
4 冑␲ ⑀ 0
␦mᐉ,mᐉ⬘ ,
共57兲
共58兲
共59兲
¯ᐉ
˜␬ = N共⑀F兲 / 共32冑⑀0⑀F兲 , ␾ᐉ = ⌫共ᐉ − 1 / 2兲 / ⌫共ᐉ + 1兲 , ␾
where
ˆ ᐉ = ⌫共2ᐉ − 3 / 2兲 / ⌫共2ᐉ + 2兲. Here
= ⌫共ᐉ − 3 / 2兲 / ⌫共ᐉ + 1兲, and ␾
⌫共x兲 is the gamma function, and ␥ᐉ,mᐉ is an angular averaged
quantity defined in AppendixB
2
Ⰷ Aᐉ,mᐉ,mᐉRᐉ,mᐉ,mᐉ, the
In strong coupling, since Bᐉ,m
ᐉ,mᐉ
i,j
sound
velocity
is
simplified
to
Cᐉ,m
ᐉ
i,j
2
1/2
⬇ 关Aᐉ,mᐉ,mᐉQᐉ,m ,m / Bᐉ,m ,m 兴 for any ᐉ. Using the expresᐉ ᐉ
ᐉ ᐉ
sions above in Eq. 共45兲, for ᐉ ⫽ 0, we obtain
i,i
Cᐉ⫽0,m
ᐉ
=
冉
ˆᐉ
15␥ᐉ,兵mᐉ其兩⌬ᐉ,mᐉ兩2␾
= vF
8M ␾ᐉ⑀0
冉
冊
冊
ˆᐉk
20␥ᐉ,兵mᐉ其冑␲␾
F
␾2ᐉ
k0
FIG. 13. Plots of reduced Goldstone 共sound兲 velocity 共C0,0兲r
= C0,0 / vF vs interaction strength 1 / 共kFa0兲 for k0 ⬇ 200kF.
i,j
In Fig. 14, we show the evolution of C1,0
as a function of
3
i,i
is strongly
1 / 共kFa1兲 for the p-wave case. Notice that C1,0
x,x
y,y
anisotropic in weak coupling, since C1,0 = C1,0 ⬇ 0.44vF and
z,z 冑
x,x
C1,0
= 3C1,0
⬇ 0.79vF, thus reflecting the order-parameter
i,i
is isotropic in strong coupling,
symmetry. In addition, C1,0
i,i
since C1,0 = vF冑3kF / 共2␲k0兲 ⬇ 0.049vF for k0 ⬇ 200kF, thus revealing the secondary role of the order-parameter symmetry
in this limit. The anisotropy is very small in the intermediate
z,z
is a monotoniregime beyond ␮1 ⬍ 0. Notice also that C1,0
3
cally decreasing function of 1 / 共kFa1兲 in the BCS side until
x,x
y,y
␮1 = 0, where it saturates. However, C1,0
= C1,0
is a nonmono3
tonic function of 1 / 共kFa1兲 , and it also saturates beyond ␮1
i,i
= 0. Therefore the behavior of C1,0
reflects the disappearance
of nodes of the quasiparticle energy E1共k兲 as ␮1 changes
sign.
These collective excitations may contribute significantly
to the thermodynamic potential, which is discussed next.
1/2
共60兲
1/2
.
共61兲
Therefore the sound velocity is also very small and its smallness is controlled by the interaction range k0 through the
diluteness condition, i.e., 共k0 / kF兲3 Ⰷ 1, for ᐉ ⫽ 0. Notice that
the sound velocity is independent of the scattering parameter
for ᐉ ⫽ 0.
Now, we turn our attention to a numerical analysis of the
phase collective modes during the evolution from weakcoupling BCS to strong-coupling BEC limits.
E. Corrections to ⍀sp
艎 due to collective modes
In this section, we analyze corrections to the saddle-point
thermodynamic potential ⍀sp
ᐉ due to low-energy collective
excitations. The evaluation of Bosonic Matsubara frequency
coll
sums in Eq. 共10兲 leads to ⍀fluct
ᐉ → ⍀ᐉ , where
冉
⍀coll
ᐉ = 兺 ⬘ Wᐉ共q兲 +
q
2
ln兵1 − exp关− ␤Wᐉ共q兲兴其
␤
冊
共62兲
is the collective-mode contribution to the thermodynamic potential. The prime on the summation indicates that a momen-
D. Evolution from BCS to BEC regime
We focus only on s-wave 共ᐉ = 0 , mᐉ = 0兲 and p-wave 共ᐉ
= 1 , mᐉ = 0兲 cases, since they may be the most relevant to
current experiments involving ultracold atoms.
In Fig. 13, we show the evolution of C0,0 as a function of
1 / 共kFa0兲 for the s-wave case. The weak-coupling AndersonBogoliubov velocity C0,0 = vF / 冑3 evolves continuously to the
strong-coupling Bogoliubov velocity C0,0 = vF冑kFa0 / 共3␲兲.
Notice that the sound velocity is a monotonically decreasing
function of 1 / 共kFa0兲, and the evolution across ␮0 = 0 is a
crossover.
FIG. 14. Plots of reduced Goldstone 共sound兲 velocity 共Cx,x
1,0兲r
z,z
共solid squares兲 and 共Cz,z
1,0兲r = C1,0 / vF 共hollow squares兲 vs
interaction strength 1 / 共kF3a1兲 for k0 ⬇ 200kF. The inset zooms into
the unitarity region.
= Cx,x
1,0 / vF
013608-12
PHYSICAL REVIEW A 74, 013608 共2006兲
NONZERO ORBITAL ANGULAR MOMENTUM¼
tum cutoff is required since a long-wavelength and lowfrequency approximation is used to derive the collectivemode dispersion. Notice that the first term in Eq. 共62兲
contributes to the ground-state energy of the interacting
Fermi system. This contribution is necessary to recover the
ground-state energy of the effective Bose system in the
strong-coupling limit.
The corrections to the saddle-point number equation
coll
zp
Ncoll
ᐉ = −⳵⍀ᐉ / ⳵␮ᐉ are due to the zero-point motion 共Nᐉ 兲 and
te
thermal excitation 共Nᐉ 兲 of the collective modes,
⳵
兺⬘Wᐉ共q兲,
⳵␮ᐉ q
Nzp
ᐉ =−
⬘
Nte
ᐉ =−兺
q
k
冋
ᐉ
ᐉ
ᐉ
ᐉ
−1
Lᐉ,m
ᐉ,mᐉ⬘
共q兲 =
␦mᐉ,mᐉ⬘
4␲V ␭ᐉ
−1
−兺
共64兲
册
tanh关␰ᐉ共k兲/共2Tc,ᐉ兲兴
1
−
.
2⑀共k兲
2␰ᐉ共k兲
k
共67兲
This is the generalization of the ᐉ = 0 case to ᐉ ⫽ 0, where
␰± = ␰ᐉ共k ± q / 2兲. From Sfluct
ᐉ , we obtain the thermodynamic
sp
fluct
sp
potential ⍀gauss
=
⍀
+
⍀
ᐉ
ᐉ
ᐉ , where ⍀ᐉ is the saddle-point
and
⍀fluct
contribution
with
⌬ᐉ共k兲 = 0,
ᐉ =
1
− ␤ 兺q ln det关Lᐉ共q兲 / ␤兴 is the fluctuation contribution.
We evaluate the bosonic Matsubara frequency 共iv j兲 sums
by using contour integration, and determine the branch cut
and pole terms. We use the phase shift ␸fluct
ᐉ 共q , w兲
= Arg关det Lᐉ共q , iv j → w + i0+兲兴 to replace det Lᐉ共q兲, leading to
=−兺
⍀fluct
ᐉ
q
共66兲
k,s
where nF共x兲 = 1 / 关exp共␤x兲 + 1兴 is the Fermi distribution. Notice that the summation over spins 共s兲 is not present in the
SHS case. It is important to emphasize that the inclusion of
around Tᐉ = Tc,ᐉ is essential to produce the qualitatively
Nfluct
ᐉ
correct physics with increasing coupling, as discussed next.
冕
⬁
−⬁
dw
nB共w兲˜␸fluct
ᐉ 共q,w兲,
␲
共68兲
fluct
fluct
where ˜␸fluct
and nB共x兲
ᐉ 共q , w兲 = ␸ᐉ 共q , w兲 − ␸ᐉ 共q , 0兲
= 1 / 关exp共␤x兲 − 1兴 is the Bose distribution. Notice that this
equation is the generalization of the s-wave 共ᐉ = 0兲 case
关11,12兴 for ᐉ ⫽ 0. Furthermore, the phase shift can be written
␸sc
␸bs
where ˜␸sc
as ˜␸fluct
ᐉ 共q , w兲 = ˜
ᐉ 共q , w兲 + ˜
ᐉ 共q , w兲,
ᐉ 共q , w兲
*
= ˜␸ᐉ共q , w兲⌰共w − wq兲, is the branch cut 共scattering兲 and
˜␸bs
ᐉ 共q , w兲 is the pole 共bound state兲 contribution. Here, ⌰共x兲 is
the Heaviside function, wq* = wq − 2␮ᐉ with wq = 兩q兩2 / 共4M兲 is
the branch frequency and ␮ᐉ is the Fermionic chemical potential.
The branch cut 共scattering兲 contribution to the thermody⬁ dw
namic potential becomes ⍀sc
␸sc
ᐉ = −兺q兰−⬁ ␲ nB共w兲˜
ᐉ 共q , w兲. For
each q, the integrand is nonvanishing only for w ⬎ wq* since
˜␸sc
ᐉ 共q , w兲 = 0 otherwise. Thus the branch cut 共scattering兲 consc
tribution to the number equation Nsc
ᐉ = −⳵⍀ᐉ / ⳵␮ᐉ is given by
Nsc
ᐉ =兺
共65兲
This expression is independent of mᐉ since the interaction
amplitude ␭ᐉ depends only on ᐉ. Similarly, the saddle point
number equation reduces to
ᐉ
1 − n F共 ␰ +兲 − n F共 ␰ −兲
␰+ + ␰− − iv j
ᐉ
⳵Wᐉ共q兲
nB关Wᐉ共q兲兴.
⳵␮ᐉ
Nsp
ᐉ = 兺 nF关␰ᐉ共k兲兴,
ᐉ
ᐉ
*
In this section, we concentrate on physical properties near
critical temperatures T = Tc,ᐉ. To calculate Tc,ᐉ, the selfconsistency 共order-parameter and number兲 equations have to
be solved simultaneously. At T = Tc,ᐉ, then ⌬ᐉ,mᐉ = 0, and the
saddle-point order-parameter equation 共19兲 reduces to
= 兺 ⌫2ᐉ共k兲
−1
= 共Fᐉ,m ,m⬘兲11 is the element of the fluctuation propagator
ᐉ ᐉ
given by
⌫2ᐉ共k兲Y ᐉ,mᐉ共k̂兲Y ᐉ,m⬘共k̂兲.
VI. BCS TO BEC EVOLUTION NEAR T = Tc,艎
4␲k2ᐉ
0 aᐉ
To evaluate the Gaussian contribution to the thermodynamic potential, we sum over the Fermionic Matsubara Frequencies in Eq. 共10兲, and obtain the action Sfluct
ᐉ
−1
−1
†
= ␤兺q,mᐉ,m⬘⌳ᐉ,m
共q兲Lᐉ,m ,m⬘共q兲⌳ᐉ,m⬘共q兲, where Lᐉ,m ,m⬘共q兲
共63兲
Here nB共x兲 = 1 / 关exp共␤x兲 − 1兴 is the Bose distribution. For ᐉ
= 0, the last equation can be solved to obtain Nte
0=
2
兲, which vanishes at T = 0. Here
−6共⳵C0,0 / ⳵␮0兲␨共4兲T4 / 共␲2C0,0
te
␨共x兲 is the zeta function. Similarly, Nᐉ⫽0
has a power-law
dependence on T, and therefore vanishes at T = 0 since the
collective modes are not excited. Nzp
ᐉ gives small contributions to the number equation in weak and strong couplings,
but may lead to significant contributions in the intermediate
regime for all ᐉ. The impact of Nzp
ᐉ on the order parameter
and chemical potential in the intermediate regime may require a careful analysis of the full fluctuation contributions
关40兴.
Until now, we discussed the evolution of superfluidity
from the BCS to the BEC regime at zero temperature. In the
rest of the manuscript, we analyze the evolution of superfluidity from the BCS to the BEC limit at finite temperatures.
MV
A. Gaussian fluctuations
q
冕 冋
⬁
0
册
dw ⳵nB共w̃兲
⳵
˜␸ᐉ共q,w̃兲, 共69兲
− nB共w̃兲
␲
⳵␮ᐉ
⳵␮ᐉ
where w̃ = w + wq* .
When aᐉ ⬍ 0, there are no bound states above Tc,ᐉ and Nsc
ᐉ
represents the correction due to scattering states. However,
when aᐉ ⬎ 0, there are bound states represented by poles at
w ⬍ wq* . The pole 共bound-state兲 contribution to the number
equation is
Nbs
ᐉ = − 兺 nB关Wᐉ共q兲兴␩ᐉ关q,Wᐉ共q兲兴,
q
where Wᐉ共q兲 corresponds to the poles of L−1
ᐉ 共q兲 and
013608-13
共70兲
M. ISKIN AND C. A. R. SÁ DE MELO
再
␩ᐉ关q,Wᐉ共q兲兴 = Res
⳵ det L−1
ᐉ 关q,Wᐉ共q兲兴/⳵␮ᐉ
det L−1
ᐉ 关q,Wᐉ共q兲兴
冎
PHYSICAL REVIEW A 74, 013608 共2006兲
共71兲
is the residue. Heavy numerical calculations are necessary to
find the poles as a function of q for all couplings. However,
in sufficiently strong coupling, when nF共␰±兲 Ⰶ 1 in Eq. 共67兲,
the pole 共bound-state兲 contribution can be evaluated analytically by eliminating ␭ᐉ in favor of the two-body bound-state
energy Ẽb,ᐉ in vacuum. Notice that Ẽb,ᐉ is related to the Eb,ᐉ
obtained from the T-matrix approach, where multiple scattering events are included. However, they become identical in
the dilute limit.
A relation between ␭ᐉ and Ẽb,ᐉ can be obtained by solving
the Schroedinger equation for two fermions interacting via a
pairing potential V共r兲. After Fourier transforming from real
to momentum space, the Schroedinger equation for the pair
wave function ␺共k兲 is
2⑀共k兲␺共k兲 +
1
V共k,k⬘兲␺共k⬘兲 = Ẽb␺共k兲.
V兺
k
共72兲
⬘
Using the Fourier expansion of V共k , k⬘兲 given in Eq. 共2兲 and
choosing only the ᐉth angular momentum channel, we obtain
⌫2ᐉ共k兲
1 1
= 兺
.
␭ᐉ V k 2⑀共k兲 − Ẽ
b,ᐉ
共73兲
This expression relates Ẽb,ᐉ ⬍ 0 to ␭ᐉ in order to express Eq.
共74兲 in terms of binding energy Ẽb,ᐉ ⬍ 0. Notice that, this
equation is similar to the order-parameter equation in the
strong-coupling limit 共␮ᐉ ⬍ 0 and 兩␮ᐉ兩 Ⰷ Tc,ᐉ兲, where ␭1ᐉ
⌫2ᐉ共k兲
= V1 兺k 2⑀共k兲−2␮ᐉ . Therefore ␮ᐉ → Ẽb,ᐉ / 2 as the coupling increases.
Substitution of Eq. 共73兲 in Eq. 共71兲 yields the pole contribution which is given by Wᐉ共q兲 = wq + Ẽb,ᐉ − 2␮ᐉ, and the
residue at this pole is ␩ᐉ关q , Wᐉ共q兲兴 = −2兺mᐉ. Therefore the
bound-state contribution to the phase shift in the sufficiently
strong-coupling limit is given by ˜␸bs
ᐉ 共q , w兲 = ␲⌰共w − wq
+ ␮B,ᐉ兲, which leads to the bound-state number equation
Nbs
ᐉ = 2 兺 nB关wq − ␮B,ᐉ兴,
共74兲
q,mᐉ
where ␮B,ᐉ = 2␮ᐉ − Ẽb,ᐉ 艋 0 is the chemical potential of the
Bosonic molecules. Notice that Eq. 共74兲 is only valid for
interaction strengths where ␮B,ᐉ 艋 0. Thus this expression
can not be used over a region of coupling strengths where
␮B,ᐉ is positive.
sp
sc
FIG. 15. Fractions of unbound Fsp
0 = N0 / N0, scattering F0
sc
bs
bs
= N0 / N0, bound F0 = N0 / N0 fermions at T = Tc,0 vs 1 / 共kFa0兲 for
k0 ⬇ 200kF.
bs
= 0兲 case. Notice that Nsp
0 共N0 兲 dominates in weak 共strong兲
sc
coupling, while N0 is the highest for intermediate couplings.
Thus all fermions are unbound in the strictly BCS limit 共not
shown in the figure兲, while all fermions are bound in the
strictly BEC limit.
In Fig. 16, we present plots of different contributions to
the number equation as a function of 1 / 共kF3a1兲 for the p-wave
bs
共ᐉ = 1, mᐉ = 0兲 case. Notice also that Nsp
1 共N1 兲 dominates in
sc
weak 共strong兲 coupling, while N1 is the highest for intermediate couplings. Thus again all fermions are unbound in the
strictly BCS limit, while all fermions are bound in the strictly
BEC limit.
bs
Therefore the total fluctuation contribution Nsc
ᐉ + Nᐉ is
sp
negligible in weak coupling and Nᐉ is sufficient. However,
the inclusion of fluctuations is necessary for strong coupling
to recover the physics of BEC. However, in the vicinity of
the unitary limit 关1 / 共kF2ᐉ+1aᐉ兲 → 0兴, our results are not strictly
applicable and should be regarded as qualitative.
Next, we discuss the chemical potential and critical temperature. In weak coupling, we introduce a shell about the
Fermi energy 兩␰ᐉ共k兲兩 艋 wD, such that ␮ᐉ ⬇ ⑀F Ⰷ wD Ⰷ Tc,ᐉ.
Then, in Eq. 共65兲, we set tanh关兩␰ᐉ共k兲兩 / 共2Tc,ᐉ兲兴 = 1 outside the
shell and treat the integration within the shell as usual in the
BCS theory. In strong coupling, we use that min关␰ᐉ共k兲兴
= 兩␮ᐉ兩 Ⰷ Tc,ᐉ and set tanh关␰ᐉ共k兲 / 共2Tc,ᐉ兲兴 = 1. Therefore, in
strictly weak and strong coupling, the self-consistency equations are decoupled, and play reversed roles. In weak
共strong兲 coupling the order-parameter equation determines
Tc,ᐉ 共␮ᐉ兲 and the number equation determines ␮ᐉ 共Tc,ᐉ兲.
In weak coupling, the number equation Nᐉ ⬇ Nsp
ᐉ leads to
B. Critical temperature and chemical potential
To obtain the evolution from BCS to BEC, the number
sc
bs
= Nsp
Nᐉ ⬇ Ngauss
ᐉ
ᐉ + Nᐉ + Nᐉ and order-parameter 关Eq. 共65兲兴
equations have to be solved self-consistently for Tc,ᐉ and ␮ᐉ.
First, we analyze the number of unbound, scattering, and
bound fermions as a function of the scattering parameter for
the s-wave 共ᐉ = 0兲 and p-wave 共ᐉ = 1兲 cases.
In Fig. 15, we plot different contributions to the number
equation as a function of 1 / 共kFa0兲 for the s-wave 共ᐉ = 0, mᐉ
sp
sc
FIG. 16. Fractions of unbound Fsp
1 = N1 / N1, scattering F1
bs
bs
3
= Nsc
/
N
,
bound
F
=
N
/
N
fermions
at
T
=
T
vs
1
/
共k
a
兲
for
1
1
c,1
1
1
1
F 1
k0 ⬇ 200kF.
013608-14
PHYSICAL REVIEW A 74, 013608 共2006兲
NONZERO ORBITAL ANGULAR MOMENTUM¼
␮ᐉ ⬇ ⑀F
冉 冊
共75兲
Tc,ᐉ
⑀F
for any ᐉ. In strong coupling, the order parameter equation
leads to
␮0 = −
␮ᐉ⫽0 = −
1
2Ma20
共76兲
,
冑␲
共77兲
,
Mk2ᐉ−1
a ᐉ␾ ᐉ
0
where ␾ᐉ = ⌫共ᐉ − 1 / 2兲 / ⌫共ᐉ + 1兲 and ⌫共x兲 is the Gamma function. This calculation requires a0k0 ⬎ 1 for ᐉ = 0, and
aᐉ␾ᐉ ⬎ 共ᐉ + 1兲冑␲ for ᐉ ⫽ 0 for the order-parameter equak2ᐉ+1
0
tion to have a solution with ␮ᐉ ⬍ 0. Furthermore, we assume
兩␮ᐉ兩 Ⰶ ⑀0 = k20 / 共2M兲 to obtain Eqs. 共76兲 and 共77兲. Notice that
␮ᐉ = Eb,ᐉ / 2 in this limit.
On the other hand, the solution of the order-parameter
equation in weak coupling is
Tc,0 =
冋
冋冉 冊
册
册
8
␲ kF
␲
⑀F exp ␥ − 2 +
−
,
␲
2 k0 2kF兩a0兩
k0
kF
Tc,ᐉ ⬃ ⑀F exp tᐉ
␲
2ᐉ−1
−
2kF2ᐉ+1兩aᐉ兩
,
共78兲
共79兲
where ␥ ⬇ 0.577 is the Euler’s constant, t1 = ␲ / 4 and tᐉ⬎1
= ␲2ᐉ+1共2ᐉ − 3兲!! / ᐉ!. These expressions are valid only when
the exponential terms are small. In strong coupling, the number equation Nᐉ ⬇ Nbs
ᐉ leads to
THS
Tc,ᐉ
=
2␲
M B,ᐉ
冋 册
nᐉ
2/3
␨共3/2兲
兺
m
=
0.218
冉兺 冊
2/3 ⑀F ,
共80兲
mᐉ
ᐉ
where M B,ᐉ = 2M is the mass of the Bosonic molecules. Here,
nᐉ = kF3 / 共3␲2兲 is the density of fermions. For THS Fermi
gases, we conclude that the BEC critical temperature of
s-wave superfluids is the highest, and this temperature is reduced for higher angular momentum states. However, for
SHS Fermi gases
SHS
=
Tc,ᐉ⫽0
2␲
M B,ᐉ
冋 册
nᐉ
␨共3/2兲
兺
m
ᐉ
2/3
=
0.137⑀F
冉兺 冊
2/3 ,
=
␲/共kF2ᐉ+1a*ᐉ兲
1
1
共2 − 2−ᐉ+3/2兲⌫ ᐉ + ␨ ᐉ +
2
2
冉 冊冉 冊
共82兲
to order Tc,ᐉ / ⑀0, where ⑀0 = k20 / 共2M兲 Ⰷ Tc,ᐉ. Notice that this
relation depends on k0 only through a*ᐉ.
On the other hand, if temporal fluctuations are neglected,
the solution for T0,ᐉ from the saddle-point self-consistency
equations is 兩Ẽb,ᐉ兩 = 2T0,ᐉ ln关3冑␲共T0,ᐉ / ⑀F兲3/2 / 4兴 and ␮ᐉ
= Ẽb,ᐉ / 2 which leads to
T0,ᐉ ⬃
兩Ẽb,ᐉ兩
共83兲
2 ln共兩Ẽb,ᐉ兩/⑀F兲3/2
up to logarithmic accuracy. Therefore T0,ᐉ grows without
bound as the coupling increases. Within this calculation, the
normal state for T ⬎ T0,ᐉ is described by unbound and nondegenerate fermions since ⌬ᐉ共k兲 = 0 and 兩␮ᐉ兩 / T0,ᐉ
⬃ ln共兩Ẽb,ᐉ兩 / ⑀F兲3/2 Ⰷ 1. Notice that the saddle-point approximation becomes progressively worse with increasing coupling, since the formation of bound states is neglected.
We emphasize that there is no phase transition across T0,ᐉ
in strong coupling. However, this temperature is related to
the pair breaking or dissociation energy scale. To see this
connection, we consider the chemical equilibrium between
nondegenerate unbound fermions 共f兲 and bound pairs 共b兲
such that b ↔ f ↑ + f↓ for THS singlet states and b ↔ f ↑ + f↑
for SHS triplet states.
Notice that T0,ᐉ is sufficiently high that the chemical potential of the bosons and the fermions satisfy 兩␮b兩 Ⰷ T and
兩␮f兩 Ⰷ T at the temperature T of interest. Thus both the unbound fermions 共f兲 and molecules 共b兲 can be treated as classical ideal gases. The equilibrium condition ␮b = 2␮f for
these nondegenerate gases may be written as
T ln兵nb关2␲ / 共M bT兲兴3/2其 − Ẽb,ᐉ = 2T ln兵nf关2␲ / 共M fT兲兴3/2其, where
nb 共nf兲 is the boson 共fermion兲 density, M b 共M f兲 is the boson
共fermion兲 mass, and Ẽb,ᐉ is the binding energy of a Bosonic
molecule. The dissociation temperature above which some
fraction of the bound pairs 共molecules兲 are dissociated, is
then found to be
共81兲
Tdissoc,ᐉ ⬇
mᐉ
where nᐉ = kF3 / 共6␲2兲 and ␨共x兲 is the zeta function. Here, the
summation over mᐉ is over degenerate spherical harmonics
involved in the order parameter of the system, and can be at
most 兺mᐉ = 2ᐉ + 1. For SHS states, we conclude that the BEC
critical temperature of p-wave superfluids is the highest, and
this temperature is reduced for higher angular momentum
states.
For completeness, it is also possible to relate aᐉ and Tc,ᐉ
when chemical potential vanishes. When ␮ᐉ = 0, the solution
of number equation is highly nontrivial and it is difficult to
find the value of the scattering parameter a*ᐉ at ␮ᐉ = 0. However, the critical temperature in terms of a*ᐉ can be found
analytically from Eq. 共65兲 as
ᐉ+1/2
兩Ẽb,ᐉ兩
ln共兩Ẽb,ᐉ兩/⑀F兲3/2
,
共84兲
where we dropped a few constants of order unity. Therefore
the logarithmic term is an entropic contribution which favors
broken pairs and leads to a dissociation temperature considerably lower than the absolute value of binding energy 兩Ẽb,ᐉ兩.
The analysis above gives insight into the logarithmic factor
appearing in Eq. 共83兲 since T0,ᐉ ⬃ Tdissoc,ᐉ / 2. Thus T0,ᐉ is essentially the pair dissociation temperature of bound pairs
共molecules兲, while Tc,ᐉ is the phase coherence temperature
corresponding to BEC of bound pairs 共Bosonic molecules兲.
In Fig. 17, we show Tc,0 for the s-wave 共ᐉ = 0, mᐉ = 0兲
case. Notice that Tc,0 grows from an exponential dependence
in weak coupling to a constant in strong coupling with increasing interaction. Furthermore, the mean field T0,0 and
013608-15
PHYSICAL REVIEW A 74, 013608 共2006兲
M. ISKIN AND C. A. R. SÁ DE MELO
FIG. 17. Plot of reduced critical temperature Tr = Tc,0 / ⑀F vs interaction strength 1 / 共kFa0兲 at T = Tc,0 for k0 ⬇ 200kF.
FIG. 19. Plot of reduced critical temperature Tr = Tc,1 / ⑀F vs interaction strength 1 / 共kF3a1兲 at T = Tc,1 for k0 ⬇ 200kF.
Gaussian Tc,0 are similar only in weak coupling, while T0,0
increases without bound as T0,0 ⬃ 1 / 关共Ma20兲兩ln共kFa0兲兩兴 in
strong coupling. When ␮0 = 0, we also obtain analytically
Tc,0 / ⑀F ⬇ 2.15/ 共kFa*0兲2 from Eq. 共82兲. The hump in Tc,0
around 1 / 共kFa0兲 ⬇ 0.5 is similar to the those in Ref. 关12兴, and
might be an artifact of the approximations used here. Thus a
more detailed self-consistent numerical analysis is needed to
determine if this hump is real.
In Fig. 18, we show ␮0 for the s-wave case, where it
changes from ⑀F in weak coupling to Eb,0 / 2 = −1 / 共2Ma20兲 in
strong coupling. Notice that ␮0 at Tc,0 is qualitatively similar
to ␮0 at T = 0, however, it is reduced at Tc,0 in weak coupling.
Furthermore, ␮0 changes sign at 1 / 共kFa0兲 ⬇ 0.32.
In Fig. 19, we show Tc,1 for the p-wave 共ᐉ = 1, mᐉ = 0兲
case. Tc,1 grows from an exponential dependence in weak
coupling to a constant in strong coupling with increasing
interaction. For completeness, we present the limiting ex␲k
pressions
Tc,1 = ␲8 ⑀F exp关␥ − 38 + 4kF0 − 2k3␲兩a 兩 兴,
and
Tc,1
For any given ᐉ, mean-field and Gaussian theories lead to
similar results for Tc,ᐉ and T0,ᐉ in the BCS regime, while they
are very different in the BEC side. In the latter case, T0,ᐉ
increases without bound, however, Gaussian theory results in
a constant critical temperature which coincides with the BEC
temperature of bosons. Notice that the pseudogap region
Tc,ᐉ ⬍ T ⬍ T0,ᐉ for ᐉ = 0 state is much larger than ᐉ ⫽ 0 states
since T0,ᐉ⫽0 grows faster than T0,ᐉ⫽0. Furthermore, similar
humps in Tc,ᐉ around 1 / 共kF2ᐉ+1aᐉ兲 = 0 are expected for any ᐉ
as shown for the s-wave and p-wave cases, however, whether
these humps are physical or not may require a fully selfconsistent numerical approach.
As shown in this section, the frequency 共temporal兲 dependence of fluctuations about the saddle point is crucial to describe adequately the Bosonic degrees of freedom that
emerge with increasing coupling. In the next section, we derive the TDGL functional near Tc,ᐉ to emphasize further the
importance of these fluctuations.
1
= M2␲B,1 关 ␨共3/2兲
兴 = 0.137⑀F in the weak- and strong-coupling
limits, respectively. Furthermore, the mean field T0,1 and
Gaussian Tc,1 are similar only in weak coupling, while T0,1
increases without bound as T0,1 ⬃ 1 / 关共Mk0a1兲兩ln共kF2k0a1兲兩兴 in
strong coupling. When ␮1 = 0, we also obtain analytically
Tc,1 / ⑀F ⬇ 1.75/ 共kF3a*1兲2/3 from Eq. 共82兲. The hump in the intermediate regime is similar to the one found in fermionboson model 关36兴. But to determine if this hump is real, it
may be necessary to develop a fully self-consistent numerical calculation.
In Fig. 20, we show ␮1 for the p-wave case, where it
changes from ⑀F in weak coupling to Eb,1 / 2 = −1 / 共Mk0a1兲 in
strong coupling. Notice that ␮1 at Tc,1 is both qualitatively
and quantitatively similar to ␮1 at T = 0. Furthermore, ␮1
changes sign at 1 / 共kF3a1兲 ⬇ 0.02.
n
2/3
F
1
FIG. 18. Plot reduced chemical potential ␮r = ␮0 / ⑀F 共inset兲 vs
interaction strength 1 / 共kFa0兲 at T = Tc,0 for k0 ⬇ 200kF.
VII. TDGL FUNCTIONAL NEAR Tc,艎
Our basic motivation here is to investigate the lowfrequency and long-wavelength behavior of the order parameter near Tc,ᐉ. To study the evolution of the time-dependent
Ginzburg-Landau 共TDGL兲 functional near Tc,ᐉ, we need to
expand the effective action Seff
ᐉ in Eq. 共5兲 around ⌬ᐉ,mᐉ = 0
leading to
sp
gauss
Seff
+
ᐉ = Sᐉ + Sᐉ
␤
2
兺兵q ,m
n
†
⫻共q1兲⌳ᐉ,mᐉ 共q2兲⌳ᐉ,m
ᐉ
2
ᐉn其
3
†
bᐉ,兵mᐉ 其共兵qn其兲⌳ᐉ,m
n
ᐉ1
共q3兲⌳ᐉ,mᐉ 共q1 − q2 + q3兲.
4
Here, ⌳ᐉ,mᐉ共q兲 is the pairing fluctuation field.
FIG. 20. Plot of reduced chemical potential ␮r = ␮1 / ⑀F 共inset兲 vs
interaction strength 1 / 共kF3a1兲 at T = Tc,1 for k0 ⬇ 200kF.
013608-16
PHYSICAL REVIEW A 74, 013608 共2006兲
NONZERO ORBITAL ANGULAR MOMENTUM¼
−1
共q兲, and expand
We first consider the static part of Lᐉ,m
it
in
powers
i,j
of
to
qi
ᐉ,mᐉ⬘
−1
Lᐉ,m ,m⬘共q , 0兲 = aᐉ,mᐉ,m⬘
ᐉ
ᐉ ᐉ
get
q iq j
+ 兺i,jcᐉ,m ,m⬘ 2M + ¯ . Next, we consider the time dependence
ᐉ ᐉ
of the TDGL equation, where it is necessary to expand
−1
−1
Lᐉ,m ,m⬘共0 , iv j兲 − Lᐉ,m ,m⬘共0 , 0兲 in powers of w after analytic
ᐉ ᐉ
ᐉ ᐉ
continuation iv j → w + i0+.
In the x = 共x , t兲 representation, the calculation above leads
to the TDGL equation
兺
m
ᐉ2
冋
i,j
aᐉ,mᐉ ,mᐉ − 兺 cᐉ,m
1
2
i,j
ᐉ1,mᐉ2
ⵜ iⵜ j
+ 兺 bᐉ,兵mᐉ 其
n
2M mᐉ ,mᐉ
3
†
⫻共0兲⌳ᐉ,m
共x兲⌳ᐉ,mᐉ 共x兲 − idᐉ,mᐉ ,mᐉ
ᐉ3
4
1
4
2
册
which is the generalization of the TDGL equation to higher
momentum channels of THS singlet and SHS triplet states.
Notice that, for THS triplet states, there may be extra gradient mixing textures and fourth-order terms in the expansion
关39兴, which are not discussed here. All static and dynamic
expansion coefficients are presented in Appendix B. The
condition det aᐉ = 0 with matrix elements aᐉ,mᐉ ,mᐉ is the
1
2
Thouless criterion, which leads to the order-parameter equai,j
tion. The coefficient cᐉ,m ,m reflects a major difference beᐉ2
i,j
= c0,0,0␦i,j is isotropic
tween ᐉ = 0 and ᐉ ⫽ 0 cases. While c0,0,0
i,j
in space, cᐉ⫽0,m ,m is anisotropic, thus characterizing the
ᐉ1
ᐉ2
anisotropy of the order parameter. The coefficient bᐉ,兵mᐉ 其共0兲
n
is positive and guarantees the stability of the theory. The
coefficient dᐉ,mᐉ ,mᐉ is a complex number. Its imaginary part
1
2
reflects the decay of Cooper pairs into the two-particle continuum for ␮ᐉ ⬎ 0. However, for ␮ᐉ ⬍ 0, imaginary part of
dᐉ,mᐉ ,mᐉ vanishes and the behavior of the order parameter is
1
2
propagating reflecting the presence of stable bound states.
Next, we present the asymptotic forms of aᐉ,mᐉ ,mᐉ ;
1
2
i,j
bᐉ,兵mᐉ 其共0兲; cᐉ,m
,m and dᐉ,mᐉ ,mᐉ which are used to recover
n
ᐉ1
ᐉ2
1
2
the usual Ginzburg-Landau 共GL兲 equation for BCS superfluids in weak coupling and the Gross-Pitaevskii 共GP兲 equation
for a weakly interacting dilute Bose gas in strong coupling.
A. Weak-coupling (BCS) regime
The weak-coupling BCS regime is characterized by ␮ᐉ
⬎ 0 and ␮ᐉ ⬇ ⑀F Ⰷ Tc,ᐉ. For any given ᐉ, we find the following values for the coefficients:
冉 冊
aᐉ,mᐉ ,mᐉ = ␬w
ᐉ ln
1
2
bᐉ,兵mᐉ 其共0兲 =
n
i,j
cᐉ,m
=
ᐉ1,mᐉ2
T
␦mᐉ ,mᐉ ,
1
2
Tc,ᐉ
␬w␨共3兲
7␥ᐉ,兵mᐉ 其 ᐉ 2
n
8Tc,ᐉ
i,j
7␣ᐉ,m
ᐉ1,mᐉ2
1
冉冊
⑀F
⑀0
共86兲
冊
1
␲
+i
␦mᐉ ,mᐉ ,
1
2
4⑀F 8Tc,ᐉ
,
共89兲
1
ᐉ1
2
ᐉ2
angular averaged quantities defined in Appendix B. Notice
that the critical transition temperature is determined by
det aᐉ = 0.
In the particular case, where only one of the spherical
harmonics Y ᐉ,mᐉ共k̂兲 is dominant and characterizes the order
parameter, we can rescale the pairing field as
w
共x兲 =
⌿ᐉ,m
ᐉ
冑
bᐉ,兵mᐉ其共0兲
␬wᐉ
⌳ᐉ,mᐉ共x兲
to obtain the conventional TDGL equation
冋
w
GL 2 2
GL
− ␧ᐉ + 兩⌿ᐉ,m
兩2 − 兺 共␰ᐉ,m
兲i ⵜi + ␶ᐉ,m
ᐉ
i
ᐉ
ᐉ
共90兲
册
⳵
w
⌿ᐉ,m
= 0.
ᐉ
⳵t
共91兲
with
兩␧ᐉ兩 Ⰶ 1,
共␰ᐉ,mᐉ兲2i 共T兲
Here,
␧ᐉ = 共Tc,ᐉ − T兲 / Tc,ᐉ
i,i
GL 2
= cᐉ,m ,m / 关2Maᐉ,mᐉ,mᐉ兴 = 共␰ᐉ,m 兲i / ␧ᐉ is the characteristic GL
ᐉ ᐉ
ᐉ
GL
length and ␶ᐉ,mᐉ = −idᐉ,mᐉ,mᐉ / aᐉ,mᐉ,mᐉ = ␶ᐉ,m
/ ␧ᐉ is the characᐉ
teristic GL time.
In this limit, the GL coherence length is given by
GL
i,i
兲i = 关7␣ᐉ,m
␨共3兲 / 共4␲2兲兴1/2共⑀F / Tc,ᐉ兲, which makes
kF共␰ᐉ,m
ᐉ
ᐉ,mᐉ
GL
共␰ᐉ,m 兲i much larger than the interparticle spacing kF−1. There
ᐉ
is a major difference between ᐉ = 0 and ᐉ ⫽ 0 pairings regardGL
i,j
i,j
ing 共␰ᐉ,m
兲i. While c0,0,0
= c0,0,0␦i,j is isotropic, cᐉ⫽0,m
,m
ᐉ
␦i,j is
ᐉ1,mᐉ2
GL
Thus 共␰0,0
兲i
i,i
= cᐉ,m
ᐉ1
ᐉ2
in general anisotropic in space 共see Appendix
GL
B兲.
is isotropic and 共␰ᐉ⫽0,m
兲i is not.
ᐉ
GL
Furthermore, ␶ᐉ,m = −i / 共4⑀F兲 + ␲ / 共8Tc,ᐉ兲 showing that the
ᐉ
w
dynamics of ⌿ᐉ,m
共x兲 is overdamped reflecting the conᐉ
tinuum of Fermionic excitations into which a pair can decay.
In addition, there is a small propagating term since there is
no perfect particle-hole symmetry. As the coupling grows,
the coefficient of the propagating term increases while that of
the damping term vanishes for ␮ᐉ 艋 0. Thus the mode is
propagating in strong coupling reflecting the stability of the
bound states against the two particle continuum.
B. Strong-coupling (BEC) regime
The strong-coupling BEC regime is characterized by ␮ᐉ
⬍ 0 and ⑀0 = k20 / 共2M兲 Ⰷ 兩␮ᐉ兩 Ⰷ Tc,ᐉ. For ᐉ = 0, we find the following coefficients:
a0,0,0 = 2␬s0共2兩␮0兩 − 兩Ẽb,0兩兲,
ᐉ
␬wᐉ ⑀F␨共3兲
2 ,
4␲2Tc,ᐉ
2
冉
ᐉ
2
where ␬w
ᐉ = N共⑀F兲共⑀F / ⑀0兲 / 共4␲兲 with N共⑀F兲 = MVkF / 共2␲ 兲 is
the density of states per spin at the Fermi energy. Here
i,j
␦mᐉ ,mᐉ is the Kronecker delta, and ␣ᐉ,m
,m and ␥ᐉ,兵mᐉ其 are
⳵
⌳ᐉ,mᐉ 共x兲 = 0,
2
⳵t
共85兲
ᐉ1
dᐉ,mᐉ ,mᐉ = ␬w
ᐉ
共87兲
b0,兵0其共0兲 =
共88兲
013608-17
␬s0
,
8 ␲ 兩 ␮ 0兩
i,j
= ␬s0␦i,j ,
c0,0,0
共92兲
共93兲
共94兲
PHYSICAL REVIEW A 74, 013608 共2006兲
M. ISKIN AND C. A. R. SÁ DE MELO
d0,0,0 = 2␬s0 ,
共95兲
where ␬s0 = N共⑀F兲 / 共64冑⑀F兩␮0兩兲. Similarly, for ᐉ ⫽ 0, we obtain
aᐉ⫽0,mᐉ ,mᐉ = 2␬sᐉ␾ᐉ共2兩␮ᐉ兩 − 兩Ẽb,ᐉ兩兲␦mᐉ ,mᐉ ,
1
1
2
2
共96兲
␬sᐉ␾ˆ ᐉ
,
2⑀0
共97兲
= ␬sᐉ␾ᐉ␦mᐉ ,mᐉ ␦i,j ,
共98兲
dᐉ⫽0,mᐉ ,mᐉ = 2␬sᐉ␾ᐉ␦mᐉ ,mᐉ ,
共99兲
bᐉ⫽0,兵mᐉ 其共0兲 = 15␥ᐉ,兵mᐉ 其
n
i,j
cᐉ⫽0,m
ᐉ1,mᐉ2
1
n
1
2
2
1
2
s
= N共⑀F兲 / 共64冑␲⑀F⑀0兲. Here ␾ᐉ = ⌫共ᐉ − 1 / 2兲 / ⌫共ᐉ
where ␬ᐉ⫽0
ˆ ᐉ = ⌫共2ᐉ − 3 / 2兲 / ⌫共2ᐉ + 2兲, where ⌫共x兲 is the
+ 1兲 and ␾
i,j
gamma function. Notice that cᐉ⫽0,m
,m is isotropic in space
ᐉ1
ᐉ2
for any ᐉ. Thus the anisotropy of the order parameter plays a
secondary role in the TDGL theory in this limit.
In the particular case, where only one of the spherical
harmonics Y ᐉ,mᐉ共k̂兲 is dominant and characterizes the order
parameter, we can rescale the pairing field as
s
⌿ᐉ,m
共x兲 = 冑dᐉ,mᐉ,mᐉ⌳ᐉ,mᐉ共x兲,
ᐉ
GL
To calculate 共␰ᐉ,m
兲i in the strong-coupling limit, we need
ᐉ
to know ⳵␮ᐉ / ⳵T evaluated at Tc,ᐉ 共see below兲. The temperature dependence of ␮ᐉ in the vicinity of Tc,ᐉ can be obtained
by noticing that ␮B,ᐉ = ñ共T兲Uᐉ,mᐉ, where ñ共T兲 = nB,ᐉ关1
GL
兲i = 关␲2 / 共2MkFUᐉ,mᐉ兲兴1/2
− 共T / Tc,ᐉ兲3/2兴. This leads to kF共␰ᐉ,m
ᐉ
in the BEC regime. Using the asymptotic values of Uᐉ,mᐉ, we
GL
GL
兲i = 关␲ / 共8kFa0兲兴1/2 for ᐉ = 0 and kF共␰ᐉ⫽0,m
兲i
obtain kF共␰0,0
ᐉ
2
ˆ ᐉ兲兴1/2共k0 / kF兲1/2 for ᐉ ⫽ 0. Therefore
= 关␾ / 共480冑␲␥ᐉ,兵m 其␾
ᐉ
ᐉ
GL
共␰ᐉ,m
兲i is also much larger than the interparticle spacing kF−1
ᐉ
in this limit, since kFa0 → 0 for ᐉ = 0 and k0 Ⰷ kF for any ᐉ.
C. Ginzburg-Landau coherence length versus average Cooper
pair size
In the particular case, where only one of the spherical
harmonics Y ᐉ,mᐉ共k̂兲 is dominant and characterizes the order
parameter, we can define the GL coherence length as
i,i
/ 共2Maᐉ,mᐉ,mᐉ兲. An expansion of the pa共␰ᐉ,mᐉ兲2i 共T兲 = cᐉ,m
ᐉ,mᐉ
i,i
rameters aᐉ,mᐉ,mᐉ and cᐉ,m
,m in the vicinity of Tc,ᐉ leads to
共100兲
GL 2
共␰ᐉ,m
兲
ᐉ i
to obtain the conventional Gross-Pitaevskii 共GP兲 equation
冋
s
␮B,ᐉ + Uᐉ,mᐉ兩⌿ᐉ,m
兩2 −
ᐉ
册
ⵜ2
⳵
s
−i
⌿ᐉ,m
= 0 共101兲
ᐉ
2M B,ᐉ ⳵t
for a dilute gas of bosons. Here, ␮B,ᐉ = −aᐉ,mᐉ,mᐉ / dᐉ,mᐉ,mᐉ
= 2␮ᐉ − Ẽb,ᐉ
is
the
chemical
potential,
M B,ᐉ
i,i
= Mdᐉ,mᐉ,mᐉ / cᐉ,m
=
2M
is
the
mass,
and
U
ᐉ,mᐉ
ᐉ,mᐉ
2
= bᐉ,兵mᐉ其共0兲 / dᐉ,m
is the repulsive interactions of the
,m
ᐉ ᐉ
bosons. We obtain U0,0 = 4␲a0 / M
and Uᐉ⫽0,mᐉ
ˆ ᐉ␥ᐉ,兵m 其 / 共M ␾2k0兲 for ᐉ = 0 and ᐉ ⫽ 0, respec= 240␲2冑␲␾
ᐉ
ᐉ
tively. Notice that the mass of the composite bosons is independent of the anisotropy and symmetry of the order parameter for any given ᐉ. However, this is not the case for the
repulsive interactions between bosons, which explicitly depends on ᐉ.
For ᐉ = 0, U0,0 = 4␲aB,0 / M B,0 is directly proportional to
the fermion 共boson兲 scattering length a0 共aB,0兲, where aB,0
= 2a0 is the boson-boson scattering length. A better estimate
for aB,0 ⬇ 0.6a0 can be found in the literature 关44–47兴. While
for ᐉ ⫽ 0, Uᐉ,mᐉ is a constant 共independent of the scattering
parameter aᐉ兲 depending only on the interaction range k0 and
the particular 共ᐉ , mᐉ兲 state. For a finite range potential,
nB,ᐉUᐉ,mᐉ is small compared to ⑀F, where nB,ᐉ = nᐉ / 2 is the
density of bosons. In the ᐉ = 0 case nB,0U0,0 / ⑀F
= 4kFa0 / 共3␲兲 is much smaller than unity. For ᐉ ⫽ 0 and even,
ˆ ᐉ␥ᐉ,兵m 其 / ␾2共kF / k0兲. In the case of SHS
nB,ᐉUᐉ,mᐉ / ⑀F = 80冑␲␾
ᐉ
ᐉ
states
where
ᐉ⫽0
and
odd,
nB,ᐉUᐉ,mᐉ / ⑀F
2
ˆ
冑
= 40 ␲␾ᐉ␥ᐉ,兵mᐉ其 / ␾ᐉ共kF / k0兲. The results for higher angular
momentum channels reflect the diluteness condition
共kF / k0兲3 Ⰶ 1.
ᐉ
T
ᐉ
GL 2 c,ᐉ
, where the prefactor is the GL co兲
共␰ᐉ,mᐉ兲2i 共T兲 ⬇ 共␰ᐉ,m
ᐉ i Tc,ᐉ−T
herence length and given by
=
i,i
cᐉ,m
ᐉ,mᐉ
2MTc,ᐉ
冋
⳵aᐉ,mᐉ,mᐉ
⳵T
册
−1
共102兲
.
T=Tc,ᐉ
The slope of the coefficient aᐉ,mᐉ,mᐉ with respect to T is given
by
⳵aᐉ,mᐉ,mᐉ
⳵T
= 兺k
关
Yᐉ共k兲
2T2
+
共
Xᐉ共k兲
⳵␮ᐉ Yᐉ共k兲
⳵T 2T␰ᐉ共k兲 − ␰2ᐉ共k兲
兲兴
⌫2ᐉ共k兲
8␲
. Here Xᐉ共k兲 and
Yᐉ共k兲 are defined in Appendix B. Notice that while ⳵␮ᐉ / ⳵T
vanishes at Tc,ᐉ in weak coupling, it plays an important role
GL
兲i representing
in strong coupling. Furthermore, while 共␰ᐉ,m
ᐉ
the phase coherence length is large compared to interparticle
spacing in both BCS and BEC limits, it should have a minimum near ␮ᐉ ⬇ 0.
GL
兲i of the GL coherence length must be
The prefactor 共␰ᐉ,m
ᐉ
compared with the average Cooper pair size ␰pair
defined by
ᐉ
2
共␰pair
ᐉ 兲
=
具Zᐉ共k兲兩r2兩Zᐉ共k兲典
具Zᐉ共k兲兩Zᐉ共k兲典
=−
具Zᐉ共k兲兩ⵜk2 兩Zᐉ共k兲典
具Zᐉ共k兲兩Zᐉ共k兲典
,
共103兲
where Zᐉ共k兲 = ⌬ᐉ共k兲 / 关2Eᐉ共k兲兴 is the zero-temperature pair
wave function. In the BCS limit, ␰pair
ᐉ is much larger than the
interparticle distance kF−1 since the Cooper pairs are weakly
is a debound. Furthermore, for ␮ᐉ ⬍ 0, we expect that ␰pair
ᐉ
creasing function of interaction for any ᐉ, since Cooper pairs
become more tightly bound as the interaction increases.
GL
兲i and ␰pair
for s-wave 共ᐉ = 0兲 and
Next, we compare 共␰ᐉ,m
ᐉ
ᐉ
p-wave 共ᐉ = 1兲 states.
GL
In Fig. 21, a comparison between 共␰0,0
兲i and ␰pair
0 is shown
pair
for s-wave 共ᐉ = 0, mᐉ = 0兲. ␰0
changes from kF␰pair
0
= 关e␥ / 冑2␲兴共⑀F / Tc,0兲 in the BCS limit to kF␰pair
0
= 关⑀F / 共2兩␮0兩兲兴1/2 = kFa0 / 冑2 in the BEC limit as the interaction
013608-18
PHYSICAL REVIEW A 74, 013608 共2006兲
NONZERO ORBITAL ANGULAR MOMENTUM¼
VIII. CONCLUSIONS
FIG. 21. Plots of GL coherence length kF␰GL
0,0 共solid squares兲,
and zero-temperature Cooper pair size kF␰pair
0,0 共hollow squares兲 vs
interaction strength 1 / 共kFa0兲 at T = Tc,0 for k0 ⬇ 200kF.
increases. Here ␥ ⬇ 0.577 is the Euler’s constant. Further冑 1/3
冑 2
more, when ␮0 = 0, we obtain kF␰pair
0 = 7关⌫ 共1 / 4兲 / ␲兴 / 4
⬇ 1.29, where ⌫共x兲 is the Gamma function. Notice that ␰pair
0
is continuous at ␮0 = 0, and monotonically decreasing function of 1 / 共kFa0兲 with a limiting value controlled by a0 in
GL
strong coupling. However, 共␰0,0
兲i is a nonmonotonic function
of 1 / 共kFa0兲 having a minimum around 1 / 共kFa0兲 ⬇ 0.32 共␮0
GL
= 0兲. It changes from kF共␰0,0
兲i = 关7␨共3兲 / 共12␲2兲兴1/2共⑀F / Tc,0兲 in
GL
the BCS to kF共␰0,0 兲i = 关␲ / 共8kFa0兲兴1/2 in the BEC limit as the
coupling increases, where ␨共x兲 is the Zeta function. Notice
GL
兲i grows as 1 / 冑kFa0 in strong-coupling limit.
that, 共␰0,0
GL
In Fig. 22, a comparison between 共␰1,0
兲z and ␰pair
1 is shown
pair
for p wave 共ᐉ = 1, mᐉ = 0兲. Notice that, ␰1 is nonanalytic at
␮1 = 0, and is a monotonically decreasing function of
1 / 共kF3a1兲 with a limiting value controlled by kF / k0 in strong
coupling. This nonanalytic behavior is associated with the
change in E1共k兲 from gapless 共with line nodes兲 in the BCS to
GL
fully gapped in the BEC side. However, 共␰1,0
兲z is a nonmono3
tonic function of 1 / 共kFa1兲 having a minimum around
GL
GL
1 / 共kF3a1兲 ⬇ 0.02 共␮1 = 0兲. It changes from kF共␰1,0
兲x = kF共␰1,0
兲y
GL
2 1/2
冑
= kF共␰1,0 兲z / 3 = 关7␨共3兲 / 共20␲ 兲兴 共⑀F / Tc,1兲 in the BCS to
GL
kF共␰1,0
兲i = 关␲k0 / 共36kF兲兴1/2 in the BEC limit as the coupling
GL
increases. Notice that, ␰1,0
saturates in strong-coupling limit
reflecting the finite range of interactions.
GL
兲z shown in Figs.
It is important to emphasize that 共␰ᐉ,m
ᐉ
共21兲 and 共22兲 is only qualitative in the intermediate regime
around unitarity 1 / 共kF2ᐉ+1aᐉ兲 = 0 since our theory is not
strictly applicable in that region.
FIG. 22. Plots of GL coherence length kF␰GL
1,0 共solid squares兲,
and zero-temperature Cooper pair size kF␰pair
1,0 共hollow squares兲 vs
interaction strength 1 / 共kF3a1兲 at T = Tc,1 for k0 ⬇ 200kF.
In this manuscript, we extended the s-wave 共ᐉ = 0兲 functional integral formalism to finite angular momentum ᐉ including two hyperfine states 共THS兲 pseudospin singlet and
single hyperfine states 共SHS兲 pseudospin triplet channels. We
analyzed analytically superfluid properties of a dilute Fermi
gas in the ground state 共T = 0兲 and near critical temperatures
共T ⬇ Tc,ᐉ兲 from weak coupling 共BCS兲 to strong coupling
共BEC兲 as a function of scattering parameter 共aᐉ兲 for arbitrary
ᐉ. However, we presented numerical results only for THS
s-wave and SHS p-wave symmetries which may be relevant
for current experiments involving atomic Fermi gases. The
main results of our paper are as follows.
First, we analyzed the low-energy scattering amplitude
within a T-matrix approach. We found that bound states occur only when aᐉ ⬎ 0 for any ᐉ. The energy of the bound
states Eb,ᐉ involves only the scattering parameter a0 for ᐉ
= 0. However, another parameter related to the interaction
range 1 / k0 is necessary to characterize Eb,ᐉ for ᐉ ⫽ 0. Therefore all superfluid properties for ᐉ ⫽ 0 depend strongly on k0
and aᐉ, while for ᐉ = 0 they depend strongly only on a0 but
weakly on k0.
Second, we discussed the order parameter, chemical potential, quasiparticle excitations, momentum distribution,
atomic compressibility, ground-state energy, collective
modes, and average Cooper pair size at T = 0. There we
showed that the evolution from BCS to BEC is just a crossover for ᐉ = 0, while the same evolution for ᐉ ⫽ 0 exhibits a
quantum phase transition characterized by a gapless superfluid on the BCS side to a fully gapped superfluid on the
BEC side. This transition is a many body effect and takes
place exactly when chemical potential ␮ᐉ⫽0 crosses the bottom of the fermion band 共␮ᐉ⫽0 = 0兲, and is best reflected as
nonanalytic behavior in the ground-state atomic compressibility, momentum distribution, and average Cooper pair size.
Third, we discussed the critical temperature, chemical potential, and the number of unbound, scattering and bound
fermions at T = Tc,ᐉ. We found that the critical BEC temperature is the highest for ᐉ = 0. We also derived the timedependent Ginzburg-Landau functional 共TDGL兲 near Tc,ᐉ
and extracted the Ginzburg-Landau 共GL兲 coherence length
and time. We recovered the usual TDGL equation for BCS
superfluids in the weak-coupling limit, whereas in the strongcoupling limit we recovered the Gross-Pitaevskii 共GP兲 equation for a weakly interacting dilute Bose gas. The TDGL
equation exhibits anisotropic coherence lengths for ᐉ ⫽ 0
which become isotropic only in the BEC limit, in sharp contrast to the ᐉ = 0 case, where the coherence length is isotropic
for all couplings. Furthermore, the GL time is a complex
number with a larger imaginary component for ␮ᐉ ⬎ 0 reflecting the decay of Cooper pairs into the two-particle continuum. However, for ␮ᐉ ⬍ 0 the imaginary component vanishes and Cooper pairs become stable above Tc,ᐉ.
In summary, the BCS to BEC evolution in higher angular
momentum 共ᐉ ⫽ 0兲 states exhibit quantum phase transitions
and is much richer than in conventional ᐉ = 0 s-wave systems, where there is only a crossover. These ᐉ ⫽ 0 states
might be found not only in atomic Fermi gases, but also in
013608-19
PHYSICAL REVIEW A 74, 013608 共2006兲
M. ISKIN AND C. A. R. SÁ DE MELO
nuclear 共pairing in nuclei兲, astrophysics 共neutron stars兲 and
condensed-matter 共high-Tc and organic superconductors兲
systems.
i,j
Qᐉ,m
ᐉ,mᐉ⬘
=兺
k
1
4E5ᐉ共k兲
2
¨ i,j 2
兵␰¨ i,j
ᐉ Eᐉ共k兲␰ᐉ共k兲 + ⌬ᐉ Eᐉ共k兲⌬ᐉ共k兲
+ 3␰˙ iᐉ␰˙ ᐉj ⌬2ᐉ共k兲 + 3⌬˙ iᐉ⌬˙ ᐉj ␰2ᐉ共k兲 − 3共␰˙ iᐉ⌬˙ ᐉj
ACKNOWLEDGMENT
*
+ ␰˙ ᐉj ⌬˙ iᐉ兲␰ᐉ共k兲⌬ᐉ共k兲其⌫2ᐉ共k兲Y ᐉ,mᐉ共k̂兲Y ᐉ,m⬘共k̂兲,
ᐉ
We would like to thank the NSF 共Grant No. DMR0304380兲 for financial support.
共A5兲
corresponding to the qiq j term, and
APPENDIX A: EXPANSION COEFFICIENTS AT T = 0
From the rotated fluctuation matrix F̃−1
ᐉ 共q兲 expressed in
the amplitude-phase basis, we can obtain the expansion coefficients necessary to calculate the collective modes at T
= 0. In the long-wavelength 共兩q兩 → 0兲 and low-frequency
q iq j
其 Ⰶ min兵2Eᐉ共k兲其 is used.
limit 共w → 0兲 the condition 兵w , 2M
While there is no Landau damping and a well defined expansion is possible for ᐉ = 0 case for all couplings, extra care is
necessary for ᐉ ⫽ 0 when ␮ᐉ ⬎ 0 since Landau damping is
present.
In all the expressions below we use the following simpli␰˙ iᐉ = ⳵␰ᐉ共k + q / 2兲 / ⳵qi,
␰¨ i,j
fying
notation:
ᐉ = ⳵␰ᐉ共k
i,j
i
˙
¨
+ q / 2兲 / 共⳵qi⳵q j兲, ⌬ᐉ = ⳵⌬ᐉ共k + q / 2兲 / ⳵qi, and ⌬ᐉ = ⳵2⌬ᐉ共k
+ q / 2兲 / 共⳵qi⳵q j兲, which are evaluated at q = 0.
The coefficients necessary to obtain the matrix element
−1
共F̃ᐉ,m ,m⬘兲11 are
ᐉ
ᐉ
Aᐉ,mᐉ,m⬘ =
ᐉ
␦mᐉ,mᐉ⬘
4␲V−1␭ᐉ
−兺
k
␰2ᐉ
2E3ᐉ
*
⌫2ᐉ共k兲Y ᐉ,mᐉ共k̂兲Y ᐉ,m⬘共k̂兲,
Rᐉ,mᐉ,m⬘ = 兺
ᐉ
ᐉ
ᐉ,mᐉ⬘
=兺
k
Bᐉ,mᐉ,m⬘ = 兺
ᐉ
In this section, we perform a small q and iv j → w + i0+
expansion near Tc,ᐉ, where we assumed that the fluctuation
field ⌳ᐉ,mᐉ共x , t兲 is a slowly varying function of x and t.
−1
The zeroth-order coefficient Lᐉ,m ,m⬘共0 , 0兲 is diagonal in
aᐉ,mᐉ,m⬘ =
ᐉ
ᐉ
*
ᐉ
i,j
␰2ᐉ共k兲 2
*
⌫ᐉ共k兲Y ᐉ,mᐉ共k̂兲Y ᐉ,m⬘共k̂兲,
5
ᐉ
k 8Eᐉ共k兲
cᐉ,m
ᐉ
4␲V ␭ᐉ
k
1
*
⌫2共k兲Y ᐉ,mᐉ共k̂兲Y ᐉ,m⬘共k̂兲,
ᐉ
2Eᐉ共k兲 ᐉ
册
共B1兲
=
1
兺
4␲ k
i,j
再冋
Xᐉ共k兲
8␰2ᐉ共k兲
−
册
␤Yᐉ共k兲
␦mᐉ,mᐉ⬘␦i,j
16␰ᐉ共k兲
冎
␤2k2Xᐉ共k兲Yᐉ共k兲 2
⌫ᐉ共k兲,
ᐉ,mᐉ⬘
16M ␰ᐉ共k兲
共B2兲
i,j
␣ᐉ,m ,m⬘ =
ᐉ ᐉ
冕
*
dk̂k̂ik̂ jY ᐉ,mᐉ共k̂兲Y ᐉ,m⬘共k̂兲.
ᐉ
Here,
dk̂ = sin共␪k兲d␪kd␾k,
k̂x = sin共␪k兲cos共␾k兲,
k̂y
i,j
= sin共␪k兲sin共␾k兲 and k̂z = cos共␪k兲. In general, ␣ᐉ,m ,m⬘ is a
ᐉ ᐉ
fourth-order tensor for fixed ᐉ. However, in the particular
case where only one of the spherical harmonics Y ᐉ,mᐉ共k̂兲 is
i,j
dominant and characterizes the order parameter, ␣ᐉ,m ,m⬘
ᐉ
共A4兲
corresponding to the 共q = 0, w = 0兲 term,
ᐉ,mᐉ⬘
+ ␣ᐉ,m
共A3兲
ᐉ
−兺
冋
where Yᐉ共k兲 = sech2关␤␰ᐉ共k兲 / 2兴 and the angular average
corresponding to the w2 term. Here ␦mᐉ,m⬘ is the Kronecker
ᐉ
delta.
The coefficients necessary to obtain the matrix element
−1
共F̃ᐉ,m ,m⬘兲22 are
−1
ᐉ
␦mᐉ,mᐉ⬘ V
Xᐉ共k兲 2
⌫ 共k兲 ,
−兺
4␲ ␭ᐉ k 2␰ᐉ共k兲 ᐉ
ᐉ
共A2兲
corresponding to the qiq j term, and
␦mᐉ,mᐉ⬘
ᐉ
where Xᐉ共k兲 = tanh关␤␰ᐉ共k兲 / 2兴. The second-order coefficient
−1
M ⳵2Lᐉ,m ,m⬘共q , 0兲 / 共⳵qi⳵q j兲 evaluated at q = 0 is given by
⫻关2⌬2ᐉ共k兲 − 3␰2ᐉ共k兲兴其⌫2ᐉ共k兲Y ᐉ,mᐉ共k̂兲Y ᐉ,m⬘共k̂兲,
Pᐉ,mᐉ,m⬘ =
共A7兲
APPENDIX B: EXPANSION COEFFICIENTS AT T = Tc,艎
2
2
˙i ˙ j ˙ j ˙ i
˙j
˙i⌬
+⌬
ᐉ ᐉ␰ᐉ共k兲关␰ᐉ共k兲 − 4⌬ᐉ共k兲兴 + 共␰ᐉ⌬ᐉ + ␰ᐉ⌬ᐉ兲⌬ᐉ共k兲
ᐉ
共A6兲
corresponding to the w term.
¨ i,jE2共k兲␰ 共k兲⌬ 共k兲 + 5␰˙ i ␰˙ j ⌬2共k兲␰2共k兲
+ 3⌬
ᐉ
ᐉ
ᐉ ᐉ ᐉ
ᐉ ᐉ
ᐉ
ᐉ
k
␰ᐉ共k兲 2
*
⌫ᐉ共k兲Y ᐉ,mᐉ共k̂兲Y ᐉ,m⬘共k̂兲,
ᐉ
4E3ᐉ共k兲
mᐉ and mᐉ⬘, and is given by
ᐉ
␰ᐉ共k兲 ¨ i,j 2
兵␰ᐉ Eᐉ共k兲关␰2ᐉ共k兲 − 2⌬2ᐉ共k兲兴
4E7ᐉ共k兲
Dᐉ,mᐉ,m⬘ = 兺
ᐉ
ᐉ
corresponding to the 共q = 0, w = 0兲 term,
i,j
*
⌫2ᐉ共k兲Y ᐉ,mᐉ共k̂兲Y ᐉ,m⬘共k̂兲,
corresponding to the w2 term.
The coefficients necessary to obtain the matrix element
−1
共F̃ᐉ,m ,m⬘兲12 is
共A1兲
Cᐉ,m
k
1
8E3ᐉ共k兲
ᐉ
i,j
= ␣ᐉ,m
␦mᐉ,mᐉ⬘ is diagonal in mᐉ and mᐉ⬘. In this case, we use
ᐉ,mᐉ
i,j
Gaunt coefficients 关48兴 to show that ␣ᐉ,m
is also diagonal
ᐉ,mᐉ
i,j
i,i
in i and j leading to ␣ᐉ,m ,m⬘ = ␣ᐉ,m ,m ␦mᐉ,m⬘␦i,j.
ᐉ
013608-20
ᐉ
ᐉ
ᐉ
ᐉ
PHYSICAL REVIEW A 74, 013608 共2006兲
NONZERO ORBITAL ANGULAR MOMENTUM¼
The coefficient of fourth-order term is approximated at
qn = 0, and given by
bᐉ,兵mᐉ 其共0兲 =
n
␥ᐉ,兵mᐉ 其
n
4␲
兺k
冋
Xᐉ共k兲
4␰3ᐉ共k兲
−
␤Yᐉ共k兲
8␰2ᐉ共k兲
册
Qᐉ,mᐉ,m⬘共iv j兲 = −
␦mᐉ,mᐉ⬘
4␲
ᐉ
冋兺
k
Xᐉ共k兲
4␰2ᐉ共k兲
⌫2ᐉ共k兲
册
− i␲ 兺 Xᐉ共k兲␦关2␰ᐉ共k兲 − w兴⌫2ᐉ共k兲 .
k
⌫4ᐉ共k兲,
共B4兲
共B3兲
Keeping only the first-order terms in w leads to Qᐉ,mᐉ,m⬘共w
ᐉ
+ i0+兲 = −dᐉ,mᐉ,m⬘w + . . ., where
ᐉ
where the angular average
dᐉ,mᐉ,m⬘ =
ᐉ
␥ᐉ,兵mᐉ 其 =
n
冕
*
*
dk̂Y ᐉ,mᐉ 共k̂兲Y ᐉ,m
共k̂兲Y ᐉ,mᐉ 共k̂兲Y ᐉ,m
共k̂兲.
ᐉ2
1
3
␦mᐉ,mᐉ⬘
4␲
+i
ᐉ4
To extract the time dependence, we expand Qᐉ,mᐉ,m⬘共iv j兲
ᐉ
−1
−1
= Lᐉ.m ,m⬘共q = 0 , iv j兲 − Lᐉ,m ,m⬘共0 , 0兲 in powers of w after the
冋兺
k
Xᐉ共k兲
4␰2ᐉ共k兲
␲␤
N共⑀F兲
8
冑
⌫2ᐉ共k兲
␮ᐉ 2
⌫ 共␮ 兲⌰共␮ᐉ兲
⑀F ᐉ ᐉ
册
共B5兲
analytic continuation iv j → w + i0+. We use the relation
共x ± i0+兲−1 = P共1 / x兲 ⫿ i␲␦共x兲, where P is the principal value
and ␦共x兲 is the delta function, to obtain
is also diagonal in mᐉ and mᐉ⬘. Here N共⑀F兲 = MVkF / 共2␲2兲 is
the density of states per spin at the Fermi energy, ⌫2ᐉ共x兲
= 共⑀0xᐉ兲 / 共⑀0 + x兲ᐉ+1 is the interaction symmetry in terms of
energy, and ⌰共x兲 is the Heaviside function.
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