BCS-BEC crossover and nodal points contribution in p-wave resonance superfluids
We solve the Leggett equations for BCS–BEC crossover of the resonance p-wave superfluid. We calculate sound velocity, specific heat and the normal density for the BCS-domain (μ > 0), BEC-domain (μ < 0) as well as for the interesting interpolation point (μ = 0) in the triplet A₁-phase in 3D. We...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1173032017-05-22T03:03:03Z BCS-BEC crossover and nodal points contribution in p-wave resonance superfluids Kagan, M.Yu. Efremov, D.V. Электронные свойства проводящих систем We solve the Leggett equations for BCS–BEC crossover of the resonance p-wave superfluid. We calculate sound velocity, specific heat and the normal density for the BCS-domain (μ > 0), BEC-domain (μ < 0) as well as for the interesting interpolation point (μ = 0) in the triplet A₁-phase in 3D. We are especially interested in the quasiparticle contribution coming from the zeroes of the superfluid gap in the A1-phase. We discuss the spectrum of orbital waves and the superfluid hydrodynamics at temperature T → 0. In this context we elucidate the difficult problem of chiral anomaly and mass-current nonconcervation appearing in the BCS-domain. We present the different approaches to solve this problem. To clarify this problem experimentally we propose an experiment for the measurement of anomalous current in superfluid A1-phase in the presence of aerogel for ³He and in the presence of Josephson tunneling structures for the ultracold gases in magnetic traps. 2009 Article BCS-BEC crossover and nodal points contribution in p-wave resonance superfluids / M.Yu. Kagan, D.V. Efremov // Физика низких температур. — 2009. — Т. 35, № 8-9. — С. 779-788. — Бібліогр.: 20 назв. — англ. 0132-6414 PACS: 67.30.H–, 67.85.Lm, 74.20.Rp http://dspace.nbuv.gov.ua/handle/123456789/117303 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Электронные свойства проводящих систем Электронные свойства проводящих систем |
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Электронные свойства проводящих систем Электронные свойства проводящих систем Kagan, M.Yu. Efremov, D.V. BCS-BEC crossover and nodal points contribution in p-wave resonance superfluids Физика низких температур |
| description |
We solve the Leggett equations for BCS–BEC crossover of the resonance p-wave superfluid. We calculate sound velocity, specific heat and the normal density for the BCS-domain (μ > 0), BEC-domain (μ < 0) as well as for the interesting interpolation point (μ = 0) in the triplet A₁-phase in 3D. We are especially interested in the quasiparticle contribution coming from the zeroes of the superfluid gap in the A1-phase. We discuss the spectrum of orbital waves and the superfluid hydrodynamics at temperature T → 0. In this context we elucidate the difficult problem of chiral anomaly and mass-current nonconcervation appearing in the BCS-domain. We present the different approaches to solve this problem. To clarify this problem experimentally we propose an experiment for the measurement of anomalous current in superfluid A1-phase in the presence of aerogel for ³He and in the presence of Josephson tunneling structures for the ultracold gases in magnetic traps. |
| format |
Article |
| author |
Kagan, M.Yu. Efremov, D.V. |
| author_facet |
Kagan, M.Yu. Efremov, D.V. |
| author_sort |
Kagan, M.Yu. |
| title |
BCS-BEC crossover and nodal points contribution in p-wave resonance superfluids |
| title_short |
BCS-BEC crossover and nodal points contribution in p-wave resonance superfluids |
| title_full |
BCS-BEC crossover and nodal points contribution in p-wave resonance superfluids |
| title_fullStr |
BCS-BEC crossover and nodal points contribution in p-wave resonance superfluids |
| title_full_unstemmed |
BCS-BEC crossover and nodal points contribution in p-wave resonance superfluids |
| title_sort |
bcs-bec crossover and nodal points contribution in p-wave resonance superfluids |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2009 |
| topic_facet |
Электронные свойства проводящих систем |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117303 |
| citation_txt |
BCS-BEC crossover and nodal points contribution in p-wave resonance superfluids / M.Yu. Kagan, D.V. Efremov // Физика низких температур. — 2009. — Т. 35, № 8-9. — С. 779-788. — Бібліогр.: 20 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT kaganmyu bcsbeccrossoverandnodalpointscontributioninpwaveresonancesuperfluids AT efremovdv bcsbeccrossoverandnodalpointscontributioninpwaveresonancesuperfluids |
| first_indexed |
2025-07-08T12:00:25Z |
| last_indexed |
2025-07-08T12:00:25Z |
| _version_ |
1837080013487210496 |
| fulltext |
Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9, p. 779–788
BCS–BEC crossover and nodal points contribution
in p-wave resonance superfluids
M.Yu. Kagan1 and D.V. Efremov1,2
1
P.L. Kapitza Institute for Physical Problems, 2 Kosygina Str., Moscow 119334, Russia
E-mail: kagan@kapitza.ras.ru
2
Institute for Theoretical Physics, Technical University of Dresden, Dresden, Germany
Received December 22, 2008
We solve the Leggett equations for BCS–BEC crossover of the resonance p-wave superfluid. We calcu-
late sound velocity, specific heat and the normal density for the BCS-domain (μ > 0), BEC-domain (μ < 0) as
well as for the interesting interpolation point (μ = 0) in the triplet A1-phase in 3D. We are especially inter-
ested in the quasiparticle contribution coming from the zeroes of the superfluid gap in the A1-phase. We dis-
cuss the spectrum of orbital waves and the superfluid hydrodynamics at temperature T → 0. In this context
we elucidate the difficult problem of chiral anomaly and mass-current nonconcervation appearing in the
BCS-domain. We present the different approaches to solve this problem. To clarify this problem experimen-
tally we propose an experiment for the measurement of anomalous current in superfluid A1-phase in the
presence of aerogel for
3
He and in the presence of Josephson tunneling structures for the ultracold gases in
magnetic traps.
PACS: 67.30.H– Superfluid phase of
3
He;
67.85.Lm Degenerate Fermi gases;
74.20.Rp Pairing symmetries (other than s-wave).
Keywords: BCS–BEC crossover, Feshbach resonance, superfluidity,
3
He-A, chiral anomaly.
1. Introduction
The first experimental results on p-wave Feshbach res-
onance [1–3] in ultracold fermionic gases 40K and 6Li
make the field of quantum gases closer to the interesting
physics of superfluid 3He and the physics of unconven-
tional superconductors such as Sr2RuO4. In this context it
is important to try to build the bridge between the physics
of ultracold gases and the physics of quantum liquids and
to enrich both communities with the experience and
knowledge accumulated in each of these fields. The pur-
pose of the present paper is first of all to describe the tran-
sition from the weakly bound Cooper pairs with p-wave
symmetry to strongly bound local p-wave pairs (mole-
cules) and try to reveal the nontrivial topological effects
connected with the presence of the nodes in the super-
fluid gap of the triplet p-wave A1-phase. Note, that the
A1-phase symmetry is relevant both to ultracold Fermi-
gases in the regime of p-wave Feshbach resonance and
to superfluid 3He-A in the presence of a magnetic field
B > Hc ~ Tc/μB, which is large enough to destroy iso-
tropic B phase of 3He already at T = 0. We pay the special
attention to the spectrum of collective excitations and to
the superfluid hydrodynamics of the A1-phase, where the
topological effects are very pronounced, especially in the
BCS-domain. We propose the experimental verification
of the different approaches connected with the problem of
chiral anomaly and mass-current nonconservation in
superfluid A1-phase of 3He in the presence of aerogel as
well as for the A1 p-wave condensates in magnetic traps
in the presence of Josephson tunneling structures.
Our paper is organized as follows. Chapter 1 provides
an Introduction. In Chapter 2 we briefly comment on the
recent experiments on p-wave Feshbach resonance and
describe the global phase-diagram for p-wave resonance
superfluids. In Chapter 3 we describe the quasiparticle
spectrum and nodal points in A1-phase. In Chapter 4 we
solve Leggett equations for triplet superfluids with the
symmetry of A1-phase and study the behavior of super-
fluid gap Δ, chemical potential μ, and sound velocity cs in
BCS- (μ > 0) and BEC-domains (μ < 0) as well as close to
the interesting interpolation point μ = 0. In Chapter 5 we
study the temperature behavior of the normal density ρn
and specific heat C
v
in BCS-domain, in BEC-domain, and
close to μ = 0. In Chapter 6 we describe the orbital waves
spectrum in BCS- and BEC-domains of the A1-phase and
describe the problem of chiral anomaly (mass-current
nonconservation) which exists in the superfluid hydrody-
namics of A1-phase in BCS-domain at T→ 0. In Chapter
© M.Yu. Kagan and D.V. Efremov, 2009
7 we present two different approaches to the calculation
of anomalous current: first one based on supersymmetric
hydrodynamics [4] and the second one on the analogy
with Dirac equation in quantum electrodynamics (QED)
[5,6]. Note that both approaches are very general. The
first of them is based on the inclusion of fermionic gold-
stone mode in low-frequency hydrodynamic action [4]. It
can be useful for all nodal superfluids and supercon-
ductors with zeroes of the superconductive gap such as
3He-A, Sr2RuO4, UPt3, UNi2Al3, U1–xThxBe13 and so on
[7]. The second approach is connected with the appear-
ance of the Dirac-like spectrum of fermions with zero
mode [5,6] which also arises in many condensed-matter
systems such as 3He-A, chiral superconductor Sr2RuO4,
organic conductor α-(BEDT-TTF)2I3, or recently discov-
ered graphene [7–10]. In Chapter 8 we propose a scenario
according to which the chiral anomaly exists only in
high-frequency ballistic (or Knudsen) regime and is de-
stroyed by damping in low-frequency hydrodynamic re-
gime. In Chapter 9 we propose to use aerogel to increase
the damping in superfluid 3He-A1-phase and thus to make
the transition to hydrodynamic regime easier. We also
provide a brief discussion on the role of Josephson tun-
neling structure in magnetic traps. Finally in Chapter 10
we present our conclusions and acknowledgements.
2. Feshbach resonance and phase-diagram for p-wave
resonance superfluids
In the first experiments on p-wave Feshbach reso-
nance the experimentalists measure the molecule forma-
tion in the ultracold fermionic gas of 6Li-atoms close to
resonance magnetic field B0 [1,2].
In the last years the analogous experiments on p-wave
molecules formation in spin-polarized fermionic gas of
40K-atoms were started [3]. The lifetime of p-wave mole-
cules is rather short yet [1–3]. However the physicists
working in ultracold gases have started intensively to
study the huge bulk of experimental and theoretical wis-
dom accumulated in the physics of superfluid 3He in
1970-s–1980-s (see [11]).
To understand the essence of p-wave Feshbach reso-
nance we should recollect the basic formula on p-wave
scattering amplitude from [12]:
f
p
V
p
r
ip
l
p
= =
+ +
1
2
2
0
31 1
2
, (1)
where l = 1 is an orbital momentum in the p-wave chan-
nel, Vp = r0
2ap is scattering volume, ap is p-wave scat-
tering length, r0 is the range of the interaction, p is the
scattered momentum. For Feshbach resonance in fermi-
onic systems p ~ pF and usually pFr0 < 1. The scattering
length ap and hence the scattering volume Vp are divergent in
the resonance magnetic field B0 (see Fig. 1): 1/Vp = 1/ap =
= 0. The imaginary part of the scattering amplitude fp is
small, so p-wave Feshbach resonance is intrinsically nar-
row.
The first theoretical review article on p-wave Fesh-
bach resonance mostly deals with mean-field two-chan-
nel description of the resonance [13]. In our paper we will
study p-wave Feshbach resonance in the framework of
one-channel description, which is more close to the phy-
sics of superfluid 3He and captures rather well the es-
sential physics of BCS–BEC crossover in p-wave super-
fluids.
In magnetic traps (in the absence of the so-called di-
polar splitting) people usually study fully (100%) pola-
rized gas or more precisely — one hyperfine component
of the gas. On the language of 3He they study the pairs
with Stot = S z
tot = 1, or |↑↑>-pairs. In our paper we con-
sider p-wave triplet A1-phase where just Stot = S z
tot = 1.
The qualitative picture of the global phase-diagram
of the BCS–BEC crossover in A1-phase is presented in
Fig. 2.
BCS-domain where chemical potential μ > 0, occupies
on the global phase-diagram, the region of negative
values of the gas parameter λp = Vp pF
3 < 0 (or the nega-
tive values of the scattering length ap). It stretches also to
the small positive values of the inverse gas parameter
1/λp ≤ 1 and is separated from the BEC-domain (where
μ < 0 and the inverse gas parameter is large and positive
1/λp ≥ 1) by the crossover line μ(T) = 0. Usually in the re-
gime of Feshbach resonance the density of «up» spins n =
= pF
3/6π2 is fixed. Deep inside BCS-domain (for small
absolute values of the gas parameter |λp| << 1) we have
780 Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9
M.Yu. Kagan and D.V. Efremov
0
Vp
B
B0
Fig. 1. The sketch of the p-wave Feshbach resonance.
the standard BCS-like formula for the critical temperature
of the A1-phase:
TCp F
p= −
01
2
.
/ | |ε π λ
e , (2)
where the preexponential factor for the 100% polarized
A1-phase is defined by second order diagrams of Gor’kov
and Melik-Barchudarov type [14] and approximately equ-
als to 0.1εF [15].
Deep in BEC-domain (λp << 1) the well-known for-
mula of Einstein is working in principal approximation
for Bose-condensation of p-wave molecules with the den-
sity n/2 and the mass 2m:
T
n
m
Cp =
/ /
3 31
2
2
2 3
.
( )
. (3)
In the unitary limit 1/λp = 0. Hence here TCp ≈0.1εF and
we are still in BCS-regime. In the rest of the paper we will
consider low temperatures T << Tc, so we will work deep
in the superfluid parts of BCS- and BEC-domains of the
A1-phase.
3. Quasiparticle energy and nodal points in A1-phase
For standard s-wave pairing the quasiparticle spec-
trum reads:
E
p
m
p = −
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ +
2
2
0
2
2
μ Δ . (4)
It has no zeroes (no nodes), so the topology of the s-wave
pairing problem is trivial. For triplet A1-phase however:
E
p
m p
p
F
= −
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ +
⋅2
2
2
22
μ
| |Δ p
, (5)
where Δ = +Δ 0( )e ex yi is the complex order parameter
in A1-phase, Δ0 is the magnitude of the superfluid gap. In
fact: | | sin ( )Δ ⋅ = = ×p p l
2
0
2 2 2
0
2 2Δ Δp θ , where l e e= ×x y
is the unit vector of orbital momentum (see Fig. 3). Note
that pF is fixed by fixed density n. Angle θ is the angle
between momentum p and the orbital momentum quanti-
zation axis l e= z .
For μ > 0 (BCS-domain) there are two nodes in the
spectrum for p2/2m = μ and for θ = 0 or π. For μ < 0
(BEC-domain) there are no nodes. The interesting point
μ = 0 is a boundary between the totally gapped BEC-do-
main and the BCS-domain with two nodes of the quasi-
particle spectrum corresponding to the south and north
poles of Fig. 3. Sometimes this point is called the point of
topological phase transition. However, as we shall see be-
low, this phase transition does not manifest itself in the
Leggett equations for the superfluid gap Δ and the che-
mical potential μ, which are continuous functions for
μ→ ±0.
4. Leggett equations for A1-phase
The Leggett equations for the triplet A1-phase in 3D
are the evident generalization of the standard Leggett
equations for the s-wave BCS–BEC crossover. The first
equation reads:
n
p p dp
dx
E
F
r
p
p
= = −
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟∫ ∫
−
3
2
2
2
0
1
1
1
6 4
1
2
1
0
π π
ξ
/
, (6)
where
BCS–BEC crossover and nodal points contribution in p-wave resonance superfluids
Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 781
T (T) = 0μ
BEC-domain
1/λp
0 (T = 0) = 0μ
ap < 0
a >p 0
μ > 0
μ < 0
BCS-domain
Fig. 2. Qualitative picture of the BCS–BEC crossover in
A1-phase in the axis temperature T and 1/λp — inverse gas pa-
rameter for p-wave superfluids (λp = Vp pF
3
, Vp is scattering
volume).
p
θ
ex
ey
ez
l
| ·Δ Δ θp|/p = sinF 0
Fig. 3. The topology of the superfluid gap in A1-phase. θ is the
angle between momentum p and the axis of orbital momentum
quantization l e= z. There are two nodes in the quasiparticle
spectrum corresponding to the south and north poles.
ξ μp
p
m
= −
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
2
2
, E
p
p
p p
F
= +ξ θ2 0
2 2
2
2Δ
sin
is a quasiparticle spectrum, x = cos θ. This equation de-
fines the chemical potential μ for fixed density n.
The second self-consistency equation defines the mag-
nitude of the superfluid gap Δ0. It reads:
− = −
⎧
⎨
⎩
⎫
⎬
⎭= −
∫ ∫π
μ ξ
m
f
dx
p dp
El p p
r
Re
( )
/
1
2 2
1 1
1
1
1
4
0
1 0
, (7)
where
Re
( )
1
2
1 4
1 0f V
m
rl p=
= +
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟μ
μ
π
is a real part of an inverse scattering amplitude in p-wave
channel for total energy E = 2μ of colliding particles. This
energy is relevant for pairing problem.
The solution of Leggett equations yields for the
BCS-domain:
Δ 0
2
0~ ~ ;
/ | |ε μ ε−π λ
F Cp F
p Te ≈ > . (8)
The sound velocity in 3D reads:
c
n
m
d
dn
s
F= ⎛
⎝⎜
⎞
⎠⎟
=μ 1 2
3
/
v
. (9)
For 1/|λp| = 0: Δ0 ~ εF and hence unitary limit is still inside
BCS-domain.
In BEC-domain:
Δ 0 02≈ <<ε εF F Fp r for p rF 0 1<< , (10)
and chemical potential μ
μ
= − + <
| |Eb B
2 2
0, where
| |E
mr a
b
p
= π
2 0
(11)
is a binding energy of a triplet pair (molecule).
Accordingly:
μ
ε
B
F
Fp r≈
4
3
0
(12)
is a bosonic chemical potential which governs the repul-
sive interaction between two p-wave molecules.
The sound velocity in BEC-domain reads:
c
n
m
d
dn
p rs
B B
B
F
F F= ⎛
⎝
⎜
⎞
⎠
⎟ ≈ <<
2 3
1 2
0
μ /
v
v for p rF 0 1<< ,
(13)
where nB = n/2 is bosonic density.
In the interesting interpolative region where μ → 0
(more rigorously |μ| < Δ0
2/εF) we have:
Δ 0 00 2( )μ ε= = F Fp r (14)
for the magnitude of the superfluid gap. It is possible to
show that it behaves in a regular way (linearly in μ) for
small μ.
For the gas parameter λp in the point μ = 0 we have
λ μp ( )= = >0
3
4
0 . (15)
Hence the interesting point μ = 0 is effectively in
BEC-domain (in the domain of positive p-wave scattering
length ap > 0).
The sound velocity (compressibility of the system) is
also continuous close to μ = 0:
c p rs
F
F=
v
3
0 (16)
as in BEC-domain (coincides with (16)). It means that the
derivative
∂
∂
⏐
⏐
⏐ =
→±
μ π
μn
r
m0
2
0
2
is continuous close to μ = 0. The gap derivative ∂ ∂Δ 0 / μ
also behaves in a continuous manner close to μ = 0:
∂
∂
⏐
⏐
⏐ =
→±
Δ 0
0 0
3
2μ μ p rF
.
We can conclude this chapter by the statement that
inspite of nontrivial topology in A1-phase, the BCS–BEC
crossover in the framework of Leggett equations has a
standard character and does not reveal any singularities.
5. Specific heat and normal density
at temperatures T << Tc
In this chapter we study the thermodynamic functions
namely, normal density ρn and specific heat C
v
in reso-
nance p-wave superfluids with A1-symmetry at low tem-
peratures T << Tc. Our goal is to try to find nontrivial con-
tributions to ρn and C
v
from the nodal points.
5.1. Specific heat in A1-phase
The fermionic (quasiparticle) contribution to C
v
yields (see [17]):
C
n E T
T
E
dp
pv
=
∂
∂∫ 0
3
32
( / )
( )
p
π
, (17)
where
n
E
T
p
E Tp
0
1
1
⎛
⎝
⎜
⎞
⎠
⎟ =
+( )
/
e
782 Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9
M.Yu. Kagan and D.V. Efremov
is quasiparticle distribution function, Ep is quasiparticle
energy given by (5).
The results of the calculations yield:
C N
T
v
~ ( )0
3
0
2Δ
(18)
for the BCS-domain, where
N
mpF( )0
2 2
=
π
is a density of states at Fermi-surface. In the BEC region
we consider separately contribution of the fermionic and
bosonic modes. The contribution of the fermionic excita-
tions in the BEC-domain reads:
C
mT E
T
b E Tb
v
~
( ) /
| |/2
2 4
3 2 2
2
2
π
e
−
, (19)
where |Eb| is given by (11).
Finally in the interesting region of small μ and low
temperatures (|μ| < T < Δ 0
2 / εF ) we have
C
mT TF
v
�
( ) /2
2
3 2
2
0
2π
ε
Δ
. (20)
For small |μ| but intermediate temperatures |μ| < Δ 0
2 / εF <
< T < Δ0:
C
mT
v
�
( ) /2
2
3 2
π
. (21)
Bosonic contribution (contribution from the sound waves)
yields:
C
T
c
B
s
v
�
3
3 2
1
2π
, (22)
where the sound velocity cs is given by (9) in BCS-do-
main and by (13), (16) in BEC-domain and close to μ = 0.
We can see that it is possible to separate a power-law
fermionic contribution C
v
∼ T5/2 at low temperatures and
C
v
~ T3/2 at intermediate temperatures from bosonic one
C B
v
~ T3 close to the interesting point μ = 0.
5.2. Normal density in A1-phase
The quasiparticle contribution to normal density
yields (see [14]):
ρ
∂
∂ π
n
p
p
p
n E /T
E
d
= − ∫13 2
2 0
3
3
( )
( )
p
. (23)
In the BCS-domain the evaluation of ρn yields:
ρ ρn
T
~
2
0
2Δ
, (24)
where ρ = nm is a total mass-density.
In the BEC-domain:
ρ
π
n
E Tm
mT b~ ( ) / | |/
2
3 2 2
2 e
−
. (25)
Finally close to μ = 0 at low temperatures (|μ| < T < Δ 0
2 / εF ):
ρ
π
ε
n
Fm
mT
T
�
2
3 2
0
2
2( ) /
Δ
. (26)
At intermediate temperatures |μ| < Δ 0
2 / εF < T < Δ0:
ρ
π
n
m
mT�
2
3 22( ) / . (27)
Bosonic (phonon) contribution from the sound waves
yields (see Lifshitz, Pitaevskii [17]):
ρn
B
s
T
c
~
4
5
, (28)
where cs is again given by (5), (13) and (16) in BCS-,
BEC-domain and close to μ = 0, respectively. We can
again separate a fermionic (quasiparticle) contribution to
ρn (ρn ~ T5/2 at low temperatures and ρn ~ T3/2 at interme-
diate temperatures) from bosonic one (ρn ~ T4) close to
the point μ = 0.
6. Orbital waves and the chiral anomaly in A1-phase
The topological effects in A1-phase are really pro-
nounced in the spectrum of orbital waves and in the
superfluid hydrodynamics at low temperatures T → 0 es-
pecially in BCS-domain. Here by symmetry requirements
we can write the following expression for the total
mass-current:
j j jtot an= +B , (29)
where
j l l lan rot= − ⋅�
2
0
m
C ( ) (30)
is an anomalous current. In BEC-domain C0 = 0 and
anomalous current is absent. However it is a difficult
question whether C0 = 0 or not in the BCS-domain. At the
same time jB in (29) is a total mass-current in BEC-do-
main for p-wave molecules. It reads:
j
l
B S
m
= +ρ
ρ
v
�
2 2
rot , (31)
where (�ρ/ m2 ) l is the density of orbital momentum, v S is
a superfluid velocity.
Anomalous current jan violates conservation law for
total mass-current (total linear momentum) jtot since it
cannot be expressed as a divergence of some momentum
tensor ∏ik:
BCS–BEC crossover and nodal points contribution in p-wave resonance superfluids
Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 783
∂
∂
∂
∂
j
t x
i
k
ik
an ≠ − ( )Π . (32)
Thus the presence of anomalous current destroys the
superfluid hydrodynamics of the A1-phase at T → 0. Its
contribution to the equation for total linear momentum (to
∂ ∂j / ti
tot ) can be compensated only by adding the term
with a normal velocity and normal density ρn nT( )= 0 v
to the total current jtot already at T = 0 (see [5,6]). The
anomalous current also changes significantly the spec-
trum of orbital waves. This additional Goldstone branch
of collective excitations in the A1-phase is connected
with the rotation of l-vector around perpendicular axis. It
is quadratic at low frequencies (A1-phase is called an or-
bital ferromagnet). However, the coefficient in front of q2
is drastically different in BCS- and BEC-domains.
In BEC-domain for small ω and q:
ω ~
q
m
2
. (33)
At the same time in BCS-domain:
ω
ρ
ρ
~
( )
q
m C
2
0−
. (34)
If according to (11), (12) the relative difference
( )ρ
ρ ε
−
<<
C
F
0 0
2
2
1�
Δ
then the coefficient of q2 in (34) is much larger in BCS-
domain than in BEC-domain for instance in superfluid
3He-A:
Δ
0 310
ε ε
F
C
F
T
� �
−
and thus ( ) /ρ ρ− −C 0
610� .
There are two competing approaches how to deal with
the complicated problem of anomalous current in
BCS-domain at T→ 0.
7. Two different approaches to the problem of chiral
anomaly in A1-phase
The first approach [4] is based on supersymmetric hy-
drodynamics of the A1-phase.
7.1. Supersymmetric hydrodynamics of the A1-phase
The idea of [4] was to check whether the chiral anom-
aly is directly connected with the zeroes of the gap. The
authors of [4] assumed that in condensed matter system at
low frequencies the only physical reason for anomaly can
be connected with the infrared singularity. Note that ultra-
violet singularities are absent in condensed matter sys-
tems in contrast with quantum electrodynamics. Strong
(critical) fluctuations are also suppressed in 3D system.
Thus the main idea of [4] was to check the dangerous in-
frared regions where the gap is practically zero. To do that
the authors of [4] consider the total hydrodynamic action
Stot of the A1-phase for low frequencies and small q-vec-
tors as a sum of bosonic and fermionic contributions:
Stot = SB + SF , (35)
where S B S( , , )ρ l v is a bosonic action and SF is a fer-
mionic action connected with the zeroes of a superfluid
gap (see Fig. 4).
Generally speaking the idea of [4] was to describe by
the supersymmetric hydrodynamics all the zero energy
Goldstone modes including the fermionic Goldstone
mode which comes from the zeroes of the gap.
The authors of [4] were motivated by the nice paper of
Volkov and Akulov [18] who for the first time included
massless fermionic neutrino in the effective infrared
Lagrangian for electro-weak interactions.
After the integration over the fermionic variables the
authors of [4] got the effective bosonic action and che-
cked what infrared anomalies were present in it.
As a result they obtained:
S B
eff = SB + ΔSB , (36)
where a nodal contribution to the liquid–crystal like part
of the effective action, which is connected with the gradi-
ent orbital energy, reads:
784 Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9
M.Yu. Kagan and D.V. Efremov
l
SB
SF
Fig. 4. The qualitative illustration of fermionic (SF) and
bosonic (SB) contributions to the total hydrodynamic action
Stot of the A1-phase at T→ 0.
ΔS rot rotB
F l t
l
p
d x= − × + ⋅
⎡
⎣
⎢
⎢
⎤
⎦
⎥∫
2
2
4 2
2
2
2
3
v v
v2π
( ) ( )l l l l
⎥
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟ln .
l
r
MF
2
2
(37)
Note, that in weak-coupling case
v v v v vt F
F
F l F
� �
Δ 0
ε
<< , .
In Eq. (37), x = (r,t), lMF is a mean-free path, ξ 0 < <r lMF ,
ξ 0 0~ /v F Δ is a coherence length.
We can see that only weak logarithmic singularities are
present in ΔSB. They yield logarithmic renormalization of
the orbital wave spectrum:
ω�
q
m qF
2
0ln
Δ
v
(38)
for small ω and q.
However we do not observe any sign of strong singu-
larity (actually it should be δ-functional since the fermi-
onic density ρF coming from the nodal regions in SF is
small in comparison with the total density ρ). In other
words, we do not see any trace of the anomalous contribu-
tion:
j v l l l van rot⋅ = − ⋅ ⋅S S
m
C
�
2
0( ) ( ) (39)
in ΔSB.
Hence even if chiral anomaly exists in the BCS-do-
main of the A1-phase, it is not directly connected with the
dangerous regions of momentum space near zeroes of the
gap (it does not have an infra-red character).
7.2. The approach based on the formal analogy with
quantum electrodynamics
The authors of [5,6] have the different also rather nice
approach based on the formal analogy between the anom-
alous current in 3He-A and chiral anomaly in quantum
electrodynamics (QED-theory). They assume that anoma-
lous current with the coefficient C0 ~ ρ in BCS-domain
of the A1-phase is not directly connected with the zeroes
of the gap (thus it is not contained even in the super-
symmetric hydrodynamics). They assume that it is con-
nected with the global topological considerations. To
illustrate this point they solve microscopic Bogolubov–
de-Gennes (BdG) equations for fermionic quasiparticles
in a given twisted texture ( )l || lrot of the l-vector. To be
more specific they consider the case:
l l l= +0 δ , (40)
where
l l e l l Bx lz z z y y x= = = = =0 0; ;δ . (41)
In this case:
l l =⋅ = =rot constl
l
x
Bz
y∂
∂
(42)
and accordingly
j ean = −
�
2
0
m
C B z . (43)
After linearization BdG equations become equivalent
to Dirac equation in magnetic field B = l l⋅ rot . Its solution
yields the following level structure for fermionic
quasiparticles:
E p pn z z n( ) ( )
~
= ± +ξ 2 2Δ , (44)
where ξ μ( ) ;p
p
m
e
p
p
z
z z
F
= − = = ±
2
2
1 is an electric charge
and
~
| |Δ n t Fn p eB2 22= v (45)
is a gap squared vt ~ vF Δ0/εF .
For n ≠ 0 (see Fig. 5) all the levels are gapped
~
Δ n ≠ 0
and doubly degenerate with respect to pz → – pz. Their
contribution to total mass-current is zero for T→ 0.
For n = 0 there is no gap
~
Δ
0
0= and we have an asym-
metric chiral branch which exists only for pz < 0. The en-
ergy spectrum for n = 0 yields:
E p z0 = ξ( ) . (46)
We can say that there is no gap for zeroth Landau level.
Moreover in BCS-domain E0 = 0 for |pz| = pF — the
chiral level crosses the origin in Fig. 5.
Note that in BEC-domain E0 ≥ |μ| and the zeroth Lan-
dau level does not cross the origin.
The zeroth Landau level gives an anomalous contribu-
tion to the total current in BCS-domain:
BCS–BEC crossover and nodal points contribution in p-wave resonance superfluids
Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 785
pz < 0
En
n = 1
n = 2
n = 1
n = 2
n = 0
ξ z)(p0
Fig. 5. The level structure of the Dirac equation in magnetic
field B = l l⋅ rot . All the levels with n ≠ 0 are doubly degener-
ate. The zeroth level is chiral. It crosses the origin for |pz| = pF
in BCS-domain (μ > 0).
j r e l l l lan rot rot( ) ( ) ( ) (= = − ⋅ = − ⋅
<
∫0
2 22
0
0
z
z
z
p
p
d p
C
m
z
π
ξ
�
) l ,
(47)
where
( )
| |
l l⋅
= = ∫rot p
p
eB
f
dp
z
F
y
2 2 22 2 0
2
π π π
, (48)
and as a result:
C
mpF
0
3
26
≈ ≈
π
ρ (49)
in BCS-domain.
Note that f0 ( x – py /eB) in (48) is an eigenfunction for
zeroth Landau level.
It is easy to observe that the integral for C0 in (47),
(48) is governed by the narrow cylindrical tube inside the
Fermi-sphere (see Fig. 6) with the length pF parallel to
the l-vector and radius of the cylinder squared given by:
〈 〉p p eBy F
2
� | | . (50)
8. Whether the chiral anomaly can be destroyed
by damping
The authors of [4] expressed their doubts and worries
with respect to the calculation of C0 based on Dirac equ-
ation in magnetic field B = l l⋅ rot . From their point of view
the calculation of C0 from (48), (49) is an oversimli-
fication of a difficult many-particle problem. In particular
they emphasized the role of the finite damping γ = 1/τ to
destroy the chiral anomaly at low frequencies ω < γ, thus
restoring the superfluid hydrodynamics (without normal
velocity v n and normal density ρn). Indeed, if the damp-
ing γ is larger than the level spacing of the Dirac equation:
ω 0 =
⋅
v t F
F
p
p
| |l lrot
(51)
in case when ξ(pz) = 0, then the contribution from the
zeroth Landau level should be washed out by damping
(see Fig. 7) and the chiral anomaly should be destroyed.
The damping γ for chiral fermions (for fermions living
close to the nodes) in a very clean A1-phase without im-
purities is defined at T = 0 by the different decay pro-
cesses (see [17]).
It is natural to assume that the only parameter which
defines γ at T = 0 for chiral fermions is
Δ Δ0 0〈 〉 =
〈 〉⊥θ
p
pF
.
The leading term in decay processes is given by emission
of an orbital wave (see Fig. 8). It reads
γ
ε
∝
/ + −⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⊥Δ 0
2 2 2 2 2p p p pF F z F
F
v ( )
. (52)
For pz = pF (ξ(pz) = 0):
γ
ε
�
Δ 0
2 2
2
F F
p
p
⊥ . (53)
786 Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9
M.Yu. Kagan and D.V. Efremov
– kF
l e= z
0
Fig. 6. The contribution to the coefficient C0 is governed by
the narrow cylindrical tube of the length pF and the width
〈 〉p p eBy F
2
� | | inside the Fermi-sphere.
E1
E0
Fig. 7. The possible role of the damping in destruction of the
chiral anomaly at low frequencies and small k-vectors when
γ > ω0 (ω0 = E1 – E0 is level spacing).
Fig. 8. Different decay processes for damping of chiral fer-
mions at T = 0: the standard three-fermion decay process and a
decay process with the emission of orbital wave.
Note that for chiral fermions on zeroth Landau level
〈 〉
=
⋅⎛
⎝
⎜
⎞
⎠
⎟⊥
/
p
p pF F
| |l lrot
1 2
(54)
and the level spacing for ξ(pz) = 0 reads:
ω 0 0� Δ
〈 〉⊥p
pF
. (55)
Hence γ/ω0 << 1 for these two decay processes. Thus it is
not obvious could we wash out the contribution form the
zeroth Landau level by the different decay processes in
superclean 3He-A1-phase at T = 0.
9. Discussion
The inequality ω0 << γ, which is sufficient to destroy
the chiral anomaly can be definitely fulfilled in a moder-
ately clean 3He-A1-phase, that is in the presence of the
sufficient amount of aerogel.
The possible role of aerogel. Nowadays a lot of physi-
cists working in the field of superfluid 3He concentrate
their efforts on studying the phase-diagram of superfluid
3He in the presence of impurities which partially suppress
the p-wave pairing. Namely they study the superfluid 3He
in the presence of aerogel, which constitutes the system of
SiO2-cylindrical filaments (strands) with the diameter
30 � forming the network in 3He [19,20].
It is possible nowadays to create experimentally rather
substantial amount of aerogel in 3He and have the damp-
ing γ as much as 0.1TC. Note, that effectively in the pre-
sence of aerogel γ is an external parameter which depends
only on the concentration of aerogel x.
Moderately clean case means:
0 < γ < Δ0, (56)
and can be achieved experimentally. Thus a very interest-
ing experimental proposal is to check our conjecture that
an anomalous coefficient C0( 0 << γ) is small by creating
a twisted texture l l| | rot and then varying the aerogel con-
centration x. After that it is interesting to decrease the
aerogel concentration drastically and to answer experi-
mentally the question whether γ(x → 0) is larger or smal-
ler then 0 in A1-phase.
The similar project with the impurities can be also pro-
posed for magnetic traps if it will be possible experimen-
tally to get a sufficient lifetime for p-wave resonance con-
densate in BCS-domain. Note, that experimentally we can
measure either the anomalous current (30) directly or the
spectrum of orbital waves (33), (34), which is usually eas-
ier to do.
Note also, that in magnetic traps the textures of l-vec-
tor can be prepared with the help of «cutting» the conden-
sate by the laser beam on two parts and then creating the
Josephson tunneling structures for studying the circula-
tion of anomalous current along the contour which
contains one or two Josephson junctions.
10. Conclusion
In conclusion we would like to point out that in this pa-
per we solve the Leggett equations for BCS–BEC cross-
over in A1-phase of the resonance p-wave superfluid. As
a result we found the behavior of the superfluid gap Δ, the
chemical potential μ, and the sound velocity cS in
BCS-domain (μ > 0), in BEC-domain (μ < 0) and close to
the interesting interpolation point μ = 0.
We observed that fermionic (quasiparticle) contribu-
tion to the normal density ρn and the specific heat C
v
have power-law temperature dependences in BCS-do-
main, while they are exponential in BEC-domain. Close
to the interesting point μ = 0 the power-law fermionic
contributions to ρn and C
v
can be separated rather reli-
ably from the bosonic (phonon) contributions.
We review the two different approaches to the interest-
ing problem of the chiral anomaly in BCS A1-phase and
propose a scenario for the crossover from the high-fre-
quency ballistic regime with the presence of chiral anom-
aly to a low-frequency hydrodynamic regime with its ab-
sence.
Finally we stress the role of aerogel to create the dif-
ferent textures of the l-vector and increase the damping in
superfluid A1-phase of 3He.
The authors acknowledge interesting and useful dis-
cussion with A.S. Alexandrov, A.F. Andreev, D.M. Lee,
I.A. Fomin, Yu. Kagan, G.V. Shlyapnikov, V.V. Val’kov,
G.E. Volovik, L.P. Pitaevskii, W. Ketterle, V. Gurarie, W.
Halperin and are grateful to RFBR (grant # 08-02-00224)
for financial support of this work.
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