Evolution of the dynamics of neutral superconductors between BCS and BEC regimes: the variational approach
We use variational approach to analyze the evolution of the dynamics of a neutral s-wave superconductor between BCS and BEC regimes. We consider 2D case, when BCS–BEC crossover occurs already at weak coupling and is governed by the ratio of the two scales — the Fermi energy EF and the bound state en...
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Chubukov, A.V. Mozyrsky, D. 2021-02-03T19:01:53Z 2021-02-03T19:01:53Z 2018 Evolution of the dynamics of neutral superconductors between BCS and BEC regimes: the variational approach / A.V. Chubukov, D. Mozyrsky // Физика низких температур. — 2018. — Т. 44, № 6. — С. 684-690. — Бібліогр.: 42 назв. — англ. 0132-6414 PACS: 74.20.Fg, 74.25.Вt, 74.78.–w https://nasplib.isofts.kiev.ua/handle/123456789/176148 We use variational approach to analyze the evolution of the dynamics of a neutral s-wave superconductor between BCS and BEC regimes. We consider 2D case, when BCS–BEC crossover occurs already at weak coupling and is governed by the ratio of the two scales — the Fermi energy EF and the bound state energy for two fermions in a vacuum, E₀. BCS and BEC regimes correspond to EF >> E₀ and EF << E₀, respectively. We compute the spectrum of low-energy bosonic excitations and show that the velocity of phase fluctuations remains υF / √2 through BCS–BEC crossover. We also discuss the topological Aφ ̇ term in the effective action. We thank I. Aleiner, J.J. Dziarmaga, I. Martin, M. Stone, J. Sauls, D. Solenov, and G.E. Volovik for valuable discussions. The work at Los Alamos was performed under the auspices of the United States Department of Energy under Contract DE5AC52-06NA25396 and supported by LDRD and BES Grant No’s. 20170460ER and E3B7, respectively. The work by AVC was supported by the Office of Basic Energy Sciences U.S. Department of Energy under award DE-SC0014402. AVC is thankful to KITP at UCSB, where part of the work has been done. KITP is supported by NSF grant PHY-1125915. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова Evolution of the dynamics of neutral superconductors between BCS and BEC regimes: the variational approach Article published earlier |
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Evolution of the dynamics of neutral superconductors between BCS and BEC regimes: the variational approach |
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Evolution of the dynamics of neutral superconductors between BCS and BEC regimes: the variational approach Chubukov, A.V. Mozyrsky, D. Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова |
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Evolution of the dynamics of neutral superconductors between BCS and BEC regimes: the variational approach |
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Evolution of the dynamics of neutral superconductors between BCS and BEC regimes: the variational approach |
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Evolution of the dynamics of neutral superconductors between BCS and BEC regimes: the variational approach |
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Evolution of the dynamics of neutral superconductors between BCS and BEC regimes: the variational approach |
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evolution of the dynamics of neutral superconductors between bcs and bec regimes: the variational approach |
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Chubukov, A.V. Mozyrsky, D. |
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Chubukov, A.V. Mozyrsky, D. |
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Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова |
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Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова |
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2018 |
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English |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Article |
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We use variational approach to analyze the evolution of the dynamics of a neutral s-wave superconductor between BCS and BEC regimes. We consider 2D case, when BCS–BEC crossover occurs already at weak coupling and is governed by the ratio of the two scales — the Fermi energy EF and the bound state energy for two fermions in a vacuum, E₀. BCS and BEC regimes correspond to EF >> E₀ and EF << E₀, respectively. We compute the spectrum of low-energy bosonic excitations and show that the velocity of phase fluctuations remains υF / √2 through BCS–BEC crossover. We also discuss the topological Aφ ̇ term in the effective action.
|
| issn |
0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/176148 |
| citation_txt |
Evolution of the dynamics of neutral superconductors between BCS and BEC regimes: the variational approach / A.V. Chubukov, D. Mozyrsky // Физика низких температур. — 2018. — Т. 44, № 6. — С. 684-690. — Бібліогр.: 42 назв. — англ. |
| work_keys_str_mv |
AT chubukovav evolutionofthedynamicsofneutralsuperconductorsbetweenbcsandbecregimesthevariationalapproach AT mozyrskyd evolutionofthedynamicsofneutralsuperconductorsbetweenbcsandbecregimesthevariationalapproach |
| first_indexed |
2025-11-26T21:41:37Z |
| last_indexed |
2025-11-26T21:41:37Z |
| _version_ |
1850777784336515072 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6, pp. 684–690
Evolution of the dynamics of neutral superconductors
between BCS and BEC regimes: the variational approach
Andrey V. Chubukov
Department of Physics, University of Minnesota, Minneapolis MN 55455, USA
E-mail: achubuko@umn.edu
Dmitry Mozyrsky
Theoretical Division (T-4), Los Alamos National Laboratory, Los Alamos NM 87545, USA
E-mail: mozyrsky@lanl.gov
Received January 24, 2018, published online April 25, 2018
We use variational approach to analyze the evolution of the dynamics of a neutral s-wave superconductor be-
tween BCS and BEC regimes. We consider 2D case, when BCS–BEC crossover occurs already at weak coupling
and is governed by the ratio of the two scales — the Fermi energy EF and the bound state energy for two fermi-
ons in a vacuum, E0. BCS and BEC regimes correspond to EF >> E0 and EF << E0, respectively. We compute
the spectrum of low-energy bosonic excitations and show that the velocity of phase fluctuations remains / 2Fv
through BCS–BEC crossover. We also discuss the topological Aφ term in the effective action.
PACS: 74.20.Fg Nanotubes;
74.25.Вt Thermodynamic properties;
74.78.–w Superconducting films and low-dimensional structures.
Keywords: s-wave, superconductor, Berry phase.
1. Preface
It is our great pleasure to present this article for a spe-
cial issue in memory of Alexei Alexeevich Abrikosov.
Alexei Alexeevich was one of the pioneers of field-theo-
retical approach to superconductivity and the co-author of
the book [1] which educated at least two generations of
physicists. He is also the father of vortex physics in super-
conductors. Our study is based on his works, and we hope
it will be of interest to the readers of the special memorial
issue of the Low Temperatures Physics.
2. Introduction
In this paper, we discuss the evolution of the dynamic
properties of a neutral superconductor between BCS re-
gime, when bound pairs of fermions condense immediately
once they form, and Bose–Einstein condensation (BEC)
regime, when bound pairs of fermions form at a higher insT
and condense at a smaller cT . Experimental evidence for
preformed pairs has been reported for high- cT cuprates [2]
and, more recently, for Fe-based superconductor 1eSe eF Tx x−
(Ref. 3).
We consider a 2D s-wave superconductor and assume a
rotational symmetry and 2 / (2 )k m fermionic dispersion. In
2D BCS–BEC crossover can be analyzed already within
weak coupling, when calculations are under control [4–6].
Indeed, in 2D systems, two fermions form a bound state
already at arbitrary small attraction g , with energy
2/( )002 = 2 e N gE −Λ , where 0 = / (2 )N m π is the free parti-
cle density of states per spin in 2D and Λ is the upper cut-
off for the attraction [4–6]. In 3D systems, a bound state of
two fermions emerges only once the interaction exceeds a
certain cutoff, generally of the order of fermionic band-
width [8].
The evolution of the static properties of a superconduc-
tor between BCS and BEC regimes has been extensively
discussed in the condensed matter context [4,5,7–23,42]
and also for optical lattices of ultracold atoms [24,25]. The
BCS–BEC crossover is determined by the interplay be-
tween 0E and FE : the system is deep in the BCS regime
when 0FE E and deep in the BEC regime when 0 FE E .
In the BCS regime, the pairing instability temperature and
superconducting cT are 1/2
0ins ( )FcT T E E≈ , and the pair-
ing gap ∆ also scales as 1/2
0( )FE E . In the BEC regime
© Andrey V. Chubukov and Dmitry Mozyrsky, 2018
Evolution of the dynamics of neutral superconductors between BCS and BEC regimes: the variational approach
ins 0 0/ log ( / )FT E E E , c FT E , and ∆ still scales as
1/2
0( )FE E . The chemical potential 0( = 0) = FT E Eµ − , and
it changes sign from positive to negative when 0E becomes
larger than FE .
Here we focus on the evolution of the dynamical prop-
erties. We use variational approach, in which we assume
that the order parameter ( , )∆ τr slowly fluctuates around
its equilibrium value 0∆ . We obtain the quadratic form in
the variations 0( , )∆ τ − ∆r and extract from it the spectrum
of collective modes, including gapless Anderson–Bogo-
lubov–Goldstone (AGB) phase mode. In the BCS regime,
the velocity of AGB excitations, AGBv , is the same as
the velocity of a sound wave in the normal state:
= / 2AGB Fv v , where Fv is Fermi velocity. We argue
that the value AGBv remains / 2Fv also in the BEC lim-
it. Within this approach, we also identify the linear in fre-
quency term, which corresponds to the ( , )i d d Aτ φ τ∫ r r
term in the effective action. Such term is often referred to
in the literature as the Berry phase term [26–31]. We obtain
the variation of the prefactor A with respect to the variation
of ∆. This allows us to express A as = / 2A n C+ , where n
is the density (which may depend on time), up to a con-
stant C. A constant C is irrelevant if the phase φ is defined
globally, i.e., has a certain value at any point in space, be-
cause then a prefactor can be pulled out from the d dτ∫ r ,
and the remaining term ( , )d dτφ τ∫ r r reduces to an irrele-
vant boundary term, which does not affect the equation of
motion. However, if the phase is not defined globally, as in
the case of a moving Abrikosov vortex with coordinates
( )X τ and ( )Y τ , ( , ) ( )d d XY YXτφ τ ∝ −∫ r r does not reduce
to a boundary term and does affect the equation of motion.
In particular, it gives rise to an effective Magnus force act-
ing on a vortex [26–37]. To obtain A exactly, one needs an
alternative approach, in which one expands the effective
action in terms of time derivatives of the order parameter
[38]. This approach yields = 0C in the absence of impurity
scattering.
The paper is organized as follows: in the next section
we obtain, as a warm-up exercise, the expression for the
condensation energy in the crossover between BCS and
BEC regimes. In Sec. 4 we introduce the effective action
of a superconductor in terms of its fluctuating order pa-
rameter ( , )∆ τr . In Sec. 5 we derive the dispersion relation
for the AGB mode by expanding in small variations of ∆
relative to its equilibrium value. In Sec. 6 we obtain the
term linear in the time derivative of the phase of the order
parameter (the Berry phase term). We summarize our re-
sults in Sec. 7.
3. The condensation energy
We first consider the case when the order parameter
( , )∆ τr is a constant 0∆ . In this situation the pairing Hamil-
tonian has a conventional form
( )† †
0 , ,, ,= k k kk k
k
H c c c c↑ ↓↑ ↓
ε + +∑
( )† † †*
0 0 ,, , , ,kk k k
k
c c c c ↑↑ − ↓ − ↓
+ ∆ + ∆∑ (1)
where operators †
,c σk are Fourier transforms of † ( )σψ r , etc.
and = =−ξ ξ ε − µk k k , 2= / 2mεk k . The difference be-
tween the ground state energy 0H〈 〉 in a superconductor
(with the chemical potential 0= FE Eµ − ) and in a normal
state (with the chemical potential 0 = FEµ ) is the conden-
sation energy condE .
The ground state energy is often separated into kinetic
and potential energy parts, kin pot=scE E E+ , although this
splitting is a bit elusive for a quantum system of interacting
fermions as, e.g., fermionic self-energy, which is a potential
energy of a fermion in a field created by other fermions,
contributes to the fermionic density †
,,( ) = kkn k c c ααα
〈 〉∑
and, hence, to the kinetic energy. We follow earlier works
and define the kinetic energy as kin = ( )kkE n kε∑ . The
kinetic part of the ground state energy can be expressed via
the normal Green’s function as
kin 2 2 2
, 0
= 2 ,
iE
ω
ω + ξ
− ε
ω + ∆ + ξ
∑ k
k
k k
(2)
where at = 0T and in thermodynamic limit
2 3
0
,
= / (2 ) = / (2 )S d kd SN d d
ω
ω π ε ω π∑ ∫ ∫
k
,
where S is the area of a 2D sample.
The potential energy is
0
pot 0 2 2 2
, 0
= .E
ω
∆
−∆
ω + ∆ + ξ
∑
k k
(3)
For a system with weak attraction g , the self-consistent
equation on 0∆ is
0 0
2 2 2
, 0
1= .
g S ω
∆ ∆
ω + ∆ + ξ
∑
k k
(4)
Carrying out integration over the Matsubara frequency and
over kξ up to the cutoff Λ , and using 0 0= exp ( 2/ )E N gΛ − ,
we obtain the algebraic equation [4–6]
2 2
0 0= 2 ,Eµ + ∆ − µ (5)
which defines 0∆ in terms of the actual chemical potential
µ and the two-particle bound state energy 0E . The self-
consistency equation for µ follows from the condition that
the total number of fermions, including bound pairs, is
conserved [5]. This yields
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 685
Andrey V. Chubukov and Dmitry Mozyrsky
2 2
0 = 2 .FEµ + ∆ + µ (6)
The solution of Eqs. (5), (6) is
0= ,FE Eµ − 0 0= 2 .FE E∆ (7)
Like we said, BCS–BEC crossover occurs when 0E be-
comes comparable to the Fermi energy, FE . BCS behavior
is realized when the bound state energy 0E is much smaller
than FE , and BEC behavior is realized when 0 FE E . A
negative µ at 0<FE E implies that the Fermi momentum
Fk , defined as position of the minimum of the fermionic
dispersion 2 2
0= ( )k kE ε − µ + ∆ , is zero [12]. When the
particle density is small, FE is small compared to Λ , and
the crossover occurs at small 0 /E Λ, i.e., already at weak
coupling, when 0 1N g .
To obtain the condensation energy in BCS–BEC cross-
over regime, we write
2
0
0 2 2 2
0
norm 0
0
2 ( ( ))
= ,
(2 ) ( )
2= .
(2 ) ( )
sc
id dE N S
d dE N S
i
ε ω + ε − µ + ∆ε ω
−
π ω + ∆ + ε − µ
ε ω ε
π ω − ε − µ
∫
∫
(8)
Evaluating the integrals separately for 0>FE E , when
> 0µ , and for 0<FE E , when < 0µ , and in both cases
using Eq. (7) to relate µ with 0∆ , we obtain after a straight-
forward algebra that the condensation energy
2 0
cond 0 0= , ,E SN f
µµ − ∆ ∆ ∆
(9)
where
2
21 ( 1 )( , ) = .
4 2
x x xf x y y + +
+ − (10)
Using 2 1 = 2x x y+ + (Eq. (6)) we obtain
1( , ) = ( ).
4
f x y y y x+ − (11)
Using next 2 2
0 0 0( ) = ( ) / = / = 1/ 4Fy x y E E− µ µ − µ ∆ ∆ ,
we find
1 1 1( , ) = = ,
4 4 2
f x y + (12)
Substituting into (9), we obtain that
2
0
cond 0 0 0= (2 ) =
2FE SN E E SN
∆
− − (13)
in the whole BCS–BEC crossover region.
A remark is in order about the order of integration over
frequency and dispersion. The order does not matter as
long as the integral over kε is taken within finite limits,
i.e., one can do d dω ε∫ in any order and the result will be
the same. The situation gets more tricky when the integra-
tion over ε is extended to infinite limit. The most extreme
case here is BCS limit, where µ ∆
, and the integration
over =ξ ε − µ can be formally extended to d
∞
−∞
ξ∫ . In the
BCS limit, the difference between µ and 0µ is irrelevant,
to leading order in /∆ µ , and condE can be expressed as
2 2
2
cond 0 0 2 2 2 2 2
0
=
2 ( )( )
d dE SN ω ξ ω − ξ
− ∆
π ω + ξ ω + ξ + ∆∫ . (14)
At small ω and ξ , the would be divergence of the inte-
grand is regularized by 0∆ , but at large ω and ξ , the inte-
grand scales as 2 2 2 2 2( ) / ( )ω − ξ ω + ξ , i.e., the 2D integral
is marginal by power counting. Such an integral is non-
singular, but its value depends on how the integration is
performed. If we treat ω and ξ as “equivalent” variables
and evaluate the integral using polar coordinates, the inte-
gral vanishes because of 2 2ω − ξ in the numerator. If we
integrate over ω first or over ξ first, both times in infinite
limits, the result is finite, but obviously of a different sign.
Because at = 0T the integration over Matsubara frequency
ω truly goes from −∞ to +∞ , while the integration over ξ
actually holds in finite limits, which can only approximate-
ly be extended to ±∞ , the correct way to evaluate the inte-
gral in (14) is to integrate over frequency first. Integrating,
we obtain
2
2 0
cond 0 0 2 2 2 2 2
0
1=
2 (| | )
dE SN
∆ξ
− ∆
ξ + ∆ ξ + ξ + ∆
∫ . (15)
The integrand scales as 31/ | |ξ at large | |ξ , i.e., the in-
tegral converges and can be evaluated in infinite limits. By
rescaling 0= xξ ∆ , one can immediately verify that the
integral does not depend on 0∆ , and (15) reduces to
2
2 2 2
cond 0 0 0 02
0
1= 1 =
21
dxE SN x x SN
x
∞
− ∆ + − − ∆
+
∫ .
(16)
This agrees with (13).
One can further check that the variation of the energy is
exactly the same as the variation of the chemical potential
2
0 0 0 0= ( ) = 2 ( ) / 2F F F Fn n E N E E E E Nδµ µ − + − ≡ ∆ . As
a result, E N− µ (equal to the Grand potential at = 0T ) does
not change between the normal and the superconducting
state.
Note in passing that the self-consistency analysis can be
straightforwardly extended to a finite T . In the BCS re-
gime, the onset temperatures pT for the pairing and cT for
superconductivity (i.e., for pair coherence) are comparable
and both scale as 0FE E (more precisely, pT almost coin-
cides with the mean-field cT , while the actual cT of
Berezinskii–Kosterlitz–Thouless transition is smaller by a
numerical factor). In the BEC regime the two temperatures
differ strongly: 0 0/ log ( / )p F FT E E E E
, while c FT E ,
i.e., the ratio /c pT T vanishes at 0FE → (see, e.g., Ref. 6).
When 0FE E , 0 ins= 1.76T∆ , like in BCS theory, when
0FE E , 0∆ is much larger than cT , but much smaller
than pT .
686 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
Evolution of the dynamics of neutral superconductors between BCS and BEC regimes: the variational approach
4. The effective action
The effective action for an order parameter of an s-wave
superconductor can be obtained by departing from a mi-
croscopic model with local four-fermion attractive interac-
tion g− ( > 0g ) and decoupling four-fermion interaction
via Hubbard–Stratonovich transformation [39], by intro-
ducing the pairing field ( , )r∆ τ . This procedure is well
documented, and we just quote the results.
The partition function Z is expressed via the integral
over the Grassmann fields as
[ , ]= e ,SZ d d − ψ ψψ ψ∫ (17)
where = ( , )αψ ψ τr and = ( , )αψ ψ τr are spin-full coordi-
nate and time dependent Grassmann fields, and
( )[ , ] = ( , ) ( , ) [ , ] .S d d Hα τ αψ ψ τ ψ τ ∂ ψ τ + ψ ψ∫ r r r (18)
Here τ is the imaginary (Matsubara) time = itτ and
2
†[ , ] = ( , ) ( , )
2
H
mσ σ
∇
ψ ψ ψ τ − − µ ψ τ −
r r
† †( , ) ( , ) ( , ) ( , ).g ↓ ↑↑ ↓
− ψ τ ψ τ ψ τ ψ τr r r r (19)
The four-fermion interaction is decoupled by Hubbard–Stra-
tonovich transformation
22
22 1e = e .
2
y yxax
a
dy
a
− +
π ∫ (20)
In our case we introduce two Hubbard–Stratonovich fields
( , )∆ τr and *( , )∆ τr and rewrite the partition function as
** [ , , , ]= e ,SZ d d d d − ψ ψ ∆ ∆ψ ψ ∆ ∆∫ (21)
where now
*[ , , , ] =S ψ ψ ∆ ∆
2
*| ( , ) |= ( , ) ( , ) [ , , , ] ,d d H
gα τ α
∆ τ
τ ψ τ ∂ ψ τ + + ψ ψ ∆ ∆
∫
rr r r
(22)
and
2
*[ , , , ] = ( , ) ( , )
2
H
mσ σ
∇
ψ ψ ∆ ∆ ψ τ − − µ ψ τ +
r r
( , ) ( , ) ( , ) ( , ) ( , ) ( , )∗
↑ ↓ ↓ ↑
+ ∆ τ ψ τ ψ τ + ∆ τ ψ τ ψ τ r r r r r r . (23)
The action *[ , , , ]S ψ ψ ∆ ∆ can be re-expressed in a more
compact form by introducing Gorkov–Nambu spinor
†= [ , ]T↑ ↓
ψ ψ ψ . Then
2
* | ( , ) |[ , , , ] =S d d
g
∆ τ
ψ ψ ∆ ∆ τ −∫
rr
1( , ) ( , ; , ) ( , )d d d 'd G ' '−′ ′ ′− τ τ ψ τ τ τ ψ τ∫ ∫r r r r r r , (24)
where the inverse Green’s function 1( , ; , )G '− ′τ τr r is given by
1 ˆ ˆ( , ; , ) = [ ( ) ( , , )] ( ) ( ),G ' K−
τ′ ′ ′τ τ −∂ − − ∆ τ λ δ − δ τ − τr r r r r r
(25)
with
2
2
(1/ 2 ) 0ˆ ( ) = ,
0 (1/ 2 )
m
K
m
− ∇ − µ
∇ + µ
r
and
0 ( , )ˆ ( , ) = .
( , ) 0∗
∆ τ
∆ τ
∆ τ
r
r
r
Integrating over ψ and ψ we then obtain
** [ , ]= e SZ d d − ∆ ∆∆ ∆∫ (26)
and
2
* | ( , ) |[ , ] =S d d
g
∆ τ
∆ ∆ τ −∫
rr
1ˆlog ( , ; , ).d d d 'd Tr G '−′ ′− τ τ τ τ∫ ∫r r r r (27)
In Fourier space (momentum k and Matsubara frequency ω)
1 ( , )ˆ ( , ) = .
( , )
k
k
i k
G k
k i
∗
−
∗
ω − ξ ∆ ω
ω
∆ ω ω + ξ
5. The dynamics of phase fluctuations
We now expand the action in small variations of the or-
der parameter around a constant value, ( , ) = ( , )δ∆ τ ∆ τ −r r
0 ( ) ( ).− ∆ δ δ τr . The action to the second order in δ∆ has
been obtained in the BCS–BEC crossover regime in
Ref. 40, and our expression for the quadratic form in δ∆ in
the action agrees with theirs. The authors of Ref. 40, how-
ever, didn’t extract the spectrum of the ABG mode from
their data, nor they extracted the linear in Ω (i.e., φ ) term
in the action. Collecting all second-order contributions and
Fourier transforming to momentum and Matsubara fre-
quency space, we obtain the variation of in the form
( )2 2
,
1= | | | | h.c. ,
2
A A B+ + − − + −
Ω
δ δ∆ + δ∆ + δ∆ δ∆ + ∑
q
(28)
where
= ( , ) = ( , ) e ,
= ( , ) = ( , ) e
i i
i i
d d t
d d t
− + Ωτ
+
− Ωτ
−
δ∆ δ∆ Ω τ δ∆
δ∆ δ∆ − −Ω τ δ∆
∫
∫
qr
qr
q r r
q r r
(29)
and
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 687
Andrey V. Chubukov and Dmitry Mozyrsky
/2 /2
2 2 2 2 2 2
, 0 /2 0 /2
2
0
2 2 2 2 2 2
, 0 /2 0 /2
( ( / 2) )( ( / 2) )1= ,
(( / 2) )(( / 2) )
= ,
(( / 2) )(( / 2) )
k
i i
A
g
B
± − ±
±
ω ± − ±
ω + − +
ω ± Ω + ξ − ω Ω + ξ
− ε
ω ± Ω + ∆ + ξ ω Ω + ∆ + ξ
∆
ω + Ω + ∆ + ξ ω − Ω + ∆ + ξ
∑
∑
k q k q
k
k q k q
k k q k q
(30)
______________________________________________
where, we remind, 2= = / (2 )k k k mξ ε − µ − µ.
The excitation spectrum is obtained from the condition
2=A A B+ − , or, equivalently
2
= 0.
2 2 2
A A A A A AB B+ − + − + −+ + − + − −
(31)
Evaluating the frequency and momentum integrals, we
obtain
20
2 2 20 0
= 1
2 8
NA A B+ −
+ µ − Ω + +
∆ µ + ∆
22
0
2 2 2 2
0 0
1 =
m
∆µ + µ + + µ + ∆ µ + ∆
q
2
20
2
00
,
4
F
F
F
N E E
E E m
= Ω +
+∆
q
0 0
2 2 00
= 1 ... = ...,
2 2
F
F
N N EA A B
E E
+ −
+ µ + + + +
+µ + ∆
0 0
2 2 00
1 1= , = = ,
2 4 4 F
N NA A i A A
E E
+ −−
Ω − −
+µ + ∆
(32)
where dots stand for Ω and q-dependent terms in
( ) / 2A A B+ −+ + , which we do not need. Substituting this
into (31), we obtain from (31)
2 2
2 20
0 0
= 0.
16 ( ) 2
F
F
N
E E E
Ω +
+
qv
(33)
Converting this last expression into real frequencies
( iΩ → − Ω), we see that the excitation spectrum has a gap-
less mode, whose velocity remains / 2Fv , no matter
what the ratio of 0 / FE E is.
To see that this mode corresponds to phase fluctuations,
we split the complex variations +δ∆ and −δ∆ into real and
imaginary parts ±′δ∆ and ±′′δ∆ and re-express the action in
Eq. (28) as
2 2
2
,
1= ( )
2 q
B BA A
A A+ + − − −
+ +Ω
′ ′ ′δ δ∆ δ∆ + δ∆ − +
∑
2 2
2
,
1 ( ) .
2 q
B BA A
A A+ − − − −
+ +Ω
′′ ′′ ′′+ δ∆ δ∆ + δ∆ −
∑ (34)
This expression shows that there are gapped and gap-
less modes. Suppose that the equilibrium value of 0=∆ ∆
is real. Then longitudinal variation ( , )r tδ∆ in coordinate
is real and transverse one is imaginary. Using Eq. (29) we
find that for longitudinal gap fluctuations –=+′ ′δ∆ −δ∆
and =+ −′′ ′′δ∆ δ∆ , while for transverse (phase) fluctuations
–=+′ ′δ∆ −δ∆ and =+ −′′ ′′δ∆ δ∆ . Using now the fact that at
= 0Ω and = 0q , = =A A B+ − , we immediately find from (31)
that longitudinal fluctuations are gapped and phase fluctua-
tions are gapless, as they should be.
The 2q term in the integrand in Eq. (28) for δ deter-
mines the phase stiffness. Taking 2q term from A A+ −+
in (29) and expressing ( , )r∆ τ as 0( , ) = eir ∇φ∆ τ ∆ r , such
that 0( , ) = ( )qδ∆ Ω ∆ δ − ∇φq , we obtain
2 20= ( ) = ( ) .
8 4
FN End
m m
δ ∇φ ∇φ∫ r (35)
We see that the phase stiffness also does not change in
BCS–BEC crossover and remains the same as in BCS limit
[41].
The expression for δ , Eqs. (28) and (30), have been
obtained in [40], but the excitation spectrum have not been
obtained there, although the dispersion of phase fluctua-
tions at arbitrary 0 / FE E can be extracted from Eq. (26) in
that paper.
6. The term linear in τ∂ φ in the action
The prefactors A+ and A− contain the piece linear in Ω ,
with opposite signs. The corresponding term in the action is
0
linear 2 2
0
=
8
N
iδ − ×
µ + ∆
( ) ( )2 2 2 2
– –
,
( ) ( ) ( ) ( ) =
q
+ +
Ω
′ ′ ′′ ′′× Ω δ∆ − δ∆ + δ∆ − δ∆ ∑
0
2 2
08
N
i= − ×
µ + ∆
[ ]– – – –
,
( )( ) ( )( ) .
q
+ + + +
Ω
′ ′ ′ ′ ′′ ′′ ′′ ′′× Ω δ∆ + δ∆ δ∆ − δ∆ + δ∆ + δ∆ δ∆ − δ∆∑
(36)
For definiteness, let’s assume that ±δ∆ are real, i.e.,
=± +′δ∆ δ∆ . The transverse gap variation –+′ ′δ∆ − δ∆ can
be expressed as the time variation of the phase of super-
conducting gap 0 0( , ) = e = (1 ...)ir iφτ∆ τ ∆ ∆ + φτ +
, where
( , )φ = φ τr and = /φ ∂φ ∂τ . Taking the Fourier transform we
find
688 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
Evolution of the dynamics of neutral superconductors between BCS and BEC regimes: the variational approach
– 0= 2 sin .d+′ ′ ′ ′ ′δ∆ − δ∆ − φ∆ τ Ωτ τ∫ (37)
The longitudinal gap variation can be approximated by its
= 0q and = 0Ω value = ( , )d d′ ′δ∆ τδ∆ τ∫ r r . Substituting
this into Eq. (36) we obtain
0 0
linear 2 2
0
= ( , ),
2
N
iQ d d
∆
′δ τ φδ∆ τ
µ + ∆
∫ r r (38)
where
= sin .
2
d dQ
′Ω τ ′ ′Ωτ Ωτ
π∫ (39)
The integral contains the universal contribution = (2 / )Q π ×
( )sin / = 1d× Ω Ω Ω∫ and parasitic high-energy contribu-
tion, which vanishes under proper regularization. Substitut-
ing = 1Q into (38) we obtain
0 0
linear 2 2
0
= ( , ).
2
N
i d d
∆
′δ τ φδ∆ τ
µ + ∆
∫ r r (40)
We now observe that to leading order in ( , )′δ∆ τr ,
2 2
0 0( , ) /′∆ δ∆ τ µ + ∆r is the variation of 2 2| ( , ) |µ + ∆ τr
over ∆, keeping µ constant. As the consequence, Eq. (40)
can be viewed as the variation of
2 20
linear 0
( , )= ( ) = ,
2
N
i d d C iA d d∂φ τ τ µ + ∆ + µ τφ ∂τ∫ ∫
rr r
(41)
where
2 20
0= ( ) ,
2
N
A C µ + ∆ + µ
(42)
and ( )C µ is some unknown constant, which depends on µ.
Using 2 2
0 0 0= 2 =Fn N E N µ + ∆ + µ
, we can rewrite (42) as
= ( ),
2
nA C+ µ (43)
where ( )C µ is some other constant. This constant cannot
be obtained using the variational approach. In Ref. 38 we
obtained a systematic expansion of the action in terms of
the derivatives of the order parameter over the imaginary
time. In this approach we computed the A term explicitly
and found that = 0C in the limit when the energy differ-
ence between discrete levels in the vortex core well ex-
ceeds a scattering rate due to impurities.
7. Conclusion
In this paper we analyzed the evolution of the low-fre-
quency dynamics of collective excitations of an s-wave
neutral superconductor at zero temperature, between BCS
and BEC regimes. The two regimes correspond to small
and large ratio of 0 / FE E , respectively, where FE is the
Fermi energy, and 0E is the bound state energy for two
particles. In 2D, bound state develops already at weak coupl-
ing, what allows one to analyze the crossover without in-
cluding strong coupling renormalizations. We found that
the phase velocity of the collective excitations remains
/ 2Fv through the BCS–BEC crossover. The supercon-
ducting stiffness (the prefactor for 2( )∇φ term in the ac-
tion) also does not change through the BCS–BEC crosso-
ver and remains equal to 0 / (4 ) = / (8 )FN E m n m , as in
BCS limit. The action also contains the term linear in time
derivative of φ — the Berry phase term. When φ is defined
globally, i.e., has a certain value at any point in space, such
term reduces to a boundary term and does not contribute
to equation of motion, if the density is homogeneous. If
= ( )n n τ , the linear in φ term in the action is
linear = ( / 2) ( )i d d nτ τ φ∫ ∫r , where = ( , )φ φ τr . For the
case of a moving vortex with coordinates ( )X τ and ( )Y τ ,
φ is not uniquely defined at the center of the vortex core,
i.e., for x X→ and y Y→ . In this situation
( )d d d XY YXτφ ∝ τ −∫ ∫r , which does not reduce to the
boundary term. Then the Berry phase term in the action
contributes to equation of motion even when the prefactor
is a constant, and it becomes relevant to find the exact
prefactor = / 2A n C+ . The exact prefactor A has been
computed by expanding the action in time derivatives of
the order parameter [38]. This calculation yields = 0C .
Our results for the expansion of the effective action in
terms of ′δ∆ can be straightforwardly extended to other
symmetries of the order parameter and to non-Galilean-
invariant dispersion. One can also use our formulas to ob-
tain terms with higher-order derivatives (these are terms
with higher powers of Ω and q in the integrand of Eq. (28)).
Acknowledgements
We thank I. Aleiner, J.J. Dziarmaga, I. Martin, M. Stone,
J. Sauls, D. Solenov, and G.E. Volovik for valuable discus-
sions. The work at Los Alamos was performed under the
auspices of the United States Department of Energy under
Contract DE5AC52-06NA25396 and supported by LDRD
and BES Grant No’s. 20170460ER and E3B7, respectively.
The work by AVC was supported by the Office of Basic
Energy Sciences U.S. Department of Energy under award
DE-SC0014402. AVC is thankful to KITP at UCSB, where
part of the work has been done. KITP is supported by NSF
grant PHY-1125915.
________
1. A.A. Abrikosov, L.P. Gor’kov, and I.E. Dzyaloshinski,
Methods of Quantum Field Theory on Statistical Physics,
Dover Publications, New York (1975).
2. see e.g., J.C. Campuzano, M.R. Norman, and M. Randeria,
in: Physics of Superconductors, Vol. II, K.H. Bennemann
and J.B. Ketterson (eds.), Springer, Berlin (2004), p. 167; Lu
Li, Yayu Wang, and N.P. Ong, Phys. Rev. B 87, 056502 (2013)
and references therein; D.S. Dessau, private communication.
3. K. Okazaki, Sci. Rep. 4, 4109 (2014); S. Kasahara, PNAS
111, 16309 (2014); S. Rinott, Sci. Adv. 3, e1602372 (2017).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 689
https://doi.org/10.1126/sciadv.1602372
Andrey V. Chubukov and Dmitry Mozyrsky
4. K. Miyake, Prog. Theor. Phys. 69, 1794 (1983).
5. M. Randeria, J.-M. Duan, and L.-Y. Shieh, Phys. Rev. Lett.
62, 981 (1989); Phys. Rev. B 41, 327 (1990).
6. A.V. Chubukov, I. Eremin, and D.V. Efremov, Phys. Rev. B
93, 174516 (2016).
7. V.M. Loktev, R.M. Quick, and S.G. Sharapov, Phys. Rep.
349, 1 (2001).
8. C.A.R. Sa de Melo, M. Randeria, and J.R. Engelbrecht,
Phys. Rev. Lett. 71, 3202 (1993); J.R. Engelbrecht, M.
Randeria, and C.A.R. Sa de Melo, Phys. Rev. B 55, 15153
(1997).
9. P. Noziéres and S. Schmitt-Rink, J. Low Temp. Phys. 59,
195 (1985).
10. M. Drechsler and W. Zwerger, Ann. Phys. 1, 15 (1992).
11. Y. Ohashi and A. Griffin, Phys. Rev. A 67, 033603 (2003)
12. Y.L. Loh, M. Randeria, N. Trivedi, C.-C. Chang, and R.
Scalettar, arXiv:1507.05641.
13. see M.Y. Kagan, Modern Trends in Superconductivity and
Superfluidity, Lecture Notes in Physics, 874, Springer,
Heidelberg/Germany (2013) and references therein.
14. V.B. Geshkenbein, L.B. Ioffe, and A.I. Larkin, Phys. Rev. B
55, 3173 (1997).
15. B. Tobijaszewska and R. Micnas, Acta Physica Polonica A
97, 393 (2000).
16. L. Benfatto, A. Toschi, S. Caprara, and C. Castellani, Phys.
Rev. B 64, 140506(R) (2001); L. Benfatto, S. Caprara, C.
Castellani, A. Paramekanti, and M. Randeria, Phys. Rev. B
63, 174513 (2001); L. Benfatto, A. Toschi, and S. Caprara,
Phys. Rev. B 69, 184510 (2004).
17. M. Iskin and C.A.R. Sa de Melo, Phys. Rev. B 74, 144517
(2006).
18. L. Fanfarillo, L. Benfatto, S. Caprara, C. Castellani, and M.
Grilli, Phys. Rev. B 79, 172508 (2009).
19. D. Innocenti, N. Poccia, A. Ricci, A. Valletta, S. Caprara, A.
Perali, and A. Bianconi, Phys. Rev. B 82, 184528 (2010); A.
Bianconi, D. Innocenti, A. Valletta, and A. Perali, J. Phys.:
Conf. Ser. 529, 012007 (2014).
20. J. Ranninger, J.M. Robin, and M. Eschrig, Phys. Rev. Lett.
74, 4027 (1995).
21. A. Levchenko, M.R. Norman, and A.A. Varlamov, Phys.
Rev. B 83, 020506(R) (2011).
22. L. Fanfarillo, L. Benfatto, and C. Castellani, Phys. Rev. B 85,
024507 (2012).
23. A. Guidini and A. Perali, Supercond. Sci. Technol. 27,
124002 (2014).
24. Q. Chen, J. Stajic, S. Tan, and K. Levin, Phys. Rep. 412, 188
(2005); Y. Shin, C.H. Schunck, A. Schirotzek, and W. Ketterle,
Nature 451, 689693 (2007); S. Giorgini, L.P. Pitaevskii, and
S. Stringari, Rev. Mod. Phys. 80, 1215 (2008); F. Chevy and
C. Mora, Rep. Prog. Phys. 73, 112401 (2010); Z. Shen, L.
Radzihovsky, and V. Gurarie, Phys. Rev. Lett. 109, 245302
(2012); M. Randeria and E. Taylor, Ann. Rev. Condens.
Matter Phys. 5, 209232 (2014).
25. J.P. Gaebler, Nat. Phys. 6, 569 (2010).
26. G.E. Volovik, The Universe in a Helium Droplet, Clarendon
Press, Oxford (2003).
27. P. Ao and D.J. Thouless, Phys. Rev. Lett. 93, 090403-1
(2004); J.R. Aitchison, P. Ao, D.J. Thouless, and X.-M. Zhu,
Phys. Rev. B 51, 6531 (1995).
28. J. Dziarmaga, Phys. Rev. B 53, 6572 (1996).
29. A. van Otterlo, M.V. Feigelman, V.B. Geshkenbein, and G.
Blatter, Phys. Rev. Lett. 75, 3736 (1995); A. van Otterlo,
D.S. Golubev, A.D. Zaikin, and G. Blatter, Eur. Phys. J. B
10, 131 (1999);
30. E. Simanek, Phys. Rev. B 52, 10336 (1995).
31. M. Stone, Phys. Rev. B 54, 13222 (1996).
32. N.B. Kopnin, G.E. Volovik, and U. Parts, Europhys. Lett. 32,
651 (1995).
33. J. Bardeen, R. Kummel, A.E. Jacobs, and L. Tewordt, Phys.
Rev. 187, 556 (1969).
34. J. Bardeen and M.J. Stephen, Phys. Rev. 140, A 1197 (1965);
P. Nozieres and W.F. Vinen, Philos. Mag. 14, 667 (1996).
35. N.B. Kopnin and V.E. Kravtsov, JETP Lett. 23, 578 (1976);
N.B. Kopnin and M.M. Salomaa, Phys. Rev. B 44, 9667
(1991).
36. G.E. Volovik, JETP Lett. 65, 676 (1997).
37. F. Gaitan, Phys. Rev. B 51, 9061 (1995).
38. D. Mozyrsky, A.V. Chubukov, and M. Stone, to be submitted.
39. R.L. Stratonovich, Sov. Phys. Dokl. 2, 416 (1958); J. Hubbard,
J. Phys. Rev. Lett. 3, 77 (1959); A. Altland and B. Simons,
Condensed Matter Field Theory, University Press, Cam-
bridge (2006).
40. L. He, H. Lu, G. Cao, H. Hu, and X.-J. Liu, Phys. Rev. A 92,
023620 (2015).
41. D.J. Scalapino, S.R. White, and S. Zhang, Phys. Rev. B 47,
7995 (1993); L. Benfatto, A. Toschi, S. Caprara, and C.
Castellani, Phys. Rev. B 64, 140506(R) (2001); L. Benfatto,
S. Caprara, C. Castellani, A. Paramekanti, and M. Randeria,
Phys. Rev. B 63, 174513 (2001); S.G. Sharapov, V.P. Gusynin,
and H. Beck, Eur. Phys. J. B 30, 45 (2002); L. Benfatto, A.
Toschi, and S. Caprara, Phys. Rev. B 69, 184510 (2004).
42. Y. Lemonik, I.L. Aleiner, and B.L. Altshuler,
arXiv:1505.04820.
___________________________
690 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
https://doi.org/10.1143/PTP.69.1794
https://doi.org/10.1103/PhysRevLett.62.981
https://doi.org/10.1103/PhysRevB.41.327
https://doi.org/10.1103/PhysRevB.93.174516
https://doi.org/10.1016/S0370-1573(00)00114-9
https://doi.org/10.1103/PhysRevLett.71.3202
https://doi.org/10.1103/PhysRevB.55.15153
https://doi.org/10.1007/BF00683774
https://doi.org/10.1002/andp.19925040105
https://doi.org/10.1103/PhysRevA.67.033603
https://doi.org/10.1103/PhysRevB.55.3173
https://doi.org/10.1103/PhysRevB.55.3173
https://doi.org/10.12693/APhysPolA.97.393
https://doi.org/10.1103/PhysRevB.64.140506
https://doi.org/10.1103/PhysRevB.64.140506
https://doi.org/10.1103/PhysRevB.63.174513
https://doi.org/10.1103/PhysRevB.63.174513
https://doi.org/10.1103/PhysRevB.69.184510
https://doi.org/10.1103/PhysRevB.74.144517
https://doi.org/10.1103/PhysRevB.79.172508
https://doi.org/10.1103/PhysRevB.82.184528
https://doi.org/10.1088/1742-6596/529/1/012007
https://doi.org/10.1088/1742-6596/529/1/012007
https://doi.org/10.1103/PhysRevLett.74.4027
https://doi.org/10.1103/PhysRevB.85.024507
https://doi.org/10.1088/0953-2048/27/12/124002
https://doi.org/10.1016/j.physrep.2005.02.005
https://doi.org/10.1103/RevModPhys.80.1215
https://doi.org/10.1088/0034-4885/73/11/112401
https://doi.org/10.1103/PhysRevLett.109.245302
https://doi.org/10.1146/annurev-conmatphys-031113-133829
https://doi.org/10.1146/annurev-conmatphys-031113-133829
https://doi.org/10.1103/PhysRevLett.93.090403
https://doi.org/10.1103/PhysRevB.51.6531
https://doi.org/10.1103/PhysRevB.53.6572
https://doi.org/10.1103/PhysRevLett.75.3736
https://doi.org/10.1007/s100510050836
https://doi.org/10.1007/s100510050836
https://doi.org/10.1103/PhysRevB.52.10336
https://doi.org/10.1103/PhysRevB.54.13222
https://doi.org/10.1209/0295-5075/32/8/006
https://doi.org/10.1103/PhysRev.187.556
https://doi.org/10.1103/PhysRev.187.556
https://doi.org/10.1080/14786436608211964
https://doi.org/10.1103/PhysRevB.44.9667
https://doi.org/10.1134/1.567405
https://doi.org/10.1103/PhysRevB.51.9061
https://doi.org/10.1103/PhysRevLett.3.77
https://doi.org/10.1103/PhysRevA.92.023620
https://doi.org/10.1103/PhysRevB.64.140506
https://doi.org/10.1103/PhysRevB.63.174513
https://doi.org/10.1140/epjb/e2002-00356-9
https://doi.org/10.1103/PhysRevB.69.184510
1. Preface
2. Introduction
3. The condensation energy
4. The effective action
5. The dynamics of phase fluctuations
6. The term linear in in the action
7. Conclusion
Acknowledgements
|