On the Meissner Effect of the Odd-Frequency Superconductivity with Critical Spin Fluctuations: Possibility of Zero Field FFLO pairing

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by  Yuki Fuseya

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On the Meissner Effect of the Odd-Frequency Superconductivity with Critical Spin
Fluctuations:
Possibility of Zero Field FFLO pairing
Yuki Fuseya∗ and Kazumasa Miyake
Department of Materials Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531
1
1
0 Weinvestigate theinfluenceof critical spin fluctuationson electromagnetic responses in the
2 odd-frequency superconductivity. It is shown that the Meissner kernel of the odd-frequency
superconductivityisstronglyreducedbythecriticalspinfluctuationorthemasslessspinwave
n
a modeintheantiferromagneticphase.Theseresultsimplythatthesuperfluiddensityisreduced,
J andtheLondonpenetrationdepthislengthenedfortheodd-frequencypairing.Itisalsoshown

1 that thezero field Flude-Ferrell-Larkin-Ovchinnikovpairingis spontaneously realized bothfor
even- and odd-frequency in the case of sufficiently strong coupling with low lying spin-modes.

]
n
KEYWORDS: odd-frequencysuperconductivity,Meissner effect,spinfluctuation,superfluiddensity,pene-
o
tration depth,zero-field FFLO
c
-
r
p
1. Introduction Besides the studies on the possible existence of odd-ω
u
s Odd-frequency superconductivity, whose gap function SC,thethermodynamicstabilityorthesignoftheMeiss-
t. is odd in frequency, is an old but new class of super- ner response had been still remaining unclear. Within a
a conductivity. It was introduced originally by Berezin- naive calculation, the Meissner kernelof the odd-ω pair-
m
skii for triplet pairing,1 and by Balatsky and Abrahams ingseemstohaveanoppositesingfromthatoftheusual
d- for singlet pairing.2 Soon after the Balatsky-Abrahams, (even-ω )pairing,suggestingthethermodynamicallyun-
n the possibility ofodd-frequency superconductivity (odd- stable superconducting states. Very recently, this puzzle
o ω SC) was discussed in a wide variety of models, e.g., wassolvedbytheappropriatetreatmentofthegapfunc-
c the Kondo lattice model,3–7 the square-8 and triangular tionwithretardationinthepath-integralformalism.48,49
[
lattice9 Hubbard model, and the t-J model.10 According to these works, the odd-ω SC is thermody-

2 The experimental evidence on the odd-ω SC was first namicallystable,andexhibits theconventionalMeissner
v pointedoutbytheauthors.11 InCe-basedheavyfermion effect. However, these arguments are restricted only to
2
superconductors,suchasCeCu Si andCeRhIn ,several thecoherentpart(withoutthecorrelationsduetothein-
2 2 2 5
7 experimentsonthenuclearmagneticrelaxationrate12–14 coherentpartofthequasiparticle).Bycontrast,previous
3 and specific heat15,16 exhibited behaviors of gapless SC studies3–9,11,20–23demonstratedthattheodd-ωsolutions
6. near the antiferromagnetic (AF) quantum critical point are realized only when the Cooper pairs are mediated
0 (especially in the AF phase). The origin of this gap- by a certain strong spin fluctuation, strongly suggesting
0 less SC had long been a mystery. In Ref. 11, the au- the relevant contribution of the incoherent part. So we
1 thors showed that the odd-ω SC is realized due to spin- havetodiscusscarefullytheMeissnereffectoftheodd-ω
v: fluctuationneartheAFquantumcriticalpointand/orin pairingconsideringthecontributionsfromtheincoherent
i theAFstatebysolvingtheBethe-Salpeterequation17 of part,whichareneglectedinthepreviousarguments.48,49
X
SC based on the itinerant-localized duality theory.18,19 Inthispaper,weconsidertheincoherentcorrectionsdue
ar The phase diagram so obtained agrees well with the ex- to criticalspin fluctuations which are crucialingredients
perimental phase diagram, considering the fact that the for realizing the odd-ω SC.
odd-ω SCexhibitsthegaplessbehavior.Accordingly,the In 2, the ordinary arguments for the electromagnetic
§
originof the gapless SC in Ce-compounds can be under- response, where the incoherent part is neglected, are
stood as the odd-ω SC. briefly reviewed. In 3, we introduce spin fluctuations in
§
Afterthissuggestion,detailedinvestigationshavebeen ageneralformandinvestigatetheMeissnereffectbycal-
made into the triangular-type lattice,20–22 and into the culating the current-current response function with the
effective model of U-compounds,23 both of which are re- first order correction of the critical spin fluctuations or
vealed to possess the odd-ω state under some realistic spinwavemodesintheAFstate.There,weexaminethe
conditions. It now appears probable that the odd-ω SC singlet pairing in the AF phase, where the realization
can be realized in surprisingly familiar situations. The of the odd-ω is guaranteed by the previous work.11 The
concept of the odd-ω SC is so general that nowadays it results indicate that the Meissner kernel is reduced due
has expanded into various fields, e.g., magnetic or den- to the correction of such spin fluctuations. Next, in 4,
§
sitywaveorder,24,25superconductingjunctions,26–43vor- westudythetripletpairingwithferromagneticspinfluc-
tex core,44,45 proximityeffect superfluid3He,46 andcold tuations in the same manner as in 3. It is shown that
§
atoms.47 the Meissner effect is also reduced in triplet pairings. In
5, we give another argument of the electromagnetic re-
§
∗E-mail:[email protected] sponsebycalculatingthesuperconductingdensity,which
2 J.Phys.Soc.Jpn. FullPaper AuthorName
isgivenbythespatiallygradientterminthefreeenergy. Δ Δ+
Up to the first order of the critical spin fluctuations, its = + x x
result is equal to that obtained by the current-current
_
response function as expected. With this procedure, we = x
Δ
carry out the calculation of higher order corrections up
to the third order. Then, we conclude that the Meissner
kernelisstronglyreducedbothforeven-andodd-ω pair- Fig. 1. Diagrammaticexpressionsoftheexpansionofthenormal
G andanomalous GreenfunctionF.
ing,evenifwetakeintoaccounttheinfinite orderofcor-
rections.Our results also suggest that the Flude-Ferrell-
Larkin-Ovchinnikov(FFLO)pairingisspontaneouslyre-
some straightforwardmanipulations, we have
alized without a magnetic field in the case of sufficiently
p 2
strongcoupling.Finally,in 6,wediscussimplicationsof Q (0)=2e2T x
the present results. § xx m
Xp,n(cid:16) (cid:17)
2. Electromagnetic response of the coherent 2[G(p)]3G( p)+[G(p)G( p)]2 ∆ (ε )2
µ n
× − − − | |
part
(cid:8)Ne2 ∆ (ε )2 (cid:9)
µ n
In this section, we reintroduce the Meissner kernel in = πT | | >0, (6)
m ε 3
theGinzburg-Landau(GL)regionwiththecoherentpart n | n|
X
only in a convenientformfor later discussions.The rela-
which leads to the ordinary (diamagnetic) Meissner ef-
tionbetweenthecurrentdensityandthevectorpotential fect.50 (Here, we have used the relation N = p3/3π2.)
has the following form50 F
This expression, however, does not include the effect of
j(q)= Q(q)A(q) the incoherent part, or the spin fluctuations, which is
− indispensable for the realization of the odd-ω SC. Only
Ne2 2e2T from eq. (6), we cannot see how the Meissner effect and
= A(q) p[p A(q)]
− m − m2 · electromagnetic responses will be modified by the indis-
p,n
X pensable corrections of the incoherent part.
G(p+)G(p−)+F(p+)F+(p−) , (1) It should be noted here that there is no difference be-
×
tween the even- and odd-ω pairing in the calculation
where p (p,(cid:2)ε ), p p q/2, G and F((cid:3)+) are the
n ±
≡ ≡ ± of the Meissner kernel (except for the ε -dependence of
normal Green function and the anomalous Green func- n
∆(ε )), since the structures of G and F are the same.
tion,respectively.N isthenumberofquasiparticleswith n
In other words, the classification of even- and odd-ω is
a chargee anda massm. The kernelQ (q) is givenba-
xx
that with respect to the relative coordinate, p, whereas
sically by the current-current response function, which
the electromagneticresponse is to the center ofmass co-
consists of a particle-hole diagram. (Note that the spa-
ordinate,q,sothatthetheoreticalframeworkoftheelec-
tiallygradientterminthefreeenergyconsistsofparticle-
tromagnetic response is basically the same for even- and
particle diagrams and will be discussed in 5.) These
§ odd-ω pairings.
Green functions are given as
iε +ξ 3. Singlet pairings
G(p,ε )= n p , (2)
n −ε2n+ξp2+|∆µ(p,εn)|2 3.1 Model
Our purpose here is to calculate the corrections of
∆+(p,ε )
F+(p,ε )= µ n , (3) fluctuations to the Meissner effect. For this purpose, we
n ε2 +ξ2 + ∆ (p,ε )2
n p | µ n | start from a simple and essential model. Judging from
both for the even-ω (∆ ) and odd-ω (∆ ).49 In the GL thepreviousstudies,3–11,20,21 theodd-ω pairingtendsto
e o
region,i.e.,nearT ,wecanexpandthemwithrespectto be mediated by spin fluctuations, which originate from
c
∆ . Up to the second order in ∆ , we have thelocalcomponent,i.e.,theincoherentpart.Therefore,
µ µ
weintroducethe followinglow-energyeffectiveactionon
iε +ξ
G(p) G(p)+ n p ∆ (p)2 the basis of the itinerant-localized duality theory:18,19
≃ (ε2 +ξ2)2| µ |
n p
1/T
=G(p)+G( p)[G(p)]2 ∆ (p)2, (4) S = dτψ¯p,α(τ)(∂τ +ξp)ψp,α(τ)
µ
− | | p,αZ0
X
∆(+)(p)
F(p) µ = G(p)G( p)∆(+)(p), (5) 1/T 1/T
≃−ε2 +ξ2 − − µ g2 dτ dτ′D(q,τ τ′)S(q,τ) S( q,τ′),
n p − − · −
q Z0 Z0
where G is the Green function in the normal state X
(7)
G(p) = [iε ξ ]−1. The diagrammatic expressions of
n p
these expansi−ons are shown in Fig. 1. The Meissner ef- whereψ¯p,α(τ)≡eτHψp†,αe−τH,andgisacouplingcon-
fect can be demonstrated in the limit of q 0. After stant. The spin density is given by
→
1
S(q,τ)= ψ¯ (τ)σ ψ (τ), (8)
2 p+q/2,α αβ p−q/2,β
p,α,β
X
J.Phys.Soc.Jpn. FullPaper AuthorName 3
whereσdenotesthePaulimatrix.Theactualodd-ωpair- (a-1) (a-2) (a-3) (a-4)
ing,i.e.,thegaplesssuperconductivity,hasbeenreported
mainlyinthecoexistencephaseofsuperconductivityand
antiferromagnetism.13–16 For such a situation, the most GGFF G-G-FF GGFF G-G-FF
dominant fluctuation is the transverse spin-fluctuation (b-1) (b-2)
due to the spin wave as
g2T
D(q,ωm)= v2qˆ2+Nω 2, (9) -GG-FF -GG-FF
s | m| (c-1) (c-2) (d-1) (d-2)
wheregisthecouplingconstant,T expressestheenergy
N
scaleofthe AFtransitiontemperature, v is the velocity
s
of the spin-wave, and qˆ = q Q (Q is the magnetic -GG-FF -GG-FF GGFF G-G-FF
−
order vector). This type of spin fluctuation has already (e-1) (e-2) (f-1) (f-2)
been shown to realize the odd-ω SC in the AF phase.11
When we considerin the doubly foldedBrillouinzone, qˆ
is replaced by q, so that D(q,ω ) is singular for q 0
m ∼ GGGG GGGG -GG-FF -GG-FF
and ω 0. We use this transverse spin-fluctuation for
m (g-1) (g-2) (g-3) (g-4)
∼
singlet pairings.
The corrections of the spin-fluctuation to the kernel
Q are
xx GGFF GGFF GGFF GGFF
∆Q (q)=2e2T D(q′)∆ (p)2 (h-1) (i-1) (i-2)
xx µ
| |
p,q′
X
× −vpv−pG(p+)G(p+−q′)F↑+↓(p−)F↓↑(p+) (a-1) GGGG GGFF GGFF
(j-1) (k-1) (k-2)
(cid:2)v v G( p )G( p +q′)F+(p )F (p ) (a-2)
− p −p − + − + ↑↓ − ↓↑ +
v v G(p )G(p +q′)F+(p )F (p ) (a-3)
− p −p − − ↑↓ − ↓↑ + FFFF FFFF FFFF
v v G( p )G( p +q′)F+(p )F (p ) (a-4)
− p −p − − − − ↑↓ − ↓↑ +
Fig. 2. Diagrammaticexpressionsofthe current-currentcorrela-
+v v G(p )G( p )F+(p )F ( p +q′) (b-1)
p −p + − + ↑↓ − ↓↑ − + tionwithcorrectionoffluctuations.
+v v G(p )G( p )F+( p +q′)F (p ) (b-2)
p −p − − − ↑↓ − − ↓↑ +
+vpv−p+q′G(p+)G(−p−+q′)F↑+↓(p−)F↓↑(−p++q′) +vpvp−q′G(p+)G(p+−q′)F↑+↓(p−)F↓↑(−p−+q′)
(c-1) (i-2)
+vpv−p+q′G(p−)G(−p++q′)F↑+↓(−p−+q′)F↓↑(p+) +vpvp−q′F↑+↓(p−)F↓↑(p+)F↑+↓(−p++q′)F↓↑(−p−+q′)
(c-2) (j-1)
−vpv−p+q′G(p+)G(p−)F↑+↓(−p++q′)F↓↑(−p−+q′) −vpv−pF↑+↓(p−)F↓↑(p+)F↑+↓(−p++q′)F↓↑(p+) (k-1)
(d-1) v v F+(p )F (p )F+(p )F ( p +q′) ,
− p −p ↑↓ − ↓↑ + ↑↓ − ↓↑ − −
−vpv−p+q′G(−p++q′)G(−p−+q′)F↑+↓(p−)F↓↑(p+) (cid:3)(k-2)
(d-2) where v = p /m is the velocity of quasiparticles. The
p x
+v v G(p )G(p )G(p )G(p q′) (e-1) correspondingdiagramsareshownin Fig.2.Inthe limit
p p + + − +
− of q 0, we have
+v v G(p )G(p )G(p )G(p +q′) (e-2) →
p p + − − − ∆Q (0)=2e2T v2D(0)∆ (p)2
−vpvpG(p−)G(−p++q′)F↑+↓(p+)F↓↑(p+) (f-1) xx Xp,q′ p | µ |
v v G(p )G( p +q′)F+(p )F (p ) (f-2)
− p p + − − ↑↓ − ↓↑ − × 3G(p)G(p)G(p)G(p)+3G(−p)G(−p)F↑+↓(p)F↓↑(p)
+v v G(p )G(p )F+(p )F ( p +q′) (g-1) (cid:20)
p p + − ↑↓ + ↓↑ − + +2G(p)G(p)F+(p)F (p)+G(p)G(p)F+( p)F ( p)
+v v G(p )G(p )F+( p +q′)F (p ) (g-2) ↑↓ ↓↑ ↑↓ − ↓↑ −
p p − + ↑↓ − + ↓↑ + +3G(p)G(p)F+(p)F ( p)+3G(p)G(p)F+( p)F (p)
+v v G(p )G(p )F+( p +q′)F (p ) (g-3) ↑↓ ↓↑ − ↑↓ − ↓↑
p p + − ↑↓ − − ↓↑ − 2G(p)G( p)F+(p)F (p) 2G(p)G( p)F+(p)F ( p)
+v v G(p )G(p )F+(p )F ( p q′) (g-4) − − ↑↓ ↓↑ − − ↑↓ ↓↑ −
p p + + ↑↓ − ↓↑ − −− 2G(p)G( p)F+( p)F (p)
+vpvp−q′G(p+)G(p−)G(p+ q′)G(p− q′) (h-1) − − ↑↓ − ↓↑
− − +D(q′)F+(p)F (p)D(q′)F+( p)F ( p)
+vpvp−q′G(p−)G(p+−q′)F↑+↓(−p++q′)F↓↑(p+)(i-1) +D(q′)F↑↑+↓↓(p)F↓↓↑↑(p)D(q′)F↑↑+↓↓(−−p)F↓↓↑↑(p−)
4 J.Phys.Soc.Jpn. FullPaper AuthorName
+D(q′)F+(p)F (p)D(q′)F+(p)F ( p) . (10) eq. (11). For the zeroth order of D0 term (eqs. (6)), the
↑↓ ↓↑ ↑↓ ↓↑ − contribution from the ξ2-term gives
(cid:21)
H0eisreim,wpeorhtaavnetpinutthqe′ →Gin0zsbinucreg-oLnalnydDau(qr,eωgmio)n.wUitphtωomth=e vp2 −2G3pG−p+G2pG2−p O(ξ2)
p
second order of ∆(p) (eqs. (4)-(5)), X (cid:0) (cid:1)(cid:12)
a π 1(cid:12)
2
= 2 +a π =0, (16)
∆Q (0)=2e2T v2D ∆ (p)2 − 2 2 ε
xx p 0| µ | h (cid:16) (cid:17) i| n|
Xp and for the (D1) term, the contribution from the ξ2-
O 0
12G5G1 +12G4G2 6G3G3 (11) term becomes
× − p −p p −p− p −p
(cid:2)2e2T D ∆ (ε )22NFεF (cid:3) vp2 −12G5pG1−p+12G4pG2−p−6G3pG3−p O(ξ2)
0 µ n
≃ | | 3m p
n X (cid:0) (cid:1)(cid:12)
X a π a π 1 (cid:12)
π π 3π 1 = 12 2 +12 0 6 2 =0. (17)
12 +12 6 − − 16 × − 8 ε 3
×(cid:26)− (cid:16)16(cid:17) (cid:16)−4(cid:17)− (cid:18) 8 (cid:19)(cid:27)|εn|5 Conseqhuentl(cid:16)y,thec(cid:17)orrectiontoQ isie|xanc|tlyzeroupto
2N ε 1 xx
=2e2T D ∆ (ε )2 F F ( 6π) ξ2,i.e.,the presentresultdoesnotdependonthe details
0| µ n | 3m − ε 5
n | n| of the band structure.
X
e2NT D ∆ (ε )2( 6π) 1 , (12) 4. Triplet pairings
≃ m 0| µ n | − ε 5
n | n| Next, let us consider the case of triplet pairing. The
X
where N mp /2π2, ε p2/2m, and G(p)’s are tripletpairingismediatedbytheferromagneticspinfluc-
F ≡ F F ≡ F tuation for both the even- and odd-frequency. (The de-
abbreviated as G ’s. Note that ∆(ε ) includes the fac-
p n
tailed discussions for the relationship between the pair-
tor from the momentum integration of ∆(p) as ∆(ε )
n
∆(p,ε )Φ(p,ε ), where Φ(p,ε ) is some function∼, ing symmetry and the spin fluctuation are given in
p n n n
and D includes the factor from the frequency summa- Appendix.) We consider the ferromagnetic spin fluctu-
0
P ation in the form
tion (T ). At around T , the dominant contribution
q′ c
of ∆(ε ) comes only from ε = πT , so that we can ap- g2N κ2
nP 0 c D(q,ω)= F 0 , (18)
proximate as κ2+q2 iω/η(q)
−
D0 =T g2TN/vs2q2 ∼g2Tc/TN. (13) where κ and κ0 are the inverse correlation lengths with
and without magnetic correlations, respectively. η(q)
q
X
is defined as η(q) = T q using a characteristic spin-
(Here we use v q T , where q is a cut off in the sf
momentum spacse.c) F∼inaNlly, the totacl kernel becomes fluctuation temperature Tsf.52
Withthesameprocedureasforthesingletpairing,we
e2N 1 D obtain the correction to the kernel due to the ferromag-
Q (0)= πT 6 0 ∆ (ε )2. (14)
xx m ε 3 − ε 5 | µ n | netic transverse spin-fluctuation as follows:
n (cid:20)| n| | n| (cid:21)
X
From this result, it is revealed that the Meissner ker- ∆Qxx(q)=2e2T D(q′)
nel is reduced by the critical spin fluctuation both for p,q′
X
the even- and odd-ω pairing. When the spin fluctuation
is moderate, D0 . (πTc)2/6 (i.e., g . π TcTN/6), the × −vpv−pG(p+)G(p+−q′)F↑+↑(p−)F↑↑(p+) (a-1)
Meissnerkernelbecomessmall,butremainspositive.On (cid:20)
the other hand, there is a possibility thapt the Meissner v v G( p )G( p +q′)F+(p )F (p ) (a-2)
− p −p − + − + ↑↑ − ↑↑ +
kernel becomes negative when the critical spin fluctua-
tionissostrong,i.e.,D0 &(πTc)2/6.However,thisrapid −vpv−pG(p−)G(p−+q′)F↑+↑(p−)F↑↑(p+) (a-3)
reduction is relaxed in we take into account the higher v v G( p )G( p +q′)F+(p )F (p ) (a-4)
order corrections as is shown below in 5. − p −p − − − − ↑↑ − ↑↑ +
§ −vpv−p+q′G(p+)G(p−)F↓+↓(p+−q′)F↓↓(p−−q′)
3.2 Dependences on band structure (d-1)
In the derivation of eq. (12), we assume that the den-
sity of state (DOS) is constant with respect to ξp. How- −vpv−p+q′G(−p++q′)G(−p−+q′)F↑+↑(p−)F↑↑(p+)
ever,theDOSsomewhatdependsonξ ingeneral.Here, (d-2)
p
we see the validity of the above result by expanding the +v v G(p )G(p )G(p )G(p q′) (e-1)
DOS up to (ξ2) as p1 p1 1+ 1+ 1− 1+−
O +v v G(p )G(p )G(p )G(p +q′) (e-2)
2 p p + − − −
v2F(ξ)= dξN ε (1+a ξ+a ξ2+ )F(ξ).
p p 3mZ F F 1 2 ··· −vpvpG(p−)G(−p++q′)F↑+↑(p+)F↑↑(p+) (f-1)
X
(15) v v G(p )G( p +q′)F+(p )F (p ) (f-2)
− p p + − − ↑↑ − ↑↑ −
The ξ-linear term in DOS gives no contribution since
F(ξ) is an even function in ξ for the relevant quantity, +vpvp−q′G(p+)G(p−)G(p+−q′)G(p−−q′) (h-1)
J.Phys.Soc.Jpn. FullPaper AuthorName 5
v v F+(p )F (p )F+(p q′)F (p ) (k-1) see the effect of higher order corrections to the Meissner
− p −p ↑↑ − ↑↑ + ↓↓ +− ↑↑ +
effect.
v v F+(p )F (p )F+(p )F (p q′) (k-2)
− p −p ↑↓ − ↓↑ + ↑↓ − ↓↑ −−
The correspondingdiagramsarethe sameas thatofsin- 5.1 Up to 1st order
gletpairingexceptfor(b),(c),(g),and(i)typediagrams The spatial derivative term without the spin fluctua-
in Fig. 2. (Of course, the spin indices are different from tion is
the singlet case.) Up to the second order of ∆(p), ∂2 ∂2
K (q)= T G G
∆Qxx(0)=2e2T vp2D0|∆µ(p)|2 ∂q2 p-p − Xp ∂q2 p+ −p−
p
X v2
12G(p)5G( p)1+6G(p)4G( p)2 2G(p)3G( p)3 =T 2pG(p)2G(−p)2−vp2G(p)3G(−p)
× − − − − − p (cid:20) (cid:21)
X
(cid:2)2e2T D ∆ (ε )22NFεF (cid:3) N 1
0 µ n
≃ | | 3m = πT . (23)
n 4m ε 3
X n | n|
π π 3π 1 X
12 +6 2 The correctionsdue to the spin fluctuation, which is the
× − 16 −4 − 8 ε 5
(cid:26) (cid:16) (cid:17) (cid:16) (cid:17) (cid:18) (cid:19)(cid:27)| n| same one as in the previous sections, are given by the
e2N 1 FeynmandiagramsinFig.3(a)and(b).Theircontribu-
T D ∆ (ε )2( 3π) . (19)
≃ m 0| µ n | − ε 5 tions are
n
n | |
X v2
Here, D0 ∼ g2NFTcκ20/κ2. The total kernel for triplet ∆Kp′′-p(q)=T D0 2p 6G4pG2−p−4G3pG3−p
pairing is then obtained as p (cid:20)
X (cid:0) (cid:1)
e2N 1 D
Qxx(0)= m πT ε 3 −3 ε 05 |∆µ(εn)|2. (20) −2 6G5pG−p−3G4pG2−p+G3pG3−p
n (cid:20)| n| | n| (cid:21) (cid:21)
X (cid:0)N 6D (cid:1)
TheMeissnerkernelfortripletpairingisalsoreducedby πT 0 (24)
the spinfluctuationboth for even-andodd-ω , although ≃−4m εn 5
n | |
X
the coefficient of the second term of eq. (20) is smaller
for singlet pairings, and
than that for singlet pairing.
v2
5. Higher order corrections ∆Kp′′-p(q)=T Dp 2p −12G5pG−p+6G4pG2−p−2G3pG3−p
p
So far, we have shown that the coefficient of ∆Qxx is X (cid:2) (cid:3)
reduced at least up to the first order of D. However, it N 3D0
πT (25)
is naturally expected that the signs of the higher order ≃−4m ε 5
n
n | |
corrections are oscillating; when the sign of the first or- X
for triplet pairings. Then the total ρ (T) up to (D1)
der is negative,that ofthe secondorderwill be positive, s
O
are
and that of the third order will be negative etc.. Thus,
we have to check the effect of higher order corrections, ρsin(T) πT 1 1 6D0 ∆ (ε )2, (26)
especially paying attention to their convergence. Unfor- s ≃ ε 3 − ε2 | µ n |
n | n| (cid:20) | | (cid:21)
tunately, it is too complicated to proceed to the higher X
order corrections with the scheme we took in the pre- ρtri(T) πT 1 1 3D0 ∆ (ε )2, (27)
vious sections. Here, we instead take a strategy where s ≃ n |εn|3 (cid:20) − |ε|2(cid:21)| µ n |
wecalculatethesuperfluiddensitybyadifferentmethod X
for singlet and triplet pairings, respectively. These re-
and then determine the Meissner kernel.
sults are exactly equivalent to the results obtained from
The superfluid density ρ is expressed in terms of the
s thecurrent-currentresponsefunctionintheprevioussec-
Meissner kernel as
tions, eqs. (14) and (20).
N (T) m
s
ρ (T)= = Q . (21)
s N e2N xx 5.2 2nd and 3rd order
Ontheotherhand,ρs canalsobeobtainedfromthespa- Now, we have confirmed the equivalence between two
tiallygradientterm, ∆(r)2,inthe freeenergy,53,54 in different procedures for calculating ρ (T) up to the first
|∇ | s
otherwords,thecoefficientoftheq2-term.Thisspatially order of spin fluctuation, (D1). With regard to the
O
gradient term is expressed by the superconducting pair higher order calculations, we can carry out the calcula-
susceptibility, Kp-p(q), in the form tionsofthespatiallygradienttermmoreeasilythanthat
4m ∂2 ofthecurrent-currentresponsefunctions.Theresultsfor
ρ (T)= ∆ (p)2 K (q). (22)
s N | µ | ∂q2 p-p
p
X
We should have the same results for the Meissner kernel
through the calculation of K (q). In this section, we
p-p
first show that the results in the previous sections can
alsobederivedfromthecalculationofK (q).Then,we
p-p
6 J.Phys.Soc.Jpn. FullPaper AuthorName
[1st] N 168D3
= πT 0, (29)
(a) (b) −4m ε 9
n
n | |
X
forthethirdorder(Fig.3(h)-(y)).Thetotalsingletpair
[2nd] susceptibility up to third order is given by
(c) (d) (e) (f)
N 1 D D 2
K′′ (0)= πT 1 6 0 +30 0
p-p 4m n |εn|3" − |εn|2 (cid:18)|εn|2(cid:19)
X
(g)
D 3 D 3
0 0
150 18 .
− (cid:18)|εn|2(cid:19) − (cid:18)|εn|2(cid:19) #
(30)
[3rd]
(h) (i) (j) (k) Rathersurprisingly,thecoefficientsofeachtermareinte-
gers(althougheachtermisgivenasafractionalnumber,
e.g., G5G5 =35π/128ε 9, or G4G6 = 7π/32ε 9).
(l) (m) (n) (o) Moreovper,−tphey are almo|snt|given aps a−gpeom−etric pr|ognr|es-
sion, which can be written in the convergentform:
1 6(D /ε 2)
(p) (q) (r) (s) ρ πT 1 0 | n| ∆ (ε )2
s ≃ ε 3 − 1+5(D /ε 2) | µ n |
n | n| (cid:20) 0 | n| (cid:21)
X
(31)
(t) (u) (v) (w) Therefore,ρ andsotheMeissnerkernelisreducedbythe
s
spinfluctuations,evenifwetakeintoaccountthe higher
order corrections.With this geometric progressionform,
(x) (y)
we can go beyond the perturbation theory even though
we made calculations only up to the third order. The
results shouldbe valid as far as it converges,so that our
results (31) is expected to be valid for any magnitude of
Fig. 3. Diagrammatic expressions of the pair susceptibility with the spin fluctuations, since it is written in a convergent
correctionoffluctuations.
form. Then, we also conclude that the Meissner kernel
canbenegativebothforeven-andodd-ωwhentheeffect
of massless spin fluctuations is sufficiently large.
singlet pairing are as follows:
Similarestimationsofthehigherordercorrectionswill
∆(2)K′′ (q)= T D2 begivenforthetripletpairingjustbychangingthecoef-
p-p − q
p,q ficients. The result so obtained will be qualitatively the
X
sameasthatforthesingletpairing,namely,theMeissner
∂2
G3 G3 4G4 G2 +2G5 G1 kernel of the odd-ω triplet pairing also becomes positive
× ∂q2 p+ −p− − p+ −p− p+ −p−
(cid:20) with the critical spin fluctuations.
+2G5 G1 +G3 G3 6. Discussions
p+ −p− p+ −p−
(cid:21)
Summarizingthe progresshithertomade,wefindthat
N 30D2
= πT 0, (28) the Meissner kernel in the GL region is given by
4m ε 7
Xn | n| Q = e2N πT 1 6D0/|εn|2 ∆ (ε )2,
for the second order (Fig. 3(c)-(g)), and xx m |εn|3 n (cid:20) − 1+5D0/|εn|2(cid:21)| µ n |
X
(32)
∆(3)K′′ (q)= T D2
p-p − q whose D dependence is depicted in Fig. 4. This con-
p,q 0
X
clusion will be valid, even if we consider the infinite or-
∂2
× ∂q2 −G4p+G4−p− +4G5p+G3−p− +2G5p+G3−p−der in D0. The Meissner kernel is positive but strongly
(cid:20) reduced by the critical spin fluctuations both for even-
2G4 G4 +2G5 G3 2G6 G2 and odd-ω pairing. Meanwhile, our results also indicate
− p+ −p− p+ −p− − p+ −p− that the Meissner kernel can become negative both for
−2G6p+G2−p− −4G6p+G2−p− −4G6p+G2−p− even- and odd-ω pairing, when the critical fluctuations
aresufficientlylarge.Ineq.(31),the dominantcontribu-
2G4 G4 2G4 G4 +2G7 G1
− p+ −p− − p+ −p− p+ −p− tion from the εn-summation comes only from the lower
+4G7p+G1−p− +2G7p+G1−p− +2G5p+G3−p− εwnh,ensintchee ∆re(tεanr)daitsiornapoifdltyhesuipnpterreascsetidonasisεsntrionncgr.e1a1s,e55s
+2G7 G1 +2G5 G3 +2G7 G1 Then, the condition for the sign change is roughly given
p+ −p− p+ −p− p+ −p− by D & T2( D ) or g & √T T in the case of sin-
(cid:21) 0 c ≡ x c N
glet pairing in the AF phase. Therefore, we can con-
cludethatwithmoderatemagnitudeofspinfluctuations,
J.Phys.Soc.Jpn. FullPaper AuthorName 7
1 possibility of the FFLO state, resulting in the possible
up to 2 zero-field FFLO state.
0.8
Abrahams et al. (and more recently Dahal et al.) in-
0.6 troduced a composite operator which couples a Cooper
pair to a spin fluctuation and has the same sym-
Kn 0.4 up to 1 up to 8 metry as the odd-ω pairing.51,62 They showed that,
in the composite-operator condensation states, the su-
0.2
perfluid density is reduced in some models of the
3
up to
0 magnon propagators. Although the magnon propaga-
tor in Ref. 51, (q), resembles our spin fluctuation,
-0.2 D(q), their contDributions to the Meissner kernel are
0.01 0.1 1 10
different from ours. For example, in the first order of
0 x
(q) or D(q), the composite operator theory leads to
D
(q) 2G3G +G2G2 (eq. (4.3) in Ref. 51), which
Fig. 4. (Color online) Relation between the spin fluctuation D0 D − p −p p −p
has the form of the first line of eq. (6) supplemented by
and the Meissner kernel, which corresponds to the super−fl2uid the m(cid:2)agnon propagator, w(cid:3)hereas (D1)-term is given
densityρs orthesquareoftheinversepenetration depthλL . by D(q′) 12G5G1 +12G4G2 O6G3G3 (the first
− p −p p −p− p −p
line of eq. (11)) in our theory. Furthermore, in our the-
(cid:2) (cid:3)
D < D , the superconductivity with zero center-of- ory,thehigherordertermsareshowntoconvergehaving
0 x
mass momentum is stable, but the superfluid density is an almost geometric progression structure, even though
strongly reduced, while with sufficiently large spin fluc- the sign of the higher order of D is fluctuating. How-
tuations, D > D , the uniform superconductivity is no ever, in the composite operator theory, the contribution
0 x
longer stable. In such a situation of D >D , the super- of the higher order term has not been investigated as
0 x
to whether it converges or diverges; it can differ greatly
conductivity with Cooper pairs of the finite center-of-
mass momentum can be stabilized, since the coefficient from the present results. Nevertheless, both approaches
of q2-term is negative, but that of q4-term is positive. share a common aspect in which incoherent degrees of
freedom beyond the quasiparticle picture (in a sense of
Then, the FFLO pairing can be realized spontaneously
without the magnetic field. the Fermi liquid theory) play crucial roles.
In the following discussion, we make some comments Finally, we commentonthe odd-ω in the non-uniform
on the relationship between the present results and the system,suchasthenormalmetal/superconductorjunc-
previous works. The qualitative tendency of the present tions. For the Meissner effect of the pairs in the normal
metal region,we do not need to consider the corrections
results are consistent with the standard theory of a su-
perfluid Fermi liquid,56 where λ is given by due to the incoherent part, since the spin fluctuations
L
are irrelevant there.
4πNe2 1 Y(T)
λ−2(T)= (1+ 1Fs) − , (33)
L m∗c2 3 1 1+ 1FsY(T) 7. Conclusions
3 1
where m∗ is the effective mass of quasiparticles, Fs is The effects of the critical spin fluctuations, the cru-
1
cialingredientforrealizingthe odd-ω superconductivity,
the Landauparameter,andY(T)is the so-calledYosida
on the electromagnetic response of the odd-ω supercon-
function. It is noted that eq. (33) is also valid for the
ductivity have been examined. The Meissner kernel is
systems without the Galilean invariance. There, the in-
stronglyreducedbythecriticalspinfluctuationsbothfor
coherent contribution for the current-current response
the even- and odd-ω pairing. Thus the superfluid den-
function is not negligible and is incorporated into the
Fermi liquid parameterFs, the ω-limitvertex Γω, which sity will be smaller and the penetration depth will be
1
arises from incoherent processes.56 In the case of cor- longerthanthecasewithoutcorrectionsduetothecriti-
calspinfluctuations.Moreover,whenthe effectis strong
related electron systems without the translational in-
enough, the Meissner kernel becomes negative, suggest-
variance, such as the heavy fermion systems, eq. (33)
is modified as the effective mass m∗ is replaced by the ing the zero-field spontaneous FFLO state. This conclu-
dynamic effective mass m .57 (m is given as m /m = sion has been reached by calculating two different val-
d d d
ues: one is the current-current response function,50 and
[1 ∂Σ(k,ω)/∂ω] ,whereΣ(k,ω)istheselfen-
− k=kF,ω=EF the other is the spatially gradient term (the coefficient
ergy.) The spin-fluctuation increases m , so that it re-
d
duces λ−2. When the system is near the AF critical of q2-term) in the free energy.53,54 The results, obtained
L
by two different methods, completely agree with each
pointorthemetal-insulatortransition,theDrudeweight,
D = (πe2N/m∗)(1+ 1Fs), is reduced.58–61 In this case, other at least up to the first order of fluctuation. This is
λ−2 is alsoreduced, w3hic1h is consistentwith ourresults. valid both for singlet pairing with the antiferromagnetic
L fluctuation in the coexisting phase and for triplet pair-
However, in the derivation of eq. (33), the possibility of
ing with the ferromagnetic fluctuation. The validity has
the FFLO state is discarded, so that we cannot discuss
been also checked for the shape of the density of state
the possibility of FFLO state only fromeq. (33). On the
in the case of singlet pairing, and the presentresults are
other hand, in our theory, we directly calculate the co-
shown to be invariant, namely, the present conclusion is
efficient of q2-term, which corresponds to the order pa-
not affected by the detail of the band structure.
rameter of the FFLO state, so that we can discuss the
8 J.Phys.Soc.Jpn. FullPaper AuthorName
(a) (b)
0 0
Fig. A·2. Illustrationoftheω-dependentpairinginteractionΛ(ω)
andthefunctionΛ(ω)cos(πω/ω0)for(a)η∼ω0and(b)η≪ω0.
0 Thecoefficientλ0 isgivenbytheintegrationofΛ(ω)andλ1 by
thatofΛ(ω)cos(πω/ω0).Wecanseefromtheseillustrationsthat
Fig. A·1. Illustration of the frequency dependence of pairing in- λ0 ≫λ1 for the instantaneous interaction and λ0 ∼ λ1 for the
retardedinteraction.
teractionΛ(ω).
(a) ferromagnetic (b) antiferromagnetic
Acknowledgments
WearegratefultoH.Kusunoseforfruitfuldiscussions.
This work is supported by a Grant-in-Aid for Scientific
Research(No.19340099)fromTheJapanSocietyforthe
PromotionofScience(JSPS),andaGrant-in-AidforSci-
0 q 0 q
entific Research on Innovative Areas “Heavy Electrons” y y
q q
(No.20102008)fromtheMinistryofEducation,Culture, x 0 x 0
Sports, Science and Technology, Japan. One of the au-
thors (Y. F.) is supported by a Grant-in-Aid for Young Fig. A·3. Illustrationofthemomentumdependenceofthepairing
ScientistsStart-up from JSPS. interaction Φ(q)for(a) ferromagneticcases andfor(b) antifer-
romagneticcases.
Appendix: Possiblesymmetriesofodd-ωpairing
Here,we discuss the possible symmetries of the odd-ω
pairing on the basis of a general argument. It is found neous interaction only mediates the even-ω pairing. On
that the odd-ω pairing becomes competitive with the the other hand, in cases where Λ(ω) strongly depends
even-ω one when the interaction is strongly retarded. on ω, i.e., η ω0, the function Λ(ω)cos(πω/ω0) has al-
≪
WeassumethatthepairinginteractionV(q,ω)canbe most the same form as Λ(ω), so that λ1 is nearly equal
separated as to λ0. Then, Tc of the odd-ω pairing can be comparable
to that of the even-ω pairing. In fact, if the momentum
V(q,ω)=Φ(q)Λ(ω), (A1) part of odd-ω φodd(q), which is neglected so far, has a
·
where the dimensionality of the system is arbitrary. larger value than that of even-ω φeven(q), it is possible
Generally, we can assume that Λ(ω) has a maximum at that the odd-ω pairing is stabilized prevailing over the
ω = 0, and we parameterize its peak width by η as dis- even-ω pairing.In order to determine which Tc is higher
playedinFig.A1.SinceΛ(ω)is anevenfunction,itcan and what type of gap structure develops below Tc, we
be expanded as· need more data about the model.
λ πω 2πω
Λ(ω) = 0 +λ cos +λ cos + (,A2) A.1 Spin fluctuation model
1 2
2 ω ω ··· ·
0 0 Now, we consider the varieties of pairing on the basis
2 ω0 nπω ofthetwo-dimensionalspinfluctuationmodel.Letususe
λ = dωΛ(ω)cos , (A3)
n
ω ω ·
0 Z0 0 Vs(q,ω)=3Φ(q)Λ(ω), (A6)
·
with a sufficiently large cut-off ω . The pairing interac-
0
V (q,ω)= Φ(q)Λ(ω), (A7)
tion is written in the separable form t − ·
Λeven(ε ε′) = λ0 +λ1cosπεcosπε′ + (A, 4) pasaitrhinegmwoidtheltinhteermacotmioenntfuomr sidnegpleetnd(Vens)ceanodf Φtr(ipql)etgi(vVetn)
− 2 ω ω ··· ·
0 0
as
πε πε′
Λodd(ε−ε′) = λ1sinω0 sin ω0 +··· , (A·5) Φ(k−k′)=φ0+2φ1[cos(kx−kx′)+cos(ky −ky′)],
(A8)
for the pair scattering (ε, ε) (ε′, ε′). The maxi- ·
− → −
mumeigenvalueisapproximatelygivenbyλ0/2foreven- where the ferromagnetic spin fluctuations gives φ0 >
frequency pairing and λ1 for odd-frequency pairing. If φ1 > 0 and the antiferromagnetic spin fluctuation φ0 >
Λ(ω) is independent of ω, that is to say, η →∞, λ0 >0 −φ1 >0. (See Fig. A·3)
and λ = 0 for all n = 0. When Λ(ω) is a slowly vary-
n
6
ing function, i.e., η ω , λ becomes finite but very
0 1
∼
smallasisseeninFig.A2(a).Therefore,suchinstanta-
·
J.Phys.Soc.Jpn. FullPaper AuthorName 9
ferro. s-wave p-wave d-wave Similarly, for the antiferromagnetic case, φ < φ <
0 1
singlet — — — −
0, the anisotropic even-parity singlet pairing, the odd-
triplet πφ0λ1 φ1λ0 2φ1λ1 parity singlet pairing, and the isotropic even-parity
antiferro. s-wave p-wave d-wave triplet pairing is possible. The Tc for the dx2−y2-wave
singlet — 6φ1λ1 3φ1λ0 singlet is related to the eigenvalue 3φ1λ0 and that for
the p-wave singlet is related to 6φ λ . In the case of the
triplet πφ0λ1 — — 1 1
s-wavetriplet,theT isrelatedtoπφ λ .(See TableA1
c 0 1
·
for “antiferro.”.) Therefore, if λ is comparable to λ ,
Table A·1. Possible pairing symmetries and their coefficients of 1 0
the p-wave singlet and the d -wave singlet pairing
theattractivecomponentsforferromagneticcases,φ1>0(upper x2−y2
table), and antiferromagnetic cases, φ1 < 0 (lower table). The are nearly degenerate. The p-wave singlet pairing sug-
framerepresents theodd-ω gap. gested in Ref. 11 would correspond to this case.
To summarize, s-wave and d-wave triplet odd-ω pair-
ings are possible for the ferromagnetic case, and s-wave
We can express Φ(k k′) in the separable form68 tripletandp-wavesingletodd-ω pairingsarepossiblefor
−
Φeven(k k′)=φ0+2φ1(γkγk′ +ηkηk′), (A9) the antiferromagnetic case.
− ·
Φodd(k k′)=2φ (sink sink′ +sink sink′),
− 1 x x y y
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