Current-current Fermi-liquid corrections to the superconducting fluctuations on conductivity and diamagnetism

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by  L. Fanfarillo

Current-current Fermi-liquid corrections to the superconducting fluctuations on
conductivity and diamagnetism
L. Fanfarillo,1,2 L. Benfatto,2,1 and C.Castellani1,2
1Department of Physics, Sapienza University of Rome, P. le A. Moro 2, 00185 Rome, Italy
2Institute for Complex Systems (ISC), CNR, U.O.S. Sapienza,
Sapienza University of Rome, P. le A. Moro 2, 00185 Rome, Italy
(Dated: January 10, 2012)
We analyze the behavior of the superconducting-fluctuations contribution to diamagnetism and
2
1 conductivity in a model system having current-current interactions. We show that in proximity
0 to a Mott-insulating phase one recovers an overall suppression of the fluctuating contribution to
2 the conductivity with respect to diamagnetism, in close analogy with recent experiments on the
underdopedphase of cuprate superconductors.
n
a
PACSnumbers: 74.40.-n,74.25.F-,74.25.N-
J

9
Itisgenerallybelievedthat,duetotheirlowsuperfluid as cuprates). Both δσ and δχd diverge as T approaches
]
n densities and short correlation lengths, superconduct- Tc due to the increase of the correlation length ξ(T),
o ing fluctuations (SCF) in underdoped cuprates should with a T dependence that is power-law within the GL
c be relevant for transport and thermodynamic proper- approach and exponential within the KT theory. From
- ties. Such SCF have been widely highlighted in the Eq.s (1) one could express the ratio δσ/δχ as
r d
p pseudogapregionbyseveralexperimentalmeasurements,
u ranging from diamagnetism1–4 and Nernst effect3,5 to δσ = Φ20e2 ξσ2(T) (2)
s paraconductivity6–8. Inparticular,thesurvivalofalarge |δχ | 16~k Tξ2ξ2 (T)
t. Nernst signal up to temperatures much larger than the d B 0 χd
a
m superconductingtransitiontemperature Tc in the under- where ξσ, ξχd are the correlations length extracted from
doped region has been interpreted as the evidence for paraconductivity and diamagnetism measurements, re-
-
d vortex-likephase fluctuations with a Kosterlitz-Thouless spectively. Since one would expect that the same lenght
n (KT) character due to the quasi-two-dimensional (2D) scale is involved in both cases, ξσ2/ξχ2d = 1, rigth above
o nature of the system1,3. At the same time, several Tc,δσ/δχd shouldbeoforder∼105(ΩA/T)−1,whileex-
c authors6,7 claimed that paraconductivity in underdoped perimentally it turns out to be two orders of magnitude
[
cuprates simply follows the T dependence expected for smallerthanthis1,13 (seealsodiscussionbelowEq.(36)).

2 the quasi-2DAslamazov-Larkin(AL) regime of gaussian Let us notice that the abovediscussionholds regardeless
v Ginzburg-Landau (GL) SCF close to T 9. This outcome the nature of the SCF so that the the ratio between δσ
c
3
motivated various investigations in the attempt to ex- and δχd depends only on the properties of the system
6
9 plain the experimental data on Nernst signal and dia- away from Tc.
5 magnetism within a GL-like framework10–12. In this paper we show that the quantitative disagree-
. Regardless of the KT or GL character of fluctuations, mentbetweentheSCFcontributiontodiamagnetismand
7
there are two outcomes of the experiments that are not conductivity could be understood as a consequence of
0
1 expectedforconventionalsuperconductors: (i) the range current-current interactions in a doped Mott insulator.
1 oftemperatureswherethefluctuationconductivityisob- The possible relevance of this kind of interactions to the
: served does not always match the one where a sizeable physicsofcuprateshasbeensuggestedwithinseveralcon-
v
Nernstsignalhasbeenreported7,8,and(ii)theSCFcon- texts, rangingfrom the gauge-theoryformulationfor the
i
X tributiontotheconductivityisabouttwoordersofmag- t −J model15 to the theoretical approaches emphasiz-
r nitude smaller than the fluctuating diamagnetism in the ing the role of microscopic currents16,17. As a paradig-
a same system, as recently pointed out by Bilbro et al. in matic example we focus on the t−J model within the
Ref. 13. To be more precise, we recall that in 2D the slave-bosonapproach,where the Hartree-Fock(HF) cor-
contribution of SCF to the conductivity δσ and diamag- rection of the quasiparticle dispersion leads to a differ-
netism δχ can be expressed both within the GL9 and ence between the quasiparticle current and velocity in
d
KT14 theory in terms ofthe superconducting correlation the usual Landau Fermi-liquid (FL) language18. The
length ξ(T) as: effect of this dichotomy on the GL functional for SCF
can be accounted for within a general field-theory for
k T e2 ξ2(T) the t-J model15,19, which includes fluctuations both in
δχ =− B ξ2(T) δσ = (1)
d Φ2d 16~d ξ2 the particle-particle (p-p) and particle-hole (p-h) chan-
0 0
nel. BycomputingtheALcontributiontodiamagnetism
Here Φ is the flux quantum, ξ the low temperature and conductivity we find that the role of Landau FL
0 0
correlation length and d is the thickness of the effective corrections differs in the static or dynamic limit. As
2Dsystem(i.e. theinterlayerspacingforlayeredsystems a consequence the SCF contribution to diamagnetism
2
and conductivity scales with a different prefactor, lead- (in the usual Nambu notation) as
ing to the suppression of paraconductivity with respect
to fluctuation diamagnetism when the Mott insulator is Aˆk′k = − iωn+sin(k+2k′)αφsα k−k′ τˆ0
approached, in analogy with experiments in cuprates.
Let us start from the slave-boson version of the t−J + ξ(cid:2)k0 −cos(k+2k′)αφcα k−k′ τˆ3(cid:3)− ∆k−k′γd(k+2k′)(τ9ˆ1)
model, (cid:20) (cid:21) (cid:20) (cid:21)
where ξ0 is the bare dispersion
k
H =−tδ (c† c +H.c.)+J S S (3)
iσ jσ i j ξ0 =−2tδ(cosk +cosk )−µ (10)
hiX,jiσ hXi,ji k x y
where t is the electron hopping, J is the exchange inter- The Eq. (9) can be decomposed as Aˆk,k′ =−Gˆ−01δk,k′ +
action between electron spins, c† (c ) is the fermionic Σˆk−k′. HereGˆ−01δk,k′ containstheq =0saddle-pointval-
iσ iσ ues(φc,φs,∆ )oftheHSfieldsobtainedbyminimization
creation (annihilation) operator and S = Ψ†σΨ is the 0 0 0
i i 2 i of the mean-field action
spin operator, with Ψ = (ci↑). The sum is extended
i ci↓ |φc |2 |φs |2 |∆ |2
over all the hi,ji nearest-neighbor pairs on a square lat- S = αq + αq + q −Trlog(Gˆ−1),
tice, and we use units such that the lattice spacing and MF 2g 2g g 0
q (cid:18) α (cid:19)
~ = c = e = 1. In Eq. (3) the decomposition of X X (11)
thhoeloenlepcatrrotnhaospebreaetnoralirneaadfyercmariorineidc ospuitn15o,naannddobnolysotnhiec while Σˆk−k′ contains the fluctuating parts of the HS
fields. The standard GL functional above T will then
c
fermionicdegreesoffreedomhavebeenretained. Herewe
be given by the expansion of Eq. (8) around the mean-
neglectthe bosonfluctuations,andwe assume thatslave
fieldaction(11)asS = Tr[Gˆ Σˆ]n/n,with∆ =0.
bosons are always condensed, leading to the suppression GL n 0 0
As far as the saddle-point values in the p-h channels are
factor δ = 2x/(1+x) of the hopping, scaling with the concerned, one can easily Psee that φs = 0, while φc 6= 0
doping x. The interaction term of the Eq.(3) contains 0 0
satisfies the following self-consistent equation:
contributions both from the p-p and the p-h channel.
We introduce the operators 2g
φc = cosk f(βξ ) (12)
0 N α k
Φc(q) = cosk c† c , (4) Xk
α α k+q/2,σ k−q/2,σ
Xkσ where f(x) is the Fermi function, β =1/T and ξk is the
Φs(q) = sink c† c , (5) quasiparticle dispersion, given by the τˆ3 term of Eq. (9):
α α k+q/2,σ k−q/2,σ
Xkσ ξ =−(2tδ+φc)(cosk +cosk )−µ (13)
k 0 x y
Φ∆(q) = γ (k) c c (6)
d −k+q/2,↓ k+q/2,↑
As one can see, the φc value corresponds thus to the
Xk 0
standard HF correction of the quasiparticle dispersion.
where α=x,y and γ (k)=cosk −cosk , so the inter- In order to compute the contribution δχ(q) of SCF
d x y
action term reads to the electromagnetic response function we need to in-
troduce in the effective action also the electromagnetic
S S =−g 1 Φc Φc +Φs Φs +Φ∆∗Φ∆ potential A. For the model (3) this can be done via the
hXi,ji i j Xq (cid:18)2Xα αq αq αq αq(cid:19) q (7q) PfieiesrtlshesuAˆbkskt′itmutaiotrnixc†iwciit+hαt→woca†idcdi+itαioe−naiAlαic.onwtrhibicuhtimonosdi-
with g =3J/4. The RHS of Eq. (7) represents the inter-
− 2tδ sin(k+k′) Aα τˆ (14)
action in the p-h density (Φc), p-h current (Φs) and p-p 2 α k′−k 0
(Φ∆) channel, respectively20. + tδ cos(k+2k′)αAαk′−k+sAα−s τˆ3. (15)
We decouple (7) by means of the Hubbard-
Eq. (14) corresponds to the usual term −J·A, where
Stratonovich (HS) trasformation both in the p-p and in
the p-h channel. After the integration of the fermions J ≡∂ ξ0 =2tδsink (16)
the action reads: α kα k α
is the quasiparticle current. Notice that in the presence
|φc |2 |φs |2 |∆ |2 ofHFcorrectionstothequasiparticledispersion(13)the
S = 2αgq + 2αgq + gq −TrlogAˆkk′. quasiparticlecurrentJαisdifferentfromthequasiparticle
q (cid:18) α (cid:19)
X X velocity
(8)
Here φc, φs and ∆ are the HS fields, the trace acts v ≡∂ ξ =(2tδ+φc)sink (17)
over momenta, frequencies and spins, k ≡ (k,iε ), q ≡ α kα k 0 α
n
(q,iωm), andεn, ωm arethe Matsubarafermionandbo- In the usual Landau FL language, these two quanti-
sonfrequencies,respectively. TheAˆk,k′ matrixisdefined ties are related by the Landau FL F1s corrections as
3
J = v (1 + Fs/3) for an isotropic system in three explicit form of the F(ΥHS) function:
α α 1
dimensions18. As we shall see, in our approachthe anal-
ogous role of Landau FL corrections will be played by δχαβ(q)=hFα(ΥHS)Fβ(ΥHS)i (21)
jj
the HS fields related to p-h fluctuations. Neglecting the
diamagnetic contributions, that are not relevant for the
Starting from the current-current response Λαβ(q) (20)
following discussion, the leading terms in A of the ac-
the paraconductivity simply follows from the dynamic
tion S (A,ΥHS) (ΥHS = φc,φs,∆) can be written in
GL limit (q = 0,ω →0) after analytical continuation of the
a compact form as:
Matsubara frequency ω to the real frequency ω
m
1
S (A,ΥHS)=− χαβ AαAβ +AαFα(ΥHS) (18)
GL 2 jj q −q q
δσ =[Imδχαα(ω,q=0)/ω] , (22)
jj ω→0
where χαβ is the mean-field current-current correlation
jj
function while the fluctuations contribution to the diamagnetism
is connected instead to the static limit (q→0,ω =0),
χαβ(q)=− 2 (∂ ξ )(∂ ξ )f(βξk+q2)−f(βξk−q2)
jj N Xk kα 0 kβ 0 iωn+ξk+q2 −ξk−q2 δχd =− δχjtj(q,ω =0)/q2 q→0. (23)
(19)
(cid:2) (cid:3)
and Fα(ΥHS) is a function of the ΥHS fields which de- where δχt is the transverse part of the fluctuation cor-
jj
scribes the connection of the electromagnetic potential
rection to the current-current response function. In the
to the HS fields. The current-current response func-
absence offluctuations in the p-h channel,only the pair-
tionΛαβ(q)canbecomputedfromthepartitionfunction ing field is coupled to A so that F(∆) ∼ p∆2 and one
Z[A]= DΥHSe−SGL(A,ΥHS) of the model as recoversthestandardALcorrection(seeEq. (2p7)below).
In our case, one immediately sees from Eq. (9) and (14)
R
δlnZ[A] that the φs field appears in the actions with the same
Λαβ(q)= =χαβ(q)+δχαβ(q) α
δAαδAβ (cid:12) jj jj structure of the electromagnetic potential Aα, i.e. it is
q −q(cid:12)(cid:12)Aαq=Aβ−q=0 coupled to the fermionic current. Thus, we expect that
(cid:12) (20) it will contribute to the δχαβ(q) correction (20) above.
where χαβ(q) simply fo(cid:12)llows from the quadratic term of jj
jj With lengthy but straightforwardcalculations one ob-
Eq. (18), while δχαβ(q) is the contribution coming from tainsthattheeffectiveaction(18)atleadingorderinthe
jj
the fluctuating modes coupledto A, anddepends onthe gauge and HS fields is given by
1 1χαβ 1 χαβ
S (A,φs,∆) = − χαβAαAβ − jj Aαφsβ +φsαAβ + g−1− jj φsαφsβ +
GL 2 jj q −q 2 2tδ q −q q −q 2 (2tδ)2 q −q
" #
(cid:0) (cid:1)
φs
+ g−1−Π(ω)+cq2 ∆∗q∆q−c′q Aq′ + 2tqδ′ ∆∗q∆q−q′ +∆∗q+q′∆q (24)
" # (cid:18) (cid:19)
(cid:0) (cid:1)
where summation over repeated indexes is implicit and the c′q term instead one carries out a single derivative
only the terms relevant for the following discussion are ofthe fermionic bubble whichcontains alreadya current
included. The c and c′ terms in Eq. (24) follow from the insertion J, associated to each A or φs field, see Eq.s
expansion to leading order in q of the fermionic bubbles (9) and (14) and Fig. 1. Thus, we have in a short-hand
associated to the ∆2 term and to the A∆2 and φs∆2 notation:
terms, respectively (see Fig. 1). In the former case for
example one expands c∝(∂ξ )2 ∼v2, c′ ∝∂ξ ∂ξ0 ∼v J (26)
kα F kα kα F F
T
Π(q)=−N γd2(k)Gk+q/2G−k+q/2 ≃Π(ω)−cq2. where vF, JF are the average values on the Fermi sur-
face. If we do not consider the interactions in the p-h
k
X
(25) channel leading to the HF correction of the band dis-
where the quasiparticle Green’s function, G = (iω − persion we would get ξ = ξ0 (see Eq. (13)) and the
k n k k
ξ )−1, contains the full dispersion ξ . As a consequence, velocity coincides with the current, so that c = c′. In
k k
c is proportional to the second-order derivative of G, this case, as mentioned above, the electromagnetic po-
which in turn scales as (∂ξ )2 ≡ v2. In the case of tential A is coupled only with the pairing field ∆ via
kα α
4
(a) where
χ (q)/(2tδ)2
jj
Z(q)= (29)
1/g−χ (q)/(2tδ)2
jj
and
c(q)=c′(1+Z(q)). (30)
(b)
We notice that in Eq.s (28)-(30) we used a simplified
notation valid in the q → 0 limit relevant for the fol-
lowingdiscussion,sothatthecurrent-currentcorrelation
J function χ refers to its diagonal part χαα only. At fi-
jj jj
nite q the above expressions must be properly extended
to account for the full matrix structure of the response
functions. From the definition (28) and using Eq. (20)
we compute the SCF contributionto the current-current
responsefunction. Wefindacorrectiontothemean-field
FIG.1: Fermionic bubbles: (a)is theΠ(q)bubbleassociated
to the ∆2 term, while (b) is the fermionic bubble relative valuecurrent-currentresponsefunction(19)thatleadsto
bothtoA∆2 andφs∆2terms. ThesolidlinesaretheGreen’s the RPA resummation of χjj as
functionsGcontainingthefullquasiparticledispersionξ. The
χ (q)
dashedlinesarethepairingfield∆. Thewavylinerepresents χRPA(q)=χ (q)+Z(q)χ (q)= jj ,
eithertheelectromagneticpotentialAorthecurrentfieldφs, jj jj jj 1−gχjj(q)/(2tδ)2
both associated to thequasiparticle currentJ∝∂ξ0. (31)
andasecondtermthatrepresentsthe SCFcontribution,
whichgeneralizesthestandardALresult(27)withamo-
a term ∼ cAp∆2, so that the fluctuation correction to mentum and frequency dependent vertex c(q):
p
thecurrent-currentresponsefunction(20)isgivenbythe
usual AL contribution9 (see Fig.2): δχαβ(q)=T c(q)2(2p+q) (2p+q) L(p)L(p+q).
α β
p
X
δχαβ(q)=c2T (2p+q) (2p+q) L(p)L(p+q) (27) (32)
α β
From Eq. (19) one can easily see that in the dynamic
p
X limit χ = 0, so that Z = 0 in Eq. (29) and from Eq.
jj
whereL(p)=h∆∗∆ iisthepropagatoroftheSCF.Since (30)c(q)=c′. Intheoppositestaticlimitinsteadwecan
p p
c∝v2 is constant, the SCF contribution to the diamag- rewrite χjj in Eq. (19) as
F
netism (23) and conductivity (22) is in both cases pro-
2
portionaltov4,andtheratioδσ/δχ isexpectedtobeof χ = − (∂ ξ0)2∂ f(βξ )
F d jj N kα k ξ k
order O(1). Such a result changes in the presence of p-h k
X
2 J
= α(∂2 ξ0)f(βξ ) (33)
N v kα k k
α
k
X
By direct comparison with the self-consistent equation
(12)forthedensityfieldoneimmediatelyseesthatχ =
jj
2tδ(J /v )(φc/g). From Eq. (29) it then follows that
F F 0
1+Z =v /J , leading to c(q)=c inEq. (30). We then
FIG. 2: (color online) AL contribution of the SCF to the F F
find that the vertices relative to the SCF contribution
current-current correlation function. The dashed lines repre-
to the conductivity anddiamagnetismarequantitatively
sent the SCF propagator, while the red dots reduce to the
constant c in the ordinary case Eq. (27), and to the momen- different:
tum and frequency dependent vertex c(q) in the presence of
c(q) ∝ v J ∼(2tδ+φc)2δt (q=0,ω →0)(34)
current-currentinteractions, see Eq. (32). F F 0
c(q) ∝ v2 ∼(2tδ+φc)2 (q→0,ω =0). (35)
F 0
interactions. The elimination via gaussian integrationof
the current field φs from Eq. (24) leads to The difference in the two limits reflects in a difference
in the overall prefactors of the SCF contribution to the
1 conductivity and diamagnetism. Indeed, from Eqs. (22)-
SGL(A,∆) = −2χαjjβ 1+Z(q) AαqAβ−q+ (23) one has that
+ g−1−(cid:0)Π(ω)+cq(cid:1)2 ∆∗q∆q + δσ ((2tδ+φc)2tδ)2
∝ 0 ∝δ2 (36)
− (cid:0)c(q)qAq′ ∆∗q∆q−q′(cid:1)+∆∗q+q′∆q (28) |δχd| (2tδ+φc0)4
(cid:0) (cid:1)
5
i.e. the fluctuation conductivity is suppressed by the depletion with doping19. Moreover,at low doping15,19,22
proximitytotheMottinsulatorbyafactorthatdepends one should also include the boson fluctuations neglected
on the doping. so far, which give a temperature T for the boson con-
B
We notice that in the presence of HF corrections the densationsmallerthantheoneforthegapopening,lead-
gauge-invariantform of the GL functional for SCF is re- ing to a suppression of the critical temperature T with
c
covered in a non trivial way. Indeed, in the usual case respect to its mean-field value T . In such a regime,
MF
the coupling to the gaugefield Acanbe obtainedby the the static limit of the 1+Z correctionin Eq. (30) above
minimal-coupling substitution q → q−2A in the q2∆2 couldnotbesimplygivenbyv /J : nonetheless,thedif-
F F
termoftheGaussianpropagator. Thisleadsimmediately ference between the static and dynamic limit still holds,
to the term linear in A, cA·q∆2, needed to compute possibly leading only to a quantitative difference with
the AL correction, see Eq.s (18)-(21). In our case by respect to the result (36). Finally, we notice that the
directinspection of Eq. (24)one finds instead two differ- experimental estimate of the SCF contribution to para-
ent coefficients c,c′ in the q2∆2 and A·q∆2 terms, as conductivity reportedin Ref. [13] is basedon the scaling
we explained above. However, by integrating out the φs of the conductivity at frequencies large enough with re-
field the coupling of SCF to the gauge field is described spect to quasiparticle dissipation. As a consequence, it
in general by a term c(q)A·q∆2, (see Eq. (28)), where is worth making a comparison with the present results,
c(q) isgivenby Eq.(30). As aconsequence,onerecovers that have been derived in the clean limit.
theminimal-couplingprescriptiononlyinthestaticlimit
(35) where c(q)=c21. Insummary,weanalyzedtheSCFcontributiontocon-
According to the discussion below Eq. (1) above, the ductivity and diamagnetism in the presence of current-
result (36) can be recast as an estimate of the ratio be- current interactions, by using as a paradigmatic exam-
tween the correlation lengths extracted experimentally ple the slave-boson formulation for the t − J model.
from paraconductivity and diamagnetism. For example By explicitly constructing the GL fluctuation func-
usingdatareportedinRef.[13]onehasthatatT ∼24K, tional in the presence of HF corrections we showed that
δσ ∼105(Ωm)−1, and δχ ∼60A/mT. Using in Eq. (1) current-currentinteractions,neededtorecoverthegauge-
d
ξ ∼1nmandd∼15 Åasappropriatesforcuprates,we invariantformoftheGLfunctional,modifythetransport
0
obtain that k T/dΦ2 = 0.053A/mT and e2/16~dξ2 = coefficientsleadingtoamomentumandfrequencydepen-
B 0 0
104(Ωm)−1, so that δσ/δχ ∝ ξ2/ξ2 ∼ 10−2. Such a dence of the vertex c(q) entering the AL expression for
d σ χd
strong suppression of the paraconductivity with respect the SCF contribution. Since different limits are involved
to diamagnetism, that has been rephrasedin Ref. [13] in in the definition of paraconductivity and diamagnetism,
terms of the vortex diffusion constant valid only in the we obtain a different prefactor in the two cases, with a
caseofKTfluctuations,canbemoregenerallyattributed suppressionof the paraconductivitybecause of the prox-
from Eq. (36) to the overallδ2 factor due to the proxim- imity to the Mott-insulating phase, as recently shown
ity to the Mott-insulating phase. We checked that the experimentally in cuprates13. Even though a mean-field
estimate of the ratio (36) within the mean-field solution approachto the t−J model is not satisfactory from the
of the t− J model is quantitatively larger by a factor quantitative point of view, the different strength of SCF
ten than the one experimentally found, since a prefac- contribution to conductivity and diamagnetism is more
tor ∼2t/φc partly compensate the δ2 suppression. Such general,sinceitisaconsequenceoftheexistenceofasize-
0
a quantitative disagreement is reminiscent of analogous ble difference betweenquasiparticlecurrentandvelocity.
limitations of the mean-field approach already discussed A quantitative comparisonwith experiments remains an
in the literaturein the contexts of otherphysicalquanti- interesting theoretical challenge, that certainly deserves
ties, as for example the scaling of the superfluid-density further investigation.
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