Coupled multiple-mode theory for $s_{\pm}$ pairing mechanism in iron based superconductors

0.77 MB·

by  M. N. Kiselev

Coupled multiple-mode theory for s pairing
±
mechanism in iron based superconductors
M. N. Kiselev1,*, D.V. Efremov2,+, S. L. Drechsler2, Jeroen van den Brink2, and K. Kikoin3
1TheAbdusSalamInternationalCentreforTheoreticalPhysics,StradaCostiera11,I-34151Trieste,Italy
2InstituteforTheoreticalSolidStatePhysics,IFWDresden,Germany
6 3SchoolofPhysicsandAstronomy,TelAvivUniversity,69978TelAviv,Israel
1 *[email protected]
0
[email protected]
2

t
c
O ABSTRACT

1
3 Weinvestigatetheinterplaybetweenthemagneticandthesuperconductingdegreesoffreedominunconventionalmulti-band
superconductorssuchasironpnictides. Forthispurposeadynamicalmode-modecouplingtheoryisdevelopedbasedonthe

] coupledBethe-Salpeterequations. Inordertoinvestigatetheregionofthephasediagramnottoofarfromthetetracriticalpoint
n
wherethemagneticspindensitywave,(SDW)andsuperconducting(SC)transitiontemperaturescoincide,wealsoconstruct
o
aGinzburg-LandaufunctionalincludingbothSCandSDWfluctuationsinacriticalregionabovethetransitiontemperatures.
c
- Thefluctuationcorrectionstendtosuppressthemagnetictransition,butinthesuperconductingchanneltheintrabandand
r interbandcontributionofthefluctuationsnearlycompensateeachother.
p
u
s
.
t Introduction
a
m
Therapidlyextendingrealmofhigh-T superconductorshasbeenenrichedrecentlybyanewclassofmaterials,socallediron
c
d- basedsuperconductors(FeSC).1–5 Likemanyotherhigh-Tc materials, thesecompoundscrystallizeinstronglyanisotropic
n lattices: onecanidentifyquasi-two-dimensionalsubsystemswhichcontaintheelectronsthataresubjecttoCooperpairing.
o InFeSCstheseplanesconsistofsquarepyramidswithalternativelyorientedapexverticesthatareoccupiedbypnictogen
c
ionsandasquarebaseformedbyironions. AspecificfeatureoftheelectronicstructureofFeSCisthemultipocketFermi
[
surface,mostlysemi-metallic(withbothholeandelectronpockets),althoughsomesuperconductingmaterialspossesseither

2 onlyelectron6,7 oronlyholepockets.8 AnothercharacteristicfeatureofFeSCistheinterplaybetweendifferentelectronic
v
instabilitiesofthepristinenormalmetalphases. Mostofthesematerialsareunstableagainstspin-densitywave(SDW)type
2
itinerantantiferromagnetism,andthephasediagramsofdopedcompoundscontainsdomainsofsuperconductor(SC)andSDW
5
5 ordering,thelattersometimesbeingaccompaniedbystructuralphasetransitionswithapotentialfororbitalordering.9–12 This
3 meansthatanyconsistenttheoryofsuperconductivityshouldtakeintoaccountthestronginterplaybetweenelectronicand
0 magneticinstabilitiesinpresenceoftrendstoorbitalorderingandsoftlatticedisplacementmodes(seeanothermorecomplex
.
1 approachfordescriptionofd-electronsinironpnictidesin[13]). Suchamoregeneralapproachtoconsidernotsimplyasingle
0 instabilityofa”normalstate”phasewithrespecttosuperconductivitybutinsteadtotreatonequalfootingthecompetition
6 (includingalsotheircoexistence)ofvariousorderingsisagenericproblemformanynon-standardsuperconductorsandrelated
1 phases.14,15 Eventhestandardcaseofstrongelectron-phononinteractionmediatedSCandaFermisurfacederivedinstability
:
v requiresstrictlyspeakingtheaccountofpossiblelatticeinstabilitiesandoftherelatedanharmonicitiesreducingthestrengthof
Xi theformer. ForthesakeofsimplicityherewewillonlyfocusontheinterplayofaspecificSDWandSC-ageneralizationto
thecaseofhighermulti-componentinstabilitiesisstraightforward.
r
a Aspecificmotivationtoconsideranddevelopamulti-modetheoryisprovidedbytheexperimentalobservationthatin
FeSCs electron or hole doping of pristine magnetic materials is crucial for suppression of itinerant SDW magnetism and
formationofsuperconductingstate. Apartfromthat,dopingbyisovalentimpuritiesoranoticeableconcentrationofintrinsic
defects(vacancies)canalsostronglymodifytheirphasediagram. Itwasnoticedinthiscontextthatconventionalunderstanding
of the role of non-magnetic and magnetic impurities in formation of Cooper pairs in BCS superconductors formulated in
classicalpapersbyA.A.Abrikosov,L.P.Gor’kovandP.W.Anderson(see,e.g.,16and17)shouldberevisitedandmodifiedfor
thesemultibandsuperconductorsbecauseinterbandcouplingplayscrucialroleinformationofsuperconductingorderinthese
nearlynested,quasi2Dmaterials. BothnonmagneticandmagneticscatteringessentiallyinfluenceT -inparticular,interband
c
scatteringduetonon-magneticimpuritiesisdestructivefors superconductivity,whichisconsideredtoberealizedinmostof
±
FeSC18–22andmagneticinterbandscatteringonthecontrarystabilizesthes couplingmechanism.18,23 Thekinematicsofthis
±
1
stabilizationmechanismissimilartothatinsocalledπ-junctionsinheterostructuressuperconductor/insulator/superconductor,
wheretheJosephsontunnellingthroughabarrierwithparamagneticimpuritiesisaccompaniedwiththesignchangeofthe
superconductingorderparameter∆.24 Anotherremarkableresponsetodopingoccurswhennominallynonmagneticdefects
suchasAsvacancies25–27orRusubstitutions28areintroducedin1111materials. Theseimpuritiestriggerastrongparamagnetic
response(see29forexplanationofthiseffect),andtheirobservedmodificationofT doesnotfitwithavailabletheories.
c
Tothisendweintroduceanddevelopamulti-modetheorythattakesintoaccountthefactthatthesuperconductivityin
thesematerialsarisesasaresultofcompetitionofatleasttwocouplingmechanisms: superconductingpairingwiththescalar
orderparameter∆andthespindensitywaveorderingwithvectororderparameterm(themagneticmoment). Thesemodes
arecoupledbytheinterbandelectron-electroninteraction,andweconsidertheeffectofimpurityscatteringonthiscoupling
mechanism.
Thepaperiswrittenasfollows. WestartwiththederivationofasystemofBethe-Salpeter(BS)equationscontaining
coupledSDWandCooperchannelsfortemperaturesabovethemagneticandsuperconductingtransitiontemperaturesinthe
nextsection. BasedonBSequationsweshowthatdynamicalfluctuationsplayanimportantroleclosetothetetracriticalpoint
andmustbetakenintoaccountforthecorrectdescriptionofthetetracriticalregion.
FurthermorewederivetheLandau-Ginzburg(LG)functionalandcalculatefluctuationcorrectionstothephasetransition
temperaturestosuperconductorandSDWphases(T andT,respectively). DetailedderivationofBSandLGequationsisgiven
c s
intheSection”Methods”.
Whereneeded,wewillbespecificandconcentrateonthemulti-valleysemi-metallicFeSCsfromthe1111(ReFeAsO)
and122(AeFe As )familiesunderelectronandholedoping(ReandAestayfortherare-earthandalkaline-earthelements)
2 2
respectively.
Multimode approach to SDW and BCS instabilities
InthisSectionweformulateamodecouplingapproachformultibandferropnictidesuperconductorswithnearlynestedhole
andelectronpocketsofFermisurface. Althoughacompleteconsensusislackingonthetypeofsuperconductorpairingin
thesesystem,itappearsthattheexperimentalargumentsinfavorofs mechanismarerathersolid. Thistypeoforderingwas
±
proposedoriginallyforsuperconductor-excitonicinstabilityinsemimetals30 andhasbeenreformulatedforsuperconductor
pairinginFeSCsinRefs.31,32 (forareviewsee5). Itisimportanttonotethatmode-modecouplingisaninherentconstituent
ofs pairing:33–35 inthetwo-dimensionalsystemswithnestingconditionsbetweenelectronandholepocketsaninterplay
±
betweentheSDWandCoopermodesresultsinadditionalcouplingwithineachchannelinducedbyallotherchannels. This
theoryisbasedonarenormalizationgroup(RG)approach,whichimplieslogarithmicrenormalizationoftherelevantvertices.
ItwasshownthattheCooperandSDWdecoupleatthescaleofFermienergyandfurtherflowgoesindependentlyinboth
channels. ThenaturallimitationofsuchapproachisthedemandofperfectnestinginSDWchannel. InrealFeSCsystemsthe
nestingconditionsaresatisfiedonlyapproximately. ItleadstothesuppressionoftheSDWchannelatsomeenergy.
Thegeneralapproachtodealwithmode-modecouplingthatweintroducehereisbasedonthemodecouplingtheoryof
criticalphenomena. Westartfromthe”hightemperature”region,wherethecriticalfluctuationsarealreadywelldeveloped,
whilelong-rangeSCandSDWorderarenotyetestablished.ThecorrespondingvertexpartsΓ (q,Ω,T)andΓ (Q+q,Ω,T),
sc sdw
arerepresentedatthistemperaturebythefluctuatingmodesD (q,Ω)andD (q,Ω)respectively:36,37
sc sdw
(0)
νΓ
νΓ (q,Ω,T)→D (q,Ω)= sc , (1)
sc sc −iΩ+τ +c q2
γsc c ∆
(0)
νΓ
νΓ (Q+q,Ω,T)→D (q,Ω)= sdw ,
sdw sdw − iΩ +τ +c q2
γsdw s m
withcriticalparameters
T−T T−T
c s
τ = , τ = , (2)
c s
T T
c s
andγ =8T/π . Hereweconsiderthelimitofsmallmomentac q2(cid:28)1,ν isadensityofstatesattheFermilevel,
sc/sdw ∆/m
c =ξ2 and c =ξ2 are superconducting and magnetic coherence lengths respectively, see Section ”Ginzburg-Landau
∆ s m m
approach”.
As the pole iΩ =γ (τ +c q2) of the vertex part Γ or Γ at q→0 tends to zero, the corresponding
sc/sdw sc/sdw c/s ∆/m sc sdw
instabilityresultsinthephasetransition. WhenapproachingthetransitiontemperatureT orT,thecorrespondingvertexpart
c s
divergesinthestaticlimit:
Γ (0,0,T)=Γ(0)τ−1, Γ (Q,0,T)=Γ(0)τ−1. (3)
sc sc c sdw sdw s
2/17
$)
&
,
%
&’ SDW
$
+
*
$
%
SC
!"#!$#%&’("#)
Figure1. (Coloronline)PhasediagramofcompetingSDWandSCstatesasafunctionofthecarrierconcentration. The
shadedareadenotesthecoexistenceregionofs superconductivityandtheSDW.ThecriticaltemperaturesofSDWandSC
±
transitionscoincideinthetetracriticalpoint. Theinsertshowszoomedinpartofthephasediagramaroundthemean-field
tetracriticalpoint: thenewtetracriticalpoint(filledcircle)isshiftedduetofluctuationscomputedinthemulti-modecoupling
theory(dashedlines).
ThemodecouplingapproachisvalidinthevicinityofthetetracriticalpointonthephasediagramshowninFig. 1wherethe
twotemperaturesT andT arecomparable. ThenthesystemofequationsforthevertexpartsatsmallΩandqcanbederived
c s
andsolved. ThesearetheBethe-Salpeterequationsforthemostrelevantvertices. Aswillbeshownbelow,thissystemcontains
atleastfourbareverticeswhichshouldbetakenintoaccount. Thedivergences(3)provideuswithacriterionofselectingthe
correspondingdiagrams. Namely,thetermscontainingpolarizationoperatorsintheelectron-holechannelwithtransmitted
momentumclosetoQ,andtheelectron-electronpolarizationoperatorsofCoopertypeshouldbecollectedinallorders. As
aresult, thesystemofBethe-Salpeterequationsacquiresaparquet-likestructure, andthesolutionofthemode-coupling
equationsobtainedatT >T maybemappedonthemean-fieldsolutionsofrenomalizationgroup(RG)equationsbelowT .33,34
c c
WeassumethatthesuperconductingtransitiontemperatureT isimposedonthesystembyadominants pairing. Inthis
c ±
Ansatztheinterbandelectron-holeCoulombinteractionisthemainsourceofCooperpairing,sothataweakBCSattractive
interactionintheelectronandholepocketswouldresultinSCinstabilitieswithT(e,h)(cid:28)T .
c c
Bethe-Salpeterequations
FollowingtheAnsatzformulatedabove(seedetailsintheSection”Methods”),wederivethesystemofBSequationsforthe
relevantvertexpartsΓ (seeFig. 2). ThestartingHamiltonianis:
i
(cid:104) (cid:105) (cid:104) (cid:105)
H = ∑ (εke−µ)fσ†kfσk+(εkh−µ)c†σkcσk + ∑ u1c†σk1fσ†(cid:48)k2fσ(cid:48)k3cσk4+u2c†σk1fσ†(cid:48)k2cσ(cid:48)k3fσk4
k,σ k1+k2=k3+k4,σσ(cid:48)
(cid:104)u u u (cid:105)
+ ∑ 23c†σk1c†σ(cid:48)k2fσ(cid:48)k3fσk4+ 24c†σk1c†σ(cid:48)k2cσ(cid:48)k3cσk4+ 25 fσ†k1fσ†(cid:48)k2fσ(cid:48)k3fσk4+h.c. , (4)
k1+k2=k3+k4,σσ(cid:48)
where f ,c standforannihilationoperatorinelectronandholebands;εe/handµ aredispersionsandchemicalpotential,
σk σk k
respectively.
BasedontheresultsofRGcalculations,33,34wechoosefourverticesrelevanttothes coupling(theverticesu ,u ,u =
± 1 3 4
u ,u =u ). Thevertexu whichhasbeenshowntobeirrelevant34isexcluded.
s 5 s 2
TheverticesΓ withi=1,2(seeFig. 2)describetheinteractionsinthedensity-waveblock,theverticesΓ describethe
1i 4,5
singletCooperpairingintheelectronandholepockets,respectively,andthevertexpartΓ includestheinteractionsresponsible
3
fortheinterband(e-h)Cooperpairing. ThesystemofBethe–Salpeterequationsmaybewritteninthesymmetricformusinga
matrixΛˆ:
(1−Λˆ)Γˆ =uˆ. (5)
3/17
Thematrix(1−Λˆ)isthesecularmatrixand
   
u Γ
1 11
 uσσ¯   Γ 
uˆ= u3σσ , Γˆ = Γ31 , (6)
 3   32 
 us   Γ4 
u Γ
s 5
ThesecularequationforthissystemofBethe-Salpeterequationsforthemodes
D−1 = 1−u Πσσ,
1 1 1
D−1 = 1−u Πσσ−u2ΠσσΠσ¯σ¯,
31 1 1 3 1 1
D−1 = 1−u Πσσ−u (Π +Π )/2,
32 1 1 s se sh
D−1 = 1−u Π ,
4 s se
D−1 = 1−u Π , (7)
5 s sh
isdet(1−Λˆ)=0. Inthesenotationsbothcouplingconstantsu =νU andthepolarizationloopsΠ(q=0,ω =0)=ln(W/T)
i i i
aredimensionless(hereW isthebandwidth). Weneglectbelowthedifferencebetweenthebanddispersionintheelectronand
holepockets,whencalculatingthepolarizationoperatorsΠ =Π =Π . Oneshouldnotethathisapproximationdoesnot
s se sh
implythatwerelyonperfectnesting. Inelectrondopedmaterialsthestrictnestingconditionsareofcourseviolated,butthe
smalldifferencebetweentheelectronandholepolarizationoperatorsisinsufficientforourtheorybecausetwocontributions
aresummedinthecriticalmodeD whichiscentraltoourscenario. Thecriticalityinthismodeisdominatedbytheterm
32
u Π . WeusethesignconventionthatbothΠ andΠ arepositive. Underthisconventionu >0isattractiveintheCooper
1 1 1 s s
channelandu >0facilitatestheinstabilityintheSDWchannel.
1
Intheexplicitformthesecularequationreads:
(cid:12)(cid:12) D−11(p,ω) 0 −u3Πσ1¯σ¯ 0 0 (cid:12)(cid:12)
(cid:12)(cid:12) 0 D−1(p,ω) 0 0 0 (cid:12)(cid:12)
det(1−Λˆ)=(cid:12)(cid:12)(cid:12) −u3Πσ1¯σ¯ 31 0 D−321(p,ω) −u3Πs −u3Πs (cid:12)(cid:12)(cid:12)=0. (8)
(cid:12)(cid:12) 0 0 −u3Πs D−41(p,ω) 0 (cid:12)(cid:12)
(cid:12) 0 0 −u Π 0 D−1(p,ω) (cid:12)
3 s 5
OnecanseethatinthisapproximationtheT tripletΓ -channeldescribedbythesecondrowofthesecularequation
±1 31
decouples from the rest, which corresponds to the T - SDW and superconducting channels. The D -mode plays a very
0 32
particularrole. Ontheonehanditcontributestothedensitywavechannels(bothCDWandSDW).Ontheotherhanditstrongly
affectsthesuperconductingchannel. IntheapproximationusedinthepresentpapertheSDWandCDWaredegenerate. The
interbandinteractionu >0withmomentumtransferQliftsoutthedegeneracyfavouringtheSDWtransition.
2
Wewillconsiderthepartofthephasediagramconcentration-temperature(c−T)closetothepointofthedegeneracyofthe
s andSDWchannels(i.e. thetetracriticalpoint)showninFig. 1. InthisregiontheSCinstabilitytakesplaceinthepresenceof
±
criticalSDWfluctuations. AboveT thefluctuationmodesariseatmomentap=Q+qclosetothenestingvectorQconnecting
s
theΓandX pointsintheBrillouinzoneandatsmallω →0.
WeassumethatthetwoCooperpropagatorsD andD arefarfromanydivergence,namelyu Π (cid:28)1atthesetemperatures
4 5 s s
(incaseofadominantinterbandpairingmechanismasinthepronounceds caseadoptedhere,thepurelyintrabandCooper
±
instabilitiesdevelopattemperaturesmuchlessthantheactualT ). Inthisapproximationtheverticesarereal,andthechannels
c
4,5arerepresentedbyasinglerowandcolumn.
Awayfromthetetracriticalpoint
We consider first the case when the temperature is less than the Fermi-energy T <ε and the doping c is away from the
F
tetracriticalpoint. Thenthedivergenceoftheverticesisstronglypeakedatparticularmomenta. Forinstance,puttingu →0
3
intheintegralequationfortheΓ (p ,p ,p ,p )(seeFig. 2a),oneimmediatelygetsthatthisvertexisdivergentonlyfor
11 1 2 3 4
p +p =Q. AsimilaranalysisshowsthatΓ divergesforp +p →0andp −p →Q. Thesplittingofmomenta,atwhich
1 2 32 1 2 1 3
theverticesdiverge,decouplesthematrixEq.(5)intothedensity-waveandCooperchannels. Tostudythepropertiesofthe
vertexfunctionsinthevicinityoftheirsingularitiesweintroduceΓ (p ,p ,p ,p )≈Γ(a)(p +p −Q)+Γ(b)(p −p +Q),
11 1 2 3 4 11 1 2 11 1 3
whereΓ(a)(k˜)andΓ(b)(p)havepolesatk,p→0correspondingly. FortheverticesΓ andΓ weusethesamedecomposition:
11 11 32 4
Γ (p ,p ,p ,p )≈Γ(a)(p +p )+Γ(b)(p −p +Q)andΓ (p ,p ,p ,p )≈Γ(a)(p +p )+Γ(b)(p −p +Q).
32 1 2 3 4 32 1 2 32 1 3 4 1 2 3 4 4 1 2 4 1 3
4/17
σ σ σ σ σ σ −σ σ
Γ = u + u Γ + Γ (a)
11 1 1 11 u 32
σ σ σ σ σ σ 3 −σ σ
−σ σ −σ σ σ
Γ = Γ (b)
12 u 31
−σ σ −σ 3 σ σ
σ σ σ σ σ u1 σ
u
Γ31 = u3 + −σ 3−σ + σ σ (c)
σ σ Γ
Γ σ 31σ
σ 12σ
σ σ σ σ σ u1 σ
u
Γ32 = u3 + −σ 3−σ + σ σ
−σ −σ Γ
Γ −σ 32−σ
−σ 11−σ
(c’)
σ σ σ σ σ σ
+ u Γ + u Γ
3 4 s 32
−σ −σ −σ −σ −σ −σ
σ σ σ σ σ σ σ σ
Γ4 = us + us Γ4 + u3 Γ32 (d)
−σ −σ −σ −σ −σ −σ −σ −σ
σ σ σ σ σ σ σ σ
−σ Γ5 −σ= us + −σus −σ Γ5 −σ +−σ u3 −σ Γ32−σ (e)
Figure2. DiagramsforthesystemofBethe-Salpeterequations. Solidanddashedlinesstandforholeandelectron
propagators,respectively. TheSDWpolarizationloopΠσσ containsoneelectronandoneholebareGreenfunction. The
1
CooperloopsΠ containtwoelectronortwoholebarepropagators,respectively.
s
ToidentifythecorrespondingtransitiontemperaturesweconsidertheBethe-Salpeterequationsinthestaticlimit. Forsmall
momentap,k˜,qthesystemEq.(5)splitsupinthelogarithmicapproximation(seeSection”Methods”)intocooperchannel:
u ±u
Γ(a)(k)±Γ(a)(k)= s 3 , (9)
4 32 1−(u ±u )Π (k)
s 3 s
anddensity-wavechannel
u ±u
Γ(a)(k)±Γ(b)(k)= 1 3 . (10)
11 32 1−(u ±u )Π (k)
1 3 1
Withtheseverticeswenowcalculatethesusceptibilityinthecooperanddensity-wavechannels. Intheparticle-particles and
±
s channeldrawninFig.3weobtain
++
χSC (k,T)=Π (k,T)+Π2(k,T)(Γ (k,T)±Γ (k,T)). (11)
±,++ s s 4 32
5/17
χSC = +
±
χDW = +
±
Figure3. Cooperanddensitywavessusceptibilities. Theshadedboxesdenotethecorrespondingcombinationofthevertices
(seethetextfordetails).
SubstitutingEq. (9)intoEq. (11),weget:
Π (k,T)
χSC (k,T)= s . (12)
±,++ 1−(u ±u )Π (k,T)
s 3 s
Thisequationhasthetypicalpolestructureforthesuperconductingsusceptibilityinthevicinitytothetransitiontemperature.
ThesusceptibilityχSC (k=0,T)increaseswithdecreasingtemperatureanddivergesatthecriticaltransitiontemperature.
±,++
Thishappensundercondition1−(u ±u )Π (k=0,T =T )=0.Thetwosolutionscorrespondtos ands superconducting
s 3 s c ++ ±
orderparameter. ForCDWandSDWchannelsweget:
Π (k,T)
χDW(k,T) = 1 . (13)
± 1−(u ±u )Π (k,T)
1 3 1
The SDW transition is determined by 1−(u +u )Π (q =0,T )=0. One can define a tetracritical point, when both
1 3 1 c
susceptibilitiesdiverge. Ithappensunderthecondition(u +u )Π (k=0,T )=(u +u )Π (k,T )→1. Inthiscaseandinthe
s 3 s c 1 3 1 c
vicinityofthetetracriticalpointoneshouldfindthedivergenceofthesusceptibilitytakingintoaccountthefullmatrixEq.(5).
Closetothetetracriticalpoint: dynamicalmulti-modecouplingtheory
Nowweconsidertheinterplaybetweenmagneticandsuperconductingdegreesoffreedominthevicinityofthetetracritical
point. InthiscasetheapproachbasedontheeffectivehydrodynamicactionismucheasierthanthedirectsolutionoftheBS
equations.Besides,thisapproachwhichalsoaccountsforthemulti-pointcorrelationfunctionsgoesbeyondtheconventionalBS
paradigmlimitedbytwo-andfour-pointGreen’sfunction.16,17 Thederivationoftheeffectivehydrodynamicactionintermsof
magnetic(cid:126)m((cid:126)x,t)andsuperconducting∆((cid:126)x,t)(i=e,h)dynamicalfluctuationsisdonebyintegratingouttheBSequationswith
i
respecttothe”fast”(withtheenergyoftheorderofthebandwidth)degreesoffreedomandispresentedinSection”Methods”.
Withinouranalysiswestartdiscussingthelimitingcaseu (cid:28)u andu Π ∼1correspondingtotheinterplaybetweentwo
s 3 3 s
fluctuatingmodes: oneisthes superconductingandanotheroneistheSDWmagnetic. Thisregimeisbelievedtobepresent
±
inmostofFeSC[e.g.,inbroadlystudiesdoped”A-122”systems(whereA=Ba,Sr,Ca)]TheLagrangianofthetwo-mode
couplingtheorytakestheform:
(cid:90) (cid:26) 1 (cid:27)
L(2) = d(cid:126)x ∆¯ L−1∆ +(cid:126)mD−1(cid:126)m+ A|∆ |4+B(cid:126)m4+(C −C )|∆ |2(cid:126)m2 (14)
− ± − m 2 − 1 2 −
HeretheFouriertransformsofsuperconductingandSDWfluctuatorscomputedonMatsubarafrequenciesiΩ read:
m
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
1 1 |Ω | 1 (cid:104)((cid:126)v ·(cid:126)q)(cid:105) 1 |Ω |
L−1(q,iΩ ,T)= −Π (q,iΩ ,T )=ν τ +ψ + m −ψ − p FSψ(cid:48)(cid:48) + m , (15)
± m u +u s m c c 2 4πT 2 2(4πT)2 2 4πT
s 3
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
1 1 |Ω | 1 (cid:104)((cid:126)v ·(cid:126)q)(cid:105) 1 |Ω |
D−1(Q+q,iΩ ,T)= −Π (q,iΩ ,T )=ν τ +ψ + m −ψ − p FSψ(cid:48)(cid:48) + m .
m m u +u 1 m c s 2 4πT 2 2(4πT)2 2 4πT
1 3
Hereψ(x)andψ(cid:48)(cid:48)(x)arethedi-gamma functionanditssecondderivativerespectively, (cid:104)...(cid:105) denotestheaveragingover
FS
theFermisurface(hereaparabolicdispersionandequalmasswereassumedforelectronandholepockets). Theanalytic
continuationofthesuperconductingandSDWfluctuatorstotheupperhalf-planeiΩ →Ω+i0+forΩ(cid:28)(T ,T)isgivenby
m c s
(cid:20) (cid:21)
iΩ
L−1(q,Ω,T)=ν − +τ +c q2 , (16)
± γ c ∆
sc
(cid:20) (cid:21)
iΩ
D−1(Q+q,Ω,T)=ν − +τ +c q2 .
m γ s m
sdw
6/17
∆¯h ∆h∆¯e ∆e m! m!
∆h ∆¯h ∆e ∆¯e m! m!
’"# ("# )"#
∆¯h ∆h m! m! m! ∆¯e∆e m!
m! m! ∆e ∆¯e ∆h m! m! ∆¯h
&"# !"# $"# %"#
Figure4. (Coloronline)Interactionbetweenfluctuations: wavylinesrepresentthesuperconductingfluctuatorsL (q,Ω,T),
e/h
brokenlinesstandformagnetic(SDW)fluctuatorD (q,Ω,T). Diagrams(a,b)defineAcoefficient,(c)-B-coefficient,(d,e)-
m
C -coefficient,(f,g)-C -coefficient. NotationsforsolidanddashedlinesarethesameasinFig.2
1 2
Noticethatwenormalizethefluctuatorsbythedensityofstatestomakethemdimensionless. Theequationsdefiningthe
coefficientsA,B,C ,C andc arederivedinSection”Ginzburg-Landauapproach”.Thestaticq-independentpartofthe
1 2 ∆/m
LagrangiancorrespondstotheLandauexpansionofthefreeenergy. InclusionofthegradienttermsgeneralizestheLandau
theorytotheGinzburg-Landaufunctional(seeSection”Ginzburg-Landauapproach”). TheeffectiveLagrangiandescribes
effectivenon-lineartheoryofinteractingmodeandthereforegoesbeyondlinearBethe-Salpeterapproach.Hiddennon-linearity
ofBethe-Salpeterequationsisassociatedwithcurved(”non-flat”)phasespace: Γ entersbothSDWandsuperconducting
32
equationsbeingsingularbothatsmalltotalmoment/energyandsmalldeviatingfromQmomentumtransfer. TheLagrangian
(14)isU(1)×SU(2)symmetric.
Althoughthesituationin1111and122pnictidesisdescribedintheframeworkabove,weconsiderforthesakeofgenerality
alsotheoppositelimitu (cid:28)u andu Π ∼1. Inthiscaseonedealswiththethree-modecouplingtheory: twosuperconducting
3 s s s
fluctuatingmodes∆ and∆ andoneSDW(cid:126)mmagneticmode:
e h
(cid:90)
L(3)= d(cid:126)x(cid:8)∆¯ L−1∆ +∆¯ L−1∆ +(cid:126)mD−1(cid:126)m+A(cid:0)|∆ |4+|∆ |4(cid:1)+B(cid:126)m4+C (cid:0)|∆ |2+|∆ |2(cid:1)(cid:126)m2+C (cid:0)∆¯ ∆ +∆¯ ∆ (cid:1)(cid:126)m2(cid:9), (17)
e e e h h h m e h 1 e h 2 e h h e
with
(cid:20) (cid:21)
iΩ
L−1(q,Ω,T) = L−1(q,Ω,T)=ν − +τ +c q2 . (18)
e h γ c ∆
sc
Asinthestaticcase,weneglectthedifferencebetweenthecontributionofelectronandholepocketsinthefluctuationmodes.
Inprinciple,theSDWandCDWmodesaredegeneratewhenu Π (cid:28)1andoneneedstoconsiderafour-modecouplingtheory.
3 1
Wehowever,assumethattheSDW-CDWdegeneracyisliftedoutbyadditionalinter-bandprocessesu (seeEq. (4)andfurther
2
detailsinRefs.33,34),omittedthroughthederivationoftheBSequations,T <T andrestrictourselvesbythethree-mode
CDW s
couplingtheory.
TheLagrangian(17)atC =0(Fig. 4a-e)describesU(1)×U(1)×SU(2)gaugetheorywhereeachU(1)correspondsto
2
thegaugesymmetryofthesuperconducting(e-andh-)sectorsandSU(2)representsthemagnetic(SDW)sectoroftheeffective
model. Themostimportantobservationisthatmagneticfluctuations(Fig. 4f-g)breakU(1)×U(1)→U(1)andmediatethe
cross-talkbetweentwodifferentsuperconducting(e-h)sectors.
FluctuationcorrectionstoT andT
c s
Inordertofindthefluctuationcorrectiontothesuperconductingtransitiontemperaturegivenbythemean-fieldanalysisin
thevicinityofthetetracriticalpoint,weintegrateouttheremainingslowmagneticfluctuationsin(14)andfinallyobtainthe
effectiveLagrangiandescribingthesuperconductingsystem:
(cid:90)
L(2) = d(cid:126)x(cid:8)∆¯ L˜−1∆ +A˜|∆ |4(cid:9), (19)
∆ − ± − −
where
L˜−1=L−1+K +K , (20)
± ± 1 2
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isthesuperconductingfluctuatorrenormalizedbymagneticfluctuations. Asaresult,wefindafluctuationcorrectiontoT :
c
T =T0·(1−ν−1(K +K )), (21)
c c 1 2
whereT0isthetransitiontemperaturedefinedbythestaticsolutionofBSequations(mean-fieldtheory)and
c
(cid:90) d(cid:126)kd(cid:126)q
K = T2∑ G0(k,iε )G0(k,iε )G0(−k,−iε )D (q,iΩ )G0(k−q,iε −iΩ ), (22)
1 (2π)2d e n e n e n m m h n m
nm
1 (cid:90) d(cid:126)kd(cid:126)q
K = − T2∑ G0(k,iε )G0(−k,−iε )D (q,iΩ )G0(k−q,iε −iΩ )G0(−k+q,−iε +iΩ ). (23)
2 2 (2π)2d e n e n m m h n m h n m
nm
HereG0 (k,iε )=[iε −(εe/h(k)−µ)]−1arebareGreen’sFunctionsforelectron/holebandsrespectively. Weobservethat
e/h n n
therearetwotypesofcompetingprocesses: theonewith+C leadstoK >0andresultsinsuppressionofT whilethesecond
1 1 c
onewith−C correspondstoK <0andthereforeenhancesT .
2 2 c
Similarly,forthesecondcasedescribedbytheLagrangian(17)characterizedbymagneticfluctuationbrokenU(1)×U(1)
wegetthefollowingeffectiveLagrangian
L∆(3) = (cid:90) d(cid:126)x(cid:26)(cid:0) ∆¯e ∆¯h (cid:1)(cid:18) L˜Ke−1 L˜K−21 (cid:19)(cid:18) ∆∆e (cid:19)+A˜(cid:0)|∆e|4+|∆h|4(cid:1)(cid:27), (24)
2 h h
wherethediagonal∆ −∆ and∆ −∆ inversefluctuatorsL˜−1+K andL˜−1+K aregivenbythediagramsFig. 5(a,b)
e e h h e 1e h 1h
whiletheoff-diagonal∆ −∆ couplingK isdefinedbythediagramsFig. 5(c,d). Thecorrespondingequationsforthed-
e h 2
dimensionalcases(d=2,3)aregivenby:
(cid:90) d(cid:126)kd(cid:126)q
K = T2∑ G0(k,iε )G0(k,iε )G0(−k,−iε )D (q,iΩ )G0(k−q,iε −iΩ ), (25)
1e (2π)2d e n e n e n m m h n m
nm
(cid:90) d(cid:126)kd(cid:126)q
K = T2∑ G0(k,iε )G0(k,iε )G0(−k,−iε )D (q,iΩ )G0(k−q,iε −iΩ ), (26)
1h (2π)2d h n h n h n m m e n m
nm
(cid:90) d(cid:126)kd(cid:126)q
K = T2∑ G0(k,iε )G0(−k,−iε )D (q,iΩ )G0(k−q,iε −iΩ )G0(−k+q,−iε +iΩ ). (27)
2eh (2π)2d e n e n m m h n m h n m
nm
WhilethediagramsFig. 5(a,b)alwaysreducetheeffectivetemperatureofthesuperconductingtransition,thediagramsFig. 5
(c,d)liftthedegeneracybetweene-htransitiontemperatures. Asaresult,assumingthatK =K =K ,K =K wegetthe
1e 1h 1 2eh 2
fluctuationcorrectiontothecriticaltemperatureT :
c
T =T0·(1+ν−1(|K |−|K |)). (28)
c c 2 1
ToconstructtheeffectivefieldtheoryfortheinfluenceofsuperconductingfluctuationsontheSDWdynamics,weintegrate
outtheslowsuperconductingfluctuationsin(17)andfinallyobtaintheeffectiveLagrangianforparamagneticSDWfluctuations:
(cid:90)
L(cid:126)m = d(cid:126)x(cid:8)(cid:126)mD˜−1(cid:126)m+B˜(cid:126)m4(cid:9) (29)
eff m
wherefortwo-modecouplingtheoryD˜−1=D−1+K˜ +K˜ with
m m 1 2
(cid:90) d(cid:126)kd(cid:126)q
K˜ = T2∑ G0(k,iε )G0(k,iε )G0(k,iε )L (q,iΩ )G0(q−k,iΩ −iε ), (30)
1 (2π)2d e n e n h n ± m e m n
nm
1 (cid:90) d(cid:126)kd(cid:126)q
K˜ = − T2∑ G0(k,iε )G0(−k,−iε )L (q,iΩ )G0(k−q,iε −iΩ )G0(−k+q,−iε +iΩ ), (31)
2 2 (2π)2d e n e n ± m h n m h n m
nm
and
T =T0·(1−ν−1(K˜ +K˜ )) (32)
s s 1 2
whileforthree-modecouplingtheoryD˜−1=D−1+K˜ +K˜ with
m m 1e 1h
(cid:90) d(cid:126)kd(cid:126)q
K˜ = T2∑ G0(k,iε )G0(k,iε )G0(k,iε )L (q,iΩ )G0(q−k,iΩ −iε ), (33)
1e (2π)2d e n e n h n e m e m n
nm
(cid:90) d(cid:126)kd(cid:126)q
K˜ = T2∑ G0(k,iε )G0(k,iε )G0(k,iε )L (q,iΩ )G0(q−k,iΩ −iε ). (34)
1h (2π)2d h n h n e n h m h m n
nm
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∆e ∆¯e∆h ∆¯h ∆e ∆¯h ∆h ∆¯e
!"# $"# %"# &"#
m! m! m! m!
’"# ("#
Figure5. (Coloronline)SDWfluctuationcorrectionsto: e-andh-superconductingfluctuators(a,b),cross-couplede-hterms
(c,d). SCfluctuationcorrectionstotheSDWfluctuators(e,f).
ThediagramsdefiningthefluctuationcorrectionstoT areshownonFig. 5(e,f).
s
Computingthecorrectingtermsforthethree-modecouplingtheoryweobtainthatthefluctuationcorrectiontotheSDW
transitiontemperaturealwaysleadstoitsreduction:
T =T0·(1−2ν−1|K˜ |). (35)
s s 1
ExactlyatthetetracriticalpointT0=T0=T ,theconcentrationc=c ,D−1=L−1=L−1. Duetotheassumedparticle-hole
c s 0 0 m e h
symmetrywegetK˜ =K . Finally,combining(28)and(35)weget
1 1
T −T =T ·ν−1·(|K |+|K |)>0, (36)
c s 0 1 2
andthereforethenewtetracriticalpointcorrespondstolowervaluesoftheconcentrationc<c (seeFig. 1).
0
Letusillustrate,asanexample,thecalculationofthefluctuationcorrectionsdiagramsK containingthesuperconducting
1,2
fluctuatorL (q,Ω)(theevaluationofthediagramscontainingtheSDWfluctuatorcanbeperformedinasimilarfashion). The
e/h
evaluationoftheMatsubarasumsandintegralscanbedoneinseveralsteps:
i)First,themaincontributiontothesumoverΩ isgivenbythem=0term. Thereforewecanreplace
m
T∑L (q,iΩ )G0(q−k,iΩ −iε )→T L (q,0)G0(q−k,−iε ),
e m e m n 0 e e n
m
T∑L (q,iΩ )G0(q−k,iΩ −iε )→T L (q,0)G0(q−k,−iε ). (37)
h m h m n 0 h h n
m
√
ii)Second,wenoticethattheintegraloverqisdeterminedbythesmallq: qξ ∼ τ (cid:28)1whereξ isthesuperconducting
s c s
coherencelength,ξ2 =(7ζ(3)v2)/(16dπ2T2)ford-spatialdimensionsd=2,3. Wecanthereforeneglecttheq-dependence
s,d F 0
inthee/hGreen’sfunctions:
G0 (q−k,−iε )→G0 (−k,−iε ).
e/h n e/h n
iii)Next,thesumoverfermionicMatsubarafrequencyε andsummationoverkinK andK isperformedinthesamewayas
n 1 2
explainedbelowintheSection”Ginzburg-Landauapproach”(seeSection”Methods”). Asaresultweobtain:
(cid:90) ddk (cid:90) ddk 7ζ(3)ν
T∑ [G(0)(−k,−iε )]·[G(0)(k,iε )]3=T∑ [G(0)(−k,−iε )]2·[G(0)(k,iε )]2→ (38)
(2π)d n n (2π)d n n 16π2T2
n n 0
Notice,thataftertheseapproximationsthereisnodifferencebetweenK andK .
1 2
iv)TheremainingintegralovermomentumqisperformedusingthestaticOrnstein-ZernikecorrelatorL (q,0)(see36)
e/h
∑L (q,0)→V (cid:90) ξs−,d1 qd−1dq ν−1 , (39)
q e/h d 0 (2π)d c∆q2+τc
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hereV denotestheunitvolume. Weseethatford=3theintegral(39)isconvergent,
d
1 (cid:112)
ν−1K ∼ ∼ Gi(3). (40)
1,2 νT ξ3
0 s,3
HereweintroducedtheGinzburgnumberGi(d) definedastherangeofreducedtemperaturesδT/T wherethefluctuation
c
correctionstothespecificheatarecomparabletoitsjumpatT . Ford=3theGinzburgnumberGi(3)∼(T /ε )4.
c 0 F
Ford=2theintegral(39)divergeslogarithmically:
1 (cid:16) (cid:17) (cid:16) (cid:17)
ν−1K ∼ ln c ξ−2/τ ∼Gi(2)ln c ξ−2/τ . (41)
1,2 νT ξ2 ∆ s,2 c ∆ s,2 c
0 s,2
v)Finallywereplaceτ in(41)byitslowbound-theGinzburgnumber(recall,thatbyitsdefinitionGi(d)(cid:28)τ (cid:28)1)and
c c
substituteGi(2)∼T /ε (see36wheresimilarcalculationshavebeenperformedforasingleU(1)-modetheorybymeansof
0 F
RenormalizationGrouptechnique).
PerformingsimilarcalculationsfortherenormalizationoftheSDWcriticaltemperatureT weobtain:
s
T −T (cid:16) (cid:17) T (cid:18)ε (cid:19)
2D : c s ∼−Gi(2)ln Gi(2) ∼ 0 ln F ,
T ε T
0 F 0
T −T (cid:112) (cid:18)T (cid:19)2
3D : c s ∼ Gi(3)∼ 0 . (42)
T ε
0 F
WeemphasizethattheconstantsK andK characterizingthefluctuationalshiftofthecriticaltemperaturearefoundtodepend
1 2
ontheGinzburgnumberGi(d)andthedimensionalityofthesystemd only.
Eqs. (42)representthecentralresultofthepaper: fluctuationcorrectionsareresponsibleforthecompetitionbetween
the spin density wave and superconducting critical modes in the vicinity of the tetracritical point. We have obtained this
resultsbycalculatingthefluctuationcorrectionstotheGLfunctional,butthesameresultmaybeobtainedintheframework
ofrenormalizationgroupapproach. SincetheU(1)×U(1)symmetryisaprioribrokeninthesuperconductingsectorofthe
model,theinfluenceofthemagneticfluctuationsontothesuperconductingtransitionistwo-fold:first,theintra-bandcorrections
tempttoreducethetransitiontemperature;second-theinter-bandcorrectionsdoexactlytheopposite-splitthetwotransition
temperaturesandeffectivelyfacilitatethesuperconductingtransition. Ontheotherhand,thecontributionofsuperconducting
fluctuationsintheSU(2)magneticsectorofthemodeldoesresultonlyinasuppressionoftheSDWtransition. Thecontribution
of”eigen”fluctuations,namelythesuperconducting-superconductingandmagnetic-magneticisalikeandthereforeitisdropped
offfromthedifferenceofthecriticaltemperatures. Theasymmetriccharacterofthefluctuationcorrectionsresultsintheshift
oftetracriticalpointtowardslowervaluesofthecarriersconcentration. Forthe”dirty”limitofthemulti-modecouplingtheory
theGinzburgnumbersshouldbeupdatedaccordingly.36
ThestrongrenormalizationsofT andT areexpectedinironbasedsuperconductors,whicharerathertwo-dimensionaland
c s
manyofthemhaverathersmallFermienergy. InthesematerialsthevalueoftheGinzburgnumbermaybequitelarge. For
instance,thecharacteristicvalueoftheFermienergyin1111and11compoundsis100meV,38–40whilethesuperconducting
criticaltemperatureisabout20-50K.WiththisnumberstheonefindslargeGinzburgnumberGi(2)∼T /ε ∼0.02−0.05and
c F
considerablerenormalizationsofthecriticaltemperatures(T −T)/T ∼0.1.
c s c
Discussion
Wehavedevelopedahigh-temperatureapproachtotheproblemofinterplaybetweenmagneticandsuperconductingorderingin
multi-bandsystems. Bothstaticanddynamical(fluctuationrelated)contributiontothemode-modecouplingarediscussed. It
isshownthatthefluctuationcorrectionsinthevicinityofthetetracriticalpointreducethemagnetictransitioninaccordance
withEqs. (32),(36). Onthesuperconductorsideofthephasediagramthesituationisdifferent. Theintrabandandinterband
contributionsofspinfluctuationstotheGinzburg-Landaufunctionalnearlycompensateeachother.
Theapproachthatwehaveintroducedismoregeneralthanforinstanceamean-fielddescriptionofcompetingphases4,5as
itaccountsfordynamicalfluctuationsandgoesbeyondthestaticscalingparadigm. Apartfromthis,theframeworkpresented
herehastheadvantagethatitcanbegeneralizedtoothermulti-moderegimesinastraightforwardmanner,whichcaninclude
e.g. thepresenceofanematicinstability,thecompetitionbetweens ands and/orsinglet/tripletpairings. Besides,itis
± ++
highlyappealingtodiscussinaframeworkofthisapproachtheinfluenceofconventionalmagneticdefects(transitionmetal
ionssubstitutingforFe)andweakmagneticdefectsproducedbyZnimpuritiesandAs-vacanciesonthesuperconducting
transitiontemperature. Thisworkisnowinprogress.
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