Table Of ContentSU(2)symmetryinarealisticspin-fermionmodelforcupratesuperconductors
T. Kloss1,2, X. Montiel1,2, C. Pépin1
1IPhT, L’Orme des Merisiers, CEA-Saclay, 91191 Gif-sur-Yvette, France and
2IIP,UniversidadeFederaldoRioGrandedoNorte,Av.OdilonGomesdeLima1722,59078-400Natal,Brazil
(Dated:May22,2015)
WeconsiderthePseudo-Gap(PG)stateofhigh-Tc superconductorsinformofacompositeorderparameter
fluctuating between 2p -charge ordering and superconducting (SC) pairing. In the limit of linear dispersion
F
and at the hotspots, both order parameters are related by a SU(2) symmetry and the eight-hotspot model of
Efetov et al. [Nat. Phys. 9, 442 (2013)] is recovered. In the general case however, curvature terms of the
5
dispersionwillbreakthissymmetryandthedegeneracybetweenbothstatesislifted.Takingthefullmomentum
1
dependenceoftheorderparameterintoaccount,wemeasurethestrengthofthisSU(2)symmetrybreakingover
0
thefullBrillouinzone. ForrealisticdispersionrelationsincludingcurvaturewefindgenericallythattheSU(2)
2
symmetry breaking is small and robust to the fermiology and that the symmetric situation is restored in the
y large paramagnon mass and coupling limit. Comparing the level splitting for different materials we propose
a ascenariothatcouldaccountforthecompetitionbetweenthePGandtheSCstatesinthephasediagramof
M
high-Tcsuperconductors.
2 PACSnumbers:74.40.Kb74.20.-z74.25.Dw74.72.Kf
2
] I. INTRODUCTION attempttosuchaunificationhasbeentheSO(5)theorywhich
n
relatesthed-waveSCstatetotheAForder30.Notlessfamous
o
c istheSU(2)symmetrywhichrelatestheSCd-waveorderto
Reflecting our rather poor understanding of the physics
- the π-flux phase of orbital currents1. In both cases, the key
r of cuprate superconductors, two kinds of theories are still
p question was to argue that the energy splitting between the
debating whether the final solution for this problem will
u two orders was small enough so that thermal effects would
be a “bottom-up” approach based on a strong coupling
s restore the symmetry above T and below T∗, which are the
. theory1–3 or rather a “top-down” approach, where symme- c
t SCandthePGcriticaltemperatures.Thesamequestionholds
a triesandproximity toaQuantumCritical Point(QCP)plays
here for the SU(2) symmetry relating d-wave and QDW or-
m a dominant role4–7. The recently proposed Eight Hot Spots
der. Although the EHS model in its linearized version ver-
- (EHS)modelisapromising“top-down”approachtocuprate
d superconductors8,9. ItreducestheFermisurfacetoonlyeight ifies the symmetry exactly, it is not clear if a more realistic
n Fermisurface,includingcurvatureandthewholebandstruc-
pointsontheanti-ferromagnetic(AF)zoneboundaryandtak-
o ture,willinducesmallorlargeenergysplitting.Moreover,the
ing long-range AF fluctuations between them into account.
c EHSmodelreliesonlong-rangeAFfluctuationswhichmedi-
[ When the dispersion is linearized at the hot spots, one ob-
ate the interactions, but the experimental observation points
serves surprisingly that an SU(2) symmetry relating the d-
2 outtoshort-rangeAFcorrelationswhichpotentiallywillgap
wave SC channel (Cooper pairing) to the d-wave bond or-
v a whole part of the Fermi surface, as depicted in Fig. 1. A
der, or Quadrupolar Density Wave (QDW), (charge channel)
4 theoryfor“hotregions”insteadof“hotspots”isthusneeded.
2 ispresent. Moreover,animposantpre-emptiveinstability(of
3 order of 0.6J, where J is the AF energy scale) in the form
Inthispaperweaddresscarefullyalltheseissuebyevaluat-
5 ofacompositeSU(2)orderparameteremerges,thathasbeen
ingtheSU(2)splittingonrealisticFermisurfaces,forthetwo
0 identifiedasagoodcandidateforthepseudo-gap(PG)stateof
distinctcomponentsofthecompositeorderparameter: thed-
1. thosecompounds9–12. Motivatedbyanimpressivesetofnew
waveχ-fieldinthechargesector(whichformstheQDWorder
0 experimentalresults13–22,thistheorypointsouttotheemerg-
intheEHSmodel)andthed-wave∆-fielddescribingtheSC
5 ing idea that charge order is most certainly a key player in
pairingsector. WefindthatthesplittingoftheSU(2)symme-
1
thephysicsofcupratesuperconductors,inadditiontoAFor-
: tryincreaseswiththemassoftheparamagnons,butdecreases
v der, d-wave SC state and the Mott insulator phase. Angle-
withthestrengthofthecouplingconstantbetweenAFfluctu-
i resolved photoemission spectroscopy (ARPES) experiments
X ationsandconductionelectrons. Thisopensawideregimeof
confirm as well the presence of modulations in the SC state
parameters where the splitting is minimal – of the order of a
ar forunderdopedBi220123,24andhasbeeninterpretedeitherin fewpercents–andwheretheSU(2)symmetryisexpectedto
termsofchargeorderorPairingDensityWave(PDW)inside
berecoveredthroughthermaleffectsinaregimeoftempera-
thePGphase25–28. Finally, weliketomentiontherecentin- tures T <T <T∗. Above the PG temperature T∗ all traces
c
terpretationofRamanresonancesbyacollectiveSU(2)mode
of the short-range charge and SC field have disappeared. Of
whichisthefirstexperimentalresultthatsupportstheideaof
course the SU(2) symmetry holds in the whole temperature
acompositePG29. range between T and T∗, hence subleading charge instabil-
c
TheideaofanemergingSU(2)symmetrybelongstoawide ities which occur below T∗ do have their SU(2) partners in
class of theories which explain the PG phase of the cuprates theformofPairingDensityWaves(PDW)31,32. Wealsostudy
throughthenotionofdegeneratesymmetrystatesbetweenthe the effects of the Fermi surface shape in breaking the SU(2)
d-waveSCorderandanotherpartner.Maybethemostfamous symmetry–whichisonlypreservedintheEHSmodelwitha
2
300
(cid:114) YBCO
(cid:54) YBCO
(cid:114) Bi2201
(cid:54) Bi2201
200 (cid:114) Bi2212
(cid:54)| (cid:54)(cid:114) HBig21221021
(cid:114)|, | (cid:54)(cid:114) eHl.g d1o2p0e1d
|100 (cid:54) el. doped
0
0.001 0.01 0.05 0.1
a) m
1
FIG.1. (Coloronline)SchematicFermisurfaceofhole-dopedsu-
YBCO
perconductorsinthefirstBrillouinzoneofasquarelattice.Theorder 0.8 Bi2201
parametersspatiallyextentoverhotregions,thatarecenteredaround (cid:54)| Bi2212
Hg1201
cthoeuhpolitnsgpoQtstpooospitpioonssedanrdegwiohnics.harecoupledbythe2pFandtheAFM (cid:54) | / |0.6 el. dope
(cid:60)
(cid:114) 0.4
|
0.2
0
0.001 0.01 0.05 0.1
b) m
FIG.3. (Coloronline)Panela): Variationofthemaximumvalue
ofthegapfunctions|χ|and|∆|asafunctionofthemassm. Note
thatinallmaterialsthe2pFpairingintermsof|χ|vanishesabruptly,
whereastheSCpairingintermsof|∆|approacheszeroasymptoti-
callywhentheparamagnonmassmisincreased.Panelb):Variation
of the level splitting as a function of the mass m. In both panels
FIG. 2. Schematic phase diagram of hole-doped cuprate super-
λ =44.
conductors. Theeffectivemassoftheparamagnonpropagatorm
eff
serves as as measure for the distance to the quantum critical point
QCP.IntheSCphasem ispositiveandvanishesattheQCPtobe-
eff
comenegativeintheAFMphase.Thebaremassm isdefinedfar Thefermionicfieldψ describestheelectronswhicharecou-
bare
away from criticality at some higher temperature, indicated by the pled via Lφ to spin waves described by the bosonic field φ.
dashedline. Theeffectivespin-wavepropagatorisD−1=γ|ω|+|q|2+m
q
wheremistheparamagnonmasswhichvanishesattheQCP
andγ aphenomenologicalcouplingconstant, whichweesti-
linearizeddispersion. Wefindthatthesplittingissmallaway matefromitsformintheEHSmodel9tobeoftheorder10−5.
from the points where χ and ∆ are maximal. In the physi- Fornotationalreasonswealsowriteq≡(iω,q). Neglecting
calsituationofalargeparamagnonmassandalsostrongcou- the spinwave interaction (u=0) one can formally integrate
pling,themaximumof χ movestowardsthezoneedgelead- outthebosonicdegreesoffreedom. Inthespinbosonmodel,
ing to bond order parallel to the x-y axes. All these findings thisgeneratesaneffectivespin-spininteractionoftheform
pointtotherealizationthat,whilebeingasecondaryinstabil-
itytoAFordering,chargeorderisakeyplayerinthephysics S =−∑J¯(cid:126)S (cid:126)S . (2)
int q q −q
ofthePGphaseofthecuprates. q
It is convenient to use a fermionic representation of the spin
operator(cid:126)Sandconsiderinthefollowingonlytheparamagnet-
II. MODEL icallyorderedphaseinz-directio´n,sothatJ¯=3/2J. Thepar-
tition function then writes Z = D[Ψ]exp(−S −S ) with
0 int
Westartfromthespin-fermionmodel8,9,31withLagrangian
L=L +L ,where
ψ φ
S =∑Ψ†G−1Ψ , (3a)
0 k 0,k k
Lψ =ψ∗(∂τ+εk+λφσ)ψ , (1a) k,σ
L = 1φD−1φ+u(cid:0)φ2(cid:1)2 . (1b) Sint =− ∑ Jqψk†,σψk+Q+q,σ¯ψk†(cid:48),σ¯ψk(cid:48)−q−Q,σ. (3b)
φ 2 2 k,k(cid:48),q,σ
3
800 withk¯ =k+QandthematrixMˆ is
(cid:114), m = 0.001
600 (cid:54)(cid:114)(cid:54),,, mmm === 000...000000515 Mˆk=(cid:18)mˆ†k mˆk(cid:19), mˆk=(cid:18)−−∆χ†k+kp −χ−∆kk(cid:19). (8)
(cid:54)| (cid:114)(cid:54),, mm == 00..0011 ThefermionsinEq.(6)cannowbeintegratedoutsothatthe
(cid:114)|, |400 (cid:114)(cid:54),, mm == 00..0055 partitionfunctionbecomes
| ˆ
(cid:104) 1 1 (cid:105)
200 Z= D[Mˆ]exp − ∑TrJ−1Mˆ Mˆ + ∑TrlogGˆ−1 ,
4 q k¯+q k 2 k
k,q k
(9)
0 40 80 120 with Gˆ−1=Gˆ−01−Mˆ. After functional differentiation of the
a) (cid:104) freeenergyF =−TlnZwithrespecttoMˆk weobtaintheMF
800 equationsinmatrixform
(cid:114) YBCO
(cid:54)(cid:114) BYiB2C20O1 Mˆk=∑Jk¯−k(cid:48)Gˆk(cid:48). (10)
600 (cid:54) Bi2201 k(cid:48)
(cid:114) Bi2212
(cid:54)| (cid:54)(cid:114) HBig21221021 The matrix equation can now be projected onto the different
(cid:114)|, |400 (cid:54)(cid:114) eHl.g d1o2p0e1d coordmepropnaeranmts.etWereswwihllicchoncasindenrohtebreenthoen-czaesreooaftttwheoscaommeppeotiinngt
| (cid:54) el. doped
in k space. Therefore, we consider the equation for ∆ with
200 χ =0andviceversa. Thegapequationsfollowas
0 40 80 120 ∆k=T ∑ Jk¯−k(cid:48)∆2 +ε∆2k(cid:48)+ω(cid:48)2, (11a)
ω(cid:48),k(cid:48) k(cid:48) k(cid:48)
b) (cid:104)
FIG.4. (Coloronline)Panela):Variationofthemaximumvalueof χk=−ℜTω∑(cid:48),k(cid:48)Jk¯−k(cid:48)(iω(cid:48)−εk(cid:48))(iωχ(cid:48)k−(cid:48) εk(cid:48)+p)−χk2(cid:48). (11b)
thegapfunctions|χ|and|∆|inYBCOasafunctionofthecoupling
λ fordifferentmassesm. Panelb):Variationofthemaximumvalue To solve these equations numerically, εk is parametrized
ofthegapfunctions|χ|and|∆|fordifferentcompoundsasafunction in tight-binding approximation with the following parame-
ofthecouplingλ forthemassm=0.5. ters: YBCO33 (parameter set tb2), Bi220134, Bi221235 and
Hg120136 and for electron doped cuprates37. The momen-
tumsumsinEq.(11)arethencarriedoutbydiscretizingthe
wherethebarepropagatoris k-spacebyrectangularandequidistantgrids. Tokeepthenu-
merical computations tractable we neglect the frequency de-
Gˆ0−k1=diag(iω−εk,iω+ε−k−p,iω−εk+p,iω+ε−k), (4) pendence of χ and ∆38. The Matsubara sums are then car-
ried out exactly in the limit T →0 and the momentum sums
and the spinor field Ψ = (ψ ,ψ† ,ψ ,ψ† )T. are performed over 200×200 points and over one Brillouin
k k,σ −k−p,σ¯ k+p,σ −k,σ¯
Furthermore, J−1 = 4D−1/3λ2, σ ∈ {↑,↓} labels the spin, Zone (BZ). Moreover, note that the 2pF vector which con-
q q
Q=(π,π)T istheAFMorderingvectorandpstandsforthe nectstwoopposedFSpointsat±pF dependsontheexternal
momentumkinEq.(11)andisonlyproperlydefinedonthe
2p vector, asdepictedinFig.1. Notethatthechemicalpo-
F FS.Sinceweexpectthatthemaincontributiontothemomen-
tential µ is implicitly subtracted from the dispersion ε . We
k tumsuminEq.(11)comeshoweverfromthehot-spotregion,
selecttheSCandthe2p channelbyintroducingthetwoorder
F we make the approximation to take the 2p vector constant
parameters F
and take the 2p from the hotspot for arbitrary points in the
F
firstBZ,asdepictedinFig.1. Throughoutthisarticle,weuse
∆ =(cid:104)ψ† ψ† (cid:105), χ =(cid:104)ψ† ψ (cid:105). (5)
k k,σ −k,σ¯ k k,σ k+p,σ 10%holefilling(respectively10%electronfillingintheelec-
tron doped case) and the bandgap is 104K. To evaluate the
TheinteractionS isnowdecoupledbymeansofaHubbard-
1
strength of the SU(2) symmetry, we study the level splitting
Stratonovichtransformation. Thepartitionfunctionbecomes
|χ−∆|/∆. Thisparameteraffordthestudyoftherelativeam-
(uptoanormalizationfactor)
plitudebetweentheQDWandSCorderparameter, χ and∆.
ˆ
ItvanishesforaperfectSU(2)symmetryandbecomescloser
Z= D[Ψ]D[∆,χ]exp[−S −S ]. (6)
0 1,eff to one for a complete SU(2) symmetry breaking. From the
Lagragian we find that φλ has dimension of an energy and
Theeffectiveinteractionis φ ∼m−1/2. To estimate the coupling strength, we evaluate
an effective energy via E =φλ ∼m−1/2λ and away from
(cid:104) (cid:105) eff
S1,eff = ∑ Jq−1χk†χk¯+q+Jq−1∆†k∆k¯+q −∑Ψ†kMˆkΨk, (7) criticalityE0∼m0−1/2λ wherewetakem0=1asbaremass,
k,q,σ k,σ compareFig.2.
4
0.4 0.4 0.6 0.4
0.4
Π0 0.2 Π0 0.2 Π0 Π0 0.2
0.2
0
0Π 0Π 0Π 0Π
FIG.5. (Coloronline)Gapfunctions|χ|,|∆|and|χ−∆|/|∆max|(fromuptodown)fordifferentmaterialsinthefirstBZ.Thecompounds
are(fromlefttoright):YBCO,Bi2201,Bi2212andHg1201. Thefigurescorrespondtothesmallmassandsmallcouplinglimit(λ =40and
massm=10−3;notethatthebarecouplingissmallw.r.t.therenormalizedoneE (cid:39)40KandE =1265K)thatismostunfavorableforthe
0 eff
SU(2)symmetry.
III. RESULTSANDDISCUSSION becomesoforderoneareclearlyseen. Withinthenonlinear
σ-modelassociatedtothepresenttheory9, theSU(2)regime
isasignatureofthePGofthesystem,whiletheenergysplit-
A. Massandcouplingdependenceonlevelsplitting
tingofthetwolevelsisassociatedtothesuperconductingT .
c
WeseethatFig.3a)mimicsthegenericphasediagramofthe
cuprateswherethePGlineT∗abruptlyplungesinsidetheSC
Inordertotesttheeffectofthecurvatureontheleveldegen-
domeatsomevalueofoxygendoping.
eracy,wehaveplottedinFig.3a)thevariationofthemaxima
of χ and∆withtheparamagnonmassmforafixedvalueof
thecouplingconstantλ. Weobserveasimilaritybetweenthe
various compounds that we have tested. In a wide range at AlthoughitisveryencouragingtoseethattheSU(2)regime
low value of the mass, the SU(2) degeneracy between χ and has a non-zero probability to exist, one can wonder whether
∆ is verified within a few percents. The existence of such a theparamagnonmassinholedopedcupratesuperconductors
regime is an indication that a PG driven by SU(2) symme- is small, since typically the AF correlation length is of few
tryispossibleincupratesuperconductors. Asthemassisin- lattice constants40. The issue is addressed in Fig. 4a), where
creased we progressively lose the level degeneracy with the the values of χ and ∆ are shown for a fixed mass as a func-
parameter χ abruptly dropping down while the paring ∆ is tionofthecouplingconstantλ. Hereagainagenericpattern
asymptotically going down to zero when the mass increases. emerges. Forsmallλ theSU(2)symmetryisbroken,butsur-
ItisinterestingtoseethattheSU(2)symmetryisweakforthe prisingly,aboveacertainthresholdofλ,theSU(2)symmetry
electrondopedandHg1201compoundwhichexperimentally isalmostcompletelyrestored. AsseeninFig.4a),thebigger
show much weaker signs of charge order22,39. We also find themassis,thestrongerthecouplingconstantneedstobefor
that the compound Bi2212 behaves sightly different than the the symmetry to be restored. Figure 4b) shows that this be-
othercompoundsinFig.3and4,althoughitisnotclearatthe haviorisquitegeneralamongthedifferentcompounds. Note
currentstagewherethisdeviationcomesfrom. InFig.3b)the however that the electron -doped compound is less sensitive
levelsplittingisdirectlyshownforallthecompoundsandthe totheeffectofincreasingthecouplingconstant,comparedto
two regimes, the one at low mass where the SU(2) symme- the other hole-doped ones, for which the SU(2) symmetry is
try is obtained and the higher mass regime where |χ−∆|/∆ restoredforlargeenoughλ.
5
B. Spacialdependenceofthesplitting
The SU(2) symmetry is not only broken due to the curva-
ture of the Fermi surface at the hot spots, but it is typically
broken in the BZ away from the eight hot spots. Figure 5
shows the typical shape of |χ| and |∆| for four compounds
underinvestigationforthemostunfavorablecaseforthesym-
metry, thatisforsmallvaluesofthemassandcouplingcon-
stant. The level splitting is shown as a density plot in the
bottom. It is rather small almost everywhere in the BZ and 0.1
atthehotspotpositionswithmaximaoftheorderof20-40%
aroundthe“shadow”Fermisurface. Themainlearningfrom
Π0 0.05
theseplotsisthatthevariationsoftheFermisurfacegeometry
givesarathersmalldepartingfromtheSU(2)-degeneracyfor
0
a various range of compounds. In all cases, the SU(2) sym- 0Π
metryiswellrespectedatthehotspotpositions.
FIG. 6. (Color online) Generic picture of the gap functions |χ|,
In Fig. 6 we place ourselves in the strong coupling and |∆|and|χ−∆|/|∆max|inthefirstBZforhole-dopedcuprates,here
strong mass regime and plot the variation of |χ| , |∆| and explicitlyshownforYBCO.Thefigurescorrespondtothelargemass
|χ−∆|/∆. The level splitting is also shown in Fig. 6 and andlargecouplinglimit(λ =160andmassm=0.5, sothatE (cid:39)
0
foundtobemuchsmallerthanthepreviouscaseinFig.5,and 160KandEeff=226K)wheretheSU(2)symmetryiswellrespected.
hasdroppedtoanorderof5-10%41. Interestingly,thetypical
shapeof|χ|and|∆|intheBZhaschangedcomparedtoFig.
5, withmaximanowaroundthezoneedge. Thisisthejusti-
ficationthat“hotregions”insteadof“hotspots”isthecorrect
descriptionofholedopedcupratesuperconductorswithinthe
spin-fermion model. Note that since the maximum of |χ| is
now at the zone edge, the wave vector corresponding to the
associatedchargeorderisnowparalleltothex/yaxesofthe
system,insimilaritywiththefindingsofRef.[12].
0.4
Π0
C. Globaltrendsincupratesuperconductors 0.2
The interplay between mass and coupling allows us to re- 0Π
lateglobaltrendsinthephasediagramforcupratesupercon-
FIG. 7. (Color online) Generic picture of the gap functions |χ|,
ductors with the strength of the SU(2) symmetry breaking.
|∆|and|χ−∆|/|∆max|inthefirstBZforelectrondopedmaterials.
In the electron doped compounds the coupling between AF
The figures correspond to the small mass and small coupling limit
modesandconductionelectronsisbelievedtobeweakerthan (λ =40andmassm=10−3,sothatE (cid:39)40KandE =1265K)
0 eff
fortheholedopedcase. Fromtheaboveanalysiswefindthat thatismostunfavorablefortheSU(2)symmetry.
theSU(2)symmetryislessrespectedinthatcaseleadingtoa
smallerPGdomeandoverallsmallerSC,seeFig.7. Forthe
same reasons La compounds where the coupling is also be-
IV. CONCLUSION
lievedtobesmallbehavesimilarly. Ontheotherhand,hole-
doped cuprates like YBCO live in the large mass and large
Inconclusion,thispapergivesfirmgroundtotheintuition
couplingregime. Thisresultsinbroadgappedregionsinthe
thatthechargesectorisakeyplayerinthephysicsofcuprate
BZwherethesymmetryiswellrespectedsothatboththePG
superconductors. While the main instability is still the AF
andSCdomearelarge,asshowninFig.6.
ordering, the d-wave bond order relates to the d-wave pair-
Finally, let us mention that a major effect of a magnetic ing through an SU(2) symmetry. We have shown that there
field is to invert the order of the level splitting between the exists a wide range of parameters where the SU(2) degen-
CDWandtheSCcomponents10,11. Thiswillfavorthecharge eracy is fulfilled, which gives a natural explanation for the
ordercomparedtotheSCpairing. WebelievethatUmklapp large PG regime observed in certain compounds. We argue
scatteringcanhaveasimilareffecttoinvertthelevelordering, that compounds like electron doped cuprates or the La com-
butleavedetailedinvestigationforfurtherstudies. pounds are outside the regime of SU(2) degeneracy, and the
6
morepronouncedenergysplittingisthereasonfortheweaker Carvalho. WethanktheKITP,SantaBarbaraandtheIIP,Na-
PGregime. tal for hospitality during the elaboration of this work. This
workwassupportedbyLabExPALM(ANR-10-LABX-0039-
PALM),oftheANRprojectUNESCOSANR-14-CE05-0007,
V. ACKNOWLEDGMENTS as well as the grant Ph743-12 of the COFECUB which en-
abledfrequentvisitstotheIIP,Natal. Numericalcalculations
WeacknowledgediscussionswithA.Chubukov, S.Kivel- werecarriedoutwiththeaidoftheComputerSystemofHigh
son, H. Alloul, P. Bourges, Y. Sidis, A. Sacuto, and V. S. de PerformanceoftheIIPandUFRN,Natal,Brazil.
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verysimilarandthereforenotshowninFig.6.
