Pairing mechanism for high temperature superconductivity in the cuprates: what can we learn from the two-dimensional t-J model?

0.11 MB·English

by  Huan-Qiang Zhou

Pairingmechanism forhigh temperature superconductivity inthecuprates: whatcanwelearn
from the two-dimensional t− J model?
Huan-Qiang Zhou1
1Centre for Modern Physics and Department of Physics,
Chongqing University, Chongqing 400044, The People’s Republic of China
Morethantwentyyearshavepassedsincehightemperaturesuperconductivityinthecopperoxides(cuprates)
wasdiscoveredbyJ.G.BednorzandK.A.Mu¨llerin1986[1]. Althoughintensetheoreticalandexperimental
0 effortshavebeendevotedtotheinvestigationofthisfascinatingclassofmaterials,thepairingmechanismre-
1 sponsibleforunprecedented hightransitiontemperaturesT remainselusive. Theoretically, thedifficultylies
c
0
inthefactthatthisclassofmaterials,asdopedMott-Hubbardinsulators[2],involvestrongelectroniccorrela-
2
tions,whichrendersconventionaltheoreticalapproachesunreliable. Recentprogressinnumericalsimulations

n ofstronglycorrelatedelectronsystemsinthecontextoftensornetworkrepresentations[3,4]makesitpossible
a togetaccesstoinformationencodedintheground-statewavefunctionsofthetwo-dimensionalt−Jmodel-a
J minimalmodel,aswidelybelieved,tounderstandelectronicpropertiesofdopedMott-Hubbardinsulators[5–
9 8]. Inthisregard, anintriguingquestioniswhetheror notthetwo-dimensional t−J modelholdsthekeyto
1 understandinghightemperaturesuperconductivityinthecuprates.Asitturnsout,suchakeyliesinasupercon-
ductingstatewithmixedspin-singletd+s−waveandspin-triplet p (p )-wavesymmetriesinthepresenceof
x y
] ananti-ferromagneticbackground[9].Here,thed+s-wavecomponentinthespin-singletchannelbreaksU(1)
n
symmetryinthechargesector,whereasboththeanti-ferromagneticorderandthespin-tripletp (p )-wavecom-
o x y
ponentbreaksSU(2)symmetryinthespinsector. Therefore,fourgaplessGoldstonemodesoccur. However,
c
evenifweresorttotheKosterlitz-Thoulesstransition[10],onlythed+s-wavesuperconductingcomponentsur-
-
r vivesthermalfluctuations. ThisturnsthreegaplessGoldstonemodes,arisingfromSU(2)symmetrybreaking,
p
intotwo degenerate soft modes, withtwicethe spin-triplet p (p )-wave superconducting energy gap as their
u x y
characteristicenergyscale: oneisaspin-tripletmodeobservedasaspinresonancemodeininelasticneutron
s
. scattering,theotherisaspin-singletmodeobservedasaA1gpeakinelectronicRamanscattering.Thescenario
at allowsustopredictthatpairingisofd+s-wavesymmetry, withthetwodegenerate softmodesasthelong-
m soughtkeyingredientsindeterminingthetransitiontemperatureTc,thusofferingapossiblewaytoresolvethe
controversyregardingtheelusivemechanismforhightemperaturesuperconductivityinthecuprates.
-
d
n PACSnumbers:74.20.-z,74.20.Mn,74.20.Rp
o
c
[ Imagine if we would have been able to solve a model etersareindependentofsitesonthelattice.

system describing dopedMott-Hubbardinsulators on a two- Now let us switch on thermal fluctuations. Suppose we
1
dimensionalsquarelattice,whoseground-statewavefunction restrict ourselves to a strict two dimensional system. Then,
v
8 isasuperconductingstatewithmixedspin-singletd+s−wave eveniftheKosterlitz-Thoulesstransition[10]isinvoked,only
5 andspin-tripletp (p )−wavesymmetriesinthepresenceofan spin-singletd+s-wavesuperconductingcomponentsurvives
x y
3 anti-ferromagneticbackground,withtheorderparametersfor thermal fluctuations. However, the non-abelian SU(2) sym-
3
the s-wave, d-wave, and p (p )-wave superconductingcom- metry is not allowed to be broken at any finite tempera-
. x y
1 ponents, together with the anti-ferromagnetic order parame- ture [11, 12]. This immediately implies that the Goldstone
0 ter, shown in Fig. 1, in a properdoping range. Note that ∆ modes arising from the spontaneous symmetry breaking of
0 d
and ∆ are, respectively, the spin-singlet d-wave and s-wave SU(2) in the spin sector have to be turned into degenerate
1 s
: superconducting energy gaps, whereas ∆p is the spin-triplet softmodes,withtwicethespin-tripletpx(py)-wavesupercon-
v
p (p )-wave superconducting energy gap and N is the anti- ductingenergygapastheircharacteristicenergyscale: oneis
i x y
X ferromagneticNe´elorderparameter. A fewpeculiarfeatures aspin-tripletmodeassociatedwiththeanti-ferromagneticor-
r of this state are: (i) Both the 90 degree (four-fold) rotation der,withthemomentumtransfer(π,π),andtheotherisaspin-
a symmetryandthetranslationsymmetryunderone-siteshifts singletmodeassociatedwiththespin-triplet p (p )-wavesu-
x y
are spontaneously broken on the square lattice. (ii) Spin- perconductingcomponent,withthemomentumtransfer(0,0).
rotation symmetry SU(2) is spontaneously broken, due to Ontheotherhand,thereisnothingtopreventfromthebreak-
the simultaneousoccurrenceof boththe p (p )−wave super- ingofthediscretefour-foldrotationsymmetryonthesquare
x y
conductingcomponentandtheanti-ferromagneticorder. (iii) lattice. Actually,thisbrokensymmetrynotonlymanifestsit-
U(1)symmetryinthechargesectorisspontaneouslybroken, selfintheadmixtureofasmalls-wavecomponenttothedom-
due to pairing in both spin-singlet and spin-triplet channels. inantd-wavesuperconductingstate(seeFig.1,leftpanel),but
Here, we emphasize that the symmetry mixing of the spin- also protectsthe spin-singletsoftmodethat is unidirectional
singletandspin-tripletchannelsarisesfromthespin-rotation asitarisesfromthep (p )-wavesuperconductingcomponent.
x y
symmetrybreaking,simplybecausespinisnotagoodquan- Our argument leads to a scenario that, at any finite tem-
tumnumber. (iv)Allsuperconductingcomponentsarehomo- perature, the pairing is of d + s-wave symmetry, with two
geneous,inthesensethattheirsuperconductingorderparam- degeneratesoftmodesactingasthe keyingredientsin deter-
2
mining the transition temperature Tc. Actually, two distinct 2∆ 2∆
ewniethrgtyhesccaolensv2en∆t∗ioannadlEsurepsearcreonindvuocltvoerds:,i2n∆a∗marairskeesdfrcoomntrtahset 2∆ 2∆ds ∆∆/sd 2∆ 2∆p*
anti-ferromagneticNe´elorderparameterN, whichisrespon- δ
sibleforpairing,withitscouplingstrengthdecreasingalmost
linearly with doping, whereas E = 2∆ , which is respon-
res p
sible for condensation. Therefore, E must scale with the δ δ
res
superconductingtransition temperature T , i.e., E ∼ k T ,
c res B c
withk beingtheBoltzmannconstant[seeFig.1,rightpanel]. FIG. 1: (color online) The doping dependence of the order pa-
B
Similarly, 2∆∗ scales as 2∆∗ ∼ k T∗, with T∗ being the so- rameters for a model system describing doped Mott-Hubbard in-
B
calledpseudogaptemperature[13,14]. Inaddition,onemay sulators on a two-dimensional square lattice, whose ground-state
expect that E < 2∆ , simply due to the fact that the pre- wave function is a superconducting state with mixed spin-singlet
res d d+s−waveandspin-tripletp (p )−wavesymmetriesinthepresence
dominant d-wave superconductingcomponentsurvives ther- x y
ofananti-ferromagneticbackground, withtheorderparametersfor
mal fluctuations. Considering that both the superconducting
the s-wave,d-wave,and p (p )-wavesuperconductingcomponents,
gap∆d andthetransitiontemperatureTc characterizethesu- togetherwiththeanti-ferroxmaygneticorderparameterN. Leftpanel:
perconductivity,theyshouldtrackeachotherintheentiredop- Thesuperconductinggaps∆ and∆ forthespin-singletd-waveand
d s
ingrange, implying∆d ∼ kBTc. Infact, forthe t− J model, s-wavecomponentsasafunctionofdopingδ,withtheirratio∆s/∆d
aolulorwsismuuslattoioenstiinmdaicteateasutnhiavteErsraesl c≈oe1ffi.2c5i∆endt[κ9].≈T5h.3is7inintuthrne sahnodwEnreisn=th2e∆inpsfeot.rRthieghatnptia-nfeerlr:oTmhaegenneetircgyorsdcearleasn2d∆th∗e∼spkBinT-t∗ri∼plNet
p (p )−wavesuperconductingorderasafunctionofdopingδ,with
scalingrelation: E =κk T . x y
res B c Nbeingtheanti-ferromagneticorder,andk theBoltzmannconstant.
B
Notethatthetwodistinctenergyscalesintheunderdoped Acrossover fromtheBose-Einsteincondensation (BEC)regimeto
regimearesplitofffromonesingleenergyscaleinthe(heav- theBardeen-Cooper-Schrieffer (BCS)regimeoccurs, whenthetwo
ily) overdoped regime. This naturally results in a crossover energy scales merge into one single energy scale in the (heavily)
from the Bose-Einstein condensation (BEC) regime to the overdoped regime. Note that, in general, Eres < 2∆d. Indeed,
Bardeen-Cooper-Schrieffer (BCS) regime, as conjectured in Eres ≈1.25∆d,aspredictedfromthetwo-dimensionalt−Jmodel.In
addition,E scaleswiththesuperconductingtransitiontemperature
Ref.[15],whichinturnisessentiallyequivalenttothephase res
T : E ∼k T
fluctuation picture proposed by Emery and Kivelson [16]. c res B c
However, there is an importantdifference: the superconduc-
tivityweakensintheheavilyunderdopedregime,notonlybe-
causeofthelossofphasecoherence,butalsobecauseofthe (heavily)overdopedregimeasaconsequenceoftheevolution
decreaseofthesuperconductinggap∆ withunderdoping. of the Fermiarcs in the underdopedregimeto a large Fermi
d
surface in the (heavily) overdopedregime [30, 31]. We em-
Nowafundamentalquestioniswhetherornotsucha sce-
phasizethatthepseudogapneartheantinodalregiondoesnot
narioisreallyrelevanttothehighT problem. Thisbringsus
c
characterize a precursor to the superconducting state, in the
tothephenomenologyofthehightemperaturecupratesuper-
conductors. sensethatthepseudogapsmoothlyevolvesintothesupercon-
First, let us focus on the two distinct energy scales 2∆∗ ducting gap at Tc [13, 30, 31]. Instead, it coexists with the
superconductinggap of the d + s-wave symmetry in the su-
and E . Physically, the two distinct energy scales mea-
res
perconductingstate. More likely, a precursorpairing occurs
sure, respectively, the pairing strength and the coherence of
thesuperfluidcondensate. Thisnaturallyleadsto twodiffer- inthenodalregion[32],withitsonsettemperaturelowerthan
T∗, but above T , which may be identified with the Nernst
entphases: oneischaracterizedbyincoherentpairing,which c
regime[33, 34]. As observed, the superconductinggapnear
may be identified with the pseudogapphase; the other is as-
the nodes scales as k T [29]. This makes a strong case for
sociated with the emergenceof a coherentcondensateof su- B c
our argument, if one takes into account the smallness of the
perconductingpairs,whichmaybeidentifiedwiththesuper-
conductingphaseofd+ s-wave symmetry[see Fig.2]. Evi- s-wave superconductinggap. On the other hand, ample evi-
dencehasbeenaccumulated,overtheyears,fortheuniversal
denceforthetwodistinctenergyscaleswasreportedinangle-
scalingrelationE = κ k T ,validforbothsoftmodes,i.e.,
resolved photoemission spectra [17–20], electronic Raman res B c
thespin-tripletresonancemodeininelasticneutronscattering
spectra [21–24], scanning tunneling microscopy [25], c-axis
experiments[35–41]andthespin-singletmodeobservedasa
conductivity[26],Andreevreflection[27],magneticpenetra-
A peakinelectronicRamanscattering[21,42–46],respec-
tiondepth[28],andotherprobes(forareview,see,Ref.[29]. 1g
tively. The experimentally determined κ is around 6, quite
Actually,thesestudiesindicatethatthegapneartheantinodal
closetoourtheoreticalestimate. Thispresentsapossibleres-
region, which is identified as the pseudogap, does not scale
olutionto the mysterious A problem[42]: the A modeis
with T in the underdopedregime, whereas the gap near the 1g 1g
c
a charge collective mode as a bound state of (quasiparticle)
nodal region may be identified as the superconductingorder
parameterinthecuprates[17–23]. Thisidentificationoffersa singletpairsoriginatingfromthefluctuating px(py)-wavesu-
perconductingorder.
naturalexplanationwhythe twodistinctenergyscalesin the
underdopedregimemergeintoonesingleenergyscaleinthe Second, is the pairing symmetry really of d + s-wave na-
3
if the stripe states (for reviews, see, e.g., Ref. [63–66]) were
nottouchedupon. Surprisingly,astripe-likestate,i.e.,astate
with charge and spin density wave order, coexisting with a
spin-tripletp (p )−wavesuperconductingstate,doesoccuras
x y
T agroundstateinthet−Jmodelfordopingsuptoδ≈0.18[9],
N
* with J/t = 0.4. Again, all symmetries, including the four-
T T
PG foldrotationandtranslationlatticesymmetry,SU(2)spinro-
tationandU(1)chargesymmetry,arespontaneouslybroken.
AF
Asimilarlineofreasoningyieldsthat,onlythechargedensity
T wave order survives thermal fluctuations, with the concomi-
C SC tantoccurrenceofthesoftmodes: theyarisefromtheSU(2)
symmetry breaking of the spin density wave order and the
δ spin-triplet p (p )−wave superconducting order, with twice
x y
the spin-triplet p (p )-wave superconducting energy gap as
x y
their characteristic energyscale. Althoughit remains uncer-
FIG.2: (coloronline)Aschematicphasediagramofthehole-doped tain whether or not such a commensurate stripe-like state is
high temperature cuprate superconductors plotted as a function of an artifact of our choice of the unit cell for the tensor net-
temperatureT andholedopingδ. Here, SC,PG,andAFstandfor work representation of quantum states, an important lesson
thesuperconducting, pseudogap, andanti-ferromagneticphases,re- wehavelearnedfromoursimulationisthat, thet− J model
spectively. T ,T∗,andT represent,respectively,thesuperconduct-
c N exhibitsastrongtendencytowardsastripestateintheunder-
ingtransitiontemperature,thepseudogaptemperature,andtheanti-
doped regime, consistent with the density matrix renormal-
ferromagnetic Ne´el temperature. In our scenario, the PG phase is
characterizedbyincoherentpreformedpairs,withapseudogapopen- izationgroup[67]foramorerealisticstripepatternatdoping
ing near the anti-nodal region, leaving the remnant gapless Fermi 1/8 and J/t = 0.35. From this we conclude that (i) static
arcsnear thenodes inthemomentum space. Inaddition, thefour- charge density wave order is compatible with the supercon-
foldrotationsymmetryandthetranslationsymmetryarebrokenon ductivity,soitspossibleroleisdeservedtobeexploredinthe
thesquarelattice. Superconducivityoccurs,whenlongrangephase
formationof the pseudogap; (ii) static spin density wave or-
coherencedevelopsatTc. derisdetrimentaltothed+s-wavesuperconductivity,because
nod+ s-wave superconductingcomponentcoexistswith the
charge and spin density order in the ground state; (iii) fluc-
ture? Many theoretical proposals, mainly based on the res- tuatingspindensitywaveorder,togetherwiththefluctuating
onating valence bond scenario [2], predicted a pure d-wave spin-triplet p (p )−wavesuperconductingorder,equallywell
x y
superconductivityinthecuprates[6,47–49]. Althoughavari- account for the Bose-Einstein condensation in our scenario.
etyofexperimentshavedemonstratedapredominantd-wave Therefore,the fluctuatingstripe orderis intrinsic to many, if
gap [50–52], strong evidence points to an admixture of an notall,familiesofthehightemperaturecupratesuperconduc-
s-wave component to the dominant d-wave superconductiv- tors.
ity in muon spin rotation studies [53, 54], electronic Raman
Fourth,doesourpredictionaboutthespontaneousbreaking
measurements[55], angle-resolvedelectron tunnelingexper-
of the four-fold rotation symmetry and the translation sym-
iments [56], and neutron crystal-field spectroscopy experi-
metryunderone-siteshifts in thepseudogapphase represent
ments [57]. In addition, a universal scaling relation of the
aphysicalreality? Inourscenario,thePGphaseischaracter-
superfluiddensity,ρ (0),atabsolutezero,withtheproductof
s ized by incoherentpreformedpairs, which occur in the anti-
the dc conductivity σ (T ), measured at T , and the transi-
dc c c nodal regime in the momentum space, leaving the remnant
tion temperature T , indicates that a pure d-wave supercon-
c gaplessFermiarcsinthenodalregime. Inaddition,thefour-
ductivityisrealizedinthecuprates[58]. Thisscalingrelation
foldrotationsymmetryandthetranslationsymmetryarebro-
may be regardedas a modified form of the Uemura relation
ken on the square lattice. In the superconductingphase, the
between the superfluid density ρ (0) and the transition tem-
s four-fold rotation symmetry breaking manifests itself in the
peratureT [59,60],whichworksreasonablywellintheun-
c admixture of an s-wave component to the dominant d-wave
derdoped regime. However, a significant deviation from the
state. Inthepseudophase,thebrokensymmetryprotectsthe
scalingrelationwassubsequentlyobserved[61],withasalient
spin-singletsoft mode that is unidirectionalas it arises from
featurethatthedeviationincreaseswithdoping. Thisfeature
the p (p )-wave superconductingfluctuations, as well as the
x y
stronglysuggeststhatthediscrepancyshouldbeaccountedfor
fluctuatingspindensityorderandpossiblystatic chargeden-
by removing an extra contribution from the s-wave compo- sity order. One may expect that measurable physical effects
nentinthecontextofthed+s-wavepairingsymmetry. This
arise from the coupling of electrons with the unidirectional
issue has been addressed recently [62], thus supporting our
spin-singletsoftmode. Indeed,alargein-planeanisotropyof
scenario. theNernsteffectinYBa Cu O wasreportedthatsetsinex-
2 3 y
Third, any theory regarding the underlying mechanism of actlyatthepseudogaptemperatureT∗[68]. Thisbrokenfour-
high temperature superconductivity would not be complete, fold rotation symmetry was also detected by resistivity [69]
4
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