Spin-triplet s-wave local pairing induced by Hund's rule coupling

0.28 MB·English

by  J. E. Han

Spin-triplet s-wave local pairing induced by Hund’s rule coupling
J. E. Han
Department of Physics, The Pennsylvania State University, University Park, PA 16802-6300, USA
Department of Physics, State University of New York at Buffalo, Buffalo, NY 14260, USA
(Dated: February 2, 2008)
WeshowwithinthedynamicalmeanfieldtheorythatlocalmultipletinteractionssuchasHund’s
rulecouplingproducelocalpairingsuperconductivityinthestronglycorrelatedregime. Spin-triplet
4
superconductivity driven by the Hund’s rule coupling emerges from the pairing mediated by local
0
fluctuationsin pairexchange. Incontrast totheconventionalspin-triplet theories, thelocal orbital
0
degrees of freedom has theanti-symmetric part of the exchangesymmetry, leaving thespatial part
2
as fully gapped and symmetric s-wave.

n
a PACSnumbers: 74.20.-z,74.20.Rp,74.70.Wz
J

7
I. INTRODUCTION localorbitalcorrelationsarecrucialforthesuperconduc-

] tivity and, in particular, predict the Hund’s rule (HR)
l
e For the last two decades, discovery of unconventional induced superconductivity. We discuss the local pairing
- asmediatedbyself-generatedfluctuationswithintheon-
r superconductors have fueled intensive research for novel
t site interactions and show that the Coulomb interaction
s pairing mechanisms. Of particular interest have been
. the cuprate1 andheavy-fermion2 systemswherethe very and multiplet interaction work in fundamentally differ-
t
a existence of superconductivity is quite surprising due to entwaythandescribedinperturbativetheories. TheHR
m superconductivity has the spin-triplet pairing state. In
the strong Coulomb repulsion within the framework of
- the conventional Eliashberg-Migdal theory3. Recently contrast to the well-known non-s-wave triplet pairing15,
d theanti-symmetricpartoftheexchangesymmetryofthe
discovered class of superconductors close to the ferro-
n
o magneticphasesuchasSr2RuO44,UGe25demandbetter pairingwavefunctionistakenupbythelocalorbitalvari-
ables, leaving the spatial part as the symmetric s-wave.
c understandingbetweensymmetryandsuperconductivity
[ which goes beyond the s-wave BCS theory. Wewilldiscusspossibleconsequencesofthisnewpairing

One of the most puzzling aspects of the supercon- structure.
1
This paper is organized as follows. In section II, we
v ductivity in the strongly correlated regime is the role
define the problem in models of multiplet interactions
4 of the repulsive Coulomb interaction. Common belief
0 is that the pairing mechanism is driven or assisted by for Hund’s rule and Jahn-Teller el-ph couplings. In the
1 the Coulomb repulsion to an attractive interaction be- following section III, the pairing mechanismis explained
1 in a perturbation theory of a simplified dynamical mean
tweenelectronswhichresultsinspatialortemporalstruc-
0 field theory (DMFT). Results from full DMFT calcula-
tures in the superconducting wavefunction to avoid di-
4
tionsarepresentedinthesectionIVandarediscussedin
rect Coulomb repulsion. Even some conventionals-wave
0
detail. In Appendices Aand B, we describe the calcula-
/ phonon-mediated superconductors such as alkali-doped
t tions for the localenergy levels and perturbation theory.
a fullereneshavealsoshownextraordinarilyhightransition
m temperatures6despitethestrongCoulombrepulsionnear Finally, a new Hubbard-Stratonovichdecoupling scheme
- the Mott transition7,8. for the Hund’s rule coupling which bypasses the sign-
d Even before the onset of superconductivity, direct problem is introduced in Appendix C.
n
Coulomb interaction is heavily renormalized away by
o
electronsmovinginacorrelatedmannertoformanarrow
c II. MODELS
: resonance near chemical potential. The superconductiv-
v
ityinsuchregimeshouldbeunderstoodintermsofthose
Xi correlatedelectronicbasis,typicallycalledquasi-particles Weconsidercasesofdoublydegeneratebandelectrons
at half-filling which are locally coupled via Hund’s rule
(QP).Theproblemofmutualrenormalizationeffectsbe-
r
a tween electron-electron and electron-phonon (el-ph) has (HR) and Jahn-Teller (JT) phonons. We model the
multi-band electronic system with the on-site Coulomb
received a great deal of interest resulting in different in-
terpretations, ranging from the conventional Coulomb repulsion of strength U as
pseudopotential3,9, field-theoretic approaches10,11 to nu-
merical analysis12. Recently, it has been pointed out13 H = − tijd†imσdjmσ −µ Ni
thatthesymmetryoftheel-phinteractioncriticallyinflu- iXjmσ Xi
ences the pairing interaction where the two interactions U
+ N (N 1)+ H (i), (1)
i i mult
are so interwoven that the effective pairing interaction 2 −
i i
cannot be reduced into a form similar to the McMillan X X
formula14. where d†imσ(dimσ) is the electron creation (annihilation)
Weshowhereonthebasisofgenerichamiltoniansthat operator acting on site i, orbital m(= 1,2), spin σ, µ is
2
the chemical potential and N = n . The hop- 100
i mσ imσ
pingintegralt ischosentogiveasemi-ellipticaldensity
ij Exact
of states with the half-bandwidthPW/2 = 116. The local 10-2 -30t4/KU*2
multiplet interactionis taken as H =H or H 13, q
mult HR JT 10-4
-Ep10-6
H = K n n n n 2 n n
HR Xm m↑ m↓−mX6=m′ m↑ m′↓−σ,mX<m′mσ m′σ 10-8 Ep000...000021 4tq2/U ~ K
+ K d†m↑d†m↓dm′↓dm′↑+d†m↑d†m′↓dm↓dm′↑ ,(2) 10-10 --00..0021
mX6=m′(cid:16) (cid:17) -0.030.0 0.1 0.2 0.3 0.4 0.5
1 10-12
HJT = 2 ϕ˙ν2+ωp2hϕ2ν 10-3 10-2 t 10-1 100
q
ν
gX(cid:0) (cid:1)
+√ωphXσ hϕ1(n1σ −n2σ)+ϕ2(d†1σd2σ+h.c.)i, (3) EFIqG..(51):withPatihrecHorurnedla’tsiornuleencoerugpylinEg.p Efopr istwaot-tsriatectimveodfoerl
a wide range of hopping integral to the quasi-particle bath,
where the site index i has been omitted for brevity. K tq. Coupling to quasi-particle bath (thick line) produced a
is the HR coupling constant and ϕν (ν = 1,2) are the good agreement with the perturbation theory, −30t4q/KU∗2
phonon fields with the bare frequency ω , and g the (U∗ =U+2K). Theoptimalinteractionisat4t2/U ∼K for
ph q
el-ph coupling constant. In the anti-adiabatic limit of themaximum pair attraction Ep ∝−K.
phonons (ω )17, H maps to the same form
ph JT
→ ∞
as H but with a negative (fictitious) K = 2∆
HR JT
(∆ ≡ g2/ωph). Therefore, the el-ph superconduct−ivity with c†mσ = Ns−1 kc†mkσ, Nc = mkσnmkσ and ǫq
is suppressed as the physical HR coupling is turned up. the QP energpy (ǫq ≈P0), where the nParrow QP band is
We will show that, as K increases further, superconduc- represented by a single energy level. Heff has the simi-
tivity re-emerges in a spin-triplet channel. lar construction as the exact diagonalization implemen-
tationoftheDMFT,wherethebathspectrumissampled
withafinite numberofenergylevels16. Representingthe
QPstateswithseveraldiscretelevelsoverasmallenergy
III. LOCAL PAIRING MECHANISM
range made little change. This procedure is expected to
be validaslongasthe electronsslowdownbythe strong
Wedescribemainideasbehindthelocalpairingbefore
Coulombinteractionsothatthelocalpictureismeaning-
presentingresultsfromthefulldynamicalmeanfieldthe-
ful and there is a QP reservoir which supplies particles
ory. In the strongly correlated regime, QP states within
at an infinitesimal energy cost.
the energy band [ zW,zW] (z the QP renormalization
− We solve the two-site problem Eq. (5) using the exact
factor,W non-interactingbandwidth)areusuallyrespon-
diagonalization method. We analyze the solution as a
sibleforlow-energymanybodyphenomena. Theeffectof
perturbation expansion in t 18. We estimate the pairing
U isimplicitintherenormalizationfactorzandthetrun- q
correlationbycomputingtheenergygainE forcreating
cated high energy states. In such limit, we simplify the p
two extra electrons on a half-filled impurity site,
DMFT by singling out the QP band and ignore incoher-
ent high energy excitations as
E =[E(N +2) E(N )] 2[E(N +1) E(N )], (6)
p d d d d
− − −
t
Heff =Hint+mkσ(cid:20)ǫqknmkσ+√Nqs(c†mkσdmσ+h.c.)(cid:21), twrhoenrseaEn(dn)Nids tihsetghreouonrbditsatlatdeeegneenregryacwyit(hNnd t=ota2l)e.lecIt-
X
(4) is crucial to have multiple electrons per site for the lo-
where c†mkσ(cmkσ) is creation (annihilation) of a QP cal multiplets to induce the local pairing. As shown in
state, ǫq is QP energy (absorbing chemical potential), Ref.13, the local pairing has a strong filling dependency.
k
t ( zW) is hopping integral on and off the impurity Numerical results shown for the HR problem are with
q
∼
and Ns is the number of sites. Contributions from the parameters of U =4 and K =0.1. For small tq in a HR
high energy states are correctly considered in the full system, Ep is negative (thick line inFig.1), i.e., itis en-
DMFT calculations. H is the interacting part of the ergeticallyfavorabletocreatemutuallyinteractingquasi-
int
originalhamiltonian. Note thatthe interactiononlyacts particles than to create two independent quasi-particles.
onthe impuritysite. ForsmallQPbandwidthswithzW The impurity groundstates arealwayspredominantly of
smaller than other energy scales, we further simplify the filling Nd(= 2) with the total spin S = 1 since any ex-
above effective hamiltonian to a two-site model, tra electrons created on the impurity are repelled by the
Coulomb interaction and are immediately dissolved into
H =H +ǫ N +t (c d +h.c.), (5) theQPbath. Thereforewefirstexaminetheatomiclimit
int mult q c q †mσ mσ
(t = 0). Without H the degeneracy of two-electron
mσ q mult
X
3
N=2 comes from Eq. (B9),
√3t
|ψ1Nd+2i = U q d†1 d†2 d†1 c†2 −d†2 c†1 c†1 c†2 (9)
∗ ↑ ↑ ↓ ↓ ↑ ↑ ↓ ↓
(cid:16)
2K 4 ∆ = 4g 2 /ω + d†1↑c†2↑d†1↓d†2↓−c†1↑c†2↑d†2↓c†1↓−(1↔2) |vaci,
ph (cid:17)
S=1 S=0 with the ionization energy U /2 = U/2+K. This in-
∗
termediate state has an ionized impurity at the excited
Hund’s rule JT el−ph coupling
energy U . With Eq. (B11) and E H = U /2, one
∗ 0 0 ∗
− −
gets the second order contribution to the energy
FIG. 2: Single site energy level scheme for two on-site elec-
tTrhoensd(eNgedn=era2c)ywoitfh6Hiusnldif’tsedrulbeyanthdeJmahunlt-Tipellelterincoteurpalcintigosn. √3tq 2 12t2q
E = ( U /2) 8= , (10)
Hmult. The ground states are spin-triplet (S = 1) and spin- 2 U∗ ! − ∗ · − U∗
singlet (S = 0) for Hund’s rule. The Jahn-Teller multiplets
haveinverted level structureof theHund’srulecoupling. wherethefigure8isfromthenumberoftermsinEq.(9).
The second order correction to the wavefunction from
1 4 Eq. (B10) becomes
m=1
m=2
2√3t2 1
2 3 ψNd+2 = q S ) s )+ S ) s ) S ) s ) ,
S=1 S=0 S=1 | 2 i KU | a ⊗| a | b ⊗| b − 2| c ⊗| c
intermediate state ∗ (cid:20) (cid:21)
(11)
where S ), S ) and S ) are spin-singlet excited states
FIG.3: OneofthecontributionsinE(Nd+2)forthepaircor- | a | b | c
ontheinteractingsiteasdefinedinAppendixA. TheQP
relation energy Ep in Eq. (20). Two extra electrons created
configurations s ), s ), s ) are defined with the same
ontheinteractingsitedissolveintothequasi-particlebathto a b c
| | |
avoid Coulomb repulsion, which produces the ground state symmetry as the impurity states.
configurations shown on the left and right panel. The two Oneoftheprocessesinthesecondordercontributionis
unperturbeddegenerategroundstatescoupleviaexchangeof depicted in Fig. 3. Typical(unperturbed) basis states in
the spin-triplet electron pairs in the 4-th order perturbation thegroundstatearespin-tripletsasshownontheleftand
of tq (from the left configuration to the right). In the inter- right side. The two states couple via the electron hop-
mediatestate,thespinsontheinteractingsiteareinalocally pingwhereasecond-orderintermediatestateisshownin
excited state (SeeFig. 2).
the middle. After two hoppings (eg. the processes 1 and
2 from the left configuration), the resulting intermedi-
ate state becomes a charge-neutral spin-singlet (S = 0),
configurations is 6. The HR coupling lifts the degener-
whichcorrespondto the term S ) s ) inEq.(11)and
a a
acy with the groundstates of spin-triplet (S =1) on the | ⊗|
alocaltransitionbetweenthemultipletsinFig.2. Other
impurity site with
charge-excited intermediate states have small contribu-
tionstothetransitionamplitude(oforderK/U smaller).
1
|Tm)=(cid:26)d†1↑d†2↑,√2(cid:16)d†1↑d†2↓−d†2↑d†1↓(cid:17),d†1↓d†2↓(cid:27)|vac()7,) ergFyrobmecoEmq.es(B12), the forth order correction to the en-
with m = +1,0,−1, respectively (see FIG. 2 and Ap- 2√3t2q 2 1 2
pendix A). The 1st excitation energy is 2K for the E = 2K 2K 4K
4
HR coupling. The local energetics is described in Ap- KU∗ ! "− − − (cid:18)2(cid:19) #
pendix A. Upon adding two electrons to the interacting 2
12t2 √3t 60t4
site,theextrachargesareimmediatelypushedtotheQP q8 q q , (12)
reservoirtoavoidtheCoulombrepulsion. Throughinter- − U∗ U∗ ! ≈−KU∗2
sitehopping,thegroundstatetripletdegeneracyislifted.
forK U . ThegroundstateenergyE(N +2)becomes
The resulting leading order ground state wavefunction ∗ d
≪
has the most symmetric form (U K t >0),
q
≫ ≫
12t2 60t4
1 E(N +2)=U q q . (13)
|ψ0Nd+2i= √3{|T1)⊗|t¯1)+|T0)⊗|t0)+|T¯1)⊗|t1)}, d − U∗ − KU∗2
(8) The first term is the (unperturbed) atomic-limit. The
where the second ket-vectors for the QP states in the last term contributes to the pairing interaction. The
direct products have similar definitions as Eq. (7). We denominator in Eq. (13) comes from two ionized states
analyzethe perturbedgroundstatewavefunctionandits (with charging energy U after the hoppings 1 and 3 in
∗
energy in terms of the perturbation theory developed in Fig.3for example)andone HR-excitedstate (with exci-
Appendix B. Thefirstorderperturbedwavefunctionbe- tation energy 2K after the hopping 2).
∼
4
Ground state for one extra (spin-up) electron can be U ∆ t , which retains the similar form as Eq. (20)
q
≫ ≫
obtained similarly. For hopping integrals smaller than for the HR coupling. Leading order wavefunctions and
otherenergyscales,the interactingsite is predominantly theirperturbedenergiesarewrittendowninAppendixB.
of filling 2 and of spin-triplet. Therefore the ground On the other hand, for a pure Hubbard model without
state consists of T ) c ) and T ) c ) (c ) themultipletterms(H =0),E =344t4/U3 >0sug-
c vac)). Dueto| t0he⊗ho|pmp↑inginte|ra1ct⊗ion|,mst′a↓tes|omftσot≡al gesting no supercondumctuilvtity, whicph is alsoqsupportedby
†mσ|
spins S = 3/2 do not appear in the ground state. An fullDMFTcalculations16. WithH =0,theelectrons
z mult
±
explicit calculation gives the unperturbed groundstate, only interact with charge fluctuations making the local
internal fluctuations irrelevant. The above perturbation
1
ψNd+1 = T ) c )+√2T ) c ) . (14) results are summarized in Table 1.
| 0 i √3 | 0 ⊗| 1↑ | 1 ⊗| 2↓
n o
K =∆=0 K >0 ∆>0
The larger coefficient for T ) c ) reflects more paths
1 2
for hopping transitions fo|r th⊗is|co↓nfiguration. Similar U =0 0 - −2∆
calculations as those with Nd+2 electrons give t2q/U ≪∆,K 344t4q/U3 −30t4q/KU2 −7t4q/∆U2
|ψ1Nd+1i= √26tUq∗ 3d†1↑d†2↑d†2↓+3d†1↑c†2↑c†2↓ T2-AbBanLdEmI:odTehlewpitahirHcuonrrde’lsatriuolne (eKner>gy0)foarnadntheffeeJcatihvne-T2-eslilteer
n el-ph (∆ > 0) coupling. For large U, both of the couplings
−2d†2 c†1 c†2 −d†2 c†2 c†1 +c†1 c†2 d†2 |vaci (1p5r)oduceattractive pairing energy. (∆=g2/ωph.)
↑ ↑ ↓ ↑ ↑ ↓ ↑ ↑ ↓
√3t2 So) c )
|ψ2Nd+1i= KUq∗ (cid:20)|Sa)⊗|c2↑)+|Sb)⊗|c1↑)− | c ⊗2| 1(↑116(cid:21))sAusggtehset,ptahieriCngoucloormreblaitnitoenraecnteiorngyanEdq.th(e20m)ualntidplTetabinle-
8t2 15t4
q q teraction (including the JT el-ph interaction) cannot be
E(N +1) = U . (17)
d − U − KU 2 separatedintoaformsimilartoλ µ asintheMcMillan
∗ ∗ ∗
−
formula14,wheretheCoulombinteractionrepresentedas
The state without extra electrons results in no energy
theCoulombpseudopotentialµ interactadditivelywith
gain in the 4-th order, ∗
themultipletinteraction(eg. el-phcoupling)represented
by the mass-renormalizationfactor λ.
2t
|ψNdi = 1+ U∗q nσ c†nσdnσ!|Tm)⊗|vac) (18)
X
4t2 IV. PAIRING INSTABILITY: DMFT RESULTS
q
E(N ) = U . (19)
d
− U
∗
Although the 2-site model is suggestive of the nature
Finally, the pairing correlation Eq. (6) becomes
of the pairing, we show the existence of superconduc-
E = 30t4/KU 2, (20) tivity from a lattice calculation. The lattice hamilto-
p − q ∗ nian Eq. (1) is solved within the DMFT which approx-
which is in excellent agreement with the exact diagonal- imates one-particle self-energies and vertex corrections
ization results in the t K limit. It is interesting to as momentum-independent and maps a lattice model to
q
≪
note that E becomes smaller for stronger K due to the an effective quantum impurity problem. We solve the
p
higher multiplet excitation energy. The exact diagonal- impurity problem using quantum Monte Carlo (QMC)
izationresultsshowmaximumattractivepaircorrelation technique without making any physical approximations
for 4t2/U K, where the effective hopping exchange other than discretizing the imaginary time. It has been
q ∼
energy is comparable to the internal excitation energy. known that the QMC technique often suffers from the
Upon substitution for t in Eq. (20), this relation gives sign-problem19. DMFT-impurity models with the Hub-
q
E K at the optimal pairing. bardandelectron-phononcouplingshavebeenknownnot
p
∝−
One can also interpretthe pairing as arisingfrom pair to cause any serious sign problem for most of the physi-
exchange. In Fig. 3, the spin-up (S = 1) pair on the cal parameter space13. However,inclusion of the Hund’s
z
interacting site hops out via processes 1 and 3, while rule coupling terms in Eq. (2) produced severe sign-
the spin-downpair hops backin via 2 and4. The charge problems20wheneachoftheinteractiontermsaredecou-
neutralintermediatestateforcesthetemporaloverlapbe- pled by the discrete Hubbard-Stratonovich transforma-
tweenthehoppingsofthepairs,producingtheattractive tion21,despitethatthedecoupledHRtermsresemblethe
interaction when viewed as a second order perturbation Jahn-Teller coupling. To overcome such problem, a new
in terms of the pair-basis. The same general reasoning scheme of mixing discrete21 and continuous22 Hubbard-
should apply to other types of local interactions in the Stratonovich transformations has been used. This re-
strong-U limit. movedtheproblemwiththeaveragesignslargerthan0.9
One can carry out the same perturbation analysis for for most of the runs. Details of the procedure is given
theJahn-TellerinteractionEq.(3),wherethepairingen- in Appendix C. It is speculated that the simulation is
ergy gain E is always negative. E = 7t4/∆U 2 for more stabilized by the built-in symmetry of orbitals in
p p − q ∗
5
the new transformation in contrast to the poorly pre- !n;k;1(cid:27) !m;k0;1(cid:27) !n;k;1(cid:27) !l;k00;1(cid:27)
served orbital symmetry in incomplete QMC samplings
of the old scheme. = ÆnmÆ(k(cid:0)k0) + (cid:0)
The superconducting instability is probed by comput- (cid:0)!n;(cid:0)k;2(cid:27)(cid:1)(cid:0)!m;(cid:0)k0;2(cid:27) (cid:0)(cid:2)!n;(cid:0)k;2(cid:27) (cid:3)(cid:0)!l;(cid:0)k00;2(cid:27)
ing uniform pair susceptibility χ. χ for the spin-triplet
FIG.4: Expansionofspin-tripletpairingsusceptibilityχ. The
channelcanbeexpressedinaBethe-Salpeterequationas
local index mσ (m=1,2) is for orbital and spin labels.
χ = χ +χ Γχ +χ Γχ Γχ +
0 0 0 0 0 0
···
= √χ0(I √χ0Γ√χ0)−1√χ0, (21)
−
where we have used the symmetrized form23. χ0 is the Veff
propagator of two electrons independently propagating 4
with zero net momentum, and Γ is the effective pair- 2
0
ing interaction. (See Fig. 4.) Eq. (21) is a shorthand
-2
for matrices with indices over the Matsubara frequency -4
ω . For the static uniform susceptibility, we only need -6
n
in-coming electrons of net zero momenta and net zero 2
1
frequency, as in Fig. 4. Due to the momentum and fre- -2 -1 0 ω
quency conservation at each internal vertices of Feyn- 0 -1 m
man diagrams, any internal pair of legs in the ladder ωn 1 2 -2
diagrams in Fig. 4 also have net zero momenta and fre-
quencies. The uniform (uncorrelated) 2-particle prop-
agator χ can be computed with the 1-particle Green
function 0as χ (ω ) = G(k,ω )G( k, ω ). The k- FIG. 5: The effective interaction Veff(ωn,ωm)(≡
summation is0pernformedkwithin χn du−e to−thenlocality of −Γ(ωn,ωm)/T). The attractive interaction is for small
the DMFT. Γ is approPximated by0 a local quantity Γloc energy transfer (ωn ≈ ωm). The interaction at higher
frequencies is repulsive.
obtained from a local relation similar to Eq. (21),
Γloc =[χloc] 1 [χloc] 1. (22)
0 − − − with the phonon propagator D(ω ). The attractive part
n
χloc is the local (uncorrelated) 2-particle propagator, is formed in a uniform attractive valley along the small
0
sχul0oscc(eωpnti)b=ilitGyloχcl(oωcni)sGdliorce(c−tlωync)o,manpduttehdeflroocmalQtrMipCletapsair einnteerrgayctitornans,sftehrereeffgeioctnivωenint≈eraωcmti.onFVoeffr(pωunr,eωmH)ubshboawrds
a very different profile with high energy structures24
χloc(τ1,τ2,τ3,τ4)=hT[d†1σ(τ1)d†2σ(τ2)d2σ(τ3)d1σ(τ4)]i. away from the low energy scattering ωn = ωm. The
(23) attractive interaction along the ωn = ωm line clearly
Corresponding definition of χloc for the singlet channel demonstratesthat the HR-induced localpairing is medi-
can be found in Ref.13. Since χloc are directly obtained ated by the effective low-energy coupling medium which
from the QMC measurements and χ , χloc are easily has been self-generated by the multiplet interaction.
0 0
computed from the 1-particle self-energy, we derive the Since the local spin excitation energy is 2K (see Fig. 2)
effective interaction kernel Γ( Γloc) and the uniform and the electron-spin coupling constant is of order
≡
pair susceptibility χ from Eqs. (21,22). K, a naive estimate for the effective interaction gives
The transition temperatures are determined by when K2(2/2K) K(= 0.2), as in the electron-phonon
∼ ∼
the maximum eigenvalue of the superconducting kernel theory. The numerically obtained Veff at 2 4 is about
−
√χ0Γ√χ0 in Eq.(21) approaches1. Forsuch condition, an order of magnitude larger than the simple estimate.
the effective interaction must be attractive (Γ>0). The This suggests that the renormalization effect is very
other requirement is the finite weight of quasi-particles strong and is consistent with the above observation that
at the chemical potential as in χ (ω) πzN(0)/ω, with the suppressed hopping due to the Coulomb interaction
0
N(0)non-interactingdensityofstatesa≈tthechemicalpo- enhances the local interaction25. A thorough exam-
tential. As the system becomes strongly correlated, χ ination of the renormalization effects is necessary to
0
becomes smaller as z 0 while the effective local corre- fully understand these numerical results. Diagrammatic
→
lationtends to getlargeraselectrons aremore localized. methods will prove particularly useful when backed up
The resulting T is determined by the balance between by the QMC and the exact diagonalization methods
c
χ and Γ. since the basic mechanism of the local pairing is given
0
The attractive interaction V (ω ,ω )( as in Section III.
eff n m
≡
(1/T)Γ(ω ,ω )) is shown in Fig. 5 for U/W = 0.6 Fig. 6(a) showstransition temperatures T for the HR
n m c
−
and K/W =0.1. It is quite interesting that V (ω ,ω ) couplingasafunctionoftheCoulombrepulsionU where
eff n m
resembles that for the electron-phonon coupling in the the superconductivity becomes stronger with U. The
Eliashberg theory3 where V (ω ,ω ) = g2D(ω ω ) HR superconductivity emerged only in the spin-triplet
eff n m n m
−
6
0.02 0.02
Jahn-Teller coupling
Hund’s Rule (K/W=0.15)
T /W T /W 2∆/W=0.1
c c 2∆/W=0.15
0.01
0.01
(a)
(a)
0.00
0.00
4
0.6 z )
T
V π 0.6
z 0.4 eff πT,
2(eff z 0.4
V
0.2 -
(b)
0.2
0.0 0 (b)
0 0.2 0.4 0.6 0.8 1
U/W
0.0
0 0.2 0.4 0.6 0.8 1
FIG. 6: (a) Superconducting transition temperature Tc vs U/W
Coulomb repulsion U with Hund’s rule coupling. For the
Hund’srule(HR)coupledsystem,Tc monotonicallyincreases FIG. 7: (a) Superconducting transition temperature Tc vs
with U showing local pairing mechanism. Gray symbol is Coulomb repulsion U with Jahn-Teller electron-phonon cou-
an extrapolated estimate from the superconducting kernel pling. The Jahn-Teller (JT) phonon coupling produces su-
χ0Γ. (b) Quasi-particle renormalization factor z (left axis) perconductivity with a cross-over from the conventional su-
andeffectivepairinginteractionVeff atlowest Matsubarafre- perconductivity (small U) to the local pairing regime (large
quency(right axis). z decreases as U approaches Uc. The U) for 2∆/W = 0.1. For a larger electron-phonon coupling
productzVeff fortheHRsystemincreasedasU →Uc,consis- (2∆/W = 0.15), the Mott insulator transition happened at
tent with the increasing Tc. W is the non-interacting band- smaller U. (b) Renormalization factor z as in Fig. 6.
width.
ings13. For small dopings, ferromagnetic fluctuation will
channel. Tc monotonically increased as U approached dominatethelocaltripletpairing. (4)Ascanbe inferred
the critical value Uc/W 0.75 for K/W = 0.15, a fromFig3,strongferromagneticordersuppressthepair-
≈
strong indication of the local pairing. Tc/W less than exchange. Therefore, if the pairing and ferromagnetic
0.002 (gray symbol) has been estimated by extrapolat- phases ever co-exist, the superconducting phase should
ingeigenvaluesofthe superconductingkernel√χ0Γ√χ0. be near the ferromagnetic phase boundary.
The HR-coupled superconductivity is strong only in the Jahn-Tellerphononcoupling(inFig.7)showsthesim-
local pairing regime, since at small U there exists no ex- ilar behavior of the local pairing as the HR pairing.
ternal low-energy pairing medium in the electronic HR In contrast to the HR coupling case, T is non-zero at
c
coupling. U = 0 with the pairing mediated by the external cou-
Veff(πT,πT) at the lowest Matsubara frequency at pling medium of phonons. The results for 2∆/W = 0.1
T/W = 1/120 (triangles in Fig. 6(b), to the right-hand- shows that the phonon pairing-medium provides the at-
side scale) show a growing local pairing attraction as tractive pairing for the small U and as U is increased
z 025. The product zVeff(πT,πT)( χ0Γ) increases further the superconductivity makes a cross-over to the
→ ∝
as U Uc, consistent with the increasing Tc. We em- local pairing regime. At large electron-phonon coupling
→
phasize that the local pairing becomes apparent even at (2∆/W = 0.15) the metallic region shrunk and showed
moderately correlated regime of z 0.5, although the no signs of the cross-over. Given the same coupling
∼
heuristic argument in section III was given for z 0 for strength(comparingK and K =2∆),theHRsystem
→ | ph|
the sake of clarity. has weaker superconductivity than the el-ph systems.
We can make some predictions based on generalprop- Onesignificantdifferencebetweenthismodelofdoubly
erties of the local spin-triplet pairing. (1) The exchange degenerateorbitals(commonlycalledase Emodel)and
anti-symmetry is in the orbital space, in contrast to the the C model (t H model13) is that th×e e E model
60
× ×
p-wave spin-triplet theory15. Due to the discreteness of producesstrongerenhancementofT inthe localpairing
c
orbitals,thequasi-particleexcitationisgapped. Thespa- regime. Authors inRef.13 foundno signs forthe up-turn
tial part has the s-wave symmetry. (2) For large orbital ofT neartheMotttransitionforanystudiedvaluesof∆
c
degeneracy (N > 2), coupling between different pairs in the C -model, although the enhancement of T over
d 60 c
of orbital indices will likely result in multiple gaps. (3) the conventionaltheorieswasevident. Itis probablydue
The local pairing is strong for high electron/hole dop- to the easier lowspin formationfor evenfillings, since at
7
odd number of fillings (as 3 in the C model) unpaired APPENDIX A: ATOMIC LEVEL SCHEME OF
60
spinstendtofluctuateviatheintersitehoppingandmake HUND’S RULE COUPLING
the low spin configuration less effective.
Since the HR and JT couplings induce superconduc- For 2 electrons on site, there are 6 possible configura-
tivities separately but in different symmetries, they are tions
expected to complete when both are present. For in-
stance, by turning up the JT coupling constant at a d†1 d†2 , d†1 d†2 , d†1 d†2 ,d†2 d†1 ,d†1 d†1 ,d†2 d†2 |vaci.
↑ ↑ ↓ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓
fixedfiniteHRcoupling,onecansuppressthespin-triplet n o (A1)
pairingandeventuallycrossesovertoaspin-singletpair- ThesestatesaredegenerateforHmult =0. Hmult =HHR
ing. It can be also anticipated from the mapping of the of Eq. (2) can be rewritten in the above basis set as
JT coupling to a negative HR hamiltonian in the anti-
2K 0 0 0 0 0
adiabatic limit. Fullerene systems are known to have −
0 2K 0 0 0 0
sizable strengths for both interactions, although the JT  0 −0 K K 0 0 
interaction is more dominant for alkali-doped region. A − , (A2)
0 0 K K 0 0
recent density functional calculation study26 has sug-  0 0 0 −0 K K 
gested that the fullerene in the hole-doped region may  
 0 0 0 0 K K 
have stronger Hund’s rule interaction, with possible ap-  
 
plications to cation-doped fullerenes27. the solutions of which become
Finally,wemakeanobservationofaninterestingprop-
d†1 d†2
esprtiyn aonfdthoerbliotaclalinidnitceersaicntiothneEJqT.c(o3u)p.linIfg,wEeq.sw(3a)p, ntehwe E =−2K,  1/√2(d†1 d†2 −d†2 ↑d†1 ↑)|vaci
boson fields ϕ˜ν couple to electronic spin fluctuations,  ↑ ↓ d†1↑↓d†2↓↓  (A3)
ϕicA˜.ae2lll.(,dsop†mttihh↑need-rflmcuto↓ecu+rtpmuhlais.ntcigi.on)n]t.tsehNrwemohwhsiacbϕhm˜eνcmiloptemoldaneiyiaastPnethmatehr[eerϕ˜o1uel(elnencocmhtfra↑eonx−ngteepnrdanm.iar↓Tli)nlho+ge-. EE ==02,K, (1/√11//2√√(d22†1((↑ddd†1†1†1↑↑↓dd+†2†1↓↓d+−†2↑ddd†2†2†2↑↑↓dd)†1†2|↓↓v))ac)i.|vaci
groundstate now belongs in the spin-triplet and orbital- As shown in Fig. 2, the ground states are S = 1 spin-
singletspace. Thepairsusceptibilityforthesingletchan- triplet and the excited states are S = 0 spin-singlet
nelmapstothatofthetriplet-channelEq.(23)andallre- with orbitallymixed states. The abovegroundmultiplet
sults fromJT coupling automaticallyhold for the triplet states are written as |T1), |T0) and |T¯1) for the triplet
superconductivity. This model provides a robust exam- states (T) of Sz = 1,0, 1, respectively. Similarly, we
−
ple for the local spin-fluctuation superconductivity. write the above excited states as Sa), Sb) and Sc), re-
| | |
spectively, for spin-singlets (S). With one electron on
site, the Hund’s rule coupling does not lift any degener-
acy since it is a multi-electron interaction. The same is
V. CONCLUSION
true with 3 electrons,since one canview this in terms of
the hole interaction.
We have demonstrated that the local pairing mecha-
nism in multi-band systems on general grounds. In par-
ticular, the existence of Hund’s rule induced spin-triplet APPENDIX B: PERTURBATION THEORY FOR
PAIRING CORRELATION ENERGY
s-wave superconductivity is shown. The self-generated
localmultiplet fluctuations inthe strong-U limit provide
thepairingmediumevenwithapurelyelectronicinterac- The perturbation is defined as an expansion with re-
tion. This general idea should work for other extensions spect to the intersite hopping around the interacting
of the model. Further studies on the superconductor- Hamiltonian in Eq. (5),
ferromagnetphasediagramwouldclarifytherelevanceof
H = UN(N 1)/2+H (B1)
thecurrentmodeltoknownexperimentalsystems. Fluc- 0 − mult
tuationeffects beyondthe meanfieldapproximationand V = −tq (c†mσdmσ+h.c.). (B2)
singlet-triplet competition are left for future research. mσ
X
First, let us develop a general perturbation theory up to
the 4-th order. The Heisenberg equation reads
Acknowledgments H ψ =(H +V)ψ =E ψ , (B3)
0
| i | i | i
where the energy eigenvalue and wavefunction are ex-
We thank O. Gunnarsson,E. Tosatti,M. Fabrizio and panded in terms of the perturbation V as
V. H. Crespi for helpful discussions. We acknowledge
support from the National Science Foundation DMR- E = E0+E1+E2+E3+E4, (B4)
9876232and Max-Planck forschungspreis. ψ = ψ + ψ + ψ + ψ + ψ . (B5)
0 1 2 3 4
| i | i | i | i | i | i
8
The 0-th order terms give for fermion number operator n for a quantum state α
α
and coshλ = exp(∆τU/2). The continuous Hubbard-
H0|ψ0i=E0|ψ0i. (B6) Stratonovich transformation22(CHST) applies to a gen-
The n-st order equation becomes eral operator Aˆ in a gaussian integral
H ψ +V ψ = n E ψ . (B7) eAˆ2 = dx exp( πx2+√πAˆx). (C2)
0| ni | n−1i m| n−mi Z −
m=0
X
For doubly degenerate orbital systems, the interacting
Projecting the unperturbed bra-vector ψ to Eq. (B7)
h 0| Hamiltonian Eq. (2) can be rewritten as
forn=1,onegetsthefirstordercorrectiontotheenergy
U
H = [N +N ]2 (C3)
int +
E = ψ V ψ . (B8) 2 −
1 0 0
h | | i
Introducing an operator P0 which projects out the un- + K nm nm nm nm′ 2 nmσnm′σ
perturbed wavefunction ψ , one can express the wave-  ↑ ↓− ↑ ↓− 
function to the 1st order|as0,i Xm mX6=m′ σ,mX<m′
 
1 + K d†m↑d†m↓dm′↓dm′↑+d†m↑d†m′↓dm↓dm′↑ ,
|ψ1i= E H P0V|ψ0i. (B9) mX6=m′h i
0 0
−
where N = n and N = n . Previously,
The higher order contributions are computed by induc- + m m m m
each terms in the firs↑t two lin−es have been↓decoupled by
tion. The useful expressions up to the 4th order are the DHST. N+P2,N2,N+N are expPanded into nmσnm′σ′
1 terms before the D−HST E−q. (C1) is applied. The terms
ψ = P (V E )ψ (B10)
| 2i E H 0 − 1 | 1i on the third line have been decoupled by
0 0
−
E = ψ V ψ = ψ (E H )ψ (B11)
E2 = hψ0|(E| 1iHh)ψ1| 0E− ψ0 |ψ1i, (B12) e−∆τKc†αc†βcδcγ = 1 e√∆τKσ(c†αcγ−c†βcδ), (C4)
4 2 0 0 2 2 1 1 2
h | − | i− h | i σ
X
where we have used a simplification E =0 which is the
1
whereK >0andnopairsofthe indices from α,β,γ,δ
case with our models. { }
are the same. When the QMC sampling is incomplete,
The unperturbed energies with the Jahn-Teller inter-
all the symmetry inherent in the original hamiltonian is
action,Eq.(3),atthetotaloccupationnumbersN ,N +
d d
not fully recovered, which is speculated to be a source
1,N +2 are
d
of the sign-problem. This problem can be resolved by a
4t2 14t4 different decoupling scheme as follows. We rewrite the
q q
E(N +2) = U 8∆ (B13)
d − − U − ∆U2 first line as
E(Nd+1) = U −8∆− 4Ut2q − 2∆7tU4q2 (B14) U2(N++N−)2 =U(N+2 +N−2)− U2Sz2, (C5)
4t2
E(Nd) = U −8∆− Uq, (B15) wthiethfiSrszt=twNo+te−rmNs−in. FthoretsheecoonrdbiltianlediengEenqe.r(aCcy3)Ncdan=b2e,
where ∆ = 8g2/ω is the local Jahn-Teller coupling written as
ph
−
energy. Corresponding wavefunctions of the leading or-
K(n n +n n n n n n )
der are 1↑ 1↓ 2↑ 2↓− 1↑ 2↓− 2↑ 1↓
K
= (n n n +n )2
|ψ(Nd+2)i = |Sc)⊗|sc) (B16) − 2 1↑− 2↑− 1↓ 2↓
ψ(N +1) = S ) c vac) (B17) K
| d i | c ⊗ †mσ| + (N++N ) K(n1 n2 +n2 n1 ), (C6)
ψ(N ) = S ) vac), (B18) 2 − − ↑ ↑ ↓ ↓
d c
| i | ⊗|
wherethefirsttermwithsquaredtermisdecoupledwith
with notations given in Appendix A.
the CHST of a single auxiliary field. This procedure
preserves the symmetry between orbital 1 and 2. The
APPENDIX C: HUBBARD-STRATONOVICH third line in Eq. (C3) has been the source of the sign-
DECOUPLING FOR HUND’S RULE COUPLING problem20. We express this by completing square as
The discrete Hubbard-Stratonovich transforma- K(c†1↑c†1↓c2↓c2↑+c†2↑c†2↓c1↓c1↑+···)
tion21(DHST) is given by K
e−∆τUnαnβ =e−∆τ2U(nα+nβ)12 eλσ(nα−nβ), (C1) = −+K2 ((cN†1+↑c+2↑N+c)†2↑c1K↑−(nc1†1↓nc22↓+−nc2†2↓nc11↓))2. (C7)
σ 2 − − ↑ ↑ ↓ ↓
X
9
Similarly to Eq. (C6), the squared term preserves the With the old DHST scheme for all interaction terms,
orbital symmetry with a negative coefficient (K > 0). there are 6 auxiliary fields for n n pairs and 4 for
α β
Wanidthad−d2in(ng1E↑nq2s↑.+(Cn52-↓Cn71)↓,)w=e−fi(nNal+2ly+hNav−2e)+(N++N−) pc†αroc†βvecdγcsδchteermmes,h1a0s 2indtisoctraeltpeefireledaschfotrimne nslice,.nThne imin-
1 2 1 2
N2 and3continuousfieldspertimeslice. ↑Thi↑spro↓ced↓ure
Hint =3K(N++N−)+(U −2K)(N+2 +N−2) ca±n be extended to higher degeneracy N > 2. Although
U K the arrangement in Eq. (C6) is special for N = 2, they
S2 (n n n +n )2 d
−2 z − 2 1↑− 2↑− 1↓ 2↓ canbedecoupledbytheDHSTwithoutcausingthesign-
K problem. Terms like in Eq. (C7) are problematic for the
− 2 (c†1↑c2↑+c†2↑c1↑−c†1↓c2↓−c†2↓c1↓)2,(C8) sign-problems in the impurity problem. Eq. (C7) can be
readilyextendedtohigherdegeneracyandtheabovepro-
wheretheN inthefirstlineareexpandedintonmσnm′σ′ cedure will likely removethe sign-problemin the Hund’s
anddecoupl±edbytheDHST,Eq.(C1). Termsinthesec- rule coupling.
ondandthirdlinesaredecoupledbytheCHST,Eq.(C2).
1 For a recent review, see A. Damascelli, Z. Hussain, and (1963).
Z.-X. Shen,Rev.Mod. Phys. 75, 473 (2003). 16 A. Georges et al.,Rev.Mod. Phys. 68, 13 (1996).
2 F. Steglich et al.,Phys.Rev.Lett. 43, 1892 (1979). 17 M. Capone, M. Fabrizio, C. Castellani, and E. Tosatti,
3 J. R. Schrieffer, Theory of superconductivity, Addison- Science 296, 2364 (2002).
Wesley Publishing Company, NewYork (1994). 18 W. Metzner, Phys. Rev.B 43, 8549 (1991).
4 A. P. McKenzie and Y. Maeno, Rev. Mod. Phys. 75, 657 19 E. Y. Loh, Jr., J. E. Gubernatis, R. T. Scalettar, S. R.
(2003). White,D.J.Scalapino, and R.L.Sugar,Phys.RevB41,
5 K. G. Saxenaet al.,Nature406, 587 (2000). 9301 (1990).
6 A. F. Hebard et al.,Nature 350, 600 (1991). 20 J. E. Han, M. Jarrell and D. L. Cox, Phys. Rev. B 58,
7 R. W. Lof et al.,Phys. Rev.Lett. 68, 3924 (1992). R4199 (1998).
8 D.W. Murphy et al., J. Phys. Chem. Solids 53, 1321 21 R.M. Fyeand J. E. Hirsch, Phys.Rev.B 38, 433 (1988).
(1992); R.F.Kieflet al.,Phys.Rev.Lett.69,2005(1992). 22 R.Blankenbecler, D.J.Scalapino, and R.L. Sugar,Phys.
9 P. Morel and P. W. Anderson, Phys. Rev. 125, 1263 Rev.D 24, 2278 (1981).
(1962). 23 C. Owen and D. J. Scalapino, Physica (Amsterdam) 55,
10 J. H. Kim and Z. Teˇsanovi´c, Phys. Rev. Lett. 71, 4218 691 (1971).
(1993). 24 M. Jarrell, privatecommunication.
11 D.M.Newns,H.R.Krishnamurthy,P.C.Pattnaik,C.C. 25 Veff is the effective interaction in terms of non-interacting
Tsuei, and C. L. Kane, Phys.Rev.Lett. 69, 1264 (1992). basis instead of between quasi-particles in Ep,Eq. (20).
12 J. K. Freericks and M. Jarrell, Phys. Rev. Lett. 75, 2570 26 M. Lu¨ders, N. Manini, P. Gattari, and E. Tosatti, Eur.
(1995). Phys. J. B 35 57 (2003).
13 J. E. Han, O. Gunnarsson, and V. H. Crespi, Phys. Rev. 27 W.R.DatarsandP.K.Ummat,SolidStateCommun.94,
Lett. 90, 167006 (2003). 649(1995);A.M.Panich,H.-M.Vieth,P.K.Ummat,and
14 W. L. McMillan, Phys.Rev. 167, 331 (1968). W. R.Datars, Physica B 327, 102 (2003).
15 R. Balian and N. R. Werthamer, Phys. Rev. 131, 1553