Reexamination of d-wave superconductivity in the two-dimensional Hubbard model

5 Pages·0.14 MB·English

by  Takao Morinari

Reexamination of d-wave superconductivity in the two-dimensional Hubbard model
Takao Morinari
Yukawa Institute for Theoretical Physics, Kyoto University Kyoto 606-8502, Japan
(February 1, 2008)
We reexamine the possibility of d-wave superconductivity in the hole-doped two-dimensional
Hubbard model. In terms of the gauge field description of the spin fluctuations, we show that d-
2 wave superconductivity is unstable in the perturbative region with respect to the on-site Coulomb
0 repulsion, U. Whereas in the region where d-wave superconductivity is possible, there is a strong
0 constraint on the gap of superconductivity. Analysis of the localized spin moments suggests that
2 thereisanotherd-wave(dx2−y2-wave)pairingduetotheshort-rangeantiferromagnetic correlation.

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74.25.Dw, 74.20.Rp
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5
Over the past few decades a considerable number of we do not need the U(1) gauge symmetry breaking that

] studies have been made on the Hubbard model. In spite is essential for the slave-particle gauge theory. [5]
l of its simplicity, we can expect a variety of phenom- Formulation- The partition function of the Hubbard
e
- ena: antiferromagnetism, ferromagnetism, and uncon- model is written as = c cexp( S), where
r Z D D −
t ventional superconductivity. Among others, the possi-
s bility of d-wavesuperconductivity has been considerably β R
at. investigated since the discovery of the high Tc cuprates. S = dτ cj(∂τ −µ)cj −t (cicj +cjci)
m [1] Z0 Xj hXi,ji

- The theory of d-wave superconductivity in the Hub-
d bard model is based on the antiferromagnetic spin fluc- +U nj nj . (1)
n ↑ ↓
tuations. [1] The mechanism is analogous to the p-wave j
o X
pairing mechanism due to the ferromagnetic spin fluc- 
c Hereτ dependenceofallfieldsisimplicitandthesumma-
[ tuations in the 3He system. [2] In principle, the theory tion is taken overthe nearestneighbor sites. Car-
1 relys on the perturbative expansion with respect to U, rierfieldhis,jiarerepresentedinspinorrepresentation: c =T
the on-site Coulomb repulsion. P i
v c c and c = c c . The on-site Coulomb
i i i i i
8 However, the theory has some disadvantages. First of inte↑ract↓ion term can be↑rew↓ritten as, U n n =
5 all, it is hard to describe the half-filled case in which (cid:0) (cid:1) (cid:0) (cid:1) j j↑ j↓
010 lionctearlascptiionnmboetmweenetnstahreespprinodmucoemdenattseiascphrospitoertaionndalthtoe (wUh/e4re)Pthjeh(cnojm↑p+onnej↓n)t2s−of(ctjhσecjv)e2c+tor2(σnj↑=P+n(σj↓x),iσ,y,σz)
the inverse of U. Apparently, this dependence of the are the Pauli spin matrices. Introducing Stratonovich-
2
0 interaction on U cannot be reproduced by the perturba- Hubbard fields for the charge and spin degrees of free-
t/ tive expansion in U. In addition, realistic values of U dom, [7] we obtain Z = DcDcDΩDφcDφexp(−S),
a are much larger than the carrier hopping amplitude, t. where the action is given by
m Therefore, we must pay particular attention in applying R
- theperturbativeexpansioninU,especiallywhenwedeal β
d with the effect of the localized spin moments. S = dτ cj(∂τ −µ)cj −t (cicj +cjci)
on In this paper, we re-examine the possibility of d-wave Z0U Xj U hXi,ji
c superconductivity in the hole-doped two-dimensional + φ2 φ Ω c σc
: Hubbard model without relying on the perturbative ex- 4 j − 2 j j · j j
v j j
X X
i pansion with respect to U. Starting from the path-
X integral form of the partition function, we introduce the U iU
+ φ2 φ c c , (2)
r Stratonovich-Hubbardfield for the value of the localized 4 cj − 2 cj j j
a j j
spin moments. Analysis of it suggeststhat superconduc- X X
tivity basedonthe antiferromagneticspinfluctuations is up to constant. Here the vector Ω is a unit vector, φ
j j
unstable in the perturbative region. While in the region represents the value of the localized spin moments, and
whered-wavesuperconductivityispossible,weshowthat σ is the unit matrix inspinspace . The scalarφ is as-
0 cj
there is astrongconstraintonthe value ofthe supercon- sociated with the charge fluctuations. Note that φ = 1
j
ductinggap. Insteadofthed-waveCooperpairing,there at half-filling. If we consider hole doping, then φ takes
j
is another d-wave (dx2 y2-wave) pairing due to the anti- φj 1. Fortheseamplitudes,wedonotconsiderthepos-
ferromagnetic short-ra−nge correlation. This dx2 y2-wave sibi≤lityofinhomogeneousconfigurationsofthembecause
pairing state is similar to the spinon pairing in−the RVB suchconfigurationsmaycompetewithsuperconductivity.
theory. [3,4] However, the crucial difference here is that In order to focus on the possibility of superconductivity,
1
we take the uniform values, that is, φ =φ=const. and perturbative expansion in U [1]. Formally, we can re-
j
iφ = c σ c = (1 δ), with δ the doped hole con- producetheeffectiveinteractionoftheantiferromagnetic
cj j 0 j
−h i − −
centration. As a result, we may write the action in the spin fluctuation theory as follows. Expanding the action
following form (6) in powers of U, the quadratic term yields the boson
propagator. Theeffectiveinteractionduetotheexchange
β ofthebosonscorrespondstotheinteractionderivedfrom
S = dτ c (∂ µ˜)c t (c c +c c ) the antiferromagnetic spin-fluctuation theory [1] up to
j τ j i j j i
 − −
Z0 Xj hXi,ji the paramagnoncontributions that leadsto p-wavepair-
 ing. [2] Apparently, we can trust this interaction only in
φU Ω c σc + φ2UβN, (3) the perturbative region with respect to U. In addition,
− 2 j · j j 4 the interaction exists only when φ = 0, or coupling to
j 6
X the bosons is lost. Nevertheless we will show later that

φ=0 in the perturbative region in U.
where µ˜ = µ U(1 δ)/2 and N is the number of the
− − For the analysis of the system, we rely on neither the
latticesites. Notethatthelastterminthe squarebrack-
φU-expansion nor the t-expansion because both of them
ets has the form of Hund coupling between the localized
are reliable only in part of the parameter range of U/t
spin moments and the carriers’ spins.
and δ. Alternatively, we study the system by taking the
Gauge field description of the spin fluctuations- Now
continuum limit. Taking such limit is justified when the
we rotate the spin of the fermion at j-site so that it is
in the direction of Ω . Such rotation can be done by fluctuationsarelong-ranged. Sincetheantiferromagnetic
j
spinfluctuationsmaybelong-rangednearhalf-filling,we
performing the following SU(2) transformation:
may take the continuum limit. In the continuum limit,
the action (5) is reduced to
c =U f ,
j j j (4)
c =f U ,
(cid:26) j j j φU
S = dτ d2rψ(r,τ) (∂ µ˜)σ +i σ
τ 0 τ z
where, Uj = zzj↑ −zzj↓ and Uj = zzj↑ zzj↓ . Z 1 Z (cid:20) − A − 2
Here complex(cid:18)varjia↓blesj↑zjσ(cid:19)and zjσ are d(cid:18)efi−nedj↓as jΩ↑j(cid:19)= −2m(−iσ0∇+A)2 ψ(r,τ)+SA, (7)
z σz . [7] Thus, we obtain (cid:21)
j j
where the SU(2) gauge field is defined as =
µ µ
A A
aσ = iU∂ U and S is derived from
S = dτ f ∂ µ˜+U ∂ U f a=x,y,zAµ a − µ A
 j τ − j τ j j Eq. (6) in principle.
Z Xj (cid:0) (cid:1) PThe system governed by the action (7) is the fermion

t f U U f +f U U f system with the interaction due to the exchange of the
− i i j j j j i i SU(2) gauge field . Here the gauge field is as-
hXi,ji(cid:0) (cid:1) sociated with the sApµin fluctuations. The z-coAmµponent
φU φ2U z = i(z ∂ z +z ∂ z ) describes the ferromagnetic
− 2 Xj fjσzfj+ 4 βN. (5) sApiµi(nz fl∂−uµzctua↑tziµo∂nµ↑sz. )dW↓eshµcerr↓iebaesstthheeanxt-icfoemrropmonaegnntetAicxµspi=n
 −fluctu↑atio↓n−s. ↓The↑latter relationship is implied from the
The effective action of the boson fields z and z is ob-
j j analysis of the CP1 representation of the antiferromag-
tained by integrating out fermion fields f and f :
j j netic Heisenberg model. [9]
In order to consider the antiferromagnetic spin fluc-
φU
Seff =−Trln(cid:20)(cid:18)∂τ −µ˜− 2 σz(cid:19)δij ttuhaetfiioenldss, ψw˜e =foc(uψs onψth)/e√g2autgheatfideiladgoAnxµa.lizIenthteergmasugoef
φ2U charge, the±action↑is±rew↓ritten as
t U U +U ∂ U δ + βN, (6)
ij i j i τ i ij
− 4
(cid:3)
wothheerrewtisije.=IftwfeorextphaenndeaErqe.st(6n)ewigihthborresspiteecsttaondφUti,jt=hen0 S = dτ d2r( ψ˜s(r,τ)[∂τ −µ˜+isAxτ
Z Z s=
the second order term yields the RKKY interaction be- X±
1
tweenthelocalizedspinmoments.[8]Ontheotherhand, ( i +sAx)2 ψ˜s(r,τ)
−2m − ∇
the secondorderofthe expansionofEq.(6)withrespect (cid:21)
tnoiatni.j[y7,i8el]ds the antiferromagnetic Heisenberg Hamilto- −φ2U ψ˜+(r,τ)ψ˜−(r,τ)+ψ˜−(r,τ)ψ˜+(r,τ) +SAx. (8)
h i(cid:27)
Before going into the analysis of the action (5), we
discuss how the formulation is related to the antiferro- Theaction(8)hastheformoffermionscoupledwiththe
magnetic spin fluctuation theory that is based on the U(1) gauge field.
2
Analysis of the gap equation- Now we study the possi- Eq. (3) after integrating out c and c . The variation
j j
bility of spin singlet superconductivity based on the ac- with respect to φ yields
tion (8). In the following we assume that there exists
an attractive interaction induced by the exchange of the U 1 K( 1 γ2)
dγ − =1, (12)
gthaeugmeefiaenldfi.eIlnd tHhaempirletosenniacne ofofranthaettsrpaicntisvienginletterpaacitrioinng, 4π2tZ√ν2−α2 pγ2+α2
state may have the following form where α = φU/(8t), νp= µ˜/(4t), and K(ξ) =
π/2dθ(1/ 1 ξ2sin2θ)isthecompleteellipticintegral
1 0 −
HMF = 2 k c˜†k,+ c˜†k,− c˜−k,+ c˜−k,− Rotof µthyeiefilrdsstpkind. Meanwhile the variation with respect
X(cid:16) (cid:17)
ξk φU/2 0 ∆k c˜k,+ 4 1
×−φ∆0U∗k/2 −−ξ∆0k∗k φ−−U∆ξ/kk2 φ−U0ξ/k2cc˜˜c˜††−kkk,−,,+ . (9)In deriviπn2gZt√hνe2s−eαe2qduγaKtio(pns,1−weγ2h)a=ve1u−seδd. ν < 0 (a1n3d)
  − −  ν >α, which holds for the hole doped case.
| |
The gap ∆k is evaluated from the gap equation: ∆k = FromEqs.(12)and(13)wefindU/tandδ dependence
−41Ω kVkk′∆Ekk′′ tanhβ(Ek′+2φU/2) +tanhβ(Ek′−2φU/2) , oshfoφwannidn∆Fcig..T1hbeybtohuensdoalridy bcuertwvee.enInφt6=he0φa=nd0φre=gim0eis,
wherePEk = ξk2h+|∆k|2. In principle, the interactioin thereisnoattractiveinteractionduetotheabsenceofthe
Vkk′ is derived from Eqs. (6) and (7) by eliminating the antiferromagnetic spin fluctuations as mentioned above.
p
gauge field x. However, we do not need its explicit
Aµ Note that the φ = 0 regime lies in the smaller value
form.
of U/t. Apparently, this region contains the parameter
At zero temperature, the gap equation is reduced to range of U/t and δ where perturbation in U is justified.
Therefore,forthe stateswithφ=0,perturbationtheory
1 ∆k′
∆k = Vkk′ , (10) is justified. Although we cannot say whether perturba-
−2Ωk′,EkX′>φU/2 Ek′ tion theory is reliable in the φ 6= 0 regime, states with
φ = 0 are qualitatively different from those with φ = 0
NotethatinEq.(10)thesummationink-spaceistaken be6cause the former is unstable in the U/t 0 limit.
′ →
over under the constraint Ek′ > φU/2. The presence of (In the language of renormalization group theory, they
theconstraintonspinsingletpairingstatesisunderstood should belong to different fixed points.) Turning to the
asfollows. Onecanseethatthesecondterminthebraces conditions of d-wave superconductivity, the boundaries
in Eq. (8) is similar to that of the Zeeman energy term of ∆c/t = 0.20 and ∆c/t = 0.04 in Fig. 1 suggests that
produced by applying an in-plane magnetic field to the the occurrence of d-wave superconductivity is restricted
system. Infact,suchZeemanenergytermisproportional to extremely small parameter region or we need much
to the applied magnetic field times j(c†j cj +c†j cj ). stronger attractive interaction than the RKKY type in-
(Here the direction of the in-plane magnet↑ic fi↓eld is↓ch↑o- teraction.
P Another d-wave pairing- So far we discuss the possi-
sen along the x-axis.) Apparently, in the limit of the
bility of d-wave superconductivity in the carrier system.
large in-plane magnetic field, spin singlet pairing sates
are unstable. Similarly, in the large φU limit, spin sin- Now we discuss that there is another d-wave (dx2 y2-
wave) pairing associated with the localized spin −mo-
glet states are not stable. Therefore, if spin singlet su-
ments.
perconductivity is stable, then the superconducting gap
In the φ =0 regime, we may take the form of the an-
∆ should satisfy,
6
tiferromagneticHeisenbergHamiltonianfortheactionof
∆>φU/2 ∆ . (11) the localized spin moments. In the Hamiltonian formu-
≡ c lation, it is written as H = Jφ2 Ω Ω , with
spin 4 i,j i · j
In order to find the doped hole concentration depen- J = 4t2/U. [7,8] Note that the exchanghe iinteraction be-
dence of the constraint, we need to evaluate φ. For tweenthelocalizedspinsisreducedbPyfactorφ2. Sinceφ
the calculation, we assume the staggered form for Ωj ismonotonicallydecreasingfunctionofthedopedcarrier
as Ω =( 1)jeˆ , because we are concerned with the an-
j z concentration δ, this reduction suggests the relation to
−
tiferromagneticspinfluctuations.[10]Notethatnon-zero theHeisenbergantiferromagnetlikebehaviorofspinsus-
value of φ does not imply the presence of the antiferro- ceptibility observed in the doped high T cuprates. [11]
c
magneticlong-rangeorderbutitjustimplysthepresence Now we discuss another d-wave pairing. In order to
of the antiferromagnetic correlation because there is the describe the localized spin moments Ω , we can intro-
j
phase fluctuations, or the effect of the gauge field x, as
well as the quantum fluctuations. Aµ ducefermioncreationandannihilationoperators,a†j and
Now we estimate the value of φ by solving the saddle aj, as Ωj = a†jσaj. Due to the constraint |Ωj| = 1, the
point equations derived from the action obtained from fermion operators a†j and aj must satisfy,
3
a†jσajσ =1. (14) Hubbard model. We have shown that d-wave supercon-
σ ductivity is unstable in the perturbative region in U.
X
Whereas in the region where d-wave superconductivity
Note that this constraint, the fermion system is half- is possible, there is a strong constraint on the super-
filled, is independent of the doping concentrationδ. Un- conducting gap. Instead, there is another dx2 y2-wave
dertheconstraint(14),theHamiltonianHspin isreduced pairing due to the antiferromagnetic spin correl−ation.
to, up to a constant term, Acknowledgement- I wouldlike to thank M.Sigrist,Y.
Morita, and M. Tsuchiizu for helpful discussions. This
Jφ2
work was supported in part by a Grant-in-Aid from
Hspin =− 2 Di†jDij, (15) the Ministry of Education, Culture, Sports, Science and
hXi,ji Technology.
where D = a a a a is defined on each bond.
ij i j i j
↑ ↓ − ↓ ↑
Taking the mean fields hDiji andhDi†ji, we find that the
dx2 y2-wavepairing state and the extended s-wavestate
are−degenerate. If we introduce a slight hopping term
for the fermions, then the dx2 y2-wave pairing state is
stabilized [4] and the gap is of−order of φ2J. Although [1] See, for example, D. J. Scalapino, Phys. Rep. 250, 329
the origin of this dx2 y2 pairing is similar to the spinon (1995); T. Moriya and K. Ueda, Adv. Phys. 49, 555
pairing in the RVB t−heory, that is, the short-range an-
(2000).
tiferromagnetic correlation [3], the crucial difference is [2] P. W. Anderson and W. F. Brinkman, in The Helium
that we do not rely on the U(1) gauge symmetry break- Liquids, ed. by J. G. M. Armitage and I. E. Farquhar
ingthatisessentialfortheslave-particlegaugetheory.[5] (New York,Academic Press).
In addition, it should be stressed that this pairing state [3] P.W.Anderson,G.Baskaran,Z.Zou,andT.Hsu,Phys.
does not imply a superconducting state of the fermions Rev. Lett.58, 2790 (1987).
because of the constraint (14). [4] G. Kotliar and J. Liu, Phys.Rev. B 38, 5142 (1988).
This pairing state provides another dx2 y2 pairing [5] C. Nayak,Phys.Rev. Lett. 85, 178 (2000).
which is independent of dx2 y2-wave superco−nductivity. [6] J. Hubbard,Phys. Rev.B 19, 2626 (1979).
The fact that this pairing st−ate originates from the anti- [7] H. J. Schulz, Phys.Rev.Lett. 65, 2462 (1990).
ferromagneticcorrelationbetweenthe localizedspinmo- [8] P.Lacour-GayetandM.Cyrot,J.Phys.C7,400(1974).
ments suggests that it can be associated with the pseu- [9] N. Read and S.Sachdev,Phys. Rev.B 42, 4568 (1990).
[10] As shown in Ref. [7], spiral phaseis expected away from
dogap phenomenon observed in the high T cuprates. If
c
half-fillingbecausethekineticenergyofcarriersenhances
we apply the theory to the Cu site degreesof freedomin
ferromagnetic interaction betweenthelocalized spins.In
tchaenCbueOid2epnltainfieedinwtihtehctuhpartatoebss,etrhveendtbhye daxn2g−ley-2repsaoilrvinedg order to concentrate on d-wave superconductivity based
on antiferromagnetic correlations, the staggered compo-
photoemission spectroscopy. [12] Furthermore, there is
nent is taken here.
the experiment that indicates the existence of the pseu-
[11] D. C. Johnston, Phys.Rev.Lett. 62, 957 (1989).
dogap of dx2 y2 symmetry also in the insulating phase. [12] A. G. Loeser, Z.-X. Shen, D. S. Dessau, D. S. Marshall,
[13]Since our−dx2 y2 pairing exists also in the insulating C.H.Park,P.Fournier,andA.Kapitulnik,Science273,
phase,the experim−entsupports the relationshipbetween
325 (1996); H. Ding, T. Yokoya, J. C. Campuzano, T.
experimentally observed pseudogap with dx2 y2 symme- Takahashi,M.Randeria,M.R.Norman,T.Mochiku,K.
try and the dx2 y2 pairing due to the antifer−romagnetic Kadowaki, and J. Giapintzakis, Nature 382, 51 (1996).
correlation. − [13] F. Ronning, C. Kim, L. Feng, D. S. Marshall, A. G.
Summary-Tosummarize,wehavereexaminedthepos- Loeser, L. L. Miller, J. N. Eckstein, I. Bozovic, Z.-X.
sibility of d-wave superconductivity in the hole-doped Shen, Science282, 2067 (1998).
4
12
U/t
φ=0
8
4
φ=0
0
0.0 0.1 0.2 0.3
δ
FIG. 1. The boundary between φ 6= 0 and φ = 0 in the
U/t−δ plane. Theboundaryisgivenbythesolidcurve. The
dashed curve represents ∆c/t = 0.20 and the dotted curve
∆c/t=0.04.
5