Physica B 406 (2011) 1459–1465

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

n Corr

and Tec

E-m

journal homepage: www.elsevier.com/locate/physb

Superconductivity enhanced by d-density wave: A weak-coupling theory

Kim Ha a, Ri Subok a, Ri Ilmyong a,b,n, Kim Cheongsong a, Feng Yuling b

a Department of Physics, University of Science, Unjong district, Pyongyang, Democratic People’s Republic of Koreab Science College, Changchun University of Science and Technology, No. 7089 WeiXing Road, Changchun 13022, China

a r t i c l e i n f o

Article history:

Received 5 September 2010

Received in revised form

21 January 2011

Accepted 21 January 2011Available online 28 January 2011

Keywords:

High-Tc superconductivity

d-Density wave

2D Hubbard model

Nonconventional mechanisms

26/$ - see front matter & 2011 Elsevier B.V. A

016/j.physb.2011.01.048

esponding author at: Science College, Chan

hnology, No. 7089 WeiXing Road, Changchun

ail address: ilmyong@163.com (R. Ilmyong).

a b s t r a c t

Making a revision of mistakes in Ref. [19], we present a detailed study of the competition and interplay

between the d-density wave (DDW) and d-wave superconductivity (DSC) within the fluctuation-

exchange (FLEX) approximation for the two-dimensional (2D) Hubbard model. In order to stabilize the

DDW state with respect to phase separation at lower dopings a small nearest-neighbor Coulomb

repulsion is included within the Hartree–Fock approximation. We solve the coupled gap equations for

the DDW, DSC, and p-pairing as the possible order parameters, which are caused by exchange of spin

fluctuations, together with calculating the spin fluctuation pairing interaction self-consistently within

the FLEX approximation. We show that even when nesting of the Fermi surface is perfect, as in a square

lattice with only nearest-neighbor hopping, there is coexistence of DSC and DDW in a large region of

dopings close to the quantum critical point (QCP) at which the DDW state vanishes. In particular, we

find that in the presence of DDW order the superconducting transition temperature Tc can be much

higher compared to pure superconductivity, since the pairing interaction is strongly enhanced due to

the feedback effect on spin fluctuations of the DDW gap. p-pairing appears generically in the

coexistence region, but its feedback on the other order parameters is very small. In the present work,

we have developed a weak-coupling theory of the competition between DDW and DSC in 2D Hubbard

model, using the static spin fluctuation obtained within FLEX approximation and ignoring the self-

energy effect of spin fluctuations. For our model calculations in the weak-coupling limit we have taken

U/t¼3.4, since the antiferromagnetic instability occurs for higher values of U/t.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

It is one of the most interesting and important issues incondensed-matter physics to elucidate the mechanism of high-temperature superconductivity occurring in cuprates. Among anumber of mechanisms proposed for the high-temperature super-conductivity since its discovery in 1986 the spin-fluctuationmechanism has continued to be one of the most promisingmechanisms [1]. Following systematic studies based on theparameterized spin fluctuation theory, more quantitative studieswere carried out using the 2D Hubbard model within the FLEXapproximation in which spin and charge fluctuations, the single-particle spectrum, and superconducting gap function are deter-mined self-consistently [2–7]. The results of these theories seemto be successful at least qualitatively in explaining not only d-wave pairing but also a lot of unusual phenomena, which havebeen observed in high-temperature superconducting cuprates inthe overdoped region. However, existing microscopic theories

ll rights reserved.

gchun University of Science

13022, China.

have serious problems in explaining the underdoped region of thecuprates and instead many phenomenological approaches havebeen used. These problems are, in particular, the existence of apseudogap [8,9], which appears to have the same symmetry asthe superconducting gap, below a characteristic temperature Tn

that is higher than the superconducting transition temperature Tc.This is unusual because the pseudogap is present in the normalstate above Tc and coexists with the superconducting gap in thesuperconducting state below Tc. This striking pseudogap behaviorinitiated a variety of proposals [10–13] as to its origin, since theanswer to this question may be a key ingredient for the under-standing of high-temperature superconductivity. At present, thereis no agreement as to which of these proposals is correct.

A natural explanation for the pseudogap behavior involves thediscussion of the possible occurrence of a charge–density wave inthe cuprates. Remarkably, it was shown that most of the observedproperties referred to as a so-called pseudogap phenomenon can benaturally explained by introducing a charge–density wave havingd-wave symmetry or d-density wave (d-CDW or DDW) [12].Furthermore, as it is well known that the charge–density waveand superconductivity compete with each other in strongly corre-lated electron systems, such as high-temperature superconductors, apseudogap arising from a DDW state therefore has immense appeal.

K. Ha et al. / Physica B 406 (2011) 1459–14651460

Therefore, if the DDW state is responsible for the pseudogapbehavior, its effect on DSC must be understood on the samemicroscopic level as the spin fluctuations. Most of previous studiesfor this problem has been almost restricted in the phenomenologicalor mean-field level [14–17]. Dahm et al. [18] have presented thegeneral equations of the FLEX approximation for coexisting DDWand DSC gaps in the 2D Hubbard model, but these equations havenot been solved yet because they are much more complicated thanthe pure superconducting equations.

In this paper, we present a detailed study of the competitionand interplay between the DDW and DSC within the FLEXapproximation for the 2D Hubbard model. In order to stabilizethe DDW with respect to phase separation at lower dopings, asmall nearest-neighbor Coulomb repulsion V is included withinthe Hartree–Fock approximation. The DDW state is described bysinglet electron–hole pairing between the nested states above andbelow the Fermi sea and therefore the corresponding anomalousGreen’s functions contain the step function y(9ek9�9eF9), with anone-electron band energy ek and the Fermi energyeF. This stepfunction plays an important role in our theory for the competitionand interplay between the DDW and superconductivity. While inRefs. [14–17] without the step function the DDW transitiontemperature Tn as a function of doping after passing a largedoping moves to lower dopings, in our calculation with this stepfunction it decreases monotonously with increasing doping andgoes toward larger dopings. This will result in the coexistence ofthe DDW and DSC in a broader region, even when nesting of theFermi surface is perfect, as in a square lattice with only nearest-neighbor hopping t. Most importantly, we find that the presenceof a DDW leads to a large increase of the superconductingtransition temperature Tc close to the QCP at which the DDWstate vanishes, since the pairing interaction mediated by spinfluctuations is strongly enhanced due to the feedback effect of theDDW order parameter.

In Section 2 we will derive the coupled gap equations for theDDW, DSC, and p pairing as the possible order parameters, whichare induced by exchange of spin fluctuations within the FLEXapproximation. Results from solving these equations for the 2DHubbard model will be presented in Section 3. We conclude witha summary of the main results in Section 4.

2. Gap equations for competing order parameters

Our starting point is the 2D extended Hubbard model with theonsite and nearest-neighbor Coulomb repulsions U and V

H¼�tX

o ij4sðcþis cjsþcþjs cisÞþU

Xi

nimnikþVX

o ij4ssunisnjsu, ð1Þ

The cþis and cis are the usual creation and annihilationoperators for an electron with spin s at site i, respectively, andnis ¼ cþis cis is the number operator. The sum /ijS is over nearest-neighbor pairs with hopping t, leading to the one-electron banddispersion ek¼�2t(cos kx+cos ky). This dispersion has two impor-tant features: the nesting property ek¼�ek+ Q, with the antifer-romagnetic wave vector Q¼(p, p), and the Van Hove singularityin the density of states.

Due to nesting there will be a strong instability of the Fermisurface with respect to forming the DDW state with the order

parameter iDDðkÞ �/cþkþQscksSyð9ek9�9eF9Þ, which corresponds

to condensation of singlet electron–hole pairs in nested statesabove and below the Fermi level eF. The Van Hove singularity canalso result in the appearance of a superconducting state, which isdescribed by the order parameter DB(k)�/ckmc�kkS representingCooper-pair formation of electrons having opposite spins andmomenta. In the presence of two order parameters the operators

ðcþkm,cþkþQm,c�kk,c�k�QkÞ are coupled leading to the following

Green’s function matrix:

G�1ðkÞ ¼

ion�ekþm �iDDðkÞ �DBðkÞ �iDHðkÞ

�iDDðkþQ Þ ion�ekþQ þm �iDHðkþQ Þ �DBðkþQ Þ

�DBðkÞ iDHðkþQ Þ ionþe�k�m iDDð�k�Q Þ

iDHðkÞ �DBðkþQ Þ iDDð�kÞ ionþe�k�Q�m

0BBBB@

1CCCCA,

ð2Þ

where k¼(k, ion), with on¼(2n+1)pT being a fermion Matsubarafrequency, m is the chemical potential. Here, iDH(k) is thep-pairing order parameter, which appears generically in thecoexistence of the DDW and superconductivity. In Ref. [19] there

was a sign mistake in the matrix elements G�134 and G�1

43 (see

Ref. [21]).We have extended the FLEX approximation to the coexisting

DDW and DSC state. We use here the weak-coupling gap equa-tions for the competing order parameters DD,B,H(k)

iDDðkÞ ¼T

N

Xku

½Uþ2Vð0Þ�Vðk�kuÞþPsðk�kuÞþPcðk�kÞ�G12ðkuÞ,

ð3Þ

DBðkÞ ¼T

N

Xku

½UþVðk�kuÞþPsðk�kuÞ�Pcðk�kuÞ�G13ðkuÞ, ð4Þ

iDHðkÞ ¼T

N

Xku

½UþVðk�kuÞþPsðk�kuÞ�Pcðk�kuÞ�G14ðkuÞ, ð5Þ

with the static spin and charge fluctuation interactionsPs,c(q)¼Ps,c(q, inm¼0). The FLEX approximation yields

PsðqÞ ¼ 12U2 3wsðqÞ�w0

s ðqÞ� �

; ws ¼ w0s ð1�Uw0

s Þ�1, ð6Þ

PcðqÞ ¼ 12U2 wcðqÞ�w0

c ðqÞ� �

; wc ¼ w0c ð1þUw0

c Þ�1, ð7Þ

The irreducible spin and charge susceptibilities are calculatedfrom

w0s,cðqÞ ¼�

T

N

Xk

Xon

½G11ðkþqÞG11ðkÞþG12ðkþqÞG21ðkÞ

7G13ðkþqÞG31ðkÞ7G14ðkþqÞG41ðkÞ�,ð8Þ

where q¼(q, inm¼0), with nm¼2mpT being a boson Matsubarafrequency. The plus sign is for the spin susceptibility ws

0 and theminus sign is for the charge susceptibility wc

0. In Eqs. (3)–(5) wehave taken account of the nearest-neighbor Coulomb repulsionV(q)¼2V(cos qx+cos qy) within the Hartree–Fock approximation.

In Eqs. (3)–(5) the effective interactions are dominated by thespin fluctuation interaction Ps due to the fact that the system is inthe vicinity of an antiferromagnetic instability. Since the repulsivespin fluctuation interaction Ps(q) has a large peak at q¼Q, it isattractive in the dx2�y2-wave channel and all the order para-meters DD(k), DB(k), and DH(k) have dx2�y2-wave symmetry.Therefore, we assume that they have the simple form DD,B,H(k)¼DD,B,H � dk, with dk¼cos kx�cos ky. We use the fact that DSC orderparameter DB(k) can be chosen to be real. Then both the DDW andp-pairing order parameters, iDD(k) and iDH(k), must be purelyimaginary because of their dx2�y2-wave symmetry.

Eq. (2) can therefore easily be inverted and the sum overfrequencies in Eqs. (3)–(5) carried out. Then we obtain thefollowing coupled gap equations for competing dx2�y2-wave

T

0.025

0.050

0.075

0.100

0.125

U=3.4

V=0.9 T*

DDW

K. Ha et al. / Physica B 406 (2011) 1459–1465 1461

order parameters DD(k), DB(k), and DH(k):

1

VþP¼

1

N

Xk

yð9ek9�9eF9Þd2k

Ekþmk

4EkEkþtanh

Ekþ

2Tþ

Ek�mk

4EkEk�tanh

Ek�

2T

� �,

ð9Þ

1

�VþP¼

1

N

Xk

d2k

1

4Ekþtanh

Ekþ

2Tþ

1

4Ek�tanh

Ek�

2T

� �, ð10Þ

1

�VþP¼

1

N

Xk

d2k

mkþEk

mk

1

4Ekþtanh

Ekþ

2Tþmk�Ek

mk

1

4Ek�tanh

Ek�

2T

� �,

ð11Þ

with Ek7 ¼ ½ðmk7EkÞ2þD2

BðkÞ�1=2, Ek ¼ ½e2

kþD2DðkÞ�

1=2, and mk ¼

½m2þD2HðkÞ�

1=2. Here Ek7 are quasiparticle energies in the coex-istent state of the DDW and superconductivity. The couplingparameter P is given by

P¼2

p

Z 10

dn1

nN�2Xk,ku

dkImPsðk�ku,nþ i0þ Þdku, ð12Þ

which characterizes the strength of the pairing interaction in thedx2�y2-wave channel mediated by spin fluctuations. The chargefluctuation interaction Pc has been neglected from its weaknesscompared to the spin fluctuation interaction Ps. Since Im Ps has alarge peak at k�k0 ¼Q, Eq. (12) yields P40, i.e., a dx2�y2-waveattraction. Note that the nearest-neighbor Coulomb interaction V

is attractive in Eq. (9) for the DDW gap DD(k) corresponding toelectron–hole pairing, but it is repulsive in Eqs. (10) and (11) forthe superconducting gaps DB(k) and DH(k) described by electron–electron pairing.

We perform calculations as a function of hole doping d¼1�n,with n¼1/NSi/niS being the band filling. The chemical potentialm is related to the band filling n by

n¼1

N

Xk

1þmþEk

Ekþtanh

Ekþ

2Tþm�Ek

Ek�tanh

Ek�

2T

� �, ð13Þ

At half filling (d¼0) Ek +¼Ek� and m¼0, as it should.Our Eqs. (9)–(11) look similar to the equations presented in

Ref. [14] with some important differences. First, Eq. (9) containsthe step function y(9ek9�9eF9), which means that the formation ofelectron–hole pairs takes place only between the nested statesabove and below the Fermi see. Here, eF is the Fermi energyfor the noninteracting case and is calculated from the relationn¼1/NSky(eF�ek). At half filling (n¼1) eF¼0 and the step func-tion becomes unity. Second, the gap parameters DD,B,H are relatedto the coupling constant P via the irreducible susceptibilities ws

0.Thus P must be calculated self-consistently together with DD,B,H.

Our calculations are carried out as follows: starting with initialvalues of DD,B,H, the parameters Ps,c are calculated from Eqs. (6)–(8)and (12). This is used in finding the solutions of the Eqs. (9)–(11) toget the updated DD,B,H. This procedure is repeated until convergenceis achieved. It is important to note that P is recalculated in everyiteration. In contrast to the conventional mean-field theory, in whichthe coupling parameter P is fixed, it must be calculated self-consistently, taking account of the feedback effect on spin fluctua-tions of the gaps DD,B,H. This will be especially important inexplaining our mechanism of high-temperature superconductivityinduced by the DDW as discussed below.

0.0 0.2 0.4

δ

0.0000.1 0.3

Fig. 1. Phase diagram for the pure DDW state in the absence of superconductivity.

The dashed line denotes the result calculated without taking account of the step

function y(9ek9�9eF9).

3. Results and discussion

In this section we present results obtained from solving thecoupled gap equations for the DDW, DSC, and p-pairing orderparameters, which are induced by spin fluctuations in the 2DHubbard model. We will first discuss pure DDW states in the

absence of superconductivity, then the superconducting transi-tion in the DDW state, and finally the competition between orderparameters and the influence on the phase diagram of thenearest-neighbor Coulomb repulsion. All results will be presentedfor t¼1, that is, all quantities are given in units of t. We willpresent the results for a fixed U¼3.4 and several values of V.

3.1. Pure DDW state

In the absence of superconductivity, the gap equation for theDDW order parameter DD can be written as

1

VþP¼

1

N

Xk

yð9ek9�9eF9Þd2

k

4Ektanh

Ekþm2Tþtanh

Ek�m2T

� �, ð14Þ

with Ek ¼ ½e2kþD

2DðkÞ�

1=2. In the DDW state the energy dispersion

splits into two branches 7Ek with the dx2�y2-wave gap DDðkÞ ¼DDðcoskx�coskyÞ. They are connected with each other at the nodal

points (7p/2, 7p/2) and separated by the maximum gap 2DD atthe points (7p, 0) and (0, 7p). The equations Ek7m¼0 defineeffective Fermi surface of the DDW state. For hole doping theeffective Fermi surface consists of hole pockets around the nodalpoints (7p/2, 7p/2).

The DDW transition temperature Tn is defined as the tempera-ture at which DD(T)-0. In this case, Ek-9ek9, all the quantitiesbecome the same as in the normal state. Thus, Tn is found bysolving Eq. (14) with DD(T)-0 and P in the normal state. In Fig. 1we show our results for Tn as a function of the hole doping d forU¼3.4 and V¼0.9. With increasing doping d, Tn first decreasesslowly in the small doping region and then vanishes rapidly at adoping dc, which is called the quantum critical point (QCP) of theDDW state. The dashed line shows the result of previous studies(Refs. [14–16]) calculated from Eq. (14) without regard to the stepfunction. In our calculation taking account of the step function theDDW transition temperature Tn as a function of doping decreasesmonotonously with increasing doping and goes towards thelarger dopings, which is similar to the phenomenological resultof Ref. [12], while in Refs. [14–17] taking not into account the stepfunction it moves to lower dopings after passing a large doping.This step function plays an important role in our theory for thecompetition and interplay between the DDW and superconduc-tivity, for they can coexist below the dished line as shown below.

0.0 2.0ω

5

10

15

Imχ s

(q,ω

)

1.0

Fig. 4. Spin susceptibility Im ws(q, o) at a doping d¼0.21. The dashed line and

solid line correspond to T/T*¼1.5 (normal state) and T/T*¼0.1(DDW state),

respectively.

5

K. Ha et al. / Physica B 406 (2011) 1459–14651462

For comparison we discuss the pure superconducting transi-tion in the absence of DDW order. In this case the gap equation forthe DSC order parameter DB can be written as

1

�VþP¼

1

N

Xk

d2k

2½ðek�mÞ2þD2BðkÞ�

1=2tanh

½ðek�mÞ2þD2BðkÞ�

1=2

2T,

ð15Þ

This is the well-known BCS equation for the DSC gap inducedby spin fluctuations. In the limit of V¼0 and at half-filling (m¼0)two Eqs. (14) and (15) are the same with each other, yieldingTn¼Tc. For a finite repulsion V, Tn is higher than Tc at small

dopings. Therefore, the introduction of the nearest-neighborCoulomb repulsion V yields a possibility of the DDW statedominating in the small region of doping. In Fig. 2 we presentthe doping dependence of the temperatures Tn and Tc for V¼0.0and 0.9 in the uncoupled case. Tn decreases rapidly with increasein doping. For V¼0.9 Tc is lower than Tn at lower dopings dodc,whereas it decreases so rather slowly to become nonzero at largedopings d4dc where Tn

¼0. Note that with the increase of V theDDW-transition temperature Tn is much higher and the corre-sponding QCP dc shifts slightly to higher dopings.

0.0 0.2 0.4

δ

0.000

0.025

0.050

0.075

0.100

0.125

T

U=3.4 T* (V=0.9)

Tc (V=0.9)

Tc (V=0.0)

T* (V=0.0)

0.1 0.3

Fig. 2. Phase diagram for the uncoupled superconducting and DDW state for

V¼0.0 and 0.9.

0.0

T/T*

0

1

2

3

4

5

6

7

2ΔD

/kT

*

U=3.4

V=0.9

0.2 0.4 0.6 0.8 1.0

Fig. 3. Temperature dependence of the DDW gap 2DD/kT* at a doping d¼0.21.

Dashed line is the result of the conventional weak-coupling theory.

T/T*

1.5 0.0 0.5 1.00

1

2

3

4

P

DDW Normal

Fig. 5. Temperature dependence of the pairing interaction strength P.

Below Tn we calculate the DDW gap parameter DD, the spinfluctuation spectrum Im ws(q, n), and the coupling parameter P bysolving the Eqs. (14) and (12) self-consistently within the FLEXapproximation. The resulting temperature dependence of DD isshown in Fig. 3 along with the one of the conventional weak-coupling theory where the coupling parameter is constant. Wefind that the DDW gap develops much more rapidly within theFLEX approximation (solid line) as compared with the weak-coupling theory (dashed line). This behavior is associated with thefeedback effect on spin fluctuations of the DDW gap.

Fig. 4 shows a typical result for the spin fluctuation spectrumIm ws(Q, n) at the antiferromagnetic wave vector obtained withinFLEX approximation. In the normal state the spectrum is com-paratively structureless, as shown by the dashed line. The solidline shows results in the DDW state. The opening of the DDW gapleads to a suppression of spectral weight at low frequencies, butat higher frequencies a strong resonance-like peak appears inIm ws(Q, n). The appearance of such a strong peak in the spinfluctuation spectrum yields an enhancement of the couplingparameter P in Eq. (15), as shown in Fig. 5. This in turn leads toan increase of the DDW gap, resulting in a positive feedbackeffect. This is the reason for the almost jump-like increase of theDDW gap in company with the pairing interaction below Tn. Sucha strong enhancement of the spin fluctuation pairing interactiondue to feedback of the DDW gap will especially important inexplaining high-temperature superconductivity induced by DDWas shown below.

0.0 0.2

T/T*

0

2

4

6U=3.4 V=0.9

ΔB

ΔD

2Δ/kT

*

0.4 0.6 0.8 1.0

K. Ha et al. / Physica B 406 (2011) 1459–1465 1463

3.2. Superconducting transition from the DDW state

We now study the superconducting transition from the DDWstate within FLEX theory, neglecting the p-pairing order para-meter. In the DDW state the equation for the superconductingtransition temperature Tc is given from Eq. (10) as follows:

1

�VþP¼

1

N

Xk

d2k

4

1

Ekþmtanh

Ekþm2Tc

þ1

Ek�mtanh

Ek�m2Tc

� , ð16Þ

where Ek ¼ ½e2kþD

2DðkÞ�

1=2 is the quasiparticle energy in the DDWstate. Here, we use the DDW gap DD and the coupling parameter P

calculated in the pure DDW state by solving Eqs. (14) and (12)self-consistently. Since the coupling parameter P is stronglyenhanced due to the feedback effect on spin fluctuations of theDDW gap as shown above, the superconducting transition tem-perature Tc in the DDW state can be much larger than expected inthe spin fluctuation theory without DDW order.

In Fig. 6 we show the result for the Tc calculated from Eq. (16)in the DDW state. Near half-filling Tc is zero. With increasingdoping Tc starts to increase at a finite doping d1, reaching amaximum at an optimal doping dm, then decreases with furtherincrease of doping as Tc¼Tn as far as the QCP dc. After passing dc,Tc becomes the pure superconducting transition temperature inthe absence of DDW order.

In order to get better understanding of this behavior, we take alook at the integral in Eq. (16). The dominant contribution to theintegral arises from the vicinity of the effective Fermi surfacedefined by the equations Ek7m¼0. For hole doping (mo0) theFermi surface Ek+m¼0 consists of hole pockets around the nodalpoints (7p/2, 7p/2). At small dopings, since small hole pocketsare opened around the nodal points and there dk

2-0, Eq. (14)yields Tc-0. When the hole pockets are large enough, for largedoping, dk

2 has a finite value near the Fermi surface, resulting in afinite Tc. This is the reason for the increase of Tc with increasingdoping. Close to the QCP dc, where the DDW gap vanishes, thedensity of states is closer to the one for the normal state, so thatthe transition temperature Tc for the DDW-induced superconduc-tivity is much higher than expected for the pure superconductiv-ity induced by spin fluctuations because the coupling parameter P

is strongly enhanced due to the feedback effect on spin fluctua-tions of the DDW gap. A further result of this feedback effect isTc¼Tn close to the QCP dc. This is caused by the jump-like increaseof both the DDW gap and coupling parameter below Tn.

It should be pointed out that such a strong enhancement of thesuperconducting transition temperature Tc in the DDW state close

0.0 0.2 0.40.000

0.025

0.050

0.075

0.100

0.125

T

U=3.4 V=0.9

T *

Tc

Tc

δc

DDW

DDW+DSC DSC

Normal

δ0.1 0.3

Fig. 6. Phase diagram of the hole-doped 2D Hubbard model with U¼3.4 and

V¼0.9. The dashed–dotted line shows the DDW temperature T*, the solid and

dashed lines show the superconducting Tc in the presence and absence of DDW

order, respectively. dc is the QCP at which the DDW state vanishes.

to the QCP is quite a new phenomenon and to the best of ourknowledge no such theoretical demonstration exists. Our result isobtained from the following two facts. The first is the introductionof the step function y(9ek9�9eF9) in Eq. (14) for the DDW gap,which yields the coexistence region of the DDW and DSC close tothe QCP at which the DDW state vanishes. Without takingaccount of this step function, no coexistence is found betweenthe DDW and DSC because the boundary of the DDW state isgiven as the dashed line in Fig. 1. The second is the feedback effecton spin fluctuations of the DDW gap, which leads to a strongenhancement of the coupling parameter P in Eq. (16) for thesuperconducting transition temperature Tc in the DDW state.

3.3. Competition between order parameters

Here, we discuss the results obtained from self-consistentlysolving the coupled gap Eqs. (9)–(11) for competing DDW, DSC,and p-pairing as the possible order parameters within the FLEXapproximation. When the DDW and superconductivity coexist ppairing appears generically, the corresponding order parameterDH is usually tiny. The reason is that since the factor (mk�Ek)/mk

in Eq. (11) is smaller than one, Cooper pairing is dominant andsuppresses p pairing strongly. Hence the p-pairing order para-meter is very small compared to the others, such that its feedbackinto the gap equations for DD and DB can be neglected. Similarresults are obtained in the renormalized mean-field analysis ofantiferromagnetism and superconductivity [20].

In Fig. 7 we show the temperature dependence of the DDWand DSC gaps DD and DB, calculated at a fixed d¼0.21. The dottedand dashed lines show the results for uncoupled DD and DB,

Fig. 7. Temperature dependence of competing DDW and DSC order parameters DD

(two outer lines) and DB (two inner lines) calculated at d¼0.21. Dotted and dashed

lines are the corresponding results for uncoupled case.

0.0 0.2 0.4 δ

0.0

0.1

0.2

0.3

0.4

0.5U=3.4V=0.9

δc

orde

r pa

ram

eter ΔD

ΔBΔB

0.1 0.3

Fig. 8. Doping dependence of the order parameters DD and D, calculated at

T/T*¼0.02.

0.000

0.050

0.100

T

0.0 0.1 0.2 0.3 0.40.000

0.050

0.100

T

δ

0.050

0.100T

0.000

U=3.4 V=0.0

U=3.4V=0.9

U=3.4 V=1.3

DSC

Normal

Normal

Normal

DDW+DSC

DDW

DDW+ DSC

DDW+ DSC

DSC

Tc

Tc

Tc

T*

T*

Fig. 9. Evaluation of the phase diagram for U¼3.4 and different nearest-neighbor

Coulomb repulsions.

K. Ha et al. / Physica B 406 (2011) 1459–14651464

respectively. After the growth of the DSC gap below Tc, the DDWgap is slightly suppressed and remains almost constant withlowing temperature. The DDW-induced-DSC gap develops athigher temperature, but its magnitude is small compared to thepure DSC gap. Below Tc the DDW and DSC gaps coexist.

In Fig. 8 we show the doping dependence of the gap para-meters DD and DB, calculated at T/Tn

¼0.02. The uncoupled DDWgap (dotted line) first decreases monotonically with increasing d,then vanishing rapidly at the QCP dc¼0.26. The uncoupled DSCgap (dashed line) is smaller than the DDW gap at lower dopingsdodc due to the nearest-neighbor Coulomb repulsion V, but itdecreases so rather slowly to become nonzero up to large dopingsdcod2¼0.32. The solid lines show the doping dependence ofcompeting gap parameters DD and DB. Near half filling the DSCgap DB is completely suppressed due to the strong competitionwith the DDW gap DD. Upon further doping, DB first appears at adoping d1¼0.12 and increases with increasing d, coexisting withthe DDW gap up to dc. The DDW gap is slightly suppressed by theappearance of the DSC gap in the doping region d1ododc. Afterpassing dc the DDW gap is zero and only the pure DSC gap exists.Thus the QCP dc separates a pure superconducting state at largedopings from a ground state containing both the DDW andsuperconducting gaps.

Finally we discuss the dependence of the phase diagram on thenearest-neighbor Coulomb repulsion V. Fig. 9 shows the phasediagrams calculated for V¼0.0, 0.9, and 1.3. The solid line showsthe superconducting transition temperature Tc. The dashed–dotted line describes the DDW transition temperature Tn. In thecase of negligible Coulomb repulsion, i.e., V¼0.0, the supercon-ducting phase is dominant in the whole region of doping andwipes out the DDW phase as shown in the upper panel of Fig. 9.Lowering the temperature from high values one crosses the solidline and enters the pure superconducting region. Increasing V

the pure superconducting state is more and more suppressedwhereas the DDW state becomes dominant at lower dopings. Thisis shown in the middle panel of Fig. 9 for V¼0.9, which is thesame diagram as Fig. 6. Moreover, the superconducting region isnow split into two regions. In the overdoped region only a puresuperconducting phase exists, whereas in the underdoped regionthe superconducting and DDW phases coexist with each other. In

the underdoped region a pure DDW phase exists above Tc with apseudogap of d-wave symmetry in the excitation spectrum. WhenV exceeds a threshold, i.e., V41.3, the total effective interaction�V+P becomes repulsive and the pure superconducting phase iscompletely suppressed in the overdoped region, as shown in thelower panel of Fig. 9.

As shown above, it is in general necessary to include thenearest-neighbor Coulomb repulsion V in order to stabilize theDDW, which gives a good description of the pseudogap behaviorin the underdoped region. Note that the DDW-induced-super-conducting Tc remains almost constant even with increase of theCoulomb repulsion V, as shown in the middle and lower panelsof Fig. 9. This behavior may be explained as follows: the increaseof V leads to an enhancement of the coupling parameter P due tothe feedback effect of the enhanced DDW gap, which givescompensation for a decrease in Tc due to the increase of V. Theposition of the maximum Tc is close to the QCP of the DDW stateand also does hardly move with increasing V.

4. Conclusion

We have found that high-temperature superconductivity isinduced due to the feedback effect on spin fluctuations of theDDW in the 2D Hubbard model. Using the FLEX approximationwe have derived the coupled gap equations for the DDW, DSC, andp pairing as the possible order parameters induced by exchangeof spin fluctuations. In order to stabilize the DDW state withrespect to phase separation in the underdoped region, we haveincluded a small nearest-neighbor Coulomb repulsion within theHartree–Fock approximation. Calculating the strength of thepairing interaction in the dx2�y2-wave channel mediated by spinfluctuations self-consistently, the coupled gap equations forcompeting order parameters have been solved and the corre-sponding transition temperatures have been found. Our principlefindings are: (1) Even when nesting of the Fermi surface is perfect,as in a square lattice with only nearest-neighbor hopping t, due tothe feedback effect on spin fluctuations the DDW and DSC cancoexist in the relatively large region of doping close to the QCP atwhich the DDW state vanishes. (2) The superconducting transi-tion temperature Tc in the DDW state can be much highercompared to pure superconductivity, since the pairing interactionstrength is enhanced strongly due to the feedback of the DDWgap. (3) With an increase of the Coulomb repulsion the DDW-induced-superconducting Tc remains almost constant whereasthe pure Tc decreases strongly. (4) By the QCP the phase diagramis split into two regions: in the overdoped region only apure superconducting phase exists, whereas in the under- andoptimally doped region the superconducting and the DDW orderparameters coexist. Our results give a good explanation forhigh-temperature superconductivity and pseudogap phenomenain cuprates.

In the present work, we have developed a weak-couplingtheory of the competition between DDW and DSC in 2D Hubbardmodel, using the static spin fluctuation obtained within FLEXapproximation and ignoring the self-energy effect of spin fluctua-tions. For our model calculations in the weak-coupling limit wehave taken U/t¼3.4, since the antiferromagnetic instability occursfor higher values of U/t.

Acknowledgments

This work was supported by the Academy of Science of theDPR Korea and the work of K.H. by the Alexander von HumboldtFoundation.

K. Ha et al. / Physica B 406 (2011) 1459–1465 1465

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