Physica C 469 (2009) 974–978

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Physica C

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Interplay between antiferromagnetism and superconductivity in the Hubbardmodel with frustration

Kenji Kobayashi a,*, Hisatoshi Yokoyama b

a Department of Natural Sciences, Chiba Institute of Technology, Shibazono, Narashino, Chiba 275-0023, Japanb Department of Physics, Tohoku University, Sendai 980-8578, Japan

a r t i c l e i n f o

Article history:Accepted 23 February 2009Available online 31 May 2009

PACS:71.10.Fd74.20.Mn74.25.Dw

Keywords:Hubbard modelVariational Monte Carlo methodSuperconductivityCuprate

0921-4534/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.physc.2009.02.017

* Corresponding author. Address: Department of Naof Technology, 2-1-1, Shibazono, Narashino, Chiba 275454 9605.

E-mail address: koba@sun.it-chiba.ac.jp (K. Kobaya

a b s t r a c t

Coexistence of and competition between antiferromagnetism (AF) and d-wave superconductivity (SC) arestudied for a Hubbard model on the square lattice with a diagonal transfer t0, using a variational MonteCarlo method. The following improvements are introduced into the trial function: (1) Coexistence of AFand d-wave singlet gaps that allows a continuous description of their interplay, (2) band renormalizationeffect, and (3) refined doublon–holon correlation factors. Optimizing this function for a strongly corre-lated value of U=t, we construct a phase diagram in the d (doping rate)-t0=t space, and find that fort0=t P �0:15 a coexisting state is realized, whose range of d extends as t0=t increases. In contrast, fort0=t ¼ �0:3, AF and SC states are mutually exclusive, and a coexisting state does not appear. In connectionwith the ‘‘two-gap” problem, we confirm even for the present refined function that the gradient ofmomentum distribution function at the antinodal point mainly dominates the magnitude of the d-waveSC correlation function.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Since the proximity of antiferromagnetic and superconductingphases is a feature universal to all cuprate superconductors, it isappropriate to consider that the origin of the two phases shouldbe identical, probably the antiferromagnetic spin correlation. Inmost experiments of cuprates, however, these two phases aremutually exclusive and do not microscopically coexist [1]. Mean-while, a recent NMR experiment for a multi-layered cuprate ar-gued that the two phases coexist in a single CuO2 (outer) planeowing to its extreme flatness as compared with other cuprates[2]. This discrepancy concerning the coexistence gives a key tomechanism of high-Tc superconductivity (SC), and is now activelyinvestigated [3]. Another motivation of this study is the so-called‘‘two gap” problem. Although it has been an orthodox interpreta-tion that the pseudo gap is an incoherent singlet gap as a precursorof SC, recent experiments by different means found that the gap inthe underdoped regime exhibits an opposite d (hole doping rate)dependence between the nodal ðk � ðp=2; p=2ÞÞ and antinodalðk � ðp; 0ÞÞ regions [4], leading to intensive arguments whether

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tural Sciences, Chiba Institute-0023, Japan. Tel./fax: +81 47

shi).

this different d dependence is explained by a single gap or bytwo [5]. An important clue to this problem was provided by veryrecent experiments of angle-resolved photoemission spectroscopyand scanning tunneling microscopy [6], which found two kinds of(pseudo)gaps near the antinodal point. One of them is related tothe SC gap, and the other may be of other origins such as chargeinhomogeneity.

Previous theoretical studies on the coexistence using variationalMonte Carlo (VMC) methods for the Hubbard and t-J models[7–10] drew conclusions that coexisting states are stabilized forsmall doping rates ðd K 0:1Þ. On the other hand, a recent studyby cellular dynamical mean field calculations for the Hubbardmodel argued that at strong coupling ðU J 8tÞ the two phasesdo not mix [11]. Thus, it is important to clarify the stability of coex-isting states as a function of U=t; t0=t and d in theory. In this work,we consider the above two problems using a VMC method [12],which is useful to reliably treat a wide range of parameters in cor-related systems. We adopt a wave function improved on a previousone [13] to properly describe the interplay between antiferromag-netism (AF) and SC, and applied it to the Hubbard model to linkweak and strong coupling (t-J model) regimes. In a preceding pub-lication [14], the present authors have checked U=t dependence ofthe stability of coexisting state for t0=t ¼ �0:3 within the same for-mulation; for U K 2W ðW � 8t: band width), the pure AF state ismore stable than the SC state, whereas for U J 2W an area of pureSC appears in the phase diagram. In this article, we focus on

K. Kobayashi, H. Yokoyama / Physica C 469 (2009) 974–978 975

strongly correlated cases, fixing U at a typical value 30t (>2 W), andstudy the d and t0=t dependence.

2. Method

We consider the Hubbard model on a square lattice with thenext–nearest–neighbor transfer t0,

H ¼Hkin þHU ¼Xkr

eðkÞcykrckr þ UX

j

nj"nj#; ð1Þ

eðkÞ ¼ �2tðcos kx þ cos kyÞ � 4t0 cos kx cos ky:

Eq. (1) with t0=t < 0 represents the hole-doped high-Tc cuprates,and electron-doped (more-than-half-filled) ones can be treated asless-than-half-filled systems with t0=t > 0 owing to a particle-holetransformation. We use t as the unit of energy and the lattice con-stant as the unit of distance. To this model, we apply an optimiza-tion VMC (or correlated measurement) method [15], which caneffectively optimize the parameters in the whole range of U=t.

As a variational wave function, we use a Jastrow type,W ¼ PQPGU, where PG is the Gutzwiller (on-site) projector, andPQ the doublon–holon binding factor[16]. The following improve-ments are introduced into the wave function, as in the precedingwork [14]: (1) coexistence of AF and d-wave singlet gaps to directlycheck the cooperation or competition between them [7–10], (2)band renormalization effect owing to electron correlation byadjusting hopping integrals in U, and (3) refined doublon–holoncorrelation factors, which control the effect of Mott transition nearhalf-filling more precisely [16,17].

As the one-body part U in W, we use a dx2�y2 -wave singlet statewith a nearest–neighbor pairing gap eDd for Ne electrons

U ¼X

k

ukbyk"by�k#

!Ne2

j0i; ð2Þ

Fig. 1. (a) Optimized values of gap variational parameters in W as a function of d.(b) Difference in total energy between optimized gapped and non-orderedðeDd ¼ eDAF ¼ 0Þ states [Eq. (3)], and its kinetic ðDEkinÞ and interaction ðDEUÞcomponents. The data in both panels are for t0=t ¼ 0 and U=t ¼ 30.

Fig. 2. By using the optimized W, the expectation values of staggered magnetizationm (triangles) and d-wave nearest-neighbor pair correlation function PdðrÞ for thefarthermost distance r ¼ ðL=2; L=2Þ (diamonds) are plotted as a function of d forU=t ¼ 30. The values of t0=t differ among the five panels.

where

uk ¼DdðkÞ

~ek � ~fþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið~ek � ~fÞ2 þ jDdðkÞj2

q ;

976 K. Kobayashi, H. Yokoyama / Physica C 469 (2009) 974–978

and DdðkÞ ¼ eDdðcos kx � cos kyÞ: Simultaneously, an AF gap eDAF isintroduced into bykr as bykr ¼ akcykr þ rbkcykþQr and bykþQr ¼�rbkcykr þ akcykþQr [7], where

akðbkÞ ¼12

1� ðþÞ nkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2

k þ eD2AF

q0B@

1CA264

37512

;

with nk ¼ �2tðcos kx þ cos kyÞ;k 2 AF Brillouin zone, and Q ¼ðp;pÞ; t0 does not enter in nk according to the form of a mean-fieldsolution for the ðp;pÞ-AF order.

Fig. 3. Phase diagram constructed within the present wave function in d� t0=tplane for U=t ¼ 30.

Fig. 4. Momentum distribution function nðkÞ evaluated by the optimized W for U=t ¼ 30t0=t ¼ 0;�0:15;�0:3 are simultaneously plotted in each panel.

In U, variational parameters to be optimized are eDd; eDAF;~f, and~ek. The renormalized band ~ek is adjusted by optimizing the hoppingparameters therein, up to the fourth-neighbor sites [14]. Note thata finite value of eDd does not necessarily mean the formation of a SCorder in contrast with eDAF, and ~f is reduced to the chemical poten-tial for U=t ! 0.

In addition to the seven in U, we have 21 variational parametersin the projectors PQ and PG [14]. Using optimization VMC tech-niques, we minimize the energy by searching for the optimal setof these 28 parameters. As pilot calculations, we fix the system sizeðNs ¼ L� LÞ at L ¼ 12 with periodic–antiperiodic boundary condi-tions. This system size can be quantitatively insufficient to deducethe bulk limit, and we would like to check the size dependence in afuture study.

3. Results and discussions

Let us start with the first topic, stability of coexistence. InFig. 1(a), the optimized values of the gap variational parameterseDd and eDAF are plotted as a function of doping rate d for t0=t ¼ 0.For d < 0:10, both eDd and eDAF have finite values, suggesting a coex-isting state is realized, while for 0:10 < d < 0:19 the optimizedstate with eDd – 0 and eDAF ¼ 0 is indicative of pure SC. Correspond-ing to the finite optimal values of gap parameters, the total energyis reduced from the non-ordered state (projected Fermi sea) Wno, asshown in Fig. 1(b), where the total energy gain is defined by

DEtot ¼ E½Wno� � E½W�; ð3Þ

is shown for six values of d along the path ð0; 0Þ ! ðp;0Þ ! ðp;pÞ ! ð0; 0Þ. Data for

Fig. 5. Comparison between SC correlation function PdðL=2; L=2Þ and jrnðkÞj atk ¼ ðp;0Þ for t0=t ¼ 0;�0:15. The scale of jrnðkÞj is adjusted so that the maxima ofthe two quantities should be roughly the same.

K. Kobayashi, H. Yokoyama / Physica C 469 (2009) 974–978 977

with E½Wno� being the optimized energy under the conditioneDd ¼ eDAF ¼ 0. In Fig. 1(b), the two components of DEtot, kinetic partDEkin and interaction part DEU , are also plotted; the result ofDEkin > 0 and DEU < 0 means that this SC transition is kinetic-en-ergy driven, which is a common feature of SC in strongly correlatedregimes [18]. The decrease of DEtot with the increase of d is contra-dictory to the dome-shaped behavior of SC condensation energyestimated from the specific heat measurement for the cuprates[19]. Hence, DEtot, even in the pure SC regime, does not representthe energy scale of SC but rather that of the pseudo gap; the itiner-ancy of carriers is enhanced by the coherent part of eDd [20].

Because eDd includes incoherent components of the d-wave gap,we have to consider a measure directly correlating to the SC gap(coherent part of eDd). To this end, the dx2�y2 -wave SC correlationfunction for the nearest–neighbor pairing is convenient in the pres-ent method, which is defined by

PdðrÞ ¼1Ns

Xr0

Xs;s0

rðsÞrðs0ÞhDyðr0 þ r; sÞDðr0; s0Þi; ð4Þ

with

Dðr; sÞ ¼ 1ffiffiffi2p cr"crþs# � cr#crþs"ð Þ;

rðsÞ ¼ þ1 for s ¼ ð�1; 0Þ and rðsÞ ¼ �1 for s ¼ ð0; �1Þ. In Fig. 2, weplot PdðrÞ for the farthermost distance r ¼ ðL=2; L=2Þ and simulta-neously staggered magnetization m as an indicator of AF order fordifferent values of t0=tð¼ 0; �0:15; �0:3Þ. In the case of t0=t ¼ 0shown in Fig. 2(c), the AF order stabilized by the nesting conditionat half-filling ðd ¼ 0Þ becomes weak as the doping rate increases,agreeing with eDAF as mentioned above, whereas PdðrÞ increases asd increases for small d, whose behavior is opposite to that of eDd.As a whole, PdðrÞ exhibits a dome shape. Anyway, a coexisting stateof SC and AF orders is realized for small dopings ðd < 0:10Þ, and apure SC state for higher dopings.

Next, we consider t0=t dependence. The results for jt0=tj– 0, alsoplotted in Fig. 2, exhibit marked electron-hole asymmetry with re-spect to the stability of the coexisting state. On the electron-dopedside ðt0=t > 0Þ, the magnitude of PdðrÞ becomes small as jt0=tj in-creases. Because the area of PdðrÞ – 0 diminishes but that ofm – 0 extends, the area of coexistence extends to almost the wholerange of finite values of PdðrÞ and m, and the pure SC state does notappear. In contrast, on the hole-doped side (t0=t < 0), SC is en-hanced in the highly-doped regime, but tends to be weak for smalldoping, as compared with the t0=t ¼ 0 case. As a result, fort0=t ¼ �0:15 (Fig. 2(b)), the coexisting area slightly diminishes to0:01 < d < 0:08, whereas the pure SC area extends to d � 0:3. Fort0=t ¼ �0:3, however, the SC and AF orders become mutually exclu-sive and never coexist as seen in Fig. 2(a). As a summary, we draw arough phase diagram in the d� t0=t plane in Fig. 3 for U=t ¼ 30,concluding that the area of coexistence is sensitive to the valueof t0=t. The d dependence of SC and AF orders is similar to that ofthe t-J model as a whole, but the t-J model has a wide area of thecoexisting phase even for t0=t ¼ �0:3 [21]. We will check this dis-crepancy in detail in a coming publication.

In the following, we argue that the electronic state at the anti-nodal point is essential to SC. In the preceding study [13], we no-ticed within a simple d-wave singlet wave function that thegradient of momentum distribution function nðkÞ;rnðkÞ ¼@nðkÞ@kx

; @nðkÞ@ky

� �, at the antinodal point k ¼ ðp; 0Þ deeply influences

the magnitude of d-wave SC correlation function PdðrÞ. This is be-cause jrnðkÞj is a good measure of the effect of Fermi surface,and the antinodal points are concurrently van Hove singularities,which are connected by the AF pair scattering vector Q ¼ ðp;pÞ.Here, we check this thought by using the present more refinedwave function. In Fig. 4, we plot nðkÞ for several values of t0=t

simultaneously in one panel; d is varied among the six panels. Athalf-filling ðd ¼ 0Þ, the behavior of nðkÞ is smooth for every k, indi-cating insulating states, and indistinguishable among different val-ues of t0=t. On doping carriers, however, conspicuous changes ofnðkÞ appear near ðp;0Þ for t0=t > 0 and near ðp=2;p=2Þ fort0=t < 0. The former change indicates that the AF state becomesmetallic, and the latter the manifestation of d-wave SC, whichexhibits Fermi-liquid-like behavior in the node direction. Whatwe notice here is that large t0=t dependence around the antinodalpoint continues for a wide range of d, which roughly correspondsto the range of SC. In weak correlation theories, d-wave SC of theAF-correlation origin is selectively enhanced when the Fermisurface overlaps with the antinodal point, where density of statediverges. In strongly correlated systems, however, the Fermi sur-face (line) cannot be determined clearly, but we can estimate its ef-fect, which no longer works on a (Fermi) line but in a certainexpanded area of k, with jrnðkÞj [13]. In Fig. 5, we comparejrnðp; 0Þj with PdðL=2; L=2Þ for three values of t0=t as a functionof d. The behavior of jrnðp;0Þj qualitatively coincides with thatof PdðL=2; L=2Þ, which confirms that the glue of SC pairing is theAF correlation, and simultaneously means that the electronicstructure at (p; 0) mainly controls the strength of SC. The latterpoint denies a simple dichotomy of electronic roles in the k spaceinto SC and the pseudo gap.

978 K. Kobayashi, H. Yokoyama / Physica C 469 (2009) 974–978

4. Summary

With the high-Tc cuprates in mind, we have studied the inter-play between AF and SC in the Hubbard model, using the wavefunction in which AF and d-wave gaps are simultaneously intro-duced, and a band renormalization effect and refined doublon–ho-lon correlations are taken into account. For large U=t, it is foundthat the d-wave SC and AF ordered states become competing orcoexisting, according to the values of t0=t and d. The derivative ofelectron occupation at the antinodal point is closely related tothe magnitude of SC correlation.

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