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Stability of superconductivity and antiferromagnetism in two-dimensional Hubbard model with diagonal transfer
Journal of Physics and Chemistry of Solids 67 (2006) 47–49 www.elsevier.com/locate/jpcs
Stability of superconductivity and antiferromagnetism in two-dimensional Hubbard model with diagonal transfer Hisatoshi Yokoyama a,*, Yukio Tanaka b, Masao Ogata c a Department of Physics, Tohoku University, Sendai 980-8578, Japan Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan c Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan b
Abstract Using an optimization variational Monte Carlo method, we study the effect of diagonal transfer t 0 in a two-dimensional Hubbard model on stability of d-wave superconductivity (SC) and antiferromagnetism (AF). For hole doping (t 0 /t!0), the condensation energy of SC is enhanced by a couple of times, while that of AF rapidly decreases with increasing jt 0 /tj, and AF itself is intrinsically unstable for t 0 /t(0. For electron doping (t 0 /tO0), conversely, SC tends to be suppressed, whereas the AF state not only comes to have a low energy but recovers the intrinsic stability. These results well explain various aspects of cuprate superconductors. q 2005 Elsevier Ltd. All rights reserved. PACS: 74.80.Fp; 74.25.Fy Keywords: D. Superconductivity
1. Introduction The Hubbard model on a square lattice is a simple model, which probably captures the essence of cuprate superconductors [1]. Despite considerable efforts, acquired knowledge on this model is very slight, especially in the intermediate and strong coupling regimes. For example, it is still controversial (i) whether superconductivity (SC) appears in this model, (ii) whether the strong coupling regime of this model represents substantially the same physics as the t–J model with small J/t, and (iii) how longrange transfer terms work here, which are considered indispensable to describe individual properties of cuprates. Recently, we have obtained intriguing results concerning the above issues (i) and (ii) [2], using an optimization variational Monte Carlo (OVMC) method, which is highly effective to study ground-state properties with varying U/t. Steady d-wave SC appears only for large values of U/t, namely U/tT6.5, below which the condensation energy DE is very small. The properties of SC exhibit a crossover at around Uco/tZ11–13 from a conventional BCS type to a new one [See Fig. 1a], in which superconducting (SC) transition is induced * Corresponding author. Tel.: C81 22 795 6444; fax: C81 22 795 6447. E-mail address: [email protected] (H. Yokoyama).
0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.10.024
by the decrease in kinetic energy. Such a feature has been actually observed by experiments of optical conductivity for cuprates [3]. Furthermore, DE in this region is well fitted with exp (-at/J); this implies the effective attractive interaction is the exchange process J(Z4t2/U), and is consistent with the viewpoint of the t–J model [4,5]. In these proceedings, we consider the effect of the diagonal transfer t 0 on SC and antiferromagnetic (AF) states, and how it influences the properties mentioned above. It is revealed that the Hubbard model with the t 0 term consistently explains a variety of experiments on cuprates.
2. Model and method On an earlier occasion, second- (t 0 ) and third-neighbor (t 00 ) transfer terms were introduced to fit the Fermi surface obtained through ARPES [6]. Later, using the parameters thus determined, namely the t–t 0 –t 00 –J model, various aspects of ARPES spectra were successfully explained [7]. For simplicity, here we add only the diagonal transfer t 0 to the Hubbard model on a square lattice: X † X H Z Hkin C Hpot Z 3k cks cks C U nj[njY; (1) j
ks 0
with 3k ZK2tðcos kx C cos ky ÞK4t cos kx cos ky . For cuprates, the sign of t 0 =t is negative regardless of whether the carrier is
48
H. Yokoyama et al. / Journal of Physics and Chemistry of Solids 67 (2006) 47–49
(A)
(B)
(C) t' = 0 L = 12
(a)
d-wave AF n
∆/t
1
(b)
t' = 0 L = 8–16
1.0 0.9722 0.9167 0.8611 0.8056
0.5 8
U/t 10
12
d AF
0
10
20
30
0.8
0.85
U/t
0.9 n
0.95
1
Fig. 1. Behavior of a variational parameter D for the d-wave superconducting (solid) and antiferromagnetic (open) states, (a) as a function of U/t for several values of n near half filling, and (b) as a function of n for three values of U/t. As in (a), the correlation strength U=t can be classified into three regimes: (A) weak, (B) intermediate, and (C) strong correlation regimes.
hole or electron. For convenience of calculation, however, we apply a particle–hole transformation, c†js / ðK1Þrj hjs , to morethan-half-filling cases. Thereby, an electron-doped system with electron density n(O1) and t 0 /tZa(!0) is converted to a holedoped systems with hole density dZjnK1j and t 0 /tZjaj(O0). We hence consider both signs of t 0 /t for n%1. To treat the local correlation accurately, we use an OVMC method [8], which enables us to efficiently calculate the expectation values with many variational parameters. For Hubbard-type models, one needs to employ a trial function (JZPF) which allows for at least second order terms of the strong coupling expansion in the correlation factor P [9], beyond the Gutzwiller-type level. In this study, we consider up to the second order in t/U and t 0 /U [10]. As a one-body part F, we adopt the Fermi sea FF, an AF Hartree–Fock state FAF and a d-wave BCS state FSC; FAF and FSC have a parameter D, which is closely connected with DE. We use systems of L!L (LZ8–16) with periodic–antiperiodic boundary conditions, and impose the closed shell condition on the electron density.
3. Results To begin with, we briefly review the aspect for t 0 Z0 in the light of the optimal value of the parameter D. hole doping
electron doping
hole doping
(a)
(b)
0.04
AF d-wave U/t 4 8 12 16
0.03
D E/t
Paying attention to the d-wave in Fig. 1a, where D=t is plotted as a function of U/t, we actually find the crossover behavior described in Section 1. In the range (A), the d-wave SC is very weak and BCS-like; the range (B) is a transient (crossover) regime, and in (C) the decrease in Ekin(ZhHkini) induces SC, which is steady and t–J-model-like. Fig. 1b shows the n-dependence of D/t. For both JSC and JAF, D/t decreases, as n goes away from half filling. Although D still remains finite for n(0.8 for JSC, it abruptly vanishes for n(0.85 for JAF (see Fig. 4a in Ref. [2]). Now, we proceed to the case of finite t 0 . In Fig. 2a, we depict the energy difference (condensation energy), DEZ Eorder KEnormal , for the d-wave and AF as a function of t 0 /t [11] for four values of U/t. Let us summarize the main features of this graph. (1) d-wave SC. (i) For hole doping (t 0 /t! 0), DE rapidly increases as jt 0 /tj increases from zero, exhibits a maximum, which is twice or three times larger than the value at t 0 Z0, at t 0 /tw0.15, then decreases for larger values of jt 0 /tj. (ii) For electron doping (t 0 /tO0), DE gradually decreases as t 0 /t increases, but remains finite even at t 0 /tZ0.5. (iii) U/t dependence: in the range (A) (U/tZ4), DE is small, even if t 0 is introduced. In (B) (U/tZ8), t 0 works effectively to increase DE. In (C) (U/tZ12 and 16), SC is stabilized over a wide range of t 0 /t. (2) AF. (i) For hole doping, DE rapidly decreases and
0.02
L = 10 n = 0.88
0.08 AF
0.06
n d-wave AF 0.96 0.88 0.80
U/t = 12 L = 10
0.04
d-wave
0.01 0
electron doping
0.104
0.02
–0.5
0
t'/t
0.5
0
–0.5
0
0.5
t'/t
Fig. 2. Condensation energy as a function of t 0 /t. (a) U/t dependence for nZ0.88, and (b) n dependence for U/tZ12. AF does not arise for U/tZ4 in (a) and for nZ0.8 in (b). In a range marked with a double-headed arrow, the Fermi surface (occupied k-points) lies closely to the van Hove singularity points.
H. Yokoyama et al. / Journal of Physics and Chemistry of Solids 67 (2006) 47–49
hole doping
D Ekin /t, D Epot /t
4. Discussions
electron doping
0.04
D Ekin D Epot U/t
4 8 12 16
0.02
0 L=10
–0.02
n=0.88 d-wave
–0.04
–0.5
0
49
0.5
t'/t Fig. 3. Difference in kinetic (solid) and potential (shadowed) energies between the normal and SC states as a function of t 0 /t for four values of U/t. The doubleheaded arrows mean the same as Fig. 2. The scattering of data point, particularly for t 0 /tO0, is ascribed mainly to the discrete k-points.
vanishes, although it has a narrow peak around t 0 /tZK0.15 [12]. For t 0 /t(0, the AF state does not satisfy the intrinsic stability condition v2E/vn2O0 (not shown), and phase separates into the AF phase at half filling and the d-wave SC state. (ii) For electron doping, the AF state not only gains energy increasingly but recovers the intrinsic stability condition (not shown). Although these results are basically interpreted within the level of a tight-binding approximation [13,14], they adequately explain some key properties of cuprates (Section 4). In Fig. 2b, n dependence of DE is shown for U/tZ12. Most aspects discussed for Fig. 2a apply to other densities. Here, we focus on the n dependence. As n decreases from half filling, DE both for SC and AF rapidly decreases in the whole range of t 0 /t. For nw0.8, DE of SC for t 0 /tT0 is considerably small; hence, to understand the stable SC in the overdoped regime, the enhancement of DE for negative t 0 /t must be an inevitable factor. As for the case of t 0 /tT0, the stability of SC is marginal at nZ0.8, and AF also becomes unstable for n(0.85 even with a positive t 0 /t. These results correspond well with the fact that AF in electron-doped cuprates vanishes with 15% doping at largest, and SC vanishes at about dZ0.2. SC normal Finally, we consider DEkin ðZEkin KEkin Þ and DEpot. As 0 mentioned for t /tZ0, DEkin (DEpot) is negative (positive) in the regime (A) and (B), but the relation is reversed in the strong correlation regime (C). In Fig. 3, t 0 /t dependence of DEkin and DEpot is shown for four values of U/t. Roughly speaking, for hole doping, the driving force of SC transition is Epot for U/ t(10, but Ekin for U/tT10, which situation is basically the same as that for t 0 Z0. For electron doping, although the behavior for sufficiently large U/t (Z16) is the same with that for hole doping, for U/tZ8 and 12, the magnitude of these quantities is small, and the sign sometimes changes, as t 0 /t is varied. We need a further study on this point.
In this paper, we have studied the effect of diagonal transfer. The features quite different between for t 0 /tO0 and for t 0 /t!0, discussed for Fig. 2, agree well with the well-known asymmetry between hole and electron dopings in the d–T phase diagram of cuprates. Considering that the energy scale of SC, DE, sensitively depends on the value of t 0 /t (Fig. 2), it seems reasonable that the different energy scales between LSCO and BSCCO observed in ARPES spectra [15] are primarily attributed to the effect of t 0 . Furthermore, the intrinsic stability of the AF state consistently explains the distinct behaviors of the AF phase between hole-doped and electrondoped cuprates [16]. For some of the issues addressed here, parallel discussions are possible with the t–J model [13]. Extended description will be given elsewhere [14]. References [1] P.W. Anderson, Science 235 (1987) 1196. [2] H. Yokoyama, et al., J. Phys. Soc. Jpn. 73 (2004) 1119; M. Ogata et al., an article in this volume. [3] D.N. Basov, et al., Science 283 (1999) 49; H.J.A. Molegraaf, et al., Science 295 (2002) 2239. [4] H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn 57 (1988) 2482; H. Yokoyama, M. Ogata, J. Phys. Soc. Jpn. 65 (1996) 3615. [5] D.J. Scalapino, S.R. White, Phys. Rev. B 58 (1998) 8222; E. Demler, S.C. Zhang, Nature 396 (1998) 733. [6] T. Tanamoto, H. Kohno, H. Fukuyama, J. Phys. Soc. Jpn 62 (1993) 717. [7] T. Tohyama, S. Maekawa, Supercond. Sci. Tech. 13 (2000) R17. [8] C.J. Umrigar, K.G. Wilson, J.W. Wilkins, Phys. Rev. Lett. 60 (1988) 1719; T. Giamarchi, C. Lhuillier, Phys. Rev. B 43 (1991) 12943. [9] H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn. 56 (1987) 1490; H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn 59 (1990) 3669; H. Yokoyama, Prog. Theor. Phys. 108 (2002) 59. Q Q Q [10] The wave function used is written as, JQ j ½1KmQj j ½1Km 0 Qj0 j ½1K ð1KgÞnj[njYF: The three variational parameters for local correlation control the onsite double occupation (g) and the binding between a doublon and holons in nearest-neighbor (m) and second-neighbor (m0 ) sites,, respectively, For details of JQ, refer to [9,14]. In OVMC calculations, typically 2.5!105 configurations were kept in the optimization process, and data of virtually several million samples were averaged to reduce the error in total energy to w10K4 t. [11] In JF and JAF, electrons occupy the k-points with 3k!3F. Since the system is finite (hence its k-points are discrete), the chosen k-points changes abruptly, when the value of t 0 /t is varied, This is not the case with JSC for Ds0, Eventually, both DESC and DEAF become not smooth functions of t 0 /t. [12] This peak for AF at t 0 /twK0.15 arises when the Fermi surface lies closely to the van Hove singularity points (0, Gp) and (Gp, 0), which are mutually connected with the AF nesting vector (p, p). Since such a situation appears when t 0 /t and n satisfy a special condition, the width of the peak is narrow. See also K. Yamaji, et al., Physica C 392–396 (2003) 229. [13] C.T. Shih, et al., Phys. Rev. Lett. 92 (2004) 227002. T. Watanabe, et al., an article in this volume. [14] H. Yokoyama, Y. Tanaka, M. Ogata, in preparation. [15] K. Tanaka, et al., Phys. Rev. B 70 (2004) 092503. [16] M. Masuda, et al., Phys. Rev. B 65 (2002) 134515; A.N. Lavrov, et al., Phys. Rev. Lett. 87 (2001) 017007; M. Fujitha, et al., Phys. Rev. B 67 (2003) 014514.
Stability of superconductivity and antiferromagnetism in two-dimensional Hubbard model with diagonal transfer Hisatoshi Yokoyama a,*, Yukio Tanaka b, Masao Ogata c a Department of Physics, Tohoku University, Sendai 980-8578, Japan Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan c Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan b
Abstract Using an optimization variational Monte Carlo method, we study the effect of diagonal transfer t 0 in a two-dimensional Hubbard model on stability of d-wave superconductivity (SC) and antiferromagnetism (AF). For hole doping (t 0 /t!0), the condensation energy of SC is enhanced by a couple of times, while that of AF rapidly decreases with increasing jt 0 /tj, and AF itself is intrinsically unstable for t 0 /t(0. For electron doping (t 0 /tO0), conversely, SC tends to be suppressed, whereas the AF state not only comes to have a low energy but recovers the intrinsic stability. These results well explain various aspects of cuprate superconductors. q 2005 Elsevier Ltd. All rights reserved. PACS: 74.80.Fp; 74.25.Fy Keywords: D. Superconductivity
1. Introduction The Hubbard model on a square lattice is a simple model, which probably captures the essence of cuprate superconductors [1]. Despite considerable efforts, acquired knowledge on this model is very slight, especially in the intermediate and strong coupling regimes. For example, it is still controversial (i) whether superconductivity (SC) appears in this model, (ii) whether the strong coupling regime of this model represents substantially the same physics as the t–J model with small J/t, and (iii) how longrange transfer terms work here, which are considered indispensable to describe individual properties of cuprates. Recently, we have obtained intriguing results concerning the above issues (i) and (ii) [2], using an optimization variational Monte Carlo (OVMC) method, which is highly effective to study ground-state properties with varying U/t. Steady d-wave SC appears only for large values of U/t, namely U/tT6.5, below which the condensation energy DE is very small. The properties of SC exhibit a crossover at around Uco/tZ11–13 from a conventional BCS type to a new one [See Fig. 1a], in which superconducting (SC) transition is induced * Corresponding author. Tel.: C81 22 795 6444; fax: C81 22 795 6447. E-mail address: [email protected] (H. Yokoyama).
0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.10.024
by the decrease in kinetic energy. Such a feature has been actually observed by experiments of optical conductivity for cuprates [3]. Furthermore, DE in this region is well fitted with exp (-at/J); this implies the effective attractive interaction is the exchange process J(Z4t2/U), and is consistent with the viewpoint of the t–J model [4,5]. In these proceedings, we consider the effect of the diagonal transfer t 0 on SC and antiferromagnetic (AF) states, and how it influences the properties mentioned above. It is revealed that the Hubbard model with the t 0 term consistently explains a variety of experiments on cuprates.
2. Model and method On an earlier occasion, second- (t 0 ) and third-neighbor (t 00 ) transfer terms were introduced to fit the Fermi surface obtained through ARPES [6]. Later, using the parameters thus determined, namely the t–t 0 –t 00 –J model, various aspects of ARPES spectra were successfully explained [7]. For simplicity, here we add only the diagonal transfer t 0 to the Hubbard model on a square lattice: X † X H Z Hkin C Hpot Z 3k cks cks C U nj[njY; (1) j
ks 0
with 3k ZK2tðcos kx C cos ky ÞK4t cos kx cos ky . For cuprates, the sign of t 0 =t is negative regardless of whether the carrier is
48
H. Yokoyama et al. / Journal of Physics and Chemistry of Solids 67 (2006) 47–49
(A)
(B)
(C) t' = 0 L = 12
(a)
d-wave AF n
∆/t
1
(b)
t' = 0 L = 8–16
1.0 0.9722 0.9167 0.8611 0.8056
0.5 8
U/t 10
12
d AF
0
10
20
30
0.8
0.85
U/t
0.9 n
0.95
1
Fig. 1. Behavior of a variational parameter D for the d-wave superconducting (solid) and antiferromagnetic (open) states, (a) as a function of U/t for several values of n near half filling, and (b) as a function of n for three values of U/t. As in (a), the correlation strength U=t can be classified into three regimes: (A) weak, (B) intermediate, and (C) strong correlation regimes.
hole or electron. For convenience of calculation, however, we apply a particle–hole transformation, c†js / ðK1Þrj hjs , to morethan-half-filling cases. Thereby, an electron-doped system with electron density n(O1) and t 0 /tZa(!0) is converted to a holedoped systems with hole density dZjnK1j and t 0 /tZjaj(O0). We hence consider both signs of t 0 /t for n%1. To treat the local correlation accurately, we use an OVMC method [8], which enables us to efficiently calculate the expectation values with many variational parameters. For Hubbard-type models, one needs to employ a trial function (JZPF) which allows for at least second order terms of the strong coupling expansion in the correlation factor P [9], beyond the Gutzwiller-type level. In this study, we consider up to the second order in t/U and t 0 /U [10]. As a one-body part F, we adopt the Fermi sea FF, an AF Hartree–Fock state FAF and a d-wave BCS state FSC; FAF and FSC have a parameter D, which is closely connected with DE. We use systems of L!L (LZ8–16) with periodic–antiperiodic boundary conditions, and impose the closed shell condition on the electron density.
3. Results To begin with, we briefly review the aspect for t 0 Z0 in the light of the optimal value of the parameter D. hole doping
electron doping
hole doping
(a)
(b)
0.04
AF d-wave U/t 4 8 12 16
0.03
D E/t
Paying attention to the d-wave in Fig. 1a, where D=t is plotted as a function of U/t, we actually find the crossover behavior described in Section 1. In the range (A), the d-wave SC is very weak and BCS-like; the range (B) is a transient (crossover) regime, and in (C) the decrease in Ekin(ZhHkini) induces SC, which is steady and t–J-model-like. Fig. 1b shows the n-dependence of D/t. For both JSC and JAF, D/t decreases, as n goes away from half filling. Although D still remains finite for n(0.8 for JSC, it abruptly vanishes for n(0.85 for JAF (see Fig. 4a in Ref. [2]). Now, we proceed to the case of finite t 0 . In Fig. 2a, we depict the energy difference (condensation energy), DEZ Eorder KEnormal , for the d-wave and AF as a function of t 0 /t [11] for four values of U/t. Let us summarize the main features of this graph. (1) d-wave SC. (i) For hole doping (t 0 /t! 0), DE rapidly increases as jt 0 /tj increases from zero, exhibits a maximum, which is twice or three times larger than the value at t 0 Z0, at t 0 /tw0.15, then decreases for larger values of jt 0 /tj. (ii) For electron doping (t 0 /tO0), DE gradually decreases as t 0 /t increases, but remains finite even at t 0 /tZ0.5. (iii) U/t dependence: in the range (A) (U/tZ4), DE is small, even if t 0 is introduced. In (B) (U/tZ8), t 0 works effectively to increase DE. In (C) (U/tZ12 and 16), SC is stabilized over a wide range of t 0 /t. (2) AF. (i) For hole doping, DE rapidly decreases and
0.02
L = 10 n = 0.88
0.08 AF
0.06
n d-wave AF 0.96 0.88 0.80
U/t = 12 L = 10
0.04
d-wave
0.01 0
electron doping
0.104
0.02
–0.5
0
t'/t
0.5
0
–0.5
0
0.5
t'/t
Fig. 2. Condensation energy as a function of t 0 /t. (a) U/t dependence for nZ0.88, and (b) n dependence for U/tZ12. AF does not arise for U/tZ4 in (a) and for nZ0.8 in (b). In a range marked with a double-headed arrow, the Fermi surface (occupied k-points) lies closely to the van Hove singularity points.
H. Yokoyama et al. / Journal of Physics and Chemistry of Solids 67 (2006) 47–49
hole doping
D Ekin /t, D Epot /t
4. Discussions
electron doping
0.04
D Ekin D Epot U/t
4 8 12 16
0.02
0 L=10
–0.02
n=0.88 d-wave
–0.04
–0.5
0
49
0.5
t'/t Fig. 3. Difference in kinetic (solid) and potential (shadowed) energies between the normal and SC states as a function of t 0 /t for four values of U/t. The doubleheaded arrows mean the same as Fig. 2. The scattering of data point, particularly for t 0 /tO0, is ascribed mainly to the discrete k-points.
vanishes, although it has a narrow peak around t 0 /tZK0.15 [12]. For t 0 /t(0, the AF state does not satisfy the intrinsic stability condition v2E/vn2O0 (not shown), and phase separates into the AF phase at half filling and the d-wave SC state. (ii) For electron doping, the AF state not only gains energy increasingly but recovers the intrinsic stability condition (not shown). Although these results are basically interpreted within the level of a tight-binding approximation [13,14], they adequately explain some key properties of cuprates (Section 4). In Fig. 2b, n dependence of DE is shown for U/tZ12. Most aspects discussed for Fig. 2a apply to other densities. Here, we focus on the n dependence. As n decreases from half filling, DE both for SC and AF rapidly decreases in the whole range of t 0 /t. For nw0.8, DE of SC for t 0 /tT0 is considerably small; hence, to understand the stable SC in the overdoped regime, the enhancement of DE for negative t 0 /t must be an inevitable factor. As for the case of t 0 /tT0, the stability of SC is marginal at nZ0.8, and AF also becomes unstable for n(0.85 even with a positive t 0 /t. These results correspond well with the fact that AF in electron-doped cuprates vanishes with 15% doping at largest, and SC vanishes at about dZ0.2. SC normal Finally, we consider DEkin ðZEkin KEkin Þ and DEpot. As 0 mentioned for t /tZ0, DEkin (DEpot) is negative (positive) in the regime (A) and (B), but the relation is reversed in the strong correlation regime (C). In Fig. 3, t 0 /t dependence of DEkin and DEpot is shown for four values of U/t. Roughly speaking, for hole doping, the driving force of SC transition is Epot for U/ t(10, but Ekin for U/tT10, which situation is basically the same as that for t 0 Z0. For electron doping, although the behavior for sufficiently large U/t (Z16) is the same with that for hole doping, for U/tZ8 and 12, the magnitude of these quantities is small, and the sign sometimes changes, as t 0 /t is varied. We need a further study on this point.
In this paper, we have studied the effect of diagonal transfer. The features quite different between for t 0 /tO0 and for t 0 /t!0, discussed for Fig. 2, agree well with the well-known asymmetry between hole and electron dopings in the d–T phase diagram of cuprates. Considering that the energy scale of SC, DE, sensitively depends on the value of t 0 /t (Fig. 2), it seems reasonable that the different energy scales between LSCO and BSCCO observed in ARPES spectra [15] are primarily attributed to the effect of t 0 . Furthermore, the intrinsic stability of the AF state consistently explains the distinct behaviors of the AF phase between hole-doped and electrondoped cuprates [16]. For some of the issues addressed here, parallel discussions are possible with the t–J model [13]. Extended description will be given elsewhere [14]. References [1] P.W. Anderson, Science 235 (1987) 1196. [2] H. Yokoyama, et al., J. Phys. Soc. Jpn. 73 (2004) 1119; M. Ogata et al., an article in this volume. [3] D.N. Basov, et al., Science 283 (1999) 49; H.J.A. Molegraaf, et al., Science 295 (2002) 2239. [4] H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn 57 (1988) 2482; H. Yokoyama, M. Ogata, J. Phys. Soc. Jpn. 65 (1996) 3615. [5] D.J. Scalapino, S.R. White, Phys. Rev. B 58 (1998) 8222; E. Demler, S.C. Zhang, Nature 396 (1998) 733. [6] T. Tanamoto, H. Kohno, H. Fukuyama, J. Phys. Soc. Jpn 62 (1993) 717. [7] T. Tohyama, S. Maekawa, Supercond. Sci. Tech. 13 (2000) R17. [8] C.J. Umrigar, K.G. Wilson, J.W. Wilkins, Phys. Rev. Lett. 60 (1988) 1719; T. Giamarchi, C. Lhuillier, Phys. Rev. B 43 (1991) 12943. [9] H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn. 56 (1987) 1490; H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn 59 (1990) 3669; H. Yokoyama, Prog. Theor. Phys. 108 (2002) 59. Q Q Q [10] The wave function used is written as, JQ j ½1KmQj j ½1Km 0 Qj0 j ½1K ð1KgÞnj[njYF: The three variational parameters for local correlation control the onsite double occupation (g) and the binding between a doublon and holons in nearest-neighbor (m) and second-neighbor (m0 ) sites,, respectively, For details of JQ, refer to [9,14]. In OVMC calculations, typically 2.5!105 configurations were kept in the optimization process, and data of virtually several million samples were averaged to reduce the error in total energy to w10K4 t. [11] In JF and JAF, electrons occupy the k-points with 3k!3F. Since the system is finite (hence its k-points are discrete), the chosen k-points changes abruptly, when the value of t 0 /t is varied, This is not the case with JSC for Ds0, Eventually, both DESC and DEAF become not smooth functions of t 0 /t. [12] This peak for AF at t 0 /twK0.15 arises when the Fermi surface lies closely to the van Hove singularity points (0, Gp) and (Gp, 0), which are mutually connected with the AF nesting vector (p, p). Since such a situation appears when t 0 /t and n satisfy a special condition, the width of the peak is narrow. See also K. Yamaji, et al., Physica C 392–396 (2003) 229. [13] C.T. Shih, et al., Phys. Rev. Lett. 92 (2004) 227002. T. Watanabe, et al., an article in this volume. [14] H. Yokoyama, Y. Tanaka, M. Ogata, in preparation. [15] K. Tanaka, et al., Phys. Rev. B 70 (2004) 092503. [16] M. Masuda, et al., Phys. Rev. B 65 (2002) 134515; A.N. Lavrov, et al., Phys. Rev. Lett. 87 (2001) 017007; M. Fujitha, et al., Phys. Rev. B 67 (2003) 014514.