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Numerical study on pseudogap phenomena in high-Tc cuprates
Journal of Physics and Chemistry of Solids 63 (2002) 1385±1388
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Numerical study on pseudogap phenomena in high-Tc cuprates ToÃru Sakai a,*, Yoshinori Takahashi b b
a Tokyo Metropolitan Institute of Technology, 6-6 Asahigaoka, Hino, Tokyo 191-0065, Japan Faculty of Science, Himeji Institute of Technology, 3-2-1 Kouto, Kamigori-cho, Ako-gun, Hyogo 678-1297, Japan
Abstract The pseudogap phenomena observed on cuprate high-temperature superconductors are investigated based on the ®nitetemperature Lanczos method on the 16 sites 4 £ 4 cluster t±J model, which includes two holes. The results clearly show the presence of the gap-like behavior in the temperature dependence of various magnetic properties; the NMR relaxation rate, the neutron scattering intensity and the static susceptibility. The effect of the external magnetic ®eld on the pseudogap is also studied on the same model. These results are in agreement with our previous exact diagonalization studies on the one-hole system, and con®rm our proposal of the predominant role of the antiferromagnetic spin correlation on the observed pseudogap behaviors. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: A. Oxides; A. Superconductors; D. Magnetic properties; D. Nuclear magnetic resonance (NMR)
1. Introduction The high-temperature superconducting cuprate exhibits various anomalous metallic properties in its normal phase. Among them a gap-like behavior, ®rst discovered by the NMR relaxation rate measurement [1], has attracted a lot of current interest. It is characteristic of the under-doped cuprates and observed as a broad peak in the 1/T1T±T curve at a slightly higher temperature above the superconducting transition point Tc. The possible relation of the phenomena with the origin of the superconductivity underlies the reason of such interest. The behavior has also been detected now in the temperature dependence of various other physical quantities; the neutron scattering intensity [2], the magnetic susceptibility [3], the resistivity [4], the Hall coef®cient [5], and the angle-resolved photo-emission spectrum [6]. Although a lot of theoretical explanations have been proposed as its possible candidate [7±19], the ®nal answer is still far from being established. Most theories seem to have an agreement on the important role of the antiferromagnetic spin correlation on the pseudogap formation. Our previous exact diagonalization studies [14,15] on the t±J model indicated an evidence of the gap-like behavior due to the growth of the antiferromagnetic short-range order. * Corresponding author. Fax: 181-42-585-8624. E-mail address: [email protected] (T. Sakai).
The size of cluster of our model one-hole systems, however, had to be restricted up to 10 sites at most due to the lack of the memory size of computer systems. On the other hand, it is desirable to examine two-hole systems with low doping in order to con®rm the validity of our mechanism. Bulk properties are usually masked at low-temperature, because of the presence of an unpaired hole, speci®c to the ®nite size single-hole systems. We have to therefore study the twohole system of larger size to deal with the low doping cases. In this paper, we employ a numerical method developed by Jaklic and Prelovsek [20] for this purpose. 2. Origin of gap-like behaviors According to our proposal [14,15], the pseudogap behaviors arise because of the following reason. The local spin excitations are, in general, suppressed due to the development of the short-range antiferromagnetic correlation around a characteristic temperature, of the same magnitude of the antiferromagnetic coupling J, as we decrease the temperature. As a result, various magnetic quantities will show their particular T-dependence in¯uenced by the reduced local excitations. The high-temperature series expansion for the square-lattice antiferromagnetic Heisenberg model also seems to support this view. It is shown that the temperature dependence of its NMR relaxation rate 1/T1T shows a broad peak around T , J [21]. Since the hole motion tends to destroy the short-range antiferromagnetic
0022-3697/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0022-369 7(02)00041-0
1386
T. Sakai, Y. Takahashi / Journal of Physics and Chemistry of Solids 63 (2002) 1385±1388
observed in the neutron scattering experiment. In actual numerical estimations of Im x q; v; the d -function is approximated by the Lorentzian distribution with some small width. The temperature dependence of the static susceptibility is also calculated. In addition, we evaluate the temperature dependence of the Q ; p; p component of the spin correlation function, D E 1 X ^ y ^ 3 S Q 21 j2i´ x1 Szi ; Szj ; N i;j
0.08
(a)
S(Q)
0.06 0.04
J=0.4 J=0.5 J=0.6
0.02 0.00
Im χ(ω)
(b) 0.2
0.1
0.0 0.4
(c)
χ
0.3 0.2 0.1 0.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 1. Temperature dependences of (a) S(Q), (b) Im x v; and (c) x for the 10-site t±J cluster with one hole.
order, the hole doping will reduce the cross-over temperature in agreement with the observed doping dependence of the pseudogap temperature in the T± d phase diagrams (d represents the hole concentration) of most real cuprates. The pseudogap behavior is thus simply realized as the consequence of the growth of the short-range antiferromagnetic correlation. 3. Calculations for t±J model In order to examine the validity of our pseudogap mechanism, we study the standard square-lattice t±J model Hamiltonian de®ned by X ² X 1 H 2t c j;s ci;s 1 ci;²s cj;s 1 J Si ´Sj 2 ni nj : 1 4 ki;jl;s ki;jl Throughout the paper, all the energies are measured in units of t. To prove the gap-like behavior in the temperature dependence of the NMR relaxation rate, we evaluate the q-integrated dynamical susceptibility of the conduction electrons Z Im x v ; dq Im x q; v; 2 for small v . The gap-like behavior of this quantity was also
to measure the antiferromagnetic correlation [22]. p p The results of the exact diagonalization of the 10 £ 10 cluster under the periodic boundary condition [14,15] are reproduced in Fig. 1, where (a), (b), and (c) show the temperature dependences of S(Q), Im x v; and x , respectively. Among the results for J 0:4; 0.5, and 0.6, J 0:4 corresponds to the most realistic situation. All the calculated T dependences of Im x v and x exhibit either broad gap-like peaks or slight humps around the temperature, where S(Q) shows the signi®cant increase with decreasing the temperature. They are consistent with our view on the central role of the antiferromagnetic spin correlation on the gap-like behavior. The one-hole calculation, however, does not give a clear-cut evidence because of the presence of divergent behaviors of Im x v and x at lowtemperature. They are caused by the dominated effect from the ground state doublet for ®nite cluster systems. Thus, the numerical results for the two-hole system are presented later. 4. Finite-temperature Lanczos method We performed a similar numerical analysis on the 4 £ 4 t±J cluster (the 10-site system with two holes does not exhibit the pseudogap, because d 0:2 is too large). Since the available memory size of the present computer system is not so enough for the exact diagonalization of the model, we used the Lanczos algorithm in conjunction with a random sampling of the initial states developed by Jaklic and Prelovsek [20]. According to the method, the thermal average of the physical quantity A is given by kAl
N 1 X knue2bH Aunl; Z n
4
is approximated by the form kAl
N0 X M 1 X knun; mle2benm kn; muAunl; Z n m
5
where unl n 1; ¼; N0 is a set of sampled random vectors, while un; ml m 1; ¼; M represents a set of orthonormal vectors generated by M Lanczos steps applied to each unl (N0 is the number of the random sampling). They suggested that the method gives a good approximation even if N0 p N and
T. Sakai, Y. Takahashi / Journal of Physics and Chemistry of Solids 63 (2002) 1385±1388
0.2
1387
0.2
0.1
(a) S(Q)
S(Q)
(a) J=0.4 J=0.5 J=0.6
0.0 5.0
0.0
(b)
(b)
Im χ(ω)
4.0
Im χ(ω)
4.0 3.0
3.0
2.0
2.0
1.0
1.0
0.0 1.5
0.0
(c)
(c)
1.5
χ
χ
1.0
1.0
0.5
0.0
H=0 H=0.05 H=0.1
0.1
0.5 0.0
0.2
0.4
0.6
0.8
1.0
Fig. 2. Temperature dependences of (a) S(Q), (b) Im x v; and (c) x for the 16-site t±J cluster with two holes.
M p N; where N is the dimension of the Hilbert space of the original systems. In evaluating dynamical quantities, like Im x q; v; we assumed N0 , 100 and M , 5000. We checked that every quantity is about to converge within the calculation. The resultant temperature dependences of (a) S(Q), (b) Im x (v ), and (c) x for the two-hole systems d 0:125 are shown in Fig. 2. Both Im x (v ) and x exhibit peaks around the characteristic temperature, where the antiferromagnetic spin correlation shows the signi®cant increase. Thus, our mechanism is even more clearly justi®ed for the two-hole system as the bulk property of the system. There are reports exhibiting that the temperature dependence of 1/T1T and x shows peaks at different temperatures. According to our numerical study, however, they show peaks around the same temperature. Note that we are not discussing the de®nite phase transition, but the cross-over phenomena. Then, it is reasonable to assume that cross-over behaviors in general look different depending on observations. Actually, Im x v and x in Fig. 2 show peaks at slightly different temperatures. Taking into account the simpli®ed model de®ned on a ®nite-size cluster, we have to be satis®ed with the qualitative properties of the model. No detailed quantitative comparison with experiments at this moment is still meaningless.
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 3. Temperature dependences of (a) S(Q), (b) Im x v; and (c) x for the 16-site t±J cluster with two holes under magnetic ®elds H 0; 0.05, and 0.1 J 0:4:
5. Effect of magnetic ®eld Finally, we brie¯y mention our numerical studies on the effect of the external magnetic ®eld H on the pseudogap behavior. The recent NMR experiment [23,24] indicated that the characteristic pseudogap temperature Tp is almost independent of H in the under-doped region. To check our proposal from the behavior of Tp, we calculated the same quantities as earlier for the 4 £ 4 systems with two holes under the presence of the external magnetic ®eld (H 0; 0.05, and 0.1) for ®xed J 0:4: The results are shown in Fig. 3(a)±(c). We can see that calculated peak positions are almost independent of H, in good agreement with the NMR measurement in the under-doped region. It therefore gives another con®rmation of our pseudogap mechanism based on the antiferromagnetic spin correlation.
6. Summary The ®nite-temperature Lanczos calculations on the 4 £ 4 t±J cluster with two holes con®rmed our previous proposal that the observed pseudogap behaviors in the high-Tc cuprates originate from the growth of the antiferromagnetic spin correlation. We also found that the pseudogap
1388
T. Sakai, Y. Takahashi / Journal of Physics and Chemistry of Solids 63 (2002) 1385±1388
temperature is almost independent of the magnetic ®eld at least in the under-doped region, which is consistent with the recent experiment. Acknowledgements We wish to thank Dr G.-Q. Zheng for fruitful discussions. We also thank the Supercomputer Center, Institute for Solid State Physics, University of Tokyo for the facilities and the use of the HITACHI SR8000. This research was supported in part by Grant-in-Aid for the Scienti®c Research Fund from the Ministry of Education, Science, Sports and Culture (11440103). References [1] H. Yasuoka, T. Imai, T. Shimizu, in: H. Fukuyama, S. Maekawa, A.P. Malozemoff (Eds.), Strong Correlation and Superconductivity, Springer, Berlin, 1989, p. 254. [2] B.J. Sternlieb, G. Shirane, J.M. Tranquada, M. Sato, S. Shamaoto, Phys. Rev. B 47 (1993) 5320. [3] T. Nakano, M. Oda, C. Manabe, N. Momono, Y. Miura, M. Ido, Phys. Rev. B 49 (1994) 16,000. [4] K. Takenaka, K. Mizuhashi, H. Takagi, S. Uchida, Phys. Rev. B 50 (1994) 6534.
[5] T. Nishikawa, J. Takeda, M. Sato, J. Phys. Soc. Jpn 63 (1994) 1441. [6] M.R. Norman, H. Ding, M. Randeria, J.C. Campuzano, T. Yokoya, T. Takeuchi, T. Takahashi, T. Mochiku, K. Kadowaki, P. Guptasarma, D.G. Hinks, Nature 392 (1998) 157. [7] P.W. Anderson, Phys. Rev. Lett. 64 (1990) 1839. [8] T. Tanamoto, K. Kuboki, H. Fukuyama, J. Phys. Soc. Jpn 60 (1991) 3072. [9] T. Tanamoto, H. Kohno, H. Fukuyama, J. Phys. Soc. Jpn 63 (1994) 2739. [10] M. Onoda, I. Ichinose, T. Matsui, J. Phys. Soc. Jpn 67 (1998) 2606. [11] K. Miyake, O. Narikiyo, J. Phys. Soc. Jpn 63 (1994) 3821. [12] O. Narikiyo, K. Miyake, J. Phys. Soc. Jpn 63 (1994) 4169. [13] O. Narikiyo, K. Miyake, Physica C 250 (1995) 295. [14] T. Sakai, Y. Takahashi, J. Phys. Soc. Jpn 67 (1998) 2630. [15] T. Sakai, Y. Takahashi, J. Phys. Soc. Jpn 70 (2001) 272. [16] S. Koikegami, K. Yamada, J. Phys. Soc. Jpn 67 (1998) 1114. [17] Y. Yanase, K. Yamada, J. Phys. Soc. Jpn 68 (1999) 2999. [18] A. Kobayashi, A. Tsuruta, T. Matsuura, Y. Kuroda, J. Phys. Soc. Jpn 68 (1999) 2506. [19] S. Onoda, M. Imada, J. Phys. Soc. Jpn 68 (1999) 2762. [20] J. Jaklic, P. Prelovsek, Phys. Rev. Lett. 74 (1995) 3411. [21] S. Chakrabarty, R. Orbach, Phys. Rev. Lett. 64 (1990) 224. [22] T. Sakai, Y. Takahashi, Physica B 281, 282 (2000) 800. [23] G.-Q. Zheng, et al., Phys. Rev. B 60 (1999) R9947. [24] G.-Q. Zheng, et al., Phys. Rev. Lett. 85 (2000) 405.
www.elsevier.com/locate/jpcs
Numerical study on pseudogap phenomena in high-Tc cuprates ToÃru Sakai a,*, Yoshinori Takahashi b b
a Tokyo Metropolitan Institute of Technology, 6-6 Asahigaoka, Hino, Tokyo 191-0065, Japan Faculty of Science, Himeji Institute of Technology, 3-2-1 Kouto, Kamigori-cho, Ako-gun, Hyogo 678-1297, Japan
Abstract The pseudogap phenomena observed on cuprate high-temperature superconductors are investigated based on the ®nitetemperature Lanczos method on the 16 sites 4 £ 4 cluster t±J model, which includes two holes. The results clearly show the presence of the gap-like behavior in the temperature dependence of various magnetic properties; the NMR relaxation rate, the neutron scattering intensity and the static susceptibility. The effect of the external magnetic ®eld on the pseudogap is also studied on the same model. These results are in agreement with our previous exact diagonalization studies on the one-hole system, and con®rm our proposal of the predominant role of the antiferromagnetic spin correlation on the observed pseudogap behaviors. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: A. Oxides; A. Superconductors; D. Magnetic properties; D. Nuclear magnetic resonance (NMR)
1. Introduction The high-temperature superconducting cuprate exhibits various anomalous metallic properties in its normal phase. Among them a gap-like behavior, ®rst discovered by the NMR relaxation rate measurement [1], has attracted a lot of current interest. It is characteristic of the under-doped cuprates and observed as a broad peak in the 1/T1T±T curve at a slightly higher temperature above the superconducting transition point Tc. The possible relation of the phenomena with the origin of the superconductivity underlies the reason of such interest. The behavior has also been detected now in the temperature dependence of various other physical quantities; the neutron scattering intensity [2], the magnetic susceptibility [3], the resistivity [4], the Hall coef®cient [5], and the angle-resolved photo-emission spectrum [6]. Although a lot of theoretical explanations have been proposed as its possible candidate [7±19], the ®nal answer is still far from being established. Most theories seem to have an agreement on the important role of the antiferromagnetic spin correlation on the pseudogap formation. Our previous exact diagonalization studies [14,15] on the t±J model indicated an evidence of the gap-like behavior due to the growth of the antiferromagnetic short-range order. * Corresponding author. Fax: 181-42-585-8624. E-mail address: [email protected] (T. Sakai).
The size of cluster of our model one-hole systems, however, had to be restricted up to 10 sites at most due to the lack of the memory size of computer systems. On the other hand, it is desirable to examine two-hole systems with low doping in order to con®rm the validity of our mechanism. Bulk properties are usually masked at low-temperature, because of the presence of an unpaired hole, speci®c to the ®nite size single-hole systems. We have to therefore study the twohole system of larger size to deal with the low doping cases. In this paper, we employ a numerical method developed by Jaklic and Prelovsek [20] for this purpose. 2. Origin of gap-like behaviors According to our proposal [14,15], the pseudogap behaviors arise because of the following reason. The local spin excitations are, in general, suppressed due to the development of the short-range antiferromagnetic correlation around a characteristic temperature, of the same magnitude of the antiferromagnetic coupling J, as we decrease the temperature. As a result, various magnetic quantities will show their particular T-dependence in¯uenced by the reduced local excitations. The high-temperature series expansion for the square-lattice antiferromagnetic Heisenberg model also seems to support this view. It is shown that the temperature dependence of its NMR relaxation rate 1/T1T shows a broad peak around T , J [21]. Since the hole motion tends to destroy the short-range antiferromagnetic
0022-3697/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0022-369 7(02)00041-0
1386
T. Sakai, Y. Takahashi / Journal of Physics and Chemistry of Solids 63 (2002) 1385±1388
observed in the neutron scattering experiment. In actual numerical estimations of Im x q; v; the d -function is approximated by the Lorentzian distribution with some small width. The temperature dependence of the static susceptibility is also calculated. In addition, we evaluate the temperature dependence of the Q ; p; p component of the spin correlation function, D E 1 X ^ y ^ 3 S Q 21 j2i´ x1 Szi ; Szj ; N i;j
0.08
(a)
S(Q)
0.06 0.04
J=0.4 J=0.5 J=0.6
0.02 0.00
Im χ(ω)
(b) 0.2
0.1
0.0 0.4
(c)
χ
0.3 0.2 0.1 0.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 1. Temperature dependences of (a) S(Q), (b) Im x v; and (c) x for the 10-site t±J cluster with one hole.
order, the hole doping will reduce the cross-over temperature in agreement with the observed doping dependence of the pseudogap temperature in the T± d phase diagrams (d represents the hole concentration) of most real cuprates. The pseudogap behavior is thus simply realized as the consequence of the growth of the short-range antiferromagnetic correlation. 3. Calculations for t±J model In order to examine the validity of our pseudogap mechanism, we study the standard square-lattice t±J model Hamiltonian de®ned by X ² X 1 H 2t c j;s ci;s 1 ci;²s cj;s 1 J Si ´Sj 2 ni nj : 1 4 ki;jl;s ki;jl Throughout the paper, all the energies are measured in units of t. To prove the gap-like behavior in the temperature dependence of the NMR relaxation rate, we evaluate the q-integrated dynamical susceptibility of the conduction electrons Z Im x v ; dq Im x q; v; 2 for small v . The gap-like behavior of this quantity was also
to measure the antiferromagnetic correlation [22]. p p The results of the exact diagonalization of the 10 £ 10 cluster under the periodic boundary condition [14,15] are reproduced in Fig. 1, where (a), (b), and (c) show the temperature dependences of S(Q), Im x v; and x , respectively. Among the results for J 0:4; 0.5, and 0.6, J 0:4 corresponds to the most realistic situation. All the calculated T dependences of Im x v and x exhibit either broad gap-like peaks or slight humps around the temperature, where S(Q) shows the signi®cant increase with decreasing the temperature. They are consistent with our view on the central role of the antiferromagnetic spin correlation on the gap-like behavior. The one-hole calculation, however, does not give a clear-cut evidence because of the presence of divergent behaviors of Im x v and x at lowtemperature. They are caused by the dominated effect from the ground state doublet for ®nite cluster systems. Thus, the numerical results for the two-hole system are presented later. 4. Finite-temperature Lanczos method We performed a similar numerical analysis on the 4 £ 4 t±J cluster (the 10-site system with two holes does not exhibit the pseudogap, because d 0:2 is too large). Since the available memory size of the present computer system is not so enough for the exact diagonalization of the model, we used the Lanczos algorithm in conjunction with a random sampling of the initial states developed by Jaklic and Prelovsek [20]. According to the method, the thermal average of the physical quantity A is given by kAl
N 1 X knue2bH Aunl; Z n
4
is approximated by the form kAl
N0 X M 1 X knun; mle2benm kn; muAunl; Z n m
5
where unl n 1; ¼; N0 is a set of sampled random vectors, while un; ml m 1; ¼; M represents a set of orthonormal vectors generated by M Lanczos steps applied to each unl (N0 is the number of the random sampling). They suggested that the method gives a good approximation even if N0 p N and
T. Sakai, Y. Takahashi / Journal of Physics and Chemistry of Solids 63 (2002) 1385±1388
0.2
1387
0.2
0.1
(a) S(Q)
S(Q)
(a) J=0.4 J=0.5 J=0.6
0.0 5.0
0.0
(b)
(b)
Im χ(ω)
4.0
Im χ(ω)
4.0 3.0
3.0
2.0
2.0
1.0
1.0
0.0 1.5
0.0
(c)
(c)
1.5
χ
χ
1.0
1.0
0.5
0.0
H=0 H=0.05 H=0.1
0.1
0.5 0.0
0.2
0.4
0.6
0.8
1.0
Fig. 2. Temperature dependences of (a) S(Q), (b) Im x v; and (c) x for the 16-site t±J cluster with two holes.
M p N; where N is the dimension of the Hilbert space of the original systems. In evaluating dynamical quantities, like Im x q; v; we assumed N0 , 100 and M , 5000. We checked that every quantity is about to converge within the calculation. The resultant temperature dependences of (a) S(Q), (b) Im x (v ), and (c) x for the two-hole systems d 0:125 are shown in Fig. 2. Both Im x (v ) and x exhibit peaks around the characteristic temperature, where the antiferromagnetic spin correlation shows the signi®cant increase. Thus, our mechanism is even more clearly justi®ed for the two-hole system as the bulk property of the system. There are reports exhibiting that the temperature dependence of 1/T1T and x shows peaks at different temperatures. According to our numerical study, however, they show peaks around the same temperature. Note that we are not discussing the de®nite phase transition, but the cross-over phenomena. Then, it is reasonable to assume that cross-over behaviors in general look different depending on observations. Actually, Im x v and x in Fig. 2 show peaks at slightly different temperatures. Taking into account the simpli®ed model de®ned on a ®nite-size cluster, we have to be satis®ed with the qualitative properties of the model. No detailed quantitative comparison with experiments at this moment is still meaningless.
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 3. Temperature dependences of (a) S(Q), (b) Im x v; and (c) x for the 16-site t±J cluster with two holes under magnetic ®elds H 0; 0.05, and 0.1 J 0:4:
5. Effect of magnetic ®eld Finally, we brie¯y mention our numerical studies on the effect of the external magnetic ®eld H on the pseudogap behavior. The recent NMR experiment [23,24] indicated that the characteristic pseudogap temperature Tp is almost independent of H in the under-doped region. To check our proposal from the behavior of Tp, we calculated the same quantities as earlier for the 4 £ 4 systems with two holes under the presence of the external magnetic ®eld (H 0; 0.05, and 0.1) for ®xed J 0:4: The results are shown in Fig. 3(a)±(c). We can see that calculated peak positions are almost independent of H, in good agreement with the NMR measurement in the under-doped region. It therefore gives another con®rmation of our pseudogap mechanism based on the antiferromagnetic spin correlation.
6. Summary The ®nite-temperature Lanczos calculations on the 4 £ 4 t±J cluster with two holes con®rmed our previous proposal that the observed pseudogap behaviors in the high-Tc cuprates originate from the growth of the antiferromagnetic spin correlation. We also found that the pseudogap
1388
T. Sakai, Y. Takahashi / Journal of Physics and Chemistry of Solids 63 (2002) 1385±1388
temperature is almost independent of the magnetic ®eld at least in the under-doped region, which is consistent with the recent experiment. Acknowledgements We wish to thank Dr G.-Q. Zheng for fruitful discussions. We also thank the Supercomputer Center, Institute for Solid State Physics, University of Tokyo for the facilities and the use of the HITACHI SR8000. This research was supported in part by Grant-in-Aid for the Scienti®c Research Fund from the Ministry of Education, Science, Sports and Culture (11440103). References [1] H. Yasuoka, T. Imai, T. Shimizu, in: H. Fukuyama, S. Maekawa, A.P. Malozemoff (Eds.), Strong Correlation and Superconductivity, Springer, Berlin, 1989, p. 254. [2] B.J. Sternlieb, G. Shirane, J.M. Tranquada, M. Sato, S. Shamaoto, Phys. Rev. B 47 (1993) 5320. [3] T. Nakano, M. Oda, C. Manabe, N. Momono, Y. Miura, M. Ido, Phys. Rev. B 49 (1994) 16,000. [4] K. Takenaka, K. Mizuhashi, H. Takagi, S. Uchida, Phys. Rev. B 50 (1994) 6534.
[5] T. Nishikawa, J. Takeda, M. Sato, J. Phys. Soc. Jpn 63 (1994) 1441. [6] M.R. Norman, H. Ding, M. Randeria, J.C. Campuzano, T. Yokoya, T. Takeuchi, T. Takahashi, T. Mochiku, K. Kadowaki, P. Guptasarma, D.G. Hinks, Nature 392 (1998) 157. [7] P.W. Anderson, Phys. Rev. Lett. 64 (1990) 1839. [8] T. Tanamoto, K. Kuboki, H. Fukuyama, J. Phys. Soc. Jpn 60 (1991) 3072. [9] T. Tanamoto, H. Kohno, H. Fukuyama, J. Phys. Soc. Jpn 63 (1994) 2739. [10] M. Onoda, I. Ichinose, T. Matsui, J. Phys. Soc. Jpn 67 (1998) 2606. [11] K. Miyake, O. Narikiyo, J. Phys. Soc. Jpn 63 (1994) 3821. [12] O. Narikiyo, K. Miyake, J. Phys. Soc. Jpn 63 (1994) 4169. [13] O. Narikiyo, K. Miyake, Physica C 250 (1995) 295. [14] T. Sakai, Y. Takahashi, J. Phys. Soc. Jpn 67 (1998) 2630. [15] T. Sakai, Y. Takahashi, J. Phys. Soc. Jpn 70 (2001) 272. [16] S. Koikegami, K. Yamada, J. Phys. Soc. Jpn 67 (1998) 1114. [17] Y. Yanase, K. Yamada, J. Phys. Soc. Jpn 68 (1999) 2999. [18] A. Kobayashi, A. Tsuruta, T. Matsuura, Y. Kuroda, J. Phys. Soc. Jpn 68 (1999) 2506. [19] S. Onoda, M. Imada, J. Phys. Soc. Jpn 68 (1999) 2762. [20] J. Jaklic, P. Prelovsek, Phys. Rev. Lett. 74 (1995) 3411. [21] S. Chakrabarty, R. Orbach, Phys. Rev. Lett. 64 (1990) 224. [22] T. Sakai, Y. Takahashi, Physica B 281, 282 (2000) 800. [23] G.-Q. Zheng, et al., Phys. Rev. B 60 (1999) R9947. [24] G.-Q. Zheng, et al., Phys. Rev. Lett. 85 (2000) 405.