- Email: [email protected]
On the spin excitations of high-Tc cuprates
Journal of Magnetism and Magnetic Materials 177 181 0998)483-486
ELSEVIER
~ H Journalof magnetism ~ H and magnetic materials
Invited paper
On the spin excitations of high-T cuprates Hidetoshi Fukuyama*, Hiroshi Kohno Department of Physics, Faculty of Science, University of To,o, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan
Abstract
The results of theoretical analysis of the spin excitation of high-To cuprates based on the slave-boson mean-field approximation of the t J model are summarized with special emphasis on the spin gap phenomenon typically seen in NMR rate and neutron scattering. Possible consequences of the spin gap on the anomalous frequency shifts of some particular oxygen phonon modes and important roles played by disorder are indicated. @ 1998 Elsevier Science B.V. All rights reserved. Keywords: Spin excitations; Spin gap; Mean-field theory; t-J model
1. Introduction: The spin gap
Various experiments so far carried out for high-Tc cuprates indicate the phase diagram on the plane of the doping rate, 3, and the temperature, T, as shown in Fig. 1, where 6M is the characteristic doping rate where the superconducting critical temperature, To, is maximum. In this figure, Ts6, termed as the spin gap temperature, is the characteristic temperature defined by the maxima of the nuclear magnetic resonance (NMR) rate, (T1T)- 1. The remarkable fact is that Ts6 in underdoped region (6 < 6M) appears to merge smoothly to Tc for 6 ~> &a. This indicates the essential roles played by the mechanism leading to the spin gap in stabilizing high Tc superconductivity. Such Tsc, first noted by Yasuoka et al. [-1, 2] has clearly been observed in bilayer systems such as YBCO (2 4 8) [-3] and underdoped YBCO (1 2 3). Experiment on three-layer Hg (1 2 2 3) [4] has shown that (T1T)- 1 of both inner and outer-layers show similar behaviours. Similarly, it has been clarified [-5] that a single-layer Hg (1 2 0 1) also shows this spin gap phenomenon. All these experimental data have firmly established that the spin gap in ( T I T ) -1 is intrinsic to the CuO2 plane of high-To cuprates in the underdoped region. In this context it is to be noted that in La2 xSr:,CuO4 the spin gap has been difficult to be
* Corresponding author.
identified. Very recently, however, Yasuoka [6] has succeeded in its identification by paying special attention to the broad linewidth and by extracting the intrinsic relaxation processes. Though the charge excitations are also seen to show anomalies, we will mainly discuss in this paper the spin gap associated with TsG.
2. Theoretical predictions based on slave-boson mean-field theory of the extended t - J model
The low-lying excitations of high-To cuprates, whose parent (6 = 0) compounds are charge-transfer type insulators, are described by the effective single-band model because of strong mixing [7] and explicitly by the t-J model [,8-16] H = Z tija*i~ai~+ J Z S i ' S i ,
(1)
i,j,~
where the double occupancy of each lattice site is excluded, and tij and J represent the transfer integrals of the Zhang Rice singlet and the superexchange interaction, respectively. Here we note that tij is very sensitive to the local structural properties which leads to different values oftij for different compounds [10, 11]. Hence, the characteristic features of each compound having different Fermi surfaces are represented by the different spatial extent
0304-8853/98/$19.00 ~ 1998 Elsevier Science B.V. All rights reserved PII S 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 0 8 6 5 - 2
484
H. Fukuvama, H. Kohno / Journal of Magnetism and Magnetic Materials 177-181 (1998) 483 486 T
T
L
"~.' '~TD AM \\T* \
A
~M
F
~
EL
D
6
"6
Fig. 1. Phase diagram ofhigh-T cuprates deduced from experiments. T is the temperature and 6 is the hole doping rate.
Fig. 2. Schematic phase diagram of the (extended) t J model. T is the temperature and 6 is the hole doping rate.
of t u, with which Eq. (1) is termed as extended t J model [17]. Though exact solutions of the t - J model, in general, are not known, the phase diagram on the plane of 6 and J/t are numerically pursued at T = 0 for one dimension (d = 1) [18] and two dimensions (d = 2) [19, 20]. Especially, in the latter the possibility of superconductivity of dx-' ~.-' symmetry for small doping and J/t of practical interest has been indicated in Ref. [20], which is in accordance with the variational calculations [21, 22]. Moreover, the mean-field theory based on the slave boson framework, which is developed on the original idea of RVB [7, 23] has predicted [24, 25] a phase diagram as shown in Fig. 2, where AF means antiferromagnetic phase realized at low doping. In Fig. 2, TD, TRw and TB are the characteristic temperatures of the onset of the coherent motion of spinons and holons, the formation of spinon singlet pairing and the bose condensation of holons, respectively. By these characteristic temperatures, the phase diagram is decomposed into various states. For T > TB the spin and charge are separated, while for T < TR spin and charge are combined to form ordinary electron liquid (EL). The superconducting critical temperature, T~, in this theory is the lower one of TRW and TB. It is to be noted [26] that TD and the higher one of TB and TRVB are crossover temperatures and are not true phase transition temperature in contrast to T~. In the anomalous metallic (AM) state described as a uniform RVB (u-RVB) state, spinons and holons move independently and coherently. In the spin gap (SG) state, where spin and charge are also separated, the spinons form short-range singlet described as a singlet RVB (s-RVB). This singlet has the dx: ,.-~symmetry resulting in the same symmetry for the superconducting state as well.
with g0(q, ~) being that of non-interacting spinons either in u-RVB or s-RVB state [17, 27]. The qualitative difference of the spin excitations between YBCO and LSCO, e.g., commensurate (C) in the former versus incommensurate (IC) in the latter, has been understood based on Eq. (2) by the difference of the shape of the Fermi surface. Moreover, in the case of LSCO, the absence of the low-energy excitations at the IC wave vector at low temperature and the doping dependence of the degree of IC seen in neutron scattering on the high-quality sample [28] are also in good accordance with the theoretical prediction [27]. For optimally doped YBCO, another characteristic feature is the resonance-like peak at energy ~o = 41 meV around the C wave vector present only in the superconducting state [29, 30]. There exists an interesting proposal for this 41 meV peak based on the 'n-resonance' [31]. This scenario was pursued by taking more realistic band structure into account in [32], where possible importance of the flat band nature around wave vectors (n, 0), (0, n) for the n-resonance has been indicated. In addition, a remarkable feature of the phase diagram of Fig. 2 is the existence of the spin-gap (SG) state. Actually Eq. (2) predicts a strong (almost exponential even for the present d:,: r~-symmetry) suppression of the NMR rate, (T1T)-1, below T = TRVB,which can be well above Tc in underdoped systems. Hence, it is tempting to identify Tsc. seen in experiment with TRVBin Fig. 2 [27]. Very recently, a well-defined IC peaks have been observed in the superconducting state of underdoped YBCO [33]. These peaks are located at (n _+ 5,-n + 6), differently from those of LSCO, and are naturally understood as the contribution from quasiparticles near the d-wave gap nodes. Remarkably, this IC structure persists even above To, which is consistent with the presence of the d-wave spin gap above Tc and its continuity to that of superconductivity.
3. Spin excitations and spin gap The spin excitation in this model is described by the dynamical spin susceptibility, g(q, ~o), z(q, (o) = Zo(q, o~)/[l + J(q)zo(q,o~)],
(2)
J(q) - J(cos q., + cos q~.),
(3)
4. Anomalous frequency shift of oxygen phonons The above-mentioned identification of the Ts~ with the TRVBis further supported by the agreement between the experimental results of the onset of the anomalous
tL Fukuyama, tL Kohno / Journal of Magnetism and Magnetic Materials 177-181 (1998) 483 486
frequency shift of Big [34] and BEg [35] in-plane oxygen modes vibrating along the c-axis seen in Y B C O (2 4 8) and underdoped Y B C O (1 2 3) and theoretical results based on the existence of the buckling [36]. In this model, there does not exist strong coupling between A-mode phonons and s-RVB, which seems to be also in accordance with experiments.
5. Effects of disorder As indicated in Section l, the identification of the spin gap in L S C O has been made only recently. This was one of the reasons why the spin gap phenomena were ascribed to the bilayer structure of this c o m p o u n d [37]. However, at the same time, possible important roles played by disorder has been indicated [27] by noting the experimental results of the N M R and neutron scattering which demonstrated that Zn substitution in Y B C O (2 4 8) [38] and underdoped YBCO (1 2 3) [39] destroys the spin gap in N M R rate without almost any effects on the N M R shift. Actually, it has now been clarified that the spin gap is due purely to the quantum coherence and hence is very susceptible to the perturbations leading to finite amount of local magnetic moments, which naturally order in the ground-state antiferromagnetically in bipartite lattices. Correspondingly there exist various experimental indications that L S C O is intrinsically disordered [-2, 40-42]. This finding will also be relevant to the experimental result by Mendels et al. on Zn substituted Y B C O (1 2 3) [-43].
6. Summary and discussion A m o n g various anomalous behaviors observed in high-To cuprates, the spin gap seen in N M R rate is noteworthy. It has been pointed out that the phase diagram derived by the slave-boson mean-field theory for the extended t J model has qualitative correspondence at least to a part of the experimental results, and that, especially, the identification of TsG with the formation of the shortrange singlet RVB gives coherent explanation for spin excitations and some particular modes of phonons. In spite of the encouraging results so far discussed we note various important questions which need further clarifications. Above all we note the following: (a) Origin of different temperature dependence between ( T 1 T ) -1 and the Knight shift, K. (b) Effects of fluctuations (gauge field) on the present mean-field solutions. The present mean-field theory will be useful to describe the global feature of the phase diagram, including various crossover temperatures. Another important problem is
485
the phase diagram at zero temperature. While the present RVB theory focuses on the spin-liquid state, there is an antiferromagnetic (AF) state in the vicinity of half-filling. Whether there is a quantum disordered state or quantum critical point, or there is a direct first-order transition [44-46] between the two will be one of the interesting future problems.
References [1] H. Yasuoka, T. lmai, T. Shimizu, in: H. Fukuyama, S. Maekawa, A.P. Malozemoff (Eds.), Strong Correlation and Superconductivity, Springer, Berlin, 1989, p. 254. [2] H. Yasuoka, Hyperfine Interaction 105 (1997) 27. [3] K. Asayama, Y. Kitaoka, G.-q. Zheng, K. Ishida, Progr. Nucl. Magn. Resonance Spectroscopy 28 (1996) 221. [4] M.H. Julien, P. Carreta, M. Horvatic, C. Berthier, Y. Berthier, P. S6gransan, A. Carrigton, D. Colson, Phys. Rev. Lett. 76 (1996) 4238. [5] Y. ltoh, T. Machi, A. Fukuoka, K. Tanabe, H. Yasuoka, J. Phys. Soc. Japan 65 (1996) 3751. [6] H. Yasuoka, private communication. [7] P.W. Anderson, Science 235 (1987) 1196. [8] F.C. Zhang, T.M. Rice, Phys. Rev. B 37 (1988) 3759. [9] L.H. Tjeng, H. Eskes, G.A. Sawatzky, in: H. Fukuyama, S. Maekawa, A.P. Malozemoff (Eds.), Strong Correlation and Superconductivity, Springer, Berlin, 1989, p. 33. [10] H. Matsukawa, H. Fukuyama, J. Phys. Soc. Japan 58 (1989) 2845; 3687. [11] H. Matsukawa, H. Fukuyama, J. Phys. Soc. Jpn. 59 (1990) 1723. [12] T. Tohyama, S. Maekawa, J. Phys. Soc. Japan 59 (1990) 1760. [13] S. Maekawa, Y. Ohta, T. Tohyama, in: S. Maekawa, M. Sato (Eds.), Physics of High-Temperature Superconductors, Springer, Berlin, 1992, p. 29. [14] M.S. Hybertsen, E.B. Stechel, M. Schluter, D.R. Jennison, Phys. Rev. B 41 (1990) 11068. [15] G. Dopf, A. Muramatsu, W. Hanke, Physica C 185 189 (1991) 1495. [16] J. Wagner, W. Hanke, D.J. Scalapino, ibid 1617. [17] T. Tanamoto, H. Kohno, H. Fukuyama, J. Phys. Soc. Japan 62 (1993) 717. [18] M. Ogata, M. Luchini, S. Sorella, F. Assad, Phys. Rev. Lett. 66 (1991) 2388. [19] E. Dagotto, J. Riera, Phys. Rev. Lett. 70 (1993) 682. [20] Y. Ohta, T. Shimozato. R. Eder, S. Maekawa, Phys. Rev. Lett. 73 (1994) 324. [21] F.C. Zhang, C. Gross, T.M. Rice, H. Shiba, Supercond. Sci. Technol. 1 (1988) 36. [22] H. Yokoyama, M. Ogata, J. Phys. Soc. Japan 65 (1996) 3615. [23] G. Baskaran, Z. Zou, P.W. Anderson, Solid State Commun. 63 (1987) 973. [24] Y. Suzumura, Y. Hasegawa and H. Fukuyama, J. Phys. Soc. Japan 57 (1988) 401. 2768. [25] G. Kotliar, J. Liu, Phys. Rev. B 38 (1988) 5142. [26] T. Tanamoto, K. Kuboki, H. Fukuyama, J. Phys. Soc. Japan 60 (1991) 3072.
486
H. Fukuvama. H. Kohno / Journal o['Magnetism and Magnetic Materials 177-181 (1998) 483 486
[27] T. Tanamoto, H. Kohno, H. Fukuyama, J. Phys. Soc. Japan 63 (1994) 2739. 1-28] K. Yamada, S. Wakimoto, G. Shirane, C.H. Lee, M.A. Kastner, S. Hosoya, M. Greven, Y. Endoh, R.J. Birgeneau, Phys. Rev. Lett. 75 (1995) 1626. [29] H.A. Mook, M. Yethiraj, G. Aeppli, T.E. Mason, T. Armstrong, Phys. Rev. Lett. 70 (1993) 3490. [-303 H.F. Fong, B. Keimer, P.W. Anderson, D. Reznik, F. Dogan, I.A. Aksay, Phys. Rev. Lett. 75 (1995) 316. [31] E. Demler, S.C. Zhang, Phys. Rev. Lett. 75 (1995) 4126. 1-32] H. Kohno, B. Normand and H. Fukuyama, Proc. M2S HTSC, Beijing, 28 February 4 March 1997. E33] P. Dai, H.A. Mook and F. Dogan, preprint (condmat/9707112). [34] M. K~ill, A.P. Litvinchuk, P. Berastegui, L.G. Johansson, L. B6rjesson, Physica C 225 (1994) 317. [35] H. Harashina, K. Kodama, S. Shamoto, M. Sato, K. Kakurai, M. Nishi, Physica C 263 (1996) 257. [36] B. Normand, H. Kohno, H. Fukuyama, J. Phys. Rev. B 53 (1996) 856.
[37] A.J. Millis, H. Monien, Phys. Rev. Lett. 70 (1993) 2810. [38] G.-Q Zheng, T. Ogaguchi, T. Mito, Y. Kitaoka, K. Asayama, Y. Kodama, J. Phys. Soc. Japan 62 (1993) 2591. [39] K. Kakurai, S. Shamoto, T. Kiyokura, M. Sato, J.M. Tranquada, G. Shirane, Phys. Rev. B 48 (1993) 3485. [40] P.C. Hammel, A.P. Reyes, S.-W. Cheong, Z. Fisk, J.E. Schirber, Phys. Rev. Lett. 71 (1993) 440. [41] N.P. Ong, private communication. [42] See, for example, H.Y. Hwang, B. Batlogg, H. Takagi, H.L. Kao, J. Kwo, R.J. Cava, J.J. Krajewski, W.F. Peck Jr., Phys. Rev. Lett. 72 (1994) 2636. [43] P. Mendels, H. Alloul, J.H. Brewer, G.D. Morris, T.L. Duty, S. Johnston, E.J. Ansaldo, G. Collin, J.F. Marucco, C. Niedermager, D.R. Noakes, C.E. Stronach, Phys. Rev. B 49 (1994) 10035. [44] S.C. Zhang, Science 275 (1997) 1089. [45] F.F. Assaad, M. Imada and D.J. Scalapino, preprint (cond-mat/9706173). 1-46] S. Rabello, H. Kohno, E. Demle, S.C. Zhang, preprint (cond-mat/9707027).
ELSEVIER
~ H Journalof magnetism ~ H and magnetic materials
Invited paper
On the spin excitations of high-T cuprates Hidetoshi Fukuyama*, Hiroshi Kohno Department of Physics, Faculty of Science, University of To,o, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan
Abstract
The results of theoretical analysis of the spin excitation of high-To cuprates based on the slave-boson mean-field approximation of the t J model are summarized with special emphasis on the spin gap phenomenon typically seen in NMR rate and neutron scattering. Possible consequences of the spin gap on the anomalous frequency shifts of some particular oxygen phonon modes and important roles played by disorder are indicated. @ 1998 Elsevier Science B.V. All rights reserved. Keywords: Spin excitations; Spin gap; Mean-field theory; t-J model
1. Introduction: The spin gap
Various experiments so far carried out for high-Tc cuprates indicate the phase diagram on the plane of the doping rate, 3, and the temperature, T, as shown in Fig. 1, where 6M is the characteristic doping rate where the superconducting critical temperature, To, is maximum. In this figure, Ts6, termed as the spin gap temperature, is the characteristic temperature defined by the maxima of the nuclear magnetic resonance (NMR) rate, (T1T)- 1. The remarkable fact is that Ts6 in underdoped region (6 < 6M) appears to merge smoothly to Tc for 6 ~> &a. This indicates the essential roles played by the mechanism leading to the spin gap in stabilizing high Tc superconductivity. Such Tsc, first noted by Yasuoka et al. [-1, 2] has clearly been observed in bilayer systems such as YBCO (2 4 8) [-3] and underdoped YBCO (1 2 3). Experiment on three-layer Hg (1 2 2 3) [4] has shown that (T1T)- 1 of both inner and outer-layers show similar behaviours. Similarly, it has been clarified [-5] that a single-layer Hg (1 2 0 1) also shows this spin gap phenomenon. All these experimental data have firmly established that the spin gap in ( T I T ) -1 is intrinsic to the CuO2 plane of high-To cuprates in the underdoped region. In this context it is to be noted that in La2 xSr:,CuO4 the spin gap has been difficult to be
* Corresponding author.
identified. Very recently, however, Yasuoka [6] has succeeded in its identification by paying special attention to the broad linewidth and by extracting the intrinsic relaxation processes. Though the charge excitations are also seen to show anomalies, we will mainly discuss in this paper the spin gap associated with TsG.
2. Theoretical predictions based on slave-boson mean-field theory of the extended t - J model
The low-lying excitations of high-To cuprates, whose parent (6 = 0) compounds are charge-transfer type insulators, are described by the effective single-band model because of strong mixing [7] and explicitly by the t-J model [,8-16] H = Z tija*i~ai~+ J Z S i ' S i ,
(1)
i,j,~
where the double occupancy of each lattice site is excluded, and tij and J represent the transfer integrals of the Zhang Rice singlet and the superexchange interaction, respectively. Here we note that tij is very sensitive to the local structural properties which leads to different values oftij for different compounds [10, 11]. Hence, the characteristic features of each compound having different Fermi surfaces are represented by the different spatial extent
0304-8853/98/$19.00 ~ 1998 Elsevier Science B.V. All rights reserved PII S 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 0 8 6 5 - 2
484
H. Fukuvama, H. Kohno / Journal of Magnetism and Magnetic Materials 177-181 (1998) 483 486 T
T
L
"~.' '~TD AM \\T* \
A
~M
F
~
EL
D
6
"6
Fig. 1. Phase diagram ofhigh-T cuprates deduced from experiments. T is the temperature and 6 is the hole doping rate.
Fig. 2. Schematic phase diagram of the (extended) t J model. T is the temperature and 6 is the hole doping rate.
of t u, with which Eq. (1) is termed as extended t J model [17]. Though exact solutions of the t - J model, in general, are not known, the phase diagram on the plane of 6 and J/t are numerically pursued at T = 0 for one dimension (d = 1) [18] and two dimensions (d = 2) [19, 20]. Especially, in the latter the possibility of superconductivity of dx-' ~.-' symmetry for small doping and J/t of practical interest has been indicated in Ref. [20], which is in accordance with the variational calculations [21, 22]. Moreover, the mean-field theory based on the slave boson framework, which is developed on the original idea of RVB [7, 23] has predicted [24, 25] a phase diagram as shown in Fig. 2, where AF means antiferromagnetic phase realized at low doping. In Fig. 2, TD, TRw and TB are the characteristic temperatures of the onset of the coherent motion of spinons and holons, the formation of spinon singlet pairing and the bose condensation of holons, respectively. By these characteristic temperatures, the phase diagram is decomposed into various states. For T > TB the spin and charge are separated, while for T < TR spin and charge are combined to form ordinary electron liquid (EL). The superconducting critical temperature, T~, in this theory is the lower one of TRW and TB. It is to be noted [26] that TD and the higher one of TB and TRVB are crossover temperatures and are not true phase transition temperature in contrast to T~. In the anomalous metallic (AM) state described as a uniform RVB (u-RVB) state, spinons and holons move independently and coherently. In the spin gap (SG) state, where spin and charge are also separated, the spinons form short-range singlet described as a singlet RVB (s-RVB). This singlet has the dx: ,.-~symmetry resulting in the same symmetry for the superconducting state as well.
with g0(q, ~) being that of non-interacting spinons either in u-RVB or s-RVB state [17, 27]. The qualitative difference of the spin excitations between YBCO and LSCO, e.g., commensurate (C) in the former versus incommensurate (IC) in the latter, has been understood based on Eq. (2) by the difference of the shape of the Fermi surface. Moreover, in the case of LSCO, the absence of the low-energy excitations at the IC wave vector at low temperature and the doping dependence of the degree of IC seen in neutron scattering on the high-quality sample [28] are also in good accordance with the theoretical prediction [27]. For optimally doped YBCO, another characteristic feature is the resonance-like peak at energy ~o = 41 meV around the C wave vector present only in the superconducting state [29, 30]. There exists an interesting proposal for this 41 meV peak based on the 'n-resonance' [31]. This scenario was pursued by taking more realistic band structure into account in [32], where possible importance of the flat band nature around wave vectors (n, 0), (0, n) for the n-resonance has been indicated. In addition, a remarkable feature of the phase diagram of Fig. 2 is the existence of the spin-gap (SG) state. Actually Eq. (2) predicts a strong (almost exponential even for the present d:,: r~-symmetry) suppression of the NMR rate, (T1T)-1, below T = TRVB,which can be well above Tc in underdoped systems. Hence, it is tempting to identify Tsc. seen in experiment with TRVBin Fig. 2 [27]. Very recently, a well-defined IC peaks have been observed in the superconducting state of underdoped YBCO [33]. These peaks are located at (n _+ 5,-n + 6), differently from those of LSCO, and are naturally understood as the contribution from quasiparticles near the d-wave gap nodes. Remarkably, this IC structure persists even above To, which is consistent with the presence of the d-wave spin gap above Tc and its continuity to that of superconductivity.
3. Spin excitations and spin gap The spin excitation in this model is described by the dynamical spin susceptibility, g(q, ~o), z(q, (o) = Zo(q, o~)/[l + J(q)zo(q,o~)],
(2)
J(q) - J(cos q., + cos q~.),
(3)
4. Anomalous frequency shift of oxygen phonons The above-mentioned identification of the Ts~ with the TRVBis further supported by the agreement between the experimental results of the onset of the anomalous
tL Fukuyama, tL Kohno / Journal of Magnetism and Magnetic Materials 177-181 (1998) 483 486
frequency shift of Big [34] and BEg [35] in-plane oxygen modes vibrating along the c-axis seen in Y B C O (2 4 8) and underdoped Y B C O (1 2 3) and theoretical results based on the existence of the buckling [36]. In this model, there does not exist strong coupling between A-mode phonons and s-RVB, which seems to be also in accordance with experiments.
5. Effects of disorder As indicated in Section l, the identification of the spin gap in L S C O has been made only recently. This was one of the reasons why the spin gap phenomena were ascribed to the bilayer structure of this c o m p o u n d [37]. However, at the same time, possible important roles played by disorder has been indicated [27] by noting the experimental results of the N M R and neutron scattering which demonstrated that Zn substitution in Y B C O (2 4 8) [38] and underdoped YBCO (1 2 3) [39] destroys the spin gap in N M R rate without almost any effects on the N M R shift. Actually, it has now been clarified that the spin gap is due purely to the quantum coherence and hence is very susceptible to the perturbations leading to finite amount of local magnetic moments, which naturally order in the ground-state antiferromagnetically in bipartite lattices. Correspondingly there exist various experimental indications that L S C O is intrinsically disordered [-2, 40-42]. This finding will also be relevant to the experimental result by Mendels et al. on Zn substituted Y B C O (1 2 3) [-43].
6. Summary and discussion A m o n g various anomalous behaviors observed in high-To cuprates, the spin gap seen in N M R rate is noteworthy. It has been pointed out that the phase diagram derived by the slave-boson mean-field theory for the extended t J model has qualitative correspondence at least to a part of the experimental results, and that, especially, the identification of TsG with the formation of the shortrange singlet RVB gives coherent explanation for spin excitations and some particular modes of phonons. In spite of the encouraging results so far discussed we note various important questions which need further clarifications. Above all we note the following: (a) Origin of different temperature dependence between ( T 1 T ) -1 and the Knight shift, K. (b) Effects of fluctuations (gauge field) on the present mean-field solutions. The present mean-field theory will be useful to describe the global feature of the phase diagram, including various crossover temperatures. Another important problem is
485
the phase diagram at zero temperature. While the present RVB theory focuses on the spin-liquid state, there is an antiferromagnetic (AF) state in the vicinity of half-filling. Whether there is a quantum disordered state or quantum critical point, or there is a direct first-order transition [44-46] between the two will be one of the interesting future problems.
References [1] H. Yasuoka, T. lmai, T. Shimizu, in: H. Fukuyama, S. Maekawa, A.P. Malozemoff (Eds.), Strong Correlation and Superconductivity, Springer, Berlin, 1989, p. 254. [2] H. Yasuoka, Hyperfine Interaction 105 (1997) 27. [3] K. Asayama, Y. Kitaoka, G.-q. Zheng, K. Ishida, Progr. Nucl. Magn. Resonance Spectroscopy 28 (1996) 221. [4] M.H. Julien, P. Carreta, M. Horvatic, C. Berthier, Y. Berthier, P. S6gransan, A. Carrigton, D. Colson, Phys. Rev. Lett. 76 (1996) 4238. [5] Y. ltoh, T. Machi, A. Fukuoka, K. Tanabe, H. Yasuoka, J. Phys. Soc. Japan 65 (1996) 3751. [6] H. Yasuoka, private communication. [7] P.W. Anderson, Science 235 (1987) 1196. [8] F.C. Zhang, T.M. Rice, Phys. Rev. B 37 (1988) 3759. [9] L.H. Tjeng, H. Eskes, G.A. Sawatzky, in: H. Fukuyama, S. Maekawa, A.P. Malozemoff (Eds.), Strong Correlation and Superconductivity, Springer, Berlin, 1989, p. 33. [10] H. Matsukawa, H. Fukuyama, J. Phys. Soc. Japan 58 (1989) 2845; 3687. [11] H. Matsukawa, H. Fukuyama, J. Phys. Soc. Jpn. 59 (1990) 1723. [12] T. Tohyama, S. Maekawa, J. Phys. Soc. Japan 59 (1990) 1760. [13] S. Maekawa, Y. Ohta, T. Tohyama, in: S. Maekawa, M. Sato (Eds.), Physics of High-Temperature Superconductors, Springer, Berlin, 1992, p. 29. [14] M.S. Hybertsen, E.B. Stechel, M. Schluter, D.R. Jennison, Phys. Rev. B 41 (1990) 11068. [15] G. Dopf, A. Muramatsu, W. Hanke, Physica C 185 189 (1991) 1495. [16] J. Wagner, W. Hanke, D.J. Scalapino, ibid 1617. [17] T. Tanamoto, H. Kohno, H. Fukuyama, J. Phys. Soc. Japan 62 (1993) 717. [18] M. Ogata, M. Luchini, S. Sorella, F. Assad, Phys. Rev. Lett. 66 (1991) 2388. [19] E. Dagotto, J. Riera, Phys. Rev. Lett. 70 (1993) 682. [20] Y. Ohta, T. Shimozato. R. Eder, S. Maekawa, Phys. Rev. Lett. 73 (1994) 324. [21] F.C. Zhang, C. Gross, T.M. Rice, H. Shiba, Supercond. Sci. Technol. 1 (1988) 36. [22] H. Yokoyama, M. Ogata, J. Phys. Soc. Japan 65 (1996) 3615. [23] G. Baskaran, Z. Zou, P.W. Anderson, Solid State Commun. 63 (1987) 973. [24] Y. Suzumura, Y. Hasegawa and H. Fukuyama, J. Phys. Soc. Japan 57 (1988) 401. 2768. [25] G. Kotliar, J. Liu, Phys. Rev. B 38 (1988) 5142. [26] T. Tanamoto, K. Kuboki, H. Fukuyama, J. Phys. Soc. Japan 60 (1991) 3072.
486
H. Fukuvama. H. Kohno / Journal o['Magnetism and Magnetic Materials 177-181 (1998) 483 486
[27] T. Tanamoto, H. Kohno, H. Fukuyama, J. Phys. Soc. Japan 63 (1994) 2739. 1-28] K. Yamada, S. Wakimoto, G. Shirane, C.H. Lee, M.A. Kastner, S. Hosoya, M. Greven, Y. Endoh, R.J. Birgeneau, Phys. Rev. Lett. 75 (1995) 1626. [29] H.A. Mook, M. Yethiraj, G. Aeppli, T.E. Mason, T. Armstrong, Phys. Rev. Lett. 70 (1993) 3490. [-303 H.F. Fong, B. Keimer, P.W. Anderson, D. Reznik, F. Dogan, I.A. Aksay, Phys. Rev. Lett. 75 (1995) 316. [31] E. Demler, S.C. Zhang, Phys. Rev. Lett. 75 (1995) 4126. 1-32] H. Kohno, B. Normand and H. Fukuyama, Proc. M2S HTSC, Beijing, 28 February 4 March 1997. E33] P. Dai, H.A. Mook and F. Dogan, preprint (condmat/9707112). [34] M. K~ill, A.P. Litvinchuk, P. Berastegui, L.G. Johansson, L. B6rjesson, Physica C 225 (1994) 317. [35] H. Harashina, K. Kodama, S. Shamoto, M. Sato, K. Kakurai, M. Nishi, Physica C 263 (1996) 257. [36] B. Normand, H. Kohno, H. Fukuyama, J. Phys. Rev. B 53 (1996) 856.
[37] A.J. Millis, H. Monien, Phys. Rev. Lett. 70 (1993) 2810. [38] G.-Q Zheng, T. Ogaguchi, T. Mito, Y. Kitaoka, K. Asayama, Y. Kodama, J. Phys. Soc. Japan 62 (1993) 2591. [39] K. Kakurai, S. Shamoto, T. Kiyokura, M. Sato, J.M. Tranquada, G. Shirane, Phys. Rev. B 48 (1993) 3485. [40] P.C. Hammel, A.P. Reyes, S.-W. Cheong, Z. Fisk, J.E. Schirber, Phys. Rev. Lett. 71 (1993) 440. [41] N.P. Ong, private communication. [42] See, for example, H.Y. Hwang, B. Batlogg, H. Takagi, H.L. Kao, J. Kwo, R.J. Cava, J.J. Krajewski, W.F. Peck Jr., Phys. Rev. Lett. 72 (1994) 2636. [43] P. Mendels, H. Alloul, J.H. Brewer, G.D. Morris, T.L. Duty, S. Johnston, E.J. Ansaldo, G. Collin, J.F. Marucco, C. Niedermager, D.R. Noakes, C.E. Stronach, Phys. Rev. B 49 (1994) 10035. [44] S.C. Zhang, Science 275 (1997) 1089. [45] F.F. Assaad, M. Imada and D.J. Scalapino, preprint (cond-mat/9706173). 1-46] S. Rabello, H. Kohno, E. Demle, S.C. Zhang, preprint (cond-mat/9707027).