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Kinetic-energy pairing and condensation energy in cuprates
Journal of Physics and Chemistry of Solids 67 (2006) 37–40 www.elsevier.com/locate/jpcs
Kinetic-energy pairing and condensation energy in cuprates Masao Ogata a,*, Hisatoshi Yokoyama b, Youichi Yanase a, Yukio Tanaka c, Hiroki Tsuchiura d a
Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan b Department of Physics, Tohoku University, Sendai 980-8578, Japan c Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan d Department of Applied Physics, Tohoku University, Sendai 980-8579, Japan
Abstract In order to study the condensation energy in cuprates and the possibility of kinetic-energy pairing, we study the two-dimensional Hubbard model on a square lattice near half-filling using a variational Monte Carlo method for superconductivity. We modify the variational wave function by taking account of the nearest–neighbor correlation between holons and doublons. It is found that there is a qualitative difference between the weak-coupling region and the strong-coupling region. In the strong-coupling region (UTUco), the energy gain at the superconducting transition is derived from the kinetic-energy, while in the weak-coupling region (U(Uco), a conventional BCS-type potential-energy gain occurs. Condensation energy is large in the strong-coupling side and expressed as fexp(K4at/J) with a being a constant and JZ4t2/U. This result with recent experiments of optical conductivity implies that cuprate superconductors belong to the strong-coupling region. Similar but slightly different condensation energy is also obtained in the calculation using the fluctuation exchange (FLEX) approximation. q 2005 Elsevier Ltd. All rights reserved. PACS: 74.70.Kn; 74.50.Cr; 73.20.Kr
1. Introduction In the conventional BCS theory, kinetic-energy increases below the superconducting transition temperature, while correlation energy decreases. As a total, there appears a condensation energy. However, recent experiments have shown that cuprate superconductors violate the sum rule of optical conductivity, s1(u) [1], which implies the lowering of kinetic-energy (K) below Tc because the sum of s1(u) is proportional to charminus-K in a simple case [2]. Such kineticenergy-driven superconductivity [3] sharply contrasts with that of BCS-type superconductors, and thus it is very interesting and important to study whether kinetic-energy pairing occurs in the reasonable models for high-Tc superconductors. In the resonating valence bond (RVB) theory, it has been argued that the superconductivity is induced by kinetic-energy of charge carriers [4,5]. However the weak-couplingspinfluctuation theory using a Hubbard model will not explain the kinetic-energy pairing if it is similar to the conventional BCS * Corresponding author. Tel.: C81 3 5841 4184; fax: C81 3 5841 4539. E-mail address: [email protected] (M. Ogata).
0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.10.017
theory. Moreover the presence of the superconductivity itself in the Hubbard model is still very controversial. In the weakcoupling limit, RPA theory [6], fluctuation exchange (FLEX) approximation [7], renormalization-group [8], and perturbation studies [9,10] concluded dx2Ky2 -wave superconductivity, while many quantum Monte Carlo studies [11] came to negative conclusions for a relatively weak coupling regime. Variational Monte Carlo (VMC) studies using the Gutzwiller projection argued that an antiferromagnetic (AF) order prevails widely near half filling, and narrows the superconducting region [12]. In contrast, in the strong-coupling limit, namely in t–J-type models, dx2Ky2 -wave superconductivity is supported by exact diagonalization [13], and variation methods [14,15]. In order to resolve this discrepancy and study the possibility of kinetic-energy pairing, we study the two-dimensional Hubbard model near half filling from a viewpoint of variational theory [16] and FLEX approximation [17]. In the original BCS theory, a BCS variational wavefunction was used. Correspondingly, in the case of the Hubbard model, it is natural to use some variational wavefunctions in which the effects of correlation are taken into account. We study carefully the stability of the superconducting variational states with vital improvement on the Gutzwiller projection, and estimate the condensation energy as well as the difference of the kineticenergies between the normal state and superconducting state.
M. Ogata et al. / Journal of Physics and Chemistry of Solids 67 (2006) 37–40
It is found that there is a crossover as a function of U/t and the kinetic-energy pairing actually occurs even in the Hubbard model in the strong-coupling region. Condensation energy is large in the strong-coupling side and expressed as fexp(K4at/ J) with a being a constant and JZ4t2/U. This result with recent experiments of optical conductivity implies that cuprate superconductors belong to the strong-coupling region.
2. Model and method We study the Hubbard model of a square lattice, H Z Ht C HU ZKt
X X ðc†is cjs C c†js cis Þ C U nj[njY hi;jis
(1)
j
and assume some variational wavefunctions. Generally, a Jastrow-type function, JZPF, is used as a trial state, where F signifies a one-body (Hartree–Fock) state, and P a correlation factor. As a representative normal state wavefunction, we use the Fermi sea FF for F; as a superconducting wavefunction, we use a fixed-n BCS state FSC(D, z) for F, with a dx2Ky2 -wave gap DkZD(cos kxKcos ky). Here, D and z are variational parameters which correspond to a gap function and chemical potential in the mean-field approximations. For the correlationQfactor P, the Gutzwiller (onsite) projection [18], PG Z j ½1Kð1KgÞnj[njY, has been often used for its simplicity. Although PG is successful in t–J-type models [19,15], it is necessary to include longer-range correlation in the Hubbard model [20–22]. This inclusion of the longer-range correlation is also understood from the relationship between the Hubbard and t-J models: HtKJ w eiS HHub eKiS [23,16]. Explicitly, we assume JQ Z
Y ½1KmQj PG F; with 0% m% 1
3. Results First, we consider the energy difference between the normal and the superconducting states (condensation energy) defined as DEZ Enorm KESC . Let us concentrate on the doped cases. Fig. 1 shows DE as a function of U/t for nZ0.88 and nZ0.80. It is apparent that there is a crossover in the behavior of DE. Note that our wavefunction supports the presence of d-wave superconductivity even in the small U region. However DE is very small for U(6.5t (w10K4t at UZ4t), and behaves mildly like a power-law function. We think that due to this imperceptibly weak superconductivity, many QMC studies for U/tZ2–4 [11] overlooked signs of superconductivity order. At Uw6.5t, an exponential-like rapid increase occurs, which originates in JSC Q . Then DE reaches a maximum at UZUco. As we will explain shortly, superconducting properties for UOUco qualitatively differ from those for U!Uco. For UOUco, DE is nicely fitted by the form exp(KaU/t) [or exp(K4at/J)] with a being a constant, which means that the effective attractive interaction is JZ4t2/U; a viewpoint from t–J-type models is justified. Let us turn to the superconducting properties. Fig. 2 shows the difference of kinetic-energy DEt Z ðEtnorm KEtSC Þ and potential energy DEU. The mechanism of energy gain changes at UwUco. For U(Uco, the energy gain is due to the lowering of EU (DEUO0), as in the BCS superconductivity. Inversely, for 0.015 Uco
0.005
n=0.80
(2)
hJQ jHt jJQ i ; hJQ jJQ i
20
30
Fig. 1. Energy difference (condensation energy) between the d-wave and the normal states, estimated in the optimized wavefunction JQ for 12% and 20% dopings.
0.03 L = 10
E U /t
0.02 0.01 0 -0.01 Et
-0.02
(3)
using the optimized wavefunction JQ.
10 U/t
E t /t,
for partially filling Q (n!1, n: electron density), an asymmetric projection Qj Z dj t ð1KejCt Þ is used, where djZnj[njY, ej Z ð1Knj[Þð1KnjYÞ, and t runs over all the nearest–neighbor sites. This factor takes account of virtual states in which a doublon and a holon tend to reside on the nearest–neighbor sites. In order to estimate the condensation energy as well as the differences in kinetic energies, it it suitable to carry out a variational Monte Carlo method [14,24,25], which gives numerically accurate expectation values for any correlation strengths. The total variational energy and the kinetic and potential energies are simply obtained as
hJQ jHU jJQ i EU Z hJQ jJQ i
Uco USI
0
Et Z
n=0.88
0.01
j
E Z Et C E U ;
L=10
∆E / t
38
-0.03
EU
n 0.88 0.80
0
10
U/t
20
30
Fig. 2. Difference of kinetic and potential energies between the normal and superconducting states for 12% and 20% dopings. Solid symbols represent the kinetic-energy differences. Positive sign corresponds to the kinetic-energy gain.
M. Ogata et al. / Journal of Physics and Chemistry of Solids 67 (2006) 37–40
2×10-4
∆ Ek
1×10-4
10% hole-dope 15% hole-dope 20 % hole-dope
0
-1×10-4 -2×10-4 -3×10-4 -4×10-4 2.5
3
3.5
U/t
4
4.5
5
Fig. 3. Difference of kinetic energy between the normal and superconducting states for several values of dopings obtained in the FLEX approximation.
UTUco, kinetic-energy gain occurs (Et is lower in the superconducting state), agreeing with cuprates. These aspects do not depend qualitatively on n, although the magnitude of both DEt and DEU decreases as n decreases. Note that Tc is related to the total energy gain and is not connected directly to the relative weight of the kinetic-energy gain. The lowering of Et has been found also in a dynamical cluster approximation [26]. The momentum distribution function n(k) gives another evidence for the crossover. Since the dx2Ky2 -wave gap has nodes in the yZGx(GKM) direction, n(k) has a discontinuity at kZ kF in this direction [27]. The magnitude of this jump is the renormalization factor Z, which is related to the inverse of the electron effective mass. For UTUco, the superconducting state has a larger value of Z than the normal state [16]. This means that electrons are hard to move (or heavy) in the normal state, hampered by the strong correlation; in the superconducting state, the coherence somewhat relieves the suppressed mobility. Thus, the lowering in Et induces superconductivity. Recently we calculate the same quantity using FLEX approximation. Fig. 3 shows the results of DEk for various dopings as a function of U/t. Here the next–nearest neighbor hopping t 0 ZK0.25t is included and the temperature is fixed to TZ0.005t which is far below Tc. It is clear that the sign of DEk changes from negative to positive with increasing U/t. The negative sign corresponds to the conventional BCS theory. For UOUco, we have kinetic-energy gain. 4. Conclusion In summary, we have studied the two-dimensional Hubbard model near half filling, based on evolved VMC calculations. It is found that the dx2Ky2 -wave superconducting state is stabilized in the Hubbard model if the variational states are improved by including the correlation between holons and doublons. Furthermore, there is a crossover at around Uco. For U(Uco, the superconducting state is similar to the conventional BCS state, but for UTUco, the superconducting transition is caused by lowering the kinetic-energy. Recent experiments of s1(u) for cuprates [1] accord with the characteristics of this strongcoupling region. This kind of crossover from weak- to strong-coupling regimes is not restricted to the present case. The attractive
39
Hubbard model exhibits similar behavior in the s-wave superconducting transition for any electron density [28]. Hence, the kinetic-energy-driven superconductivity arises, not because the model has repulsive interaction, but the interaction is strong enough. The VMC calculations for the t–J model showed that the exchange energy is lowered by a dx2Ky2 -wave superconducting state (see Table 1 in Ref. [15], and also [29]). Since the kinetic part of the Hubbard model induces the exchange term (JZ4t2/ U), the results of t–J models are compatible with those of the Hubbard model for UTUco. The magnitude of condensation energy in experiment for cuprates [30] are not necessarily in good accordance with the present results. We think that further refinement on the wave function Jnorm reduces the normal state energy. However, the crossover between the weak-coupling and the strong-coupling will be robust. Note that Jnorm is just a renormalized Fermi liquid state, and it is strongly demanded to construct a nonFermi liquid variational state as experiments have shown. This is an interesting open question, and we expect that the further refinement on Jnorm will lead to such a state. It will be also interesting to see the effects of t 0 for the condensation energy and the kinetic-energy pairing. It has been shown that the qualitative features do not change for finite t 0 [31], although the AF instability is enhanced for the electron doped case (t 0 O0). References [1] D.N. Basov, et al., Science 283 (1999) 49; H.J.A. Molegraaf, et al., Science 295 (2002) 2239; A.F. Santander-Syro, et al., Europhys. Lett. 62 (2003) 568; A.F. Santander-Syro, et al., cond-mat/0404290. [2] P.F. Maldague, Phys. Rev. B 16 (1977) 2437. [3] J.E. Hirsch, Physica C, 199 (1992) 305; J.E. Hirsch, F. Marsiglio, Phys. Rev. B 62 (2000) 15131; P.W. Anderson, Science 268 (1995) 1154; M. Imada, S. Onoda, in: J. Bonca, et al. (Ed.), Open Problems in Strongly Correlated Electron Systems, Kluwer Academic Publication, 2001, p. 69; J.E. Hirsch, Science 295 (2002) 2226. [4] P.W. Anderson, Physica C 341–348 (2000) 9. [5] P.A. Lee, Physica C 317–318 (1999) 194. [6] D.J. Scalapino, Phys. Rep. 250 (1995) 329. [7] N.E. Bickers, D.J. Scalapino, S.R. White, Phys. Rev. Lett. 62 (1989) 961. [8] C.J. Halboth, W. Metzner, Phys. Rev. B 61 (2000) 7364; D. Zanchi, H.J. Schulz, Phys. Rev. B 61 (2000) 13609; N. Furukawa, T.M. Rice, N. Salmhofer, Phys. Rev. Lett. 81 (1998) 3195. [9] R. Hlubina, Phys. Rev. B 59 (1999) 9600. [10] J. Kondo, J. Phys. Soc. Jpn 70 (2002) 808. [11] A. Moreo, Phys. Rev. B 45 (1992) 5059; N. Furukawa, M. Imada, J. Phys. Soc. Jpn 61 (1992) 3331; M. Guerrero, G. Ortiz, J.E. Gubernatis, Phys. Rev. B 59 (1999) 1706. [12] K. Yamaji, et al., Physica C 304 (1998) 225. [13] E. Dagotto, et al., Phys. Rev. B 49 (1994) 3548. [14] H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn 57 (1988) 2482; C. Gros, Ann. Phys. (N.Y.) 189 (1989) 53. [15] H. Yokoyama, M. Ogata, J. Phys. Soc. Jpn 65 (1996) 3615. [16] H. Yokoyama, Y. Tanaka, M. Ogata, H. Tsuchiura, J. Phys. Soc. Jpn 73 (2004) 1119. [17] Y. Yanase, M. Ogata, J. Phys. Soc. Jpn 74 (2005) 1534. [18] M.C. Gutzwiller, Phys. Rev. Lett. 10 (1963) 159. [19] C. Gros, R. Joynt, T.M. Rice, Phys. Rev. B 36 (1987) 381; H. Yokoyama, M. Ogata, Phys. Rev. Lett. 67 (1991) 3610. [20] H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn 56 (1987) 1490.
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[21] T.A. Kaplan, P. Horsch, P. Fulde, Phys. Rev. Lett. 49 (1982) 889; P. Fazekas, Phys. Scr. T 29 (1989) 125; H. Otsuka, J. Phys. Soc. Jpn 61 (1992) 1645. [22] H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn 59 (1990) 3669. [23] A.B. Harris, R.V. Lange, Phys. Rev. 157 (1967) 295. [24] H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn 56 (1987) 3582. [25] D. Ceperley, G.V. Chester, K.H. Kalos, Phys. Rev. B 16 (1977) 3081; H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn 56 (1987) 1490.
[26] T.A. Maier, et al., Phys. Rev. Lett. 92 (2004) 027005. [27] A. Paramekanti, M. Randeria, N. Trivedi, Phys. Rev. Lett. 87 (2001) 217002 (and cond-mat/0305611). [28] H. Yokoyama, Prog. Theor. Phys. 108 (2002) 59 (and in preparation). [29] D.J. Scalapino, S.R. White, Phys. Rev. B 58 (1998) 8222; E. Demler, S.C. Zhang, Nature 396 (1998) 733. [30] J.W. Loram, et al., Physica C 341–348 (2000) 831. [31] H. Yokoyama, Y. Tanaka, M. Ogata, this issue.
Kinetic-energy pairing and condensation energy in cuprates Masao Ogata a,*, Hisatoshi Yokoyama b, Youichi Yanase a, Yukio Tanaka c, Hiroki Tsuchiura d a
Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan b Department of Physics, Tohoku University, Sendai 980-8578, Japan c Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan d Department of Applied Physics, Tohoku University, Sendai 980-8579, Japan
Abstract In order to study the condensation energy in cuprates and the possibility of kinetic-energy pairing, we study the two-dimensional Hubbard model on a square lattice near half-filling using a variational Monte Carlo method for superconductivity. We modify the variational wave function by taking account of the nearest–neighbor correlation between holons and doublons. It is found that there is a qualitative difference between the weak-coupling region and the strong-coupling region. In the strong-coupling region (UTUco), the energy gain at the superconducting transition is derived from the kinetic-energy, while in the weak-coupling region (U(Uco), a conventional BCS-type potential-energy gain occurs. Condensation energy is large in the strong-coupling side and expressed as fexp(K4at/J) with a being a constant and JZ4t2/U. This result with recent experiments of optical conductivity implies that cuprate superconductors belong to the strong-coupling region. Similar but slightly different condensation energy is also obtained in the calculation using the fluctuation exchange (FLEX) approximation. q 2005 Elsevier Ltd. All rights reserved. PACS: 74.70.Kn; 74.50.Cr; 73.20.Kr
1. Introduction In the conventional BCS theory, kinetic-energy increases below the superconducting transition temperature, while correlation energy decreases. As a total, there appears a condensation energy. However, recent experiments have shown that cuprate superconductors violate the sum rule of optical conductivity, s1(u) [1], which implies the lowering of kinetic-energy (K) below Tc because the sum of s1(u) is proportional to charminus-K in a simple case [2]. Such kineticenergy-driven superconductivity [3] sharply contrasts with that of BCS-type superconductors, and thus it is very interesting and important to study whether kinetic-energy pairing occurs in the reasonable models for high-Tc superconductors. In the resonating valence bond (RVB) theory, it has been argued that the superconductivity is induced by kinetic-energy of charge carriers [4,5]. However the weak-couplingspinfluctuation theory using a Hubbard model will not explain the kinetic-energy pairing if it is similar to the conventional BCS * Corresponding author. Tel.: C81 3 5841 4184; fax: C81 3 5841 4539. E-mail address: [email protected] (M. Ogata).
0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.10.017
theory. Moreover the presence of the superconductivity itself in the Hubbard model is still very controversial. In the weakcoupling limit, RPA theory [6], fluctuation exchange (FLEX) approximation [7], renormalization-group [8], and perturbation studies [9,10] concluded dx2Ky2 -wave superconductivity, while many quantum Monte Carlo studies [11] came to negative conclusions for a relatively weak coupling regime. Variational Monte Carlo (VMC) studies using the Gutzwiller projection argued that an antiferromagnetic (AF) order prevails widely near half filling, and narrows the superconducting region [12]. In contrast, in the strong-coupling limit, namely in t–J-type models, dx2Ky2 -wave superconductivity is supported by exact diagonalization [13], and variation methods [14,15]. In order to resolve this discrepancy and study the possibility of kinetic-energy pairing, we study the two-dimensional Hubbard model near half filling from a viewpoint of variational theory [16] and FLEX approximation [17]. In the original BCS theory, a BCS variational wavefunction was used. Correspondingly, in the case of the Hubbard model, it is natural to use some variational wavefunctions in which the effects of correlation are taken into account. We study carefully the stability of the superconducting variational states with vital improvement on the Gutzwiller projection, and estimate the condensation energy as well as the difference of the kineticenergies between the normal state and superconducting state.
M. Ogata et al. / Journal of Physics and Chemistry of Solids 67 (2006) 37–40
It is found that there is a crossover as a function of U/t and the kinetic-energy pairing actually occurs even in the Hubbard model in the strong-coupling region. Condensation energy is large in the strong-coupling side and expressed as fexp(K4at/ J) with a being a constant and JZ4t2/U. This result with recent experiments of optical conductivity implies that cuprate superconductors belong to the strong-coupling region.
2. Model and method We study the Hubbard model of a square lattice, H Z Ht C HU ZKt
X X ðc†is cjs C c†js cis Þ C U nj[njY hi;jis
(1)
j
and assume some variational wavefunctions. Generally, a Jastrow-type function, JZPF, is used as a trial state, where F signifies a one-body (Hartree–Fock) state, and P a correlation factor. As a representative normal state wavefunction, we use the Fermi sea FF for F; as a superconducting wavefunction, we use a fixed-n BCS state FSC(D, z) for F, with a dx2Ky2 -wave gap DkZD(cos kxKcos ky). Here, D and z are variational parameters which correspond to a gap function and chemical potential in the mean-field approximations. For the correlationQfactor P, the Gutzwiller (onsite) projection [18], PG Z j ½1Kð1KgÞnj[njY, has been often used for its simplicity. Although PG is successful in t–J-type models [19,15], it is necessary to include longer-range correlation in the Hubbard model [20–22]. This inclusion of the longer-range correlation is also understood from the relationship between the Hubbard and t-J models: HtKJ w eiS HHub eKiS [23,16]. Explicitly, we assume JQ Z
Y ½1KmQj PG F; with 0% m% 1
3. Results First, we consider the energy difference between the normal and the superconducting states (condensation energy) defined as DEZ Enorm KESC . Let us concentrate on the doped cases. Fig. 1 shows DE as a function of U/t for nZ0.88 and nZ0.80. It is apparent that there is a crossover in the behavior of DE. Note that our wavefunction supports the presence of d-wave superconductivity even in the small U region. However DE is very small for U(6.5t (w10K4t at UZ4t), and behaves mildly like a power-law function. We think that due to this imperceptibly weak superconductivity, many QMC studies for U/tZ2–4 [11] overlooked signs of superconductivity order. At Uw6.5t, an exponential-like rapid increase occurs, which originates in JSC Q . Then DE reaches a maximum at UZUco. As we will explain shortly, superconducting properties for UOUco qualitatively differ from those for U!Uco. For UOUco, DE is nicely fitted by the form exp(KaU/t) [or exp(K4at/J)] with a being a constant, which means that the effective attractive interaction is JZ4t2/U; a viewpoint from t–J-type models is justified. Let us turn to the superconducting properties. Fig. 2 shows the difference of kinetic-energy DEt Z ðEtnorm KEtSC Þ and potential energy DEU. The mechanism of energy gain changes at UwUco. For U(Uco, the energy gain is due to the lowering of EU (DEUO0), as in the BCS superconductivity. Inversely, for 0.015 Uco
0.005
n=0.80
(2)
hJQ jHt jJQ i ; hJQ jJQ i
20
30
Fig. 1. Energy difference (condensation energy) between the d-wave and the normal states, estimated in the optimized wavefunction JQ for 12% and 20% dopings.
0.03 L = 10
E U /t
0.02 0.01 0 -0.01 Et
-0.02
(3)
using the optimized wavefunction JQ.
10 U/t
E t /t,
for partially filling Q (n!1, n: electron density), an asymmetric projection Qj Z dj t ð1KejCt Þ is used, where djZnj[njY, ej Z ð1Knj[Þð1KnjYÞ, and t runs over all the nearest–neighbor sites. This factor takes account of virtual states in which a doublon and a holon tend to reside on the nearest–neighbor sites. In order to estimate the condensation energy as well as the differences in kinetic energies, it it suitable to carry out a variational Monte Carlo method [14,24,25], which gives numerically accurate expectation values for any correlation strengths. The total variational energy and the kinetic and potential energies are simply obtained as
hJQ jHU jJQ i EU Z hJQ jJQ i
Uco USI
0
Et Z
n=0.88
0.01
j
E Z Et C E U ;
L=10
∆E / t
38
-0.03
EU
n 0.88 0.80
0
10
U/t
20
30
Fig. 2. Difference of kinetic and potential energies between the normal and superconducting states for 12% and 20% dopings. Solid symbols represent the kinetic-energy differences. Positive sign corresponds to the kinetic-energy gain.
M. Ogata et al. / Journal of Physics and Chemistry of Solids 67 (2006) 37–40
2×10-4
∆ Ek
1×10-4
10% hole-dope 15% hole-dope 20 % hole-dope
0
-1×10-4 -2×10-4 -3×10-4 -4×10-4 2.5
3
3.5
U/t
4
4.5
5
Fig. 3. Difference of kinetic energy between the normal and superconducting states for several values of dopings obtained in the FLEX approximation.
UTUco, kinetic-energy gain occurs (Et is lower in the superconducting state), agreeing with cuprates. These aspects do not depend qualitatively on n, although the magnitude of both DEt and DEU decreases as n decreases. Note that Tc is related to the total energy gain and is not connected directly to the relative weight of the kinetic-energy gain. The lowering of Et has been found also in a dynamical cluster approximation [26]. The momentum distribution function n(k) gives another evidence for the crossover. Since the dx2Ky2 -wave gap has nodes in the yZGx(GKM) direction, n(k) has a discontinuity at kZ kF in this direction [27]. The magnitude of this jump is the renormalization factor Z, which is related to the inverse of the electron effective mass. For UTUco, the superconducting state has a larger value of Z than the normal state [16]. This means that electrons are hard to move (or heavy) in the normal state, hampered by the strong correlation; in the superconducting state, the coherence somewhat relieves the suppressed mobility. Thus, the lowering in Et induces superconductivity. Recently we calculate the same quantity using FLEX approximation. Fig. 3 shows the results of DEk for various dopings as a function of U/t. Here the next–nearest neighbor hopping t 0 ZK0.25t is included and the temperature is fixed to TZ0.005t which is far below Tc. It is clear that the sign of DEk changes from negative to positive with increasing U/t. The negative sign corresponds to the conventional BCS theory. For UOUco, we have kinetic-energy gain. 4. Conclusion In summary, we have studied the two-dimensional Hubbard model near half filling, based on evolved VMC calculations. It is found that the dx2Ky2 -wave superconducting state is stabilized in the Hubbard model if the variational states are improved by including the correlation between holons and doublons. Furthermore, there is a crossover at around Uco. For U(Uco, the superconducting state is similar to the conventional BCS state, but for UTUco, the superconducting transition is caused by lowering the kinetic-energy. Recent experiments of s1(u) for cuprates [1] accord with the characteristics of this strongcoupling region. This kind of crossover from weak- to strong-coupling regimes is not restricted to the present case. The attractive
39
Hubbard model exhibits similar behavior in the s-wave superconducting transition for any electron density [28]. Hence, the kinetic-energy-driven superconductivity arises, not because the model has repulsive interaction, but the interaction is strong enough. The VMC calculations for the t–J model showed that the exchange energy is lowered by a dx2Ky2 -wave superconducting state (see Table 1 in Ref. [15], and also [29]). Since the kinetic part of the Hubbard model induces the exchange term (JZ4t2/ U), the results of t–J models are compatible with those of the Hubbard model for UTUco. The magnitude of condensation energy in experiment for cuprates [30] are not necessarily in good accordance with the present results. We think that further refinement on the wave function Jnorm reduces the normal state energy. However, the crossover between the weak-coupling and the strong-coupling will be robust. Note that Jnorm is just a renormalized Fermi liquid state, and it is strongly demanded to construct a nonFermi liquid variational state as experiments have shown. This is an interesting open question, and we expect that the further refinement on Jnorm will lead to such a state. It will be also interesting to see the effects of t 0 for the condensation energy and the kinetic-energy pairing. It has been shown that the qualitative features do not change for finite t 0 [31], although the AF instability is enhanced for the electron doped case (t 0 O0). References [1] D.N. Basov, et al., Science 283 (1999) 49; H.J.A. Molegraaf, et al., Science 295 (2002) 2239; A.F. Santander-Syro, et al., Europhys. Lett. 62 (2003) 568; A.F. Santander-Syro, et al., cond-mat/0404290. [2] P.F. Maldague, Phys. Rev. B 16 (1977) 2437. [3] J.E. Hirsch, Physica C, 199 (1992) 305; J.E. Hirsch, F. Marsiglio, Phys. Rev. B 62 (2000) 15131; P.W. Anderson, Science 268 (1995) 1154; M. Imada, S. Onoda, in: J. Bonca, et al. (Ed.), Open Problems in Strongly Correlated Electron Systems, Kluwer Academic Publication, 2001, p. 69; J.E. Hirsch, Science 295 (2002) 2226. [4] P.W. Anderson, Physica C 341–348 (2000) 9. [5] P.A. Lee, Physica C 317–318 (1999) 194. [6] D.J. Scalapino, Phys. Rep. 250 (1995) 329. [7] N.E. Bickers, D.J. Scalapino, S.R. White, Phys. Rev. Lett. 62 (1989) 961. [8] C.J. Halboth, W. Metzner, Phys. Rev. B 61 (2000) 7364; D. Zanchi, H.J. Schulz, Phys. Rev. B 61 (2000) 13609; N. Furukawa, T.M. Rice, N. Salmhofer, Phys. Rev. Lett. 81 (1998) 3195. [9] R. Hlubina, Phys. Rev. B 59 (1999) 9600. [10] J. Kondo, J. Phys. Soc. Jpn 70 (2002) 808. [11] A. Moreo, Phys. Rev. B 45 (1992) 5059; N. Furukawa, M. Imada, J. Phys. Soc. Jpn 61 (1992) 3331; M. Guerrero, G. Ortiz, J.E. Gubernatis, Phys. Rev. B 59 (1999) 1706. [12] K. Yamaji, et al., Physica C 304 (1998) 225. [13] E. Dagotto, et al., Phys. Rev. B 49 (1994) 3548. [14] H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn 57 (1988) 2482; C. Gros, Ann. Phys. (N.Y.) 189 (1989) 53. [15] H. Yokoyama, M. Ogata, J. Phys. Soc. Jpn 65 (1996) 3615. [16] H. Yokoyama, Y. Tanaka, M. Ogata, H. Tsuchiura, J. Phys. Soc. Jpn 73 (2004) 1119. [17] Y. Yanase, M. Ogata, J. Phys. Soc. Jpn 74 (2005) 1534. [18] M.C. Gutzwiller, Phys. Rev. Lett. 10 (1963) 159. [19] C. Gros, R. Joynt, T.M. Rice, Phys. Rev. B 36 (1987) 381; H. Yokoyama, M. Ogata, Phys. Rev. Lett. 67 (1991) 3610. [20] H. Yokoyama, H. Shiba, J. Phys. Soc. Jpn 56 (1987) 1490.
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