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Variational Monte Carlo study of the ground state of the two-dimensional d-p model
PHYSICA® ELSEVIER
Physica C 341-348 (2000) 251-252 www.elsevier.nl/locate/physc
Variational d-p m o d e l
Monte
Carlo
study
of the ground
state
of the two-dimensional
Soh Koike ~,b, Takashi Yanagisawa ~ and Kunihiko Yamaji ~ Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba, Ibaraki 305, Japan b Japan Science and Technology Corporation, Kawaguchi, Saitama 332-0012, Japan Variational Monte Carlo calculations are carried out to study the ground state of the two-dimensional
d-p model which is the most basic one for Cu02 plane in the cuprate superconductors. The possibility of superconductivity is investigated using the Gutzwiller-projected BCS and SDW wave functions for the 6 x 6 and 8 x 8 lattices. Near half-filling the SDW state is most stable in a wide range for both hole and electron doping cases. The d-wave superconducting state turns out to be more favorable than the SDW state when the hole doping ratio is more than or almost equal to 0.2. The transfer energy between neighboring O orbitals is found to extend the region of the d-wave state near half filling. 1. I N T R O D U C T I O N The common feature of the group of the cuprate high-To superconductors (HTSC) is the existence of the Cu02 plane in their crystal structure. The model which treats both one d-orbital of Cu atoms and two p-orbitals of 0 atoms in the plane is the two-dimensional d-p model, which is hard to treat theoretically since the three atomic sites exist in the unit cell. There are two famous theoretical models, that is, the single band Hubbard model and t-J model, both of which are considered to be effective models for the abovementioned d-p model, and are derived from the d-p model approximately. However whether the essence of the superconducting (SC) mechanism is picked up correctly or not in these effective models is not clear, thus it is often an issue of the controversy. The study of the phase diagram both for the above-mentioned effective models using the variational Monte Carlo (VMC) method were (tone. The phase diagram obtained from the t-.] model [1] was closer to the phase diagram of HTSC obtained from the experiments. However the SC condensation energy obtained from the t-,] model was order of magnitude larger than that obtained from the experiments [2,3], while the condensation energy obtained from the single band Hubbard model [4] was adequate. In this paper using the VMC method we treat the d-p 0921-4534/00/$ - see front matter ~ 2000 Elsevier Science B.M PII S0921-4534(00)00465-2
model [5], i.e., the most basic model for HTSC, and examine the ground state properties. 2. R E S U L T S We obtain the lowest energy for the employed trial wave function by the calculation of the expectation value of the Hamiltonian through the optimization of the variational parameters using the Monte Carlo technique. The wave functions considered in this paper are given by the Gutzwiller-projected SDW wave function and Gutzwiller-projected BCS wave function. Here we consider that the normal state energy is given by the above-mentioned wave function with the value of the order parameter equal to zero. These types of function are standard ground state wave functions. The parameters in our calculations are the following: ed = --2, ep = 0, Ud = 8, and tpp = 0 or 0.2 or 0.4 for the 6 x 6 square lattice, and tpp = 0.2 for the 8 x 8 square lattice. Here tpp is the transfer energy between the neighboring O orbitals and we used tpd as the energy unit. We have calculated the energy gain per site A E / N in the variational BCS (SDW) state in reference to the normal state energy, which is nothing but the SC (SDW) condensation energy. The phase diagram is shown in Fig. 1 in the plane of A E and doping ratio 5 for the 6 x 6 square lattice [5]. The d-wave state exists in a region for All rights reserved.
252
£ Koike et al./Physica C 341-348 (2000) 251-252
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. . . . . .
t' J
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z
w
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0.4
0.6
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5
Figure 1. Energy gains per site A E / N for various values of tpp for 5 for 6 x 6 square lattice are displayed. Open (Closed) symbols connected with line (broken line) represent the energy gains for tpp = 0 (tpp = 0.2). The circles, squares and triangles represent the energy gains for the SDW, d-wave and extended-s wave state, respectively. The energy gains for the SDW state near half filling are above the upper limit of the figure. The crosses connected with dotted line represent the SDW energy gains in the case of tpp = 0.4.
0.2 < [5[ < 0.5. The phase diagram for the d-p model is similar to that for the Hubbard model [4], and is quite different from that for the t-J model [1]. The extended-s wave state had an energy gain in the slightly electron-doped region, however, the SDW state occurred with the extremely large SDW-condensation energy in the region for 0 < ]~[ < 0.2. Therefore, the extendeds wave state is always hidden by the SDW state [5]. The wide region where the SDW state is most stable shrinks when tpp ~ O. This is due to the modification of the Fermi surface. If Ud is extremely large, the SDW region extends near 5 ~ 0.5 for which the d-wave region hardly exist [6]. Our result indicates that the SC phase exists for the intermediate value of Ud. We also have calculated A E / N for 8 × 8 square lattice and obtained the phase diagram in Fig. 2. The decrease of the energy gain for the d-wave state near 5 = 0 is considered to be due to the large energy difference between the highest occupied state and the lowest unoccupied one, i.e., the finite size effect. The optimized value of the gap parameter for the d-wave state near half-filling is about 0.01 and the above-mentioned energy dif-
-0.2
0 5
0.2
0.4
0.6
Figure 2. Superconducting condensation energies per site A E / N are shown for 5 for 8 x 8 lattice in the case of tpp = 0.2. Closed square (triangle) symbols connected with line represent the energy gains for the d-wave (extended-s wave) state. ference is about 0.05 ~ 0.12. The optimized value of the gap parameter for the extended-s wave state in a slightly electron-doped region is about 0.07. The energy for the SDW state for 8 × 8 lattice is under calculation now. Although these figures are slightly different, they are consistent in overall. The maximum energy gain is comparable to each other. These energy gains are comparable to the value for the Hubbard model [4] and are in reasonable agreement with the experimental SC condensation energy estimated from the critical magnetic field [2] and the specific heat [3]. This research is supported by ACT-JST.
REFERENCES 1. H. Yokoyama and M. Ogata, J. Phys. Soc. Jpn 65 (1996) 3615. 2. Z. Hao et al., Phys. Rev. B 43 (1991) 2844. 3. J. W. Loram et al., Phys. Rev. Lett. 71 (1993) 1740. 4. K. Yamaji et al., Physica C 304 (1998) 225; to appear in Physica B (Proc. LT22, 1999, Helsinki). 5. T. Yanagisawa et al., to appear in Physica B (Proc. 51-st Yamada Conf. on Strongly Correlated Systems, 1999, Nagano). 6. T. Asahata et al., J. Phys. Soc. Jpn. 65 (1996) 365.
Physica C 341-348 (2000) 251-252 www.elsevier.nl/locate/physc
Variational d-p m o d e l
Monte
Carlo
study
of the ground
state
of the two-dimensional
Soh Koike ~,b, Takashi Yanagisawa ~ and Kunihiko Yamaji ~ Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba, Ibaraki 305, Japan b Japan Science and Technology Corporation, Kawaguchi, Saitama 332-0012, Japan Variational Monte Carlo calculations are carried out to study the ground state of the two-dimensional
d-p model which is the most basic one for Cu02 plane in the cuprate superconductors. The possibility of superconductivity is investigated using the Gutzwiller-projected BCS and SDW wave functions for the 6 x 6 and 8 x 8 lattices. Near half-filling the SDW state is most stable in a wide range for both hole and electron doping cases. The d-wave superconducting state turns out to be more favorable than the SDW state when the hole doping ratio is more than or almost equal to 0.2. The transfer energy between neighboring O orbitals is found to extend the region of the d-wave state near half filling. 1. I N T R O D U C T I O N The common feature of the group of the cuprate high-To superconductors (HTSC) is the existence of the Cu02 plane in their crystal structure. The model which treats both one d-orbital of Cu atoms and two p-orbitals of 0 atoms in the plane is the two-dimensional d-p model, which is hard to treat theoretically since the three atomic sites exist in the unit cell. There are two famous theoretical models, that is, the single band Hubbard model and t-J model, both of which are considered to be effective models for the abovementioned d-p model, and are derived from the d-p model approximately. However whether the essence of the superconducting (SC) mechanism is picked up correctly or not in these effective models is not clear, thus it is often an issue of the controversy. The study of the phase diagram both for the above-mentioned effective models using the variational Monte Carlo (VMC) method were (tone. The phase diagram obtained from the t-.] model [1] was closer to the phase diagram of HTSC obtained from the experiments. However the SC condensation energy obtained from the t-,] model was order of magnitude larger than that obtained from the experiments [2,3], while the condensation energy obtained from the single band Hubbard model [4] was adequate. In this paper using the VMC method we treat the d-p 0921-4534/00/$ - see front matter ~ 2000 Elsevier Science B.M PII S0921-4534(00)00465-2
model [5], i.e., the most basic model for HTSC, and examine the ground state properties. 2. R E S U L T S We obtain the lowest energy for the employed trial wave function by the calculation of the expectation value of the Hamiltonian through the optimization of the variational parameters using the Monte Carlo technique. The wave functions considered in this paper are given by the Gutzwiller-projected SDW wave function and Gutzwiller-projected BCS wave function. Here we consider that the normal state energy is given by the above-mentioned wave function with the value of the order parameter equal to zero. These types of function are standard ground state wave functions. The parameters in our calculations are the following: ed = --2, ep = 0, Ud = 8, and tpp = 0 or 0.2 or 0.4 for the 6 x 6 square lattice, and tpp = 0.2 for the 8 x 8 square lattice. Here tpp is the transfer energy between the neighboring O orbitals and we used tpd as the energy unit. We have calculated the energy gain per site A E / N in the variational BCS (SDW) state in reference to the normal state energy, which is nothing but the SC (SDW) condensation energy. The phase diagram is shown in Fig. 1 in the plane of A E and doping ratio 5 for the 6 x 6 square lattice [5]. The d-wave state exists in a region for All rights reserved.
252
£ Koike et al./Physica C 341-348 (2000) 251-252
0.003
. . . . . .
t' J
t(.-
z
w
0.002
...........
! ......
A • 0.0025 . /
"5
I
i
o.oo,s
0.003
' 6 ~ 6 u'niist
I
?~. 0.0025 ' (°D
! ' '
/
1
o.ool
/
0.002 0.0015
¢._
z
8 ;~ 8 u'ni{s:
o.ool
I..IJ
o ooo
0.0005 0
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
'
-0.6
-0.4
5
Figure 1. Energy gains per site A E / N for various values of tpp for 5 for 6 x 6 square lattice are displayed. Open (Closed) symbols connected with line (broken line) represent the energy gains for tpp = 0 (tpp = 0.2). The circles, squares and triangles represent the energy gains for the SDW, d-wave and extended-s wave state, respectively. The energy gains for the SDW state near half filling are above the upper limit of the figure. The crosses connected with dotted line represent the SDW energy gains in the case of tpp = 0.4.
0.2 < [5[ < 0.5. The phase diagram for the d-p model is similar to that for the Hubbard model [4], and is quite different from that for the t-J model [1]. The extended-s wave state had an energy gain in the slightly electron-doped region, however, the SDW state occurred with the extremely large SDW-condensation energy in the region for 0 < ]~[ < 0.2. Therefore, the extendeds wave state is always hidden by the SDW state [5]. The wide region where the SDW state is most stable shrinks when tpp ~ O. This is due to the modification of the Fermi surface. If Ud is extremely large, the SDW region extends near 5 ~ 0.5 for which the d-wave region hardly exist [6]. Our result indicates that the SC phase exists for the intermediate value of Ud. We also have calculated A E / N for 8 × 8 square lattice and obtained the phase diagram in Fig. 2. The decrease of the energy gain for the d-wave state near 5 = 0 is considered to be due to the large energy difference between the highest occupied state and the lowest unoccupied one, i.e., the finite size effect. The optimized value of the gap parameter for the d-wave state near half-filling is about 0.01 and the above-mentioned energy dif-
-0.2
0 5
0.2
0.4
0.6
Figure 2. Superconducting condensation energies per site A E / N are shown for 5 for 8 x 8 lattice in the case of tpp = 0.2. Closed square (triangle) symbols connected with line represent the energy gains for the d-wave (extended-s wave) state. ference is about 0.05 ~ 0.12. The optimized value of the gap parameter for the extended-s wave state in a slightly electron-doped region is about 0.07. The energy for the SDW state for 8 × 8 lattice is under calculation now. Although these figures are slightly different, they are consistent in overall. The maximum energy gain is comparable to each other. These energy gains are comparable to the value for the Hubbard model [4] and are in reasonable agreement with the experimental SC condensation energy estimated from the critical magnetic field [2] and the specific heat [3]. This research is supported by ACT-JST.
REFERENCES 1. H. Yokoyama and M. Ogata, J. Phys. Soc. Jpn 65 (1996) 3615. 2. Z. Hao et al., Phys. Rev. B 43 (1991) 2844. 3. J. W. Loram et al., Phys. Rev. Lett. 71 (1993) 1740. 4. K. Yamaji et al., Physica C 304 (1998) 225; to appear in Physica B (Proc. LT22, 1999, Helsinki). 5. T. Yanagisawa et al., to appear in Physica B (Proc. 51-st Yamada Conf. on Strongly Correlated Systems, 1999, Nagano). 6. T. Asahata et al., J. Phys. Soc. Jpn. 65 (1996) 365.