Journal of Applied Mathematics and Physics, 2014, 2, 72-76
Published Online April 2014 in SciRes. http://www.scirp.org/journal/jamp
http://dx.doi.org/10.4236/jamp.2014.25010
How to cite this paper: Yanagisawa, T., et al. (2014) Electron Correlation in High Temperature Cuprates. Journal of Applied
Mathematics and Physics, 2, 72-76. http://dx.doi.org/10.4236/jamp.2014.25010
Electron Correlation in High Temperature
Cuprates
Takashi Yanagisawa1, Mitake Miyazaki2, Kunihiko Yamaji1
1Electronics and Photonics Research Institute, National Institute of Advanced Industrial Science and Technology,
1-1-1 Umezono, Tsukuba, Japan
2Hakodate National College of Technology, 14-1 Tokura, Hakodate, Japan
Received January 2014
Abstract
Electron correlation plays a key role in high-temperature cuprate superconductors. Material-pa-
rameter dependence of cuprates is important to clarify the mechanism of high temperature su-
perconductivity. In this study, we examine the ground state of the three-band Hubbard model (d-p
model) that explicitly includes oxygen p orbitals. We consider the half-filled case with the large
on-site Coulomb repulsion Ud by using the variational Monte Carlo method. The ground state is
insulating when Ud is large at half-filli ng. The ground state undergoes a transition from a metal to
a Mott insulator when the level difference εp-εd is increased.
Keywords
High-Temperature Superconductor, Electron Correlation, Mott Insulator, Metal-Insulator
Transition, Charge-Transfer Insulator
1. Introduction
The study of high-temperature superconductors has been addressed extensively since the discovery of cuprate
superconductors [1]. The CuO2 plane in cuprates plays a key role for the appearance of superconductivity [2-10]
and the electron correlation in this plane is important [11-16].
Relationship between material parameters and critical temperature TC is important to clarify the mechanism of
high temperature superconductivity. We consider two kinds of material parameters. The first category includes
transfer integrals tdp, tpp and the level of d and p electrons. These parameters determine the band structure and
the Fermi surface. The tdp is the transfer integral between nearest d and p orbitals in the CuO2 plane, and tpp is
that between nearest p orbitals. The other category is concerning with the strength of interactions such as the
Coulomb interaction, Ud and Up, and the electron-phonon interaction. The transfer integrals play an important
role to obtain a finite bulk limit of the superconducting condensation energy [12,13,17]. The parameter values
were estimated in the early stage of research of high temperature cuprates [18-21].
In this paper, we investigate the ground state of the three-band d-p model in the half-filled case. When the
Coulomb interaction Ud is large, the ground state is presumably insulating. We show, in fact, that there is a tran-
sition from a metallic state to an insulating state as the level difference between d and p electrons is increased.

T. Yanagisawa et al.
73
2. Hamiltonian
The three-band Hamiltonian with d and p electrons is
/2, /2,/2,/2,/2,/2,/2,/2,
/2, /2,/2, /2,/2, /2,/2,
()[(). .]
(
diipixix iyiy dpiix iy ix iy
ii i
pp iyixiyixiyixiyi
i
Hddpppp tdpppp hc
tpp pp pp pp
σσσσσσ σσσσσ
σσ σ
σσ σσ σσ σ
σ
εε
++ ++
++++ ++−−
++++
++ +− −+−
=++++−− +
+ −−+
∑∑ ∑
∑

   
( )
( )
/2,
/2 /2/2 /2
..) '..
xd ij
ij
ddp pp p
dp
i iixixiyiy
ii
hctddhc
Unn Unnnn
σ σσ
σ
+
−
↑↓+ ↑+ ↓+ ↑+ ↓
++ +
++ +
∑
∑∑

 
(1)
where Ud and Up indicate the on-site Coulomb interaction among d and p electrons, respectively.
i
d
σ
and
i
d
σ
+
are operators for the d electrons.
/2,ix
p
σ ±
and
/2,ix
p
σ ±
+

denote operators for the p electrons at the site
/2ix
R±
,
and in a similar way
/2,iy
p
σ ±
and
/2, iy
p
σ ±
+

are defined.
d
i
n
σ
and
/ 2,
( ,)
p
i
n xy
µσ
µ
+
=

are number operators for d
and p electrons, respectively. We have introduced the parameter t’d that is the transfer integral of d electrons
between next nearest-neighbor cooper sites, where <ij>denotes a next nearest-neighbor pair of copper sites.
The energy unit is given by tdp in this paper. We use the notation Δdp = εp-εd. The number of sites is denoted as
Ns, and the total number of atoms is denoted as Na = 3Ns. Our study is within the hole picture where the lowest
band is occupied up to the Fermi energy μ. The non-interacting part is written as
( )
0
k
k xkykxk
kdkxkyk yk
xk ppk
ykpk p
d
Hdp pp
p
σ
σσσ σ
σ
σ
εµε ε
ε εµε
εεεµ
++ +


=

−


 
−−


−−

∑
(2)
where εxk = 2itdps in(kx/2), εyk = 2itdpsin(ky/2), εpk = −4tppsin(kx/2)sin(ky/2) and εdk = −4t’dcos(kx)cos(ky). pμkσ and
dkσ, are Fourier transforms of
/2, i
p
µσ +
and diσ, respectively. The eigenvectors of this matrix give the corres-
ponding weights of d and p electrons.
3. Mott State and Wave Function
3.1. Gutzwiller Function
We adopt the Gutzwiller ansatz for the wave function:
0
d
GG
P ψψ =
(3)
where PG is the Gutzwiller projection operator given by
1 (1)
d dd
Gii
i
P gnn
↑↓

= −−

∏
(4)
with the variational parameter in the range from 0 to unity: 0 ≤ g ≤ 1. The operator PG controls the on-site elec-
tron correlation on the copper site. When we take into account Up, the correlation among p electrons is also con-
sidered. In this case PG is
dp
G GG
P PP=
, (5)
where
( )( )
/2 /2/2 /2
11 11
ppp pp
Gp p
ix ixiy iy
i
Pgn ngn n
+ ↑+ ↓+ ↑+↓
 
= −−−−
 
∏ 
(6)
with the parameter gp in the range 0 ≤ gp ≤ 1. ψ0 is a one-particle wave function. We can take various kinds of
states for ψ0; for example, the Fermi sea or the Hartree-Fock state with some order parameters.
3.2. Optimized Wave Function
There are several ways to improve the Gutzwiller function. One method is to consider an optimization operator:

T. Yanagisawa et al.
74
( )
exp G K ψ λψ =
(7)
where K is the kinetic part of the total Hamiltonian H and λ is a variational parameter [9]. The ground state en-
ergy is lowered appreciably by the introduction of λ [11]. This type of wave function is an approximation to the
wave function in quantum Monte Carlo method [22-24].
We note that the Gutzwiller function ψG cannot describe an insulating state at half-filling because we have no
kinetic energy gain in the limit g→0. A wave function for the Mott state has been proposed for the single-band
Hubbard model by adopting the doublon-holon correlation factor [25]. In this paper, instead, we consider the
optimized Gutzwiller function in Equation (7) in the limit g→0 as a Mott insulating state. This is an insulator of
charge-transfer type [26] and is a metal-insulator transition in a multi-band system [27].
3.3. Mott State in the Single-Band Case
Here we examine the Mott insulating state for the single-band Hubbard model [28]. We show the ground-state
energy per site as a function of U in Figure 1, obtained by using the wave function in Equation (7). The curva-
ture of the energy, as a function of U, changes near U ~ 8 and the parameter g vanishes simultaneously. The
state with vanishing g would be an insulating state because of vanishingly small double occupancy.
3.4. Variational Parameters of the Band Structure
In the three-band case, we have additional band parameters as variational parameters in ψ0. The one-particle
state ψ0 contains the variational parameters
dp
t

,
pp
t

,
'
d
t

,
d
ε

and
p
ε

:
00
(,, ',,)
dp ppddp
ttt ψ ψεε = 
(8)
In the non-interacting case,
dp
t

,
pp
t

and
'd t

coincide with tdp, tpp and t’d, respectively. The expectation
values of physical quantities are calculated by employing the variational Monte Carlo method [6,7].
4. Mott State of Charge-Transfer Type
Our study on the Mott state of the three-band model is based on the wave function in Equation (7). The
ground-state energy per site E/Ns-εd as a function of the level difference Δdp is shown in Figure 2. The parame-
ters are tpp = 0.4, t’d = 0.0 and Ud = 8. We set Up = 0 for simplicity because Up is not important in the low doping
case and also in the half-filled case. The parameter g for the optimized function ψ vanishes at Δdp ≈ 2 while that
for the Gutzwiller function ψG remains finite even for large Δdp. The result shows that there is a transition from a
metallic state to an insulating state at the critical value of Δdp ~≈ (Δdp) c ~ 2.
We find that the curvature of the energy, as a function of Δdp, is changed near Δdp ~ 2. The energy is well fit-
Figure 1. Ground state energy of the 2D single-
band Hubbard model as a function of U at half-fill-
ing. The system size is 6 × 6.

T. Yanagisawa et al.
75
Figure 2. Ground-state energy of the 2D d-p model
per site as a function of Δdp for tpp = 0.4, t’d = 0.0
and Ud = 8 (in units of tdp) in the half-filled case on
6 × 6 lattice. The arrow indicates a transition point
where the curvature is changed. The dotted curve is
for the Gutzwiller function ψG (with λ = 0). The
dashed curve indicates that given by a constant
times 1/(εp-εd).
Figure 3. The Gutzwiller parameter g as a function
of Δdp for ψG and the optimized wave function ψ. g
for ψ decreases and vanishes as Δdp is increased.
ted by 1/Δdp when Δdp is greater than (Δdp)c. This is shown in Figure 2 where the dashed curve indicates 1/Δdp.
This shows that the most of energy gain comes from the exchange interaction between nearest neighbor d and p
electrons. This exchange interaction is given by JK:
2
11
K dp
dpd dp
Jt U

= +


∆ −∆

(9)
In the insulating state, the energy gain is proportional to JK, which is consistent with our result.
We show the Gutawiller parameter g as a function of the level difference Δdp in Figure 3. g for the Gutzwiller

T. Yanagisawa et al.
76
function ψG descreases gradually when Δdp is increased. In contrast, g for the optimized function ψ shows a rapid
decrease and almost vanishes near (Δdp)c. This is consistent with the behavior of the energy shown in Figure 2,
indicating that the ground state is an insulator when Δdp > (Δdp)c. When g vanishes, the double occupancy of d
holes is completely excluded and we have exactly one hole on the copper site. This is the insulating state of
charge-transfer type.
5. Summary
We have investigated the ground state of the three-band d-p model at half-filling by using the variational Monte
Carlo method. We have proposed the wave function for an insulating state of charge-transfer type with an opti-
mization operator on the basis of the Gutzwiller wave function. We have shown that this wave function de-
scribes a transition from a metallic state to an insulating state as the level difference Δdp is increased. The critical
value of Δdp would depend on Ud and band parameters.
Acknowledgements
We thank J. Kondo and I. Hase for useful discussions.
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