Electron Correlation in High Temperature Cuprates

Electron correlation plays a key role in high-temperature cuprate superconductors. Material-parameter dependence of cuprates is important to clarify the mechanism of high temperature superconductivity. 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-filling. The ground state undergoes a transition from a metal to a Mott insulator when the level difference εp-εd is increased.

Relationship between material parameters and critical temperature T C is important to clarify the mechanism of high temperature superconductivity.We consider two kinds of material parameters.The first category includes transfer integrals t dp , t pp and the level of d and p electrons.These parameters determine the band structure and the Fermi surface.The t dp is the transfer integral between nearest d and p orbitals in the CuO 2 plane, and t pp is that between nearest p orbitals.The other category is concerning with the strength of interactions such as the Coulomb interaction, U d and U p , 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][19][20][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 U d is large, the ground state is presumably insulating.We show, in fact, that there is a transition from a metallic state to an insulating state as the level difference between d and p electrons is increased.

Hamiltonian
The three-band Hamiltonian with d and p electrons is . .) ' . . where 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 t dp in this paper.We use the notation Δ dp = ε p -ε d .The number of sites is denoted as N s , and the total number of atoms is denoted as N a = 3N s .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 where

Gutzwiller Function
We adopt the Gutzwiller ansatz for the wave function: where P G is the Gutzwiller projection operator given by 1 (1 ) with the variational parameter in the range from 0 to unity: 0 ≤ g ≤ 1.The operator P G controls the on-site electron correlation on the copper site.When we take into account U p , the correlation among p electrons is also considered.In this case P G is with the parameter g p in the range 0 ≤ g p ≤ 1. ψ 0 is a one-particle wave function.We can take various kinds of states for ψ 0 ; for example, the Fermi sea or the Hartree-Fock state with some order parameters.

Optimized Wave Function
There are several ways to improve the Gutzwiller function.One method is to consider an optimization operator: ( ) where K is the kinetic part of the total Hamiltonian H and λ is a variational parameter [9].The ground state energy 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][23][24].We note that the Gutzwiller function ψ G 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].

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 curvature 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.

Variational Parameters of the Band Structure
In the three-band case, we have additional band parameters as variational parameters in ψ 0 .The one-particle state ψ 0 contains the variational parameters dp t  , pp t  , ' d t  , d ε and p ε : 0 0 ( , , ' , , ) In the non-interacting case, dp t  , pp t  and ' d t  coincide with t dp , t pp and t' d , respectively.The expectation values of physical quantities are calculated by employing the variational Monte Carlo method [6,7].

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/N s -ε d as a function of the level difference Δ dp is shown in Figure 2. The parameters are t pp = 0.4, t' d = 0.0 and U d = 8.We set U p = 0 for simplicity because U p 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 Δ dp ≈ 2 while that for the Gutzwiller function ψ G remains finite even for large Δ dp .The result shows that there is a transition from a metallic state to an insulating state at the critical value of Δ dp ~≈ (Δ dp ) c ~ 2.
We find that the curvature of the energy, as a function of Δ dp , is changed near Δ dp ~ 2. The energy is well fit-   ted by 1/Δ dp when Δ dp is greater than (Δ dp ) c .This is shown in Figure 2 where dashed curve indicates 1/Δ dp .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 J K : In the insulating state, the energy gain is proportional to J K , which is consistent with our result.We show the Gutawiller parameter g as a function of the level difference Δ dp in Figure 3. g for the Gutzwiller function ψ G descreases gradually when Δ dp is increased.In contrast, g for the optimized function ψ shows a rapid decrease and almost vanishes near (Δ dp ) c .This is consistent with the behavior of the energy shown in Figure 2, indicating that the ground state is an insulator when Δ dp > (Δ dp ) 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.

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 optimization operator on the basis of the Gutzwiller wave function.We have shown that this wave function describes a transition from a metallic state to an insulating state as the level difference Δ dp is increased.The critical value of Δ dp would depend on U d and band parameters.
pμ kσ and d kσ , are Fourier transforms of /2, i p µ σ +  and d iσ , respectively.The eigenvectors of this matrix give the corresponding weights of d and p electrons.

Figure 1 .
Figure 1.Ground state energy of the 2D singleband Hubbard model as a function of U at half-filling.The system size is 6 × 6.

Figure 2 .
Figure 2. Ground-state energy of the 2D d-p model per site as a function of Δ dp for t pp = 0.4, t' d = 0.0 and U d = 8 (in units of t dp ) 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 ψ G (with λ = 0).The dashed curve indicates that given by a constant times 1/(ε p -ε d ).

Figure 3 .
Figure 3.The Gutzwiller parameter g as a function of Δ dp for ψ G and the optimized wave function ψ. g for ψ decreases and vanishes as Δ dp is increased.
d and U p indicate the on-site Coulomb interaction among d and p electrons, respectively.