Magnetsim, Fluctuations and Mechanism of High-Temperature Superconductivity

We investigate the ground state of the two-dimensional d-p model (three-band Hubbard model) by using a variational Monte Carlo method. The superconducting condensation energy is evaluated for the Gutzwiller-BCS wave function. We show that there is a crossover between strongly and weakly correlated regions as the level difference between d and p orbitals increases. The gap function and the condensation energy can be large in the crossover region. This result indicates a possibility of high-temperature superconductivity in the two-dimensional d-p model.


Introduction
The research of mechanism of high-temperature superconductivity has attracted much attention since the discovery of cuprate high-temperature superconductors [1,2]. Because it has been established that the Cooper pairs of cuprate high-temperature superconductors have the d-wave symmetry, the electron correlation plays an important role for the appearance of superconductivity. It is primarily important to clarify the phase diagram of electronic states in the CuO 2 plane contained commonly in cuprate high-temperature superconductors. The CuO 2 plane consists of oxygen atoms and copper atoms. The electronic model for this plane is the d-p model (or three-band Hubbard model) [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. An interaction with a large energy scale is necessary for the realization of high-temperature superconductivity. It has been argued whether the on-site Coulomb repulsive interaction induces superconductivity for the two-dimensional single-band Hubbard model [18,19,20,21,22,23,24,25,26,27,28,29,30] and ladder model [31,32,33].
It is also expected that there is a superconducting phase in the two-dimensional d-p model. It is, however, difficult to obtain a clear evidence of superconductivity in the two-dimensional d-p model because the energy gain by introducing the superconducting gap is very small for the Gutzwiller-BCS wave function [8,13]. Does this mean that there is no superconductivity in the d-p model? The purpose of this paper is to reexamine the stability of the superconducting state in the d-p model by using the variational Monte Carlo method. Numerous works performed so far have focused on the region where the energy difference between d and p levels are located closely each other. This is because there has been a suggestion that the superconducting critical temperature T c increases as the d-p level difference decreases [34].

Hamiltonian
The three-band model that explicitly includes oxygen p and copper d orbitals contains the parameters U d , U p , t dp , t pp , t ′ d , ǫ d and ǫ p . Our study is within the hole picture and the Hamiltonian is written as d iσ and d † iσ represent the operators for the d hole. p i±x/2σ and p † i±x/2σ denote the operators for the p holes at the site R i±x/2 , and in a similar way p i±ŷ/2σ and p † i±ŷ/2σ are defined. t dp is the transfer integral between adjacent Cu and O orbitals and t pp is that between nearest p orbitals.
ij denotes a next nearest-neighbor pair of copper sites. t ′ d was introduced to reproduce the Fermi surface [35] in several cuprate superconductors such as Bi 2 Sr 2 CaCu 2 O 8+δ [36] and Tl 2 ba 2 CuO 6+δ [37]. U d is the strength of the on-site Coulomb repulsion between d holes. In this paper we neglect U p among p holes because U p is small compared to U d [38,39,40,41]. In the low-doping region, U p will be of minor importance because the p-hole concentration is small [42]. The parameter values were estimated as, for example, U d = 10.5, U p = 4.0 and U dp = 1.2 in eV [39] where U dp is the nearest-neighbor Coulomb interaction between holes on adjacent Cu and O orbitals. We neglect U dp because U dp is small compared to U d . We use the notation ∆ dp = ǫ p − ǫ d . The number of sites is denoted as N , and the total number of atoms is N a = 3N . The energy unit is given by t dp .

Superconducting wave function
We examine the superconducting ground state of the two-dimensional d-p model by using the variational Monte Carlo method. The wave function is the Gutzwiller-projected wave function given as where P G is the Gutzwiller operator to control the double occupancy of d holes: g is a variational parameter in the range from 0 to unity. n diσ = d † iσ d iσ is the number operator for d holes. ψ 0 is the Fermi sea where the lowest band is occupied up to the Fermi energy µ. To represent a superconducting state, we take ψ 0 as the BCS wave function where α † kσ indicates the creation operator of the state in the lowest band with the momentum k which is represented by a linear combination of d and p electron operators d † kσ and p † kσ . The BCS parameters u k and v k are given by the ratio v bf k /u k = ∆ k /(ξ k + ξ 2 k + ∆ 2 k ) where ξ k is the dispersion relation of the lowest band. We assume the d-wave symmetry ∆ k = ∆(cos k x −cos k y ) and we regard ∆ as the superconducting gap. ∆ is a variational parameter that is optimized to give the lowest ground energy. The Projected-BCS wave function is written as where P Ne is a projection operator which extracts only the states with a fixed total hole number.
We use the variational Monte Carlo method in the evaluation of the ground-state energy [8]. We calculate the superconducting condensation energy ∆E is the ground-state energy with the gap ∆ and ∆ opt is the value of optimized superconducting gap ∆.
We show the condensation energy as a function of ∆ in Fig.1 where calculations were carried out on 8 × 8 lattice with 192 atoms. The parameters that we used are t pp = 0.4, u d = 10, U p = 0 and t ′ d = 0 in units of t dp . We put 76 holes on the lattice and we set ∆ dp = ǫ p − ǫ d = 2, 4 and 8. As shown in Fig.1, the condensation energy ∆E becomes extremely large as the level difference ∆ dp increases. When ∆ dp = 2, ∆E is very small and thus it is not easy to determine the condensation energy by numerical calculations. In contrast, surprisingly, ∆E turns out to be very large when ∆ dp is large. This indicates that the superconducting state is more stabilized in the strongly correlated region in accordance with the phenomenon in the single-band Hubbard model [30]. The existence of high-temperature superconductivity is suggested when the level difference ∆ dp is large.
Let us investigate the behavior of ∆E when the level difference ∆ dp increases further. We show the optimized gap amplitude ∆ as a function of ∆ dp in Fig.2 where the band parameters are the same as in Fig.1. The figure indicates that there is a maximum in the optimized gap function as a function of ∆ dp in the large-∆ dp region.

Discussion
The condensation energy ∆E exhibits a maximum with a large value, indicating that there occurs a crossover between strongly and weakly correlated regions. This crossover is quite similar to that in the two-dimensional Hubbard model [30]. A large fluctuation presumably exists in the crossover region. This indicates a possibility of high-temperature superconductivity in the d-p model. A crossover from weakly to strongly coupled systems is universal phenomenon that exists ubiquitously in the world. For example, the Kondo effect exhibits a universal logarithmic anomaly that appears as a crossover when the system approaches the strong coupling region (low temperature region) from the weak coupling region (high temperature region) [44,45,46,47,48,49]. A two-impurity Kondo problem also shows a crossover [50,51,52]. There may a class of phenomena that shows a crossover between weakly and strongly interacting regions. This class may include, for example, QCD [53], BCS-BEC crossover [54], Hubbard model, sine-Gordon model [55,56,57]  ∆E ∆ ∆ dp = 8t dp ∆ dp = 8t dp ∆ dp = 8t dp ∆ dp = 4t dp 2t dp Figure 1. Superconducting condensation energy as a function of the gap function for ∆ dp = 2, 4 and 8 in units of t dp . Numerical calculations were carried on 8 × 8 lattice with 76 holes. The band parameters are t pp = 0.4, t ′ d = 0, U d = 10 and U p = 0. We used the Gutzwiller-BCS wave function in eq.(5) where the hole number is fixed to be 76. ∆ ∆ dp Figure 2. Superconducting gap as a function of the level difference ∆ dp ≡ ǫ p − ǫ d . Numerical calculations were performed on 8 × 8 lattice with 76 holes. The band parameters are t pp = 0.4, t ′ d = 0, U d = 10 and U p = 0. The upper curve was calculated by using the Gutzwiller-BCS function in eq.(5) and the lower one is by the wave function in eq.(6).

Summary
We investigated the ground state of the two-dimensional Hubbard model on the basis of the variational Monte Carlo method. The superconducting condensation energy was calculated by using the Gutzwiller-projected BCS wave function. Although the condensation energy ∆E is very small when the level difference between d and p orbitals is small, ∆E increases as the level difference increases, suggesting a possibility of high-temperature superconductivity. It would be better to examine the d-p model by using improved wave functions [30,35,58,59,60,61]. The Mott transition based on the d-p model was investigated with improved wave function [35]. A crossover for the antiferromagnetic correlation can be explored on the basis of improved wave functions.