We investigate the ground state of two-dimensional Hubbard model on the basis of the variational Monte Carlo method. We use wave functions that include kinetic correlation and doublon-holon correlation beyond the Gutzwiller ansatz. It is still not clear whether the Hubbard model accounts for high-temperature superconductivity. The antiferromagnetic correlation plays a key role in the study of pairing mechanism because the superconductive phase exists usually close to the antiferromagnetic phase. We investigate the stability of the antiferromagnetic state when holes are doped as a function of the Coulomb repulsion U. We show that the antiferromagnetic correlation is suppressed as U is increased exceeding the bandwidth. High-temperature superconductivity is possible in this region with enhanced antiferromagnetic spin fluctuation and pairing interaction.
1. Introduction
The study of high-temperature superconductivity has attracted much attention since the discovery of cuprate high-temperature superconductors [1]. It is very important to clarify the properties of electronic states in the CuO_{2} plane [2–9]. The model for the CuO_{2} plane is called the d-p model (or is called the three-band Hubbard model). We often consider the simplified model, by neglecting oxygen sites in the CuO_{2} plane, called the (single-band) Hubbard model [10–15].
It remains unresolved as to whether the two-dimensional Hubbard model has a superconducting phase or not [16, 17]. It is believed that the electron correlation between electrons plays a significant role in cuprate superconductors. It is obvious that interaction with large energy scale is responsible for realization of high-temperature superconductivity. This subject has been investigated for more than two decades by using electronic models such as the two-dimensional Hubbard model, the d-p model, and the ladder Hubbard model [18–21].
The antiferromagnetic (AF) correlation plays a primarily important role in correlated electron systems. For example, the existence of striped states [22–29] can be understood on the basis of the two-dimensional Hubbard model [9, 30, 31]. The checkerboard-like density-wave modulation, observed by scanning tunneling microscopy (STM) [32–34], is also possible in some region of the parameter space in the Hubbard model [31]. The possibility of the coexistent state of antiferromagnetism and superconductivity has been reported [35, 36], and we can show the coexistence using the Hubbard model [9]. Thus the two-dimensional Hubbard model can describe some of anomalous properties reported for cuprate superconductors. The spin fluctuation, which is one of candidates of attractive interaction for high-temperature superconductivity, comes from the antiferromagnetic spin correlation. It is thus important to examine the stability of the antiferromagnetic state.
In the mean-field theory the antiferromagnetic correlation is enhanced as U is increased. This is also the case for the wave function of simple Gutzwiller ansatz. When we consider electron correlation beyond the Gutzwiller ansatz (mean-field wave function), the nature of the antiferromagnetism changes in the strongly correlated region where U is larger than the bandwidth. The AF correlation is increased as U is increased from zero and is maximally enhanced when U is about the bandwidth. The AF correlation shows a tendency to be suppressed when U is further increased more than the bandwidth. This is because we must suppress AF correlation to get the kinetic energy gain to lower the ground state energy. This indicates that the spin fluctuation becomes large in the large-U region.
In this paper we investigate the stability of AF ordered state using the wave functions that take account of kinetic correlation and doublon-holon correlation. We show the results obtained by the variational Monte Carlo method.
2. Model and Wave Functions
The Hubbard Hamiltonian is written as (1)H=∑ijσtijciσ†cjσ+U∑ini↑ni↓,where {tij} are transfer integrals and U is the on-site Coulomb energy. The transfer integral tij for nearest-neighbor pairs ij is denoted as tij=-t and that for next-nearest-neighbor pair ij is tij=-t′. Otherwise, tij vanishes. We denote the number of sites as N and the number of electrons as Ne. The energy unit is given by t.
The simple wave function is given by the Gutzwiller function: (2)ψG=PGψ0,where PG is the Gutzwiller operator (3)PG=∏j1-1-gnj↑nj↓with the parameter g in the range of 0≤g≤1. ψ0 is a trial function of the noninteracting state. To investigate the superconducting state, we use the BCS wave function ψBCS for ψ0 with the gap parameter Δ. The condensation energy is defined as (4)Econd=EΔ=0-EΔ=Δoptfor the optimized gap function Δopt.
We take into account intersite correlation effects in the wave function by multiplying a correlation operator e-λK: (5)ψλ=e-λKψG,where K indicates the kinetic term of the Hamiltonian K=∑ijσtijciσ†cjσ and λ is a real constant which is the variational parameter to be optimized [37–39]. This wave function is a first approximation to the wave function used in quantum Monte Carlo method [40].
The other way to improve the wave function is to consider the doublon-holon correlation that is controlled by the operator given by [41] (6)Pdh=∏j1-1-η∏τdj1-ej+τ+ej1-dj+τ.dj is the operator for the doubly occupied site given as dj=nj↑nj↓ and ej is that for the empty site given by ej=(1-nj↑)(1-nj↓). η is the variational parameter in the range of 0≤η≤1. The doublon-holon wave function is written as (7)ψη=PdhψG.We evaluate physical quantities for the Gutzwiller function ψG and optimized wave functions ψλ and ψη using the variational Monte Carlo method [42, 43].
3. Antiferromagnetism and Superconductivity
We first show the superconducting condensation energy and optimized superconducting gap function Δ as a function of U in Figures 1 and 2, respectively. These results were obtained by using the Gutzwiller function with a BCS trial function. The results show that Δ increases as U is increased and will have a maximum at some U. This indicates that the superconducting state becomes more stable in the strongly correlated region for larger U. The region with U being greater than the bandwidth is called the strongly correlated region [41, 44]. It is difficult to observe a clear sign of superconductivity in weakly correlated region; for example, U is less than 6t, by numerical calculations. This is consistent with the results obtained by quantum Monte Carlo methods [17, 39, 45].
Superconducting condensation energy as a function of U for the Gutzwiller function. The system is 10×10 lattice with t′=-0.2 and the electron number is Ne=84. We impose the periodic boundary condition in one direction and antiperiodic boundary condition in the other direction.
Optimized superconducting gap function as a function of U for the BCS Gutzwiller function. The conditions are the same as in Figure 1.
This result suggests that there is some effect which induces an attractive interaction to form electron pairings in strongly correlated region. We examine the stability of AF ordered state in this region. For this purpose we employ the kinetic correlation function ψλ=e-λKψG and the doublon-holon correlation function ψη=PdhψG. We compare the ground state energy for ψG, ψη, and ψλ in Table 1. Among these wave functions, ψλ shows the lowest ground state energy. This indicates that the intersite correlation induced by the kinetic operator K is important in the correlated region. In Figure 3 we show the kinetic energy which is the expectation value of the kinetic term as well as the ground state energy as a function of λ for ψλ. The kinetic energy is lowered appreciably by the kinetic correlation operator e-λK.
Ground state energy per site for wave functions ψG, doublon-holon function ψη and ψλ. The parameters are Ne=86, U=10, t′=-0.2 on 10×10 lattice with periodic boundary conditions. ΔAF is the antiferromagnetic order parameter.
Wave function
E/N
g
ΔAF
η
λ
ψG
−0.5859
0.21
0.0
1.0
0
ψG(AF)
−0.5953
0.21
0.2
1.0
0
ψη=PdhψG
−0.6179
0.21
0.0
0.78
0
ψλ=e-λKψG
−0.6643
0.06
0.0
1.0
0.06
Ground state energy (circles) and the kinetic energy (squares) as a function of λ, evaluated using ψλ. The parameters are U=10, t′=-0.2, and Ne=100.
When U is large, there is a competition between antiferromagnetic energy gain and kinetic energy gain to lower the total energy. The AF energy gain, being proportional to the AF exchange coupling J∝t2/U, is reduced gradually as U is increased. We show this in Figure 4 where the energy gain of the AF state with reference to the normal state is shown as a function of U. When U is increased more than the bandwidth, the antiferromagnetism is suppressed and its energy gain is decreased as U is increased. This indicates that there is a large AF spin fluctuation in this region. This shows a possibility of AF fluctuation induced electron pairing in the strongly correlated region.
Antiferromagnetic condensation energy as a function of U. The system is 10×10 lattice with t′=-0.2. The electron number is Ne=86 (lower curve) and Ne=84 (upper curve). In the latter Ne=84 case we have an open shell; that is, the highest occupied electron level is partially filled. We use the periodic boundary condition in both directions.
4. Summary
We have investigated the two-dimensional Hubbard model by adopting the wave function that takes into account intersite electron correlation using the variational Monte Carlo method. The condensation energy becomes large in the strongly correlated region, suggesting a possibility of high-temperature superconductivity. The reduction of antiferromagnetic correlation suggests large spin fluctuation in this region. We expect that this spin fluctuation in strongly correlated region induces an attractive interaction which promotes superconductivity. In the weakly correlated region for U/t less than about 6, there is also spin fluctuation because the AF order is suppressed as U is decreased. In numerical calculations, however, the pairing state is not stabilized in the weakly correlated region. This indicates that only the spin fluctuation effect cannot promote electron pairing and we further need strong correlation to account for high-temperature superconductivity.
It is also important to discuss the role of other parameters in the Hubbard model. In particular, the next-nearest-neighbor transfer t′ is important since the density of states crucially depends on t′ due to the van Hove singularity of a two-dimensional system. The nearest-neighbor Coulomb interaction V is also important. The on-site Coulomb interaction U is renormalized by V and V will change the nature of antiferromagnetic correlation.
It is necessary to consider three-dimensional electronic models because real cuprate superconductors are three-dimensional layered materials. It is important to investigate the electronic state of layered cuprates because an interlayer interaction may bring about interesting phenomena. They can be regarded as a multiband superconductor and have potential for new phenomena [46–52].
It is also necessary to examine the effect of strong correlation on the electron-phonon interaction [53–56]. There is a possibility that electron-phonon and electron-electron interactions cooperate to realize a high-temperature superconductor. The electron-phonon coupling may induce long-range attractive interaction. We expect that the long-range part of attractive interaction may cooperate with the on-site repulsive interaction to induce superconductivity.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author thanks K. Yamaji and M. Miyazaki for useful discussions. This work was supported in part by grant-in-aid from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. A part of numerical calculations was performed at the Supercomputer Center of the Institute for Solid State Physics, University of Tokyo.
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