Advances in Condensed Matter Physics

Volume 2015, Article ID 141263, 5 pages

http://dx.doi.org/10.1155/2015/141263

## Superconductivity, Antiferromagnetism, and Kinetic Correlation in Strongly Correlated Electron Systems

Electronics and Photonics Research Institute, National Institute of Advanced Industrial Science and Technology, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan

Received 18 August 2015; Accepted 28 September 2015

Academic Editor: Artur P. Durajski

Copyright © 2015 Takashi Yanagisawa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 . We show that the antiferromagnetic correlation is suppressed as 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 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 is larger than the bandwidth. The AF correlation is increased as is increased from zero and is maximally enhanced when is about the bandwidth. The AF correlation shows a tendency to be suppressed when 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- 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 where are transfer integrals and is the on-site Coulomb energy. The transfer integral for nearest-neighbor pairs is denoted as and that for next-nearest-neighbor pair is . Otherwise, vanishes. We denote the number of sites as and the number of electrons as . The energy unit is given by .

The simple wave function is given by the Gutzwiller function: where is the Gutzwiller operator with the parameter in the range of . is a trial function of the noninteracting state. To investigate the superconducting state, we use the BCS wave function for with the gap parameter . The condensation energy is defined as for the optimized gap function .

We take into account intersite correlation effects in the wave function by multiplying a correlation operator : where indicates the kinetic term of the Hamiltonian and 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] is the operator for the doubly occupied site given as and is that for the empty site given by . is the variational parameter in the range of . The doublon-holon wave function is written as We evaluate physical quantities for the Gutzwiller function and optimized wave functions and using the variational Monte Carlo method [42, 43].

#### 3. Antiferromagnetism and Superconductivity

We first show the superconducting condensation energy and optimized superconducting gap function as a function of in Figures 1 and 2, respectively. These results were obtained by using the Gutzwiller function with a BCS trial function. The results show that increases as is increased and will have a maximum at some . This indicates that the superconducting state becomes more stable in the strongly correlated region for larger . The region with 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, is less than , by numerical calculations. This is consistent with the results obtained by quantum Monte Carlo methods [17, 39, 45].