1. Introduction
2. Superconductivity in Many-Electron Systems
3. Optimization Variational Monte Carlo Method
4. Phase Diagram by the Optimization Variational Monte Carlo Method
5. Superconductivity and Strong Correlation
6. Discussion
7. Conclusions
Acknowledgments
Author Contributions
Ethics Statement
Informed Consent Statement
Funding
Declaration of Competing Interest
References

Takashi Yanagisawa
^{}*
^{}

Author Information

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

*

Authors to whom correspondence should be addressed.

Received: 04 July 2024 Accepted: 22 August 2024 Published: 27 August 2024

© 2024 The authors. This is an open access article under the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/).

ABSTRACT:
It is very important to clarify the mechanism of
high-temperature superconductivity in strongly correlated electron systems. The
mechanism of superconductivity in high temperature cuprate superconductors has
been studied extensively since their discovery. We investigate the properties
of correlated electron systems and mechanism of superconductivity by using the
optimization quantum variational Monte Carlo method. The many-body wave
function is constructed by multiplying by correlation operators of exponential
type. We show that *d*-wave superconducting phase exists in the strongly
correlated region where the on-site repulsive interaction is as large as the
bandwidth or more than the bandwidth. The *d*-wave pairing
correlation function is shown as a function of lattice sites, showing that the
long-range order indeed exists.

Keywords:
High-temperature
superconductivity; Strongly correlated electron systems; Mechanism of
superconductivity; Optimization variational Monte Carlo method;
Hubbard model; Phase diagram

The physics of high-temperature superconductors have been studied intensively for more than 35 years since the discovery of high-temperature superconductivity [1]. It is still a challenging issue to clarify the mechanism of high-temperature superconductivity. Since the parent materials of high-temperature cuprates are Mott insulators when no carriers are doped, high-temperature cuprates are typical strongly correlated electron systems. The strong correlation makes it hard to elucidate the mechanism of superconductivity. Thus, it is important to understand the electronic properties of strongly correlated electron systems.
The CuO_{2} plane is commonly contained in various high temperature cuprates and consists of oxygen atoms and copper atoms. It is certain that the CuO_{2} plane plays an important role in the emergence of high-temperature superconductivity [2,3,4,5,6,7,8]. The fundamental and important model on this plane is the three-band d-p model [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. The two-dimensional (2D) Hubbard model is regarded as an effective model where we consider only *d* electrons by integrating out the freedom of *p* electrons. The 2D Hubbard model [27,28,29] is also the basic model for cuprate superconductors.
The 2D Hubbard model contains fruitful physics although it looks very simple, and it may include effective interactions that induce electron pairing to bring about high-temperature superconductivity. The Hubbard model has been studied intensively to clarify the pairing mechanism of high-temperature superconductivity [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49]. One may wonder why the effective attraction arises between electrons from the on-site repulsive Coulomb interaction. This effective pairing interaction may originate from the effective nearest-neighbor exchange coupling and the kinetic energy effect. On this subject, the ladder Hubbard model (two-chain model) has also been studied [50,51,52,53,54,55].
The Hubbard model was first introduced to understand the metal-insulator transition [27]. Recent studies indicate the possibility of the existence of a superconducting (SC) phase in the parameter space of the hole density, the strength of Coulomb interaction *U* and the next nearest-neighbor transfer integral $$\boldsymbol{t}^{\prime}$$ in the ground state [47]. These three parameters are important and give plentiful structures of the phase diagram that include the superconducting phase and the antiferromagnetic phase. The transfer $$\boldsymbol{t}^{\prime}$$ plays an essential role in determining the stability of magnetic states. For example, in the case where $$\boldsymbol{t}^{\prime}=\boldsymbol{0}$$, the antiferromagnetic state becomes unstable when holes are doped. The 2D Hubbard model is also useful to understand the appearance of inhomogeneous electronic states such as stripes [56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71] and checkerboard-like density of states [72,73,74,75]; the existence of these inhomogeneous states has indeed been reported for high-temperature cuprates.
In the study of cuprate superconductors and also iron-based superconductors, lattice and charge effects play an important role. Inhomogeneous striped states could be stabilized associated with lattice distortions [62]. Many interesting properties have been reported concerning lattice effects such as an anomalous isotope effect [76,77,78,79] and a shape resonance in a superlattice of quantum strings [80,81]. In the study of cuprate superconductors Bi_{2}Sr_{2}CaCuO_{8+y} and La_{2}CuO_{4+y} for which mobile oxygen interstitials by using local probes, a scenario has been shown that a strongly correlated Fermi liquid coexists with stripes that are made of anisotropic polarons condensed into a generalized Wigner charge density wave [82,83,84].
The relation between the Hubbard model and the d-p model was investigated in the early state of the study of high-temperature cuprates by Feiner et al. [85] They were able to reduce the d-p model into an effective one-band model by means of the cell-perturbation method. It has also been shown by numerical calculations that the Hubbard model and the three-band d-p model exhibit similar electronic properties [14,26].
In order to explore the superconducting ground state, it is favorable to suppress magnetic correlations and magnetic instabilities. For this purpose, we consider the strongly correlated region with large *U*. The strong antiferromagnetic correlation is suppressed by doped hole carriers when *U* is large. In this region we calculated superconducting properties in the 2D Hubbard model, and the existence of a superconducting phase is followed.
In Section 2, we discuss the critical temperature of superconductivity in many-electron systems. We discuss improved many-body wave functions in Section 3. In Section 4, we show the results obtained by the optimization variational Monte Carlo method. We show the SC order parameter as a function of *U* and phase diagrams when we vary the hole density *x*. We discuss the kinetic energy driven superconductivity in the strongly correlated region. We also examine the possibility of superconductivity in the nematic charge-ordered phase. In Section 5, we exhibit pair correlation function as a function of lattice sites. This shows that the pair correlation function is almost constant at long distances and the wave function indeed has long-range superconducting order in the strongly correlated region. We also discuss the duality of strong electron correlation, which means that the strong correlation can be an origin of attractive interaction of *d*-wave electron pairs and at the same time, it suppresses the pair correlation function.

It is reasonable to expect that when the energy scale of an interaction is very large, we can expect superconductivity with high critical temperature *T*_{c}. Since the energy scale of the Coulomb interaction is of the order of eV, the Coulomb interaction is one of the candidates to give high-temperature superconductivity. For materials shown in Table 1, we can confirm that the following empirical relations hold for the superconducting critical temperature:
where *t* denotes the transfer integral, and *m*^{∗} and *m*_{0} are the effective mass and bare mass of electrons, respectively. The Table 1 shows typical values of *t*, the ratio *m*^{∗}⁄*m*_{0} and *T*_{c}. The order of *T*_{c} for correlated electron materials is consistent with the formula in Equation (1). For high-temperature cuprates, the transfer integral *t* is estimated as *t* ~ 0.51 eV and *T*_{C} is of the order of 100 K. Since the transfer *t* of iron pnictides is about five times smaller than that of cuprates, iron pnictides have lower *T*_{c} than cuprate superconductors. The critical temperature *T*_{c} of heavy fermions is very low although heavy fermion materials are strongly correlated electron systems. This is due to large effective mass of *f* electrons which is as large as 100~1000 times the band (bare) mass *m*_{0}. Then the characteristic energy scale is reduced considerably so that *T*_{c} is of the order of 1 K.
**Table 1. **The transfer integral *t*, effective mass ratio *m*^{∗}⁄*m*_{0} and critical temperature *T*_{c} in correlated electron systems. For Hydrides, the Debye frequency *ω*_{ln} is shown instead of *t*. For heavy fermion materials, *t*⁄(*m*^{∗}⁄*m*_{0}) corresponds to the Kondo temperature *T*_{K}.

```latexk_BT_c\simeq0.1t/(m^*/m_0),```

```latexH=\sum_{ij\sigma}t_{ij}c_{i\sigma}^{\dagger} c_{j\sigma}+U\sum_{i}n_{i\uparrow}n_{i\downarrow},```

```latex\psi_G=P_G\psi_0,```

```latexP_G=\prod_j\bigl(1-(1-g)n_{j\uparrow}n_{j\downarrow}\bigr)```

```latex\psi_J=P_JP_G\psi_0,```

```latexP_J=\prod_j\left(1-(1-\eta)\prod_\tau[d_j(1-e_{j+\tau})+e_j(1-d_{j+\tau})]\right),```

```latex\psi_{\lambda}=e^{-\lambda K}P_{G}\psi_{0}=e^{-\lambda K}\psi_{G},```

```latexK=\sum_{ij\sigma}t_{ij}c_{i\sigma}^{\dagger}c_{j\sigma}.```

```latex\psi_{\lambda}^{(2)}=P_{G}(g^{\prime})e^{-\lambda K}P_{G}(g)\psi_{0}=P_{G}(g^{\prime})\psi_{\lambda},```

```latex\psi_{\lambda}^{(3)}=e^{-\lambda'K}P_{G}(g')e^{-\lambda K}P_{G}(g)\psi_{0}=e^{-\lambda'K}P_{G}(g')\psi_{\lambda},```

```latex\psi_{BCS}=\prod_{k}\bigl(u_{k}+v_{k}c_{k\uparrow}^{\dagger}c_{-k\downarrow}^{\dagger}\bigr)|0\rangle.```

```latex\psi_{_{G-BCS}}=P_{N_{e}}P_{G}\psi_{BCS},```

```latex\psi_{\lambda-BCS}=e^{-\lambda K}P_{G}\psi_{BCS}.```

```latex\Delta_{i,i+\hat{x}}=\Delta_{s},\quad\Delta_{i,i+\hat{y}}=-\Delta_{s}.```

```latexE_{cond}/N\simeq0.2 \mathrm{\,meV}.```

```latexE_{cond}/N_{atom}\simeq0.17-0.26 \mathrm{\,eV}```

```latex\frac{1}{\chi_{c}}=\frac{\partial^2E(N_e)}{\partial N_e^2}=\frac{E(N_e+\delta N_e)+E(N_e-\delta N_e)-2E(N_e)}{(\delta N_e)^2},```

```latex\Delta E_{kin-sc}=E_{kin}(\Delta_{s}=0)-E_{kin}\big(\Delta_{s}=\Delta_{s,\mathrm{opt}}\big),```

```latex\Delta E_{U-sc}=E_{U}(\Delta_{s}=0)-E_{U}\big(\Delta_{s}=\Delta_{s,\mathrm{opt}}\big),```

```latex\Delta E_{SC}=\Delta E_{kin-sc}+\Delta E_{U-sc}.```

```latex\Delta E_{kin-sc}<0,\Delta E_{U-sc}>0.```

```latex\Delta E_{kin-sc}>0,\Delta E_{U-sc}<0.```

```latex\Delta E_{kin}=E_{kin}(\psi_{G})-E_{kin}(\psi_{\lambda})=E_{kin}(\lambda=0)-E_{kin}(\lambda_{\mathrm{opt}}),```

```latex\rho_i=\rho\cos(\boldsymbol{Q}_c\cdot(\boldsymbol{r}_i-\boldsymbol{r}_0)),m_i=m\sin(\boldsymbol{Q}_s\cdot(\boldsymbol{r}_i-\boldsymbol{r}_0)),```

```latex\Delta_{i,i+\hat{x}}=\Delta_{s}\cdot\left(1+\alpha\cos\left(\frac{1}{2}\pi x-\frac{\pi}{4}\right)\right),\Delta_{i,i+\hat{y}}=-\Delta_{s}\cdot\left(1+\alpha\cos\left(\frac{1}{2}\pi x\right)\right),```

In this section, we examine the effect of strong correlation on superconductivity. We consider the effect of the Gutzwiller operator *P*_{G} on the superconducting correlation function. The BCS wave function *ψ*_{BCS} (Δ_{s}) clearly shows the long-range correlation. In Figure 6, we show the SC correlation function $$D_{sc}(\ell)\equiv\langle\Delta^\dagger(i)\Delta(i+\ell)\rangle $$, as a function of the lattice site for *N*_{e} = 88, *U* = 18*t* and $$t^{\prime}=0$$ on a 10 × 10 lattice. Here the pair annihilation operator ∆(*i*) at the site *i* is defined by
where
for *α* = *x* and *y*. $$\hat{\alpha}$$ stands for the unit vector in the *α*-th direction.
Figure 6 shows that the pair correlation function for *U* = 18*t* is almost constant when $$\ell$$ is large indicating that the ground state is superconducting. The values of $$D_{sc}(\ell)$$ for large $$\ell$$ are suppressed considerably compared to that for the non-interacting BCS wave function. This suppression is due to the strong correlation between electrons. This makes it rather hard to confirm the existence of the superconducting phase in numerical calculations of pair correlation functions by, for example, quantum Monte Carlo calculations. In Figure 7, we show the SC correlation function $$D_{sc}(\ell)$$ of $$P_G\psi_{BCS}(\Delta_s)$$ at the site $$\ell=R_{max}=(5,5)$$ with *i* = (1,1) as a function of 1 − *g* for ∆_{s} = 0.05*t* on a 10 × 10 lattice. *R*_{max} is the most distant point from the site *i* = (1,1). Figure 7 indicates that the pair correlation function is suppressed by the electron correlation that is now given by the Gutzwiller on-site operator. Thus, we can say that the electron correlation has duality. This means that the electron correlation is an origin of attractive interaction between electrons and at the same time suppresses pair correlation functions.
The electron correlation has also an effect on the superconducting order parameter ∆. ∆ is defined by
We show ∆ as a function of 1 − *g* in Figure 8. ∆ exhibits a similar behavior to $$D_{sc}(\ell)$$, that is, ∆ is reduced by *P*_{G}.
**Figure 6. **The pair correlation function $$D_{sc}(\ell)$$ for *N*_{e} = 88, *U* = 18*t* and $$t^{\prime}=0$$ on a 10 × 10 lattice where *i* = (1,1) and $$\ell$$ = (1,1),(1,2), (1,3), (1,4), (1,5), (2,5), (3,5), (4,5) and (5,5). The figure includes $$D_{sc}(\ell)$$ for *U* = 0 (squares), that for the BCS wave function $$\psi_{BCS}(\Delta_s)$$ with ∆_{s}= 0.05*t* (open circles), and that for *U* = 18*t* (filled circles).
**Figure 7. **The pair correlation function $$D_{sc}(\ell)$$ for $$\ell=R_{max}=(5,5)$$ of $$P_G\psi_{BCS}(\Delta_s)$$ with ∆_{s} = 0.05*t* on a 10 × 10 lattice. The parameter *g* is in the range of 0 ≤ *g* ≤ 1 and 1 − *g* = 0 corresponds to the BCS wave function.
When *g* < 1. Hence the electron correlation also leads to the reduction of the SC gap ∆. The strong electron correlation has duality, which means that the electron correlation becomes an origin of attractive interaction of *d*-wave pairing and at the same time, it suppresses SC correlation function and SC gap. One origin of this suppression is certainly the renormalization of the effective transfer integral and the effective mass. The heavy effective mass *m*^{∗}⁄*m* reduces pair correlation functions and is not favorable for superconductivity as indicated by Equation (1). The exponential factor $$e^{-\lambda K}$$ could play a role in increasing pair correlation by the kinetic energy effect.
**Figure 8. **The superconducting order parameter ∆ as a function of 1 − *g* for $$P_G\psi_{BCS}(\Delta_s)$$ with ∆_{s} = 0.05*t* on a 10 × 10 lattice.

```latex\Delta(i)=\Delta_{x}(i)+\Delta_{-x}(i)-\Big(\Delta_{y}(i)+\Delta_{-y}(i)\Big),```

```latex\Delta_\alpha(i)=c_{i\downarrow}c_{i+\widehat\alpha\uparrow}-c_{i\uparrow}c_{i+\widehat\alpha\downarrow},```

```latex\Delta=\frac{1}{N}\sum_{i}\bigl(\langle c_{i\uparrow}^{\dagger}c_{i+\hat{x}\downarrow}^{\dagger}\rangle-\langle c_{i\uparrow}^{\dagger}c_{i+\hat{y}\downarrow}^{\dagger}\rangle\bigr).```

The many-body wave function is important in the study of strongly correlated electron systems. We have constructed many-body wave functions starting from the Gutzwiller function to take into account strong correlation between electrons. The series $$\psi_{G},\psi_{\lambda}^{(1)}\equiv\psi_{\lambda},\psi_{\lambda}^{(2)},\psi_{\lambda}^{(3)},\cdots$$, will approach the exact wave function.
An instability toward magnetic ordering easily occurs in the two-dimensional Hubbard model. In particular, near the half-filled case with a small number of holes, the ground state has inevitably some magnetic or charge orders. Thus, we considered the strong correlated region where magnetic correlations and magnetic instabilities are suppressed.
Thus, we need a method of calculation by which we can evaluate physical properties in the strongly correlated region. This was the purpose of the study in this paper. We chose the value *U/t* = 18 in this paper. Since the extreme strong correlation reduces the pair correlation function, it is favorable that we can choose a moderate value of U being less than *U* = 18*t*. We expect that this value is reduced when we take account of further improved wave functions $$\psi_{\lambda}^{(3)}$$, $$\psi_{\lambda}^{(4)}$$,⋯. In fact, the antiferromagnetic correlation is suppressed for the improved wave function $$\psi_{\lambda}^{(3)}$$[48]. We expect that this will lead to a superconducting state with larger gap function.

We have investigated the correlated superconducting state in the ground state of the two-dimensional Hubbard model based on the optimization variational Monte Carlo method. First, we discussed that the SC condensation energy obtained by numerical calculations is consistent with that estimated from experimental results for high-temperature cuprate superconductors. Second, we presented the phase diagram as a function of *U* based on improved many-body wave functions. The superconducting phase exists in the strongly correlated region where *U* is larger than the bandwidth. When $$t^{\prime}=0$$, the AF correlation weakens upon hole doping in the strongly correlated region and the pure *d*-wave SC is realized. Third, we have also shown the phase diagram as a function of the carrier density *x*, where basically there are three phases: antiferromagnetic insulating phase, metallic antiferromagnetic phase and superconducting phase. Fourth, then we discussed the kinetic energy effect that would assist the appearance of superconductivity and this effect may play an important role in the realization of high-temperature superconductivity. Fifth, we investigated the cooperation of charge inhomogeneous order and superconductivity. This indicates the possibility that the superconducting critical temperature *T*_{c} will increase due to the coexistence with nematic charge ordering. Lastly, we showed the pair correlation function $$D_{sc}(\ell)$$. We discussed the effect of strong electron correlation on pair correlation function and SC order parameter. The pair correlation function is suppressed by the electron correlation operator *P*_{G}. Although the correlation function $$D_{sc}(\ell)$$ becomes small due to *P*_{G}, the long-range order still exists for *ψ*_{λ}.

The author expresses his sincere thanks to K. Yamaji and M. Miyazaki for fruitful discussions. This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan (Grant No. 17K05559). A part of computations was supported by the Supercomputer Center of the Institute for Solid State Physics, the University of Tokyo. The numerical calculations were also carried out on Yukawa-21 at the Yukawa Institute for Theoretical Physics in Kyoto University.

Formal Analysis, Investigation, Resources, Writing—Original Draft Preparation, T.Y.

Not applicable.

Not applicable.

Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan (Grant No. 17K05559).

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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