Mechanism of High-Temperature Superconductivity in Correlated-Electron Systems

Abstract

It is very important to elucidate the mechanism of superconductivity for achieving room temperature superconductivity. In the first half of this paper, we give a brief review on mechanisms of superconductivity in many-electron systems. We believe that high-temperature superconductivity may occur in a system with interaction of large-energy scale. Empirically, this is true for superconductors that have been found so far. In the second half of this paper, we discuss cuprate high-temperature superconductors. We argue that superconductivity of high temperature cuprates is induced by the strong on-site Coulomb interaction, that is, the origin of high-temperature superconductivity is the strong electron correlation. We show the results on the ground state of electronic models for high temperature cuprates on the basis of the optimization variational Monte Carlo method. A high-temperature superconducting phase will exist in the strongly correlated region.

1. Introduction

It is a challenging research subject to clarify the mechanism of high temperature superconductivity, and indeed it has been studied intensively for more than 30 years [1,2,3]. For this purpose, it is important to clarify the ground state and phase diagram of electronic models with strong correlation because high temperature cuprates are strongly correlated electron systems.

Most superconductors induced by the electron–phonon interaction have s-wave pairing symmetry. We can understand physical properties of s-wave superconductivity based on the Bardeen–Cooper–Schrieffer (BCS) theory [4,5,6]. The critical temperature $T_c$ of most of electron–phonon superconductors is very low except for exceptional compounds. Many unconventional superconductors that cannot be understood by the BCS theory have been discovered. They are, for example, heavy fermion superconductors, organic superconductors and cuprate superconductors for which the pairing mechanism is different from the electron–phonon interaction. In particular, cuprate superconductors exhibit relatively high $T_c$ and have become of great interest. A common feature in both electron–phonon systems and correlated electron systems is that critical temperature may have a strong correlation with the energy scale of the interaction that induces electron pairing.

This paper has two parts. In the first part, we give a review on mechanisms of superconductivity in the electron–phonon system and in the correlated electron system. In the second part, we mainly discuss the mechanism of high-temperature cuprates.

The model for CuO${}_2$ plane in cuprate superconductors is called the d-p model or the three-band Hubbard model [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. It is certainly a very difficult task to elucidate the phase diagram of the d-p model. Simplified models are also used to investigate the mechanism of superconductivity, for example the two-dimensional (2D) single-band Hubbard model [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50] and ladder model [51,52,53,54,55,56]. The Hubbard model was introduced to understand the metal–insulator transition [57] and was employed to understand various magnetic phenomena [58,59]. On the basis of the Hubbard model, it is possible to understand the appearance of inhomogeneous states reported for cuprates, such as stripes [60,61,62,63,64,65,66,67] and checkerboard-like density wave states [68,69,70,71]. It was also expected that the Hubbard model can account for high temperature superconductivity [72].

A variational Monte Carlo method is used to examine the ground state properties of strongly correlated electron systems, where we calculate the expectation values exactly using a numerical method [28,29,30,31,36,37,38,39,40,41]. We introduced the wave function of $exp(-S)$-type in the study of superconductivity in the Hubbard model [73,74,75]. This wave function is very excellent in the sense that the energy expectation value is lower than that of any other wave functions [50].

The paper is organized as follows. In Section 2.1, we discuss the phonon mechanism of superconductivity. In Section 2.2, we discuss the electron mechanism of superconductivity. Section 2.3 is devoted to a discussion on superconductivity in correlated electron systems. In Section 3.1, we show the model for high temperature cuprates. We present the optimization variational Monte Carlo method (OVMC) in Section 3.2. We show the results on superconductivity based on the OVMC in Section 3.3. We discuss the stability of antiferromagnetic state in Section 3.4. We show the phase diagram when the hole doping rate is changed in Section 3.5. We give a summary in Section 4.

2. Part I. Superconductivity in Many-Electron Systems

2.1. Possibility of High-$T_c$ Superconductivity

In the BCS theory, the electron–phonon interaction is assumed to induce attractive interaction between electrons and the pairing symmetry is s-wave [4,5,6]. There are many superconductors with s-wave pairing symmetry and most of them are due to the electron–phonon interaction. The BCS theory was successful to explain physical properties of these superconductors.

In the strong-coupling theory based on the Green function formulation [76,77], the critical temperature $T_c$ was estimated as [78],

$$T_c=\frac{{\theta }_D}{1.45}exp\left(-\frac{1.04(1+\lambda )}{\lambda -{\mu }^{*}(1+0.62\lambda )}\right),$$

(1)

where $\lambda$ is the electron–phonon coupling constant, ${\theta }_D=\hslash {\omega }_D/k_B$ is the Debye temperature and ${\mu }^{*}$ is the renormalized Coulomb parameter defined by

$${\mu }^{*}=\frac{\mu }{1+\mu ln({ϵ}_F/{\omega }_D)},$$

(2)

for $\mu =U/{ϵ}_F$ where U is the strength of the Coulomb interaction and ${ϵ}_F$ is the Fermi energy. ${\mu }^{*}$ is the phenomenological parameter being approximately 0.1.

The electron–phonon coupling constant $\lambda$ is expressed as

$$\lambda =2{\int }_0^{\infty }d\omega \frac{\alpha {(\omega )}^{2}F(\omega )}{\omega },$$

(3)

where $\alpha (\omega )$ is the averaged electron–phonon coupling over the Fermi surface and $F(\omega )$ indicates the product of the spectral function of phonon and the density of states. This is approximately written as

$$\lambda \simeq \frac{\rho ({ϵ}_F){\langle I\rangle }^2}{M{\omega }_D^2},$$

(4)

where $\rho ({ϵ}_F)$ is the density of states at the Fermi surface and M is the mass of an atom. McMillan predicted that $T_c$ would have a limit being of the order of 30 K from the analysis for this formula [78].

The McMillan formula was modified by replacing ${\theta }_D/1.45$ by logarithmic Debye frequency ${\omega }_{ln}$ where [79]

$${\omega }_{ln}=exp\left(\frac{2}{\lambda }{\int }_0^{\infty }d\omega \alpha {(\omega )}^{2}F(\omega )\frac{ln\omega }{\omega }\right).$$

(5)

It was predicted that high critical temperature would be possible for large $\lambda$ since $T_c\propto \sqrt{\lambda }$ for $\lambda \gg 1$. If ${\omega }_{ln}$ is large, $\lambda$ is also large, and the crystal is stable, high $T_c$ would be realized. It was predicted that high $T_c$ would be realized in hydrogen solid with high Debye temperature [80]. In fact, high temperature superconductors with $T_c$ above 200 K were discovered under extremely high pressure (160∼200 GPa) in hydrogen compounds such as H${}_3$S and LaH${}_{10}$ [81,82,83].

It is important to consider multi-band superconductors in the search for high temperature superconductors. In fact, MgB${}_2$ and iron based superconductors are multi-band superconductors. An important role of Lifshitz transition in iron based superconductors and MgB${}_2$ multi-band superconductors has been predicted [84]. An interesting point is that the possibility of high-$T_c$ superconductivity in materials where tuning the chemical potential shows a quasi-1D Fermi surface topology as in organics and hydrides [85]. A layered superconductor such as cuprate superconductor can be regarded as a multiband superconductor due to interlayer couplings. A multi-band superconductivity has been investigated as a generalization of the BCS theory since early works on the two-band superconductivity [86,87,88,89]. There will appear many interesting properties in superconductors with multiple gaps such as time-reversal symmetry breaking [90,91,92,93,94,95,96,97,98,99,100,101,102,103], the existence of massless modes [104,105,106,107,108,109], unusual isotope effect [110,111,112,113,114] and fractional-flux quantum vortices [115,116,117,118,119]. When we have multiple order parameters, there appear multiple Nambu–Goldstone bosons and Higgs bosons [104,120,121,122,123,124,125,126,127,128,129]. This will result in significant excitation modes that are unique in multi-band superconductors.

It is important to include in a theoretical picture the presence of multiple electronic components with anomalous normal state properties in the charge and spin sector, e.g., the well known Fermi arcs and charge pseudogap phenomenology. The “shape resonance” scenario of multigap BCS-BEC crossover has been proposed [130,131]. The study of the electronic structure of the cuprates superconductors Bi${}_2$Sr${}_2$CaCuO${}_{8+y}$ and La${}_2$CuO${}_{4+y}$ doped by mobile oxygen interstitials using local probes has shown a scenario made of two electronic components: a strongly correlated Fermi liquid which coexists with stripes made of anisotropic polarons condensed into a generalized Wigner charge density wave [132,133,134].

2.2. Electron Correlation and Superconductivity

We discuss the electron correlation due to the Coulomb interaction between electrons. The on-site Coulomb interaction is important in the study of the metal insulator transition and magnetic properties of materials. The Hubbard model is written as [25]

$$H=\sum _{ij\sigma }{t}_{ij}{c}_{i\sigma }^{†}{c}_{j\sigma }+U\sum _i{n}_{i\uparrow }{n}_{i\downarrow },$$

(6)

where $t_{ij}$ indicates the transfer integral and the second term denotes the Coulomb interaction with the strength U. $t_{ij}$ are chosen as follows. $t_{ij}=-t$ when i and j are nearest-neighbor pairs $\langle ij\rangle$ and $t_{ij}=-t^{\prime }$ when i and j are next-nearest neighbor pairs. In the following, N is the number of lattices, and $N_e$ denotes the number of electrons.

When two electrons spin up and down at the same site, the energy becomes higher by U where U denotes the on-site Coulomb energy. In the case of half-filling, the Mott transition occurs when $U(>0)$ is as large as the bandwidth and the ground state is an insulator. The effective Hamiltonian is derived in the limit of large $U/t$ [135,136,137], based on the canonical transformation $H_{\mathrm{eff}}=e^{iS}H{e}^{-iS}$. In the limit $U/t\to \infty$, the double occupancy is not allowed. The effective Hamiltonian is written as

$$H_{\mathrm{eff}}=H+i[S,H]+\frac{i^2}{2}[S,[S,H]]+\cdots .$$

(7)

We write the Hamiltonian as $H={\tilde{H}}_0+H_1$ where

$$\begin{array}{ccc}\hfill {\tilde{H}}_0& =& \sum _{ij\sigma }{t}_{ij}(a_{i\sigma }^{†}{a}_{j\sigma }+d_{i\sigma }^{†}{d}_{j\sigma })+U\sum _i{n}_{i\uparrow }{n}_{i\downarrow },\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill H_1& =& \sum _{ij\sigma }{t}_{ij}(a_{i\sigma }^{†}{d}_{j\sigma }+d_{j\sigma }^{†}{a}_{i\sigma }).\hfill \end{array}$$

(9)

Here, we defined $a_{i\sigma }=c_{i\sigma }(1-n_{i,-\sigma })$ and $d_{i\sigma }=c_{i\sigma }{n}_{i,-\sigma }$. $a_{i\sigma }=c_{i\sigma }(1-n_{i.-\sigma })$ is the electron operator without double occupancy. We choose S to satisfy $i[S,{\tilde{H}}_0]+H_1=0$, so that $H_{\mathrm{eff}}$ reads in the subspace of no double occupancy,

$$H_{\mathrm{eff}}=\sum _{ij\sigma }{t}_{ij}{a}_{i\sigma }^{†}{a}_{j\sigma }+\frac{i}{2}[S,H_1]+\frac{i^2}{3}[S,[S,H_1]]+\cdots .$$

(10)

When we consider only the nearest-neighbor transfer $t_{ij}=-t$, the effective Hamiltonian reads

$$\begin{array}{ccc}\hfill H_{\mathrm{eff}}& =& -t\sum _{\langle ij\rangle \sigma }{a}_{i\sigma }^{†}{a}_{j\sigma }-\frac{t^2}{U}\sum _{j\mu {\mu }^{\prime }}[a_{j+\mu \uparrow }^{†}{a}_{j\downarrow }^{†}{a}_{j\downarrow }{a}_{j+{\mu }^{\prime }\uparrow }\hfill \\ & & +a_{j\uparrow }^{†}{a}_{j+\mu \downarrow }^{†}{a}_{j+{\mu }^{\prime }\downarrow }{a}_{j\uparrow }+a_{j+\mu \uparrow }^{†}{a}_{j\downarrow }^{†}{a}_{j+{\mu }^{\prime }\downarrow }{a}_{j\uparrow }+a_{j\uparrow }^{†}{a}_{j+\mu \downarrow }^{†}{a}_{j\downarrow }{a}_{j+\mu \uparrow }],\hfill \end{array}$$

(11)

where $j+\mu$ and $j+{\mu }^{\prime }$ denote the nearest-neighbor sites in the $\mu$ and ${\mu }^{\prime }$ directions, respectively. The second term being proportional to $t^2/U$ contains the nearest-neighbor exchange interaction and also three-site terms when $\mu \ne {\mu }^{\prime }$. The three-site terms are of the same order as the exchange interaction. When we neglect the three-site terms, the effective Hamiltonian reduces to the t-J model given by

$$H_{\mathrm{eff}}=-\sum _{\langle ij\rangle \sigma }(a_{i\sigma }^{†}{a}_{j\sigma }+\mathrm{h}.\mathrm{c}.)+J\sum _{\langle ij\rangle }\left({\mathbf{S}}_i·{\mathbf{S}}_j-\frac{1}{4}{\tilde{n}}_i{\tilde{n}}_j\right),$$

(12)

where $J=4t^2/U$ and ${\tilde{n}}_i={\tilde{n}}_{i\uparrow }+{\tilde{n}}_{i\downarrow }$ with ${\tilde{n}}_{i\sigma }=a_{i\sigma }^{†}{a}_{i\sigma }$.

High-temperature cuprates and heavy fermion systems are typical correlated electron systems and many superconductors have been reported. Most of superconductors in these systems have nodes in the superconducting gap, namely, the Cooper pair is anisotropic. This indicates that superconductivity is unconventional and does not conform to the conventional BCS theory. The mechanism of superconductivity is certainly non-phonon mechanism. We show several characteristic properties of cuprate high-temperature superconductors:

• The Cooper pair has d-wave symmetry.
• The superconducting phase exists near the antiferromagnetic phase and parent materials are a Mott insulator.
• The CuO${}_2$ plane is commonly contained and the on-site Coulomb repulsive interaction works between d electrons.
• The size of Cooper pair is very small being of order of 2 Å.
• The CuO${}_2$ plane is high anisotropic and there is a weak Josephson coupling between two layers.

The small size of Cooper pair also supports the non-phonon mechanism of cuprate superconductivity [138,139,140,141]. A plausible non-phonon mechanism is due to the Coulomb interaction on the same atom. Because the energy scale of the Coulomb interaction is very large, which is of the order of eV, we can expect superconductivity with high critical temperature $T_c$. The critical temperature of heavy fermion materials is, however, very low, although superconductivity occurs due to strong Coulomb interaction between f electrons. This is because the effective mass of f electrons is very large in heavy fermion systems owing to the large self-energy correction. The effective mass enhancement of heavy fermion materials becomes as large as 100–1000, which means that the effective cutoff becomes very small. As a result, the characteristic energy scale is reduced considerably and the critical temperature $T_c$ becomes very low begin of the order of 1 K. In heavy fermion systems, the characteristic energy scale is given by the Kondo temperature $T_K$. The ratio of the effective mass $m^{*}$ to the band mass $m_0$ is approximately given as $m^{*}/m_0\simeq D/T_K$ for the bandwidth D and $T_K$. Thus, the effective bandwidth for heavy fermions is given by the Kondo temperature $T_K\simeq D/(m^{*}/m_0)$. Empirically, $T_c$ is lowered as the effective mass increases. This is expressed as follows:

$$k_B{T}_c\simeq 0.1t/(m^{*}/m_0),$$

(13)

where t denotes the transfer integral proportional to the bandwidth. The estimated values of the transfer t, the ratio $m^{*}/m_0$ and $T_c$ for several compounds are shown in

Table 1. The transfer integral t, effective mass $m^{*}$ and critical temperature $T_c$ in correlated electron systems, where $m_0$ denotes the band mass. The orders of these quantities are shown in the table. For heavy fermions, $t/(m^{*}/m_0)$ corresponds to the Kondo temperature $T_K$. For Hydrides, the Debye frequency ${\omega }_{ln}$ is shown.

	t or ${\mathit{\omega }}_{ln}$	${\mathit{m}}^{*}/{\mathit{m}}_0$	$\mathit{t}/({\mathit{m}}^{*}/{\mathit{m}}_0)$	${\mathit{T}}_{\mathit{c}}$
Cuprate superconductors	5000 K	5	1000	100 K	$t\sim 0.51$ eV [142]
Iron pnictides	1000 K	$\sim 2$	500	50 K	$t\sim 0.1$ eV [143]
Heavy fermion materials	10,000 K	100∼1000	10∼100	1∼10 K	[144,145,146]
Organic superconductors	200∼500 K	2∼5	100	10 K	[147]
Hydrides $H_{3}S$	1000 K	$\sim 1$	1000	100 K	${\omega }_{ln}$ [148]

. For cuprates, the transfer t is estimated as $t\sim 0.51$ eV. The bandwidth for iron pnictides is about five times smaller than that for cuprates. A list of typical superconductors in correlated electron systems is shown in

Table 2. Superconducting materials.

Materials	${\mathit{T}}_{\mathit{c}}$	Pair Symmetry	Crystal Structure
CeCu${}_2$Si${}_2$	0.6 K	s or d	bc tetragonal	[149,150]
UPt${}_3$	0.52 K	p or f	Hexagonal	[151]
UBe${}_{13}$	0.86 K	p	Cubic	[152]
URu${}_2$Si${}_2$	1.2 K		bc tetragonal	[153,154,155]
CeRu${}_2$	6.2 K	s	Laves Cubic	[156]
UPd${}_2$Al${}_3$	2 K	d	Hexagonal	[157,158,159]
UNi${}_2$Al${}_3$	1 K	p?	Hexagonal	[158,160]
CeCoIn${}_5$	2.3 K	d	HoCoGa${}_5$ type	[161,162]
CeRhIn${}_5$	2.1 K	d	HoCoGa${}_5$ type	[163]
	(16.3 kbar)		bc tetragonal
CeRh${}_2$Si${}_2$	0.35 K		bc tetragonal	[164]
	(9 kbar)
UGe${}_2$	0.8 K	p?	Orthorhombic	[165]
	(13.5 kbar)
URhGe	0.25 K	p?	Orthorhombic	[166]
Sr${}_2$RuO${}_4$	1.4 K	p or f	Perovskite	[167]
PrOs${}_4$Sb${}_{12}$	1.85 K	line nodes?	Skutterudite	[168]
Na${}_{\mathrm{x}}$CoO${}_{2-\mathrm{y}}$·H${}_2$O	5 K	p?	Triangular lattice	[169]
Ba${}_{1-\mathrm{x}}$K${}_{\mathrm{x}}$BiO${}_3$	30 K	s	Perovskite	[170]
MgB${}_2$	39 K	s	Hexagonal	[171]
La${}_{2-\mathrm{x}}$Sr${}_{\mathrm{x}}$CuO${}_4$	36 K	d	Perovskite
YBa${}_2$Cu${}_3$O${}_{6+\mathrm{x}}$	90 K	d	Perovskite
Tl${}_2$Ba${}_2$Ca${}_{\mathrm{n}-1}$Cu${}_{\mathrm{n}}$O${}_{2\mathrm{n}+4}$	125 K	d	Perovskite
HgBa${}_2$Ca${}_{\mathrm{n}-1}$Cu${}_{\mathrm{n}}$O${}_{2\mathrm{n}+2+\delta }$	135 K	d	Perovskite
LaO${}_{1-\mathrm{x}}$F${}_{\mathrm{x}}$FeAs	26 K		ZrCuSiAs type	[172]
NdFeAsO${}_{1-\mathrm{y}}$	54 K		ZrCuSiAs type	[173]
H${}_3$S	203 K	s	(under pressure)	[81]
LaH${}_{10}$	260 K	s	(under pressure)	[82,83,174,175]

.

2.3. Superconductivity in Strongly Correlated Electron Systems

The possibility of superconductivity in strongly correlated electron systems has been discussed intensively. The perturbative calculations such as the fluctuation-exchange approximation (FLEX) have been performed to investigate the superconducting ground state [176,177,178]. There were, however, the results by quantum Monte Carlo methods, which did not support the existence of high-temperature superconductivity in the two-dimensional Hubbard model [32,33,45]. In quantum Monte Carlo calculations, the strength of the Coulomb interaction U is not large enough because the range of accessible U is very restricted. It is now certain that there is a superconducting phase in the strongly correlated region [50]. The simplest wave function of superconducting state with strong electron correlation is the Gutzwiller-projected BCS wave function:

$${\psi }_{BCS-G}=P_G\prod _{\mathbf{k}}(u_{\mathbf{k}}+v_{\mathbf{k}}{c}_{\mathbf{k}\uparrow }^{†}{c}_{-\mathbf{k}\downarrow }^{†})|0\rangle ,$$

(14)

where $u_{\mathit{k}}$ and $v_{\mathit{k}}$ are BCS parameters and $P_G$ is the Gutzwiller operator to control the on-site electron correlation. $P_G$ is written as

$$P_G=\prod _j(1-(1-g)n_{j\uparrow }{n}_{j\downarrow }),$$

(15)

where g is a variational parameter in the range of $0\le g\le 1$. The ratio of $u_{\mathit{k}}$ and $v_{\mathit{k}}$ is given as

$$\frac{v_{\mathbf{k}}}{u_{\mathbf{k}}}=\frac{{\Delta }_{\mathbf{k}}}{{\xi }_{\mathbf{k}}+{({\xi }_{\mathbf{k}}^2+{\Delta }_{\mathbf{k}}^2)}^{1/2}},$$

(16)

where ${\xi }_{\mathit{k}}$ denotes the electron dispersion relation measured from the Fermi energy and ${\Delta }_{\mathit{k}}$ is the gap function. We use the following form for the gap function in the two-dimensional case:

$$\begin{array}{ccc}\hfill {\Delta }_{\mathbf{k}}& =& \Delta (cos{k}_x-cos{k}_y)d-\mathrm{wave},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill {\Delta }_{\mathbf{k}}& =& \Delta (cos{k}_x+cos{k}_y)\mathrm{anisotropic}-s-\mathrm{wave},\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill {\Delta }_{\mathbf{k}}& =& \Delta \mathrm{isotropic}s-\mathrm{wave}.\hfill \end{array}$$

(19)

$\Delta$ is a constant and is treated as a variational parameter. The wave function ${\psi }_{BCS-G}$ is just the wave function that Anderson proposed as a wave function of the resonate-valence-bond (RVB) state [72].

It has been shown that the ground-state energy has a minimum at finite $\Delta$ for the BCS-Gutzwiller wave function with d-wave symmetry in the two-dimensional Hubbard model by using the variational Monte Carlo method [37]. The superconducting condensation energy $E_{cond}$ per site in the limit of large system size was estimated as [37,38]

$$E_{cond}/N\simeq 0.2\mathrm{meV},$$

(20)

where the transfer integral t is set at 0.5 eV. The similar result was obtained for the three-band d-p model [18]. Thus, the condensation energy per atom is of the order of ${10}^{-4}$ eV.

The superconducting condensation energy $E_{cond}$ for cuprate high-temperature superconductors was evaluated by using the result of specific heat measurement for YBCO as 0.17–0.26 meV per Cu atom [37,179]. The estimation of $E_{cond}$ from the data of critical magnetic field gives the similar result [180]. The obtained results by theoretical calculations and experimental measurements are very close each other. This agreement is very remarkable. Thus, this value indicates the characteristic energy for cuprate high-temperature superconductors. This result may support that the superconductivity in cuprate high temperature superconductors is caused by the strong electron correlation and the 2D Hubbard model includes essential ingredients.

3. Part II. Mechanism of Superconductivity in Cuprates

We discuss the mechanism of superconductivity in this part. We show numerical results obtained by using the optimized wave functions.

3.1. Model for High-${\mathit{T}}_{\mathit{c}}$ Cuprates

The Hamiltonian of the d-p model for high-$T_c$ cuprates is

$$\begin{array}{ccc}\hfill H_{dp}& =& {ϵ}_d\sum _{i\sigma }{d}_{i\sigma }^{†}{d}_{i\sigma }+{ϵ}_p\sum _{i\sigma }(p_{i+\widehat{x}/2\sigma }^{†}{p}_{i+\widehat{x}/2\sigma }+p_{i+\widehat{y}/2\sigma }^{†}{p}_{i+\widehat{y}/2\sigma })\hfill \\ & +& t_{dp}\sum _{i\sigma }[d_{i\sigma }^{†}(p_{i+\widehat{x}/2\sigma }+p_{i+\widehat{y}/2\sigma }-p_{i-\widehat{x}/2\sigma }-p_{i-\widehat{y}/2\sigma })+\mathrm{h}.\mathrm{c}.]\hfill \\ & +& t_{pp}\sum _{i\sigma }[p_{i+\widehat{y}/2\sigma }^{†}{p}_{i+\widehat{x}/2\sigma }-p_{i+\widehat{y}/2\sigma }^{†}{p}_{i-\widehat{x}/2\sigma }-p_{i-\widehat{y}/2\sigma }^{†}{p}_{i+\widehat{x}/2\sigma }+p_{i-\widehat{y}/2\sigma }^{†}{p}_{i-\widehat{x}/2\sigma }+\mathrm{h}.\mathrm{c}.]\hfill \\ & +& t_d^{\prime }\sum _{\langle \langle ij\rangle \rangle \sigma }{ϵ}_{ij}(d_{i\sigma }^{†}{d}_{j\sigma }+\mathrm{h}.\mathrm{c}.)+U_d\sum _i{d}_{i\uparrow }^{†}{d}_{i\uparrow }{d}_{i\downarrow }^{†}{d}_{i\downarrow }\hfill \\ & +& U_p\sum _i(n_{i+\widehat{x}/2\uparrow }^p{n}_{i+\widehat{x}/2\downarrow }^p+n_{i+\widehat{y}/2\uparrow }^p{n}_{i+\widehat{y}/2\downarrow }^p).\hfill \end{array}$$

(21)

Since we use the hole picture in this paper, $d_{i\sigma }$ and $d_{i\sigma }^{†}$ represent the operators for the d hole. $p_{i\pm \widehat{x}/2\sigma }$ and $p_{i\pm \widehat{x}/2\sigma }^{†}$ denote the operators for the p holes at the site $R_{i\pm \widehat{x}/2}$, and in a similar way $p_{i\pm \widehat{y}/2\sigma }$ and $p_{i\pm \widehat{y}/2\sigma }^{†}$ are defined. $n_{i+\widehat{x}/2\sigma }^p$ and $n_{i+\widehat{y}/2\sigma }^p$ are the number operators of p holes at $R_{i+\widehat{x}/2}$ and $R_{i+\widehat{y}/2}$, respectively. $t_{dp}$ is the transfer integral between adjacent Cu and O orbitals and $t_{pp}$ is that between nearest p orbitals. $t_d^{\prime }$ indicates that between d orbitals where $\langle \langle ij\rangle \rangle$ denotes a next nearest-neighbor pair of copper sites. ${ϵ}_{ij}$ takes the values $\pm 1$ (See Figure 1). This value is determined from the sign of the transfer integral between next nearest-neighbor d orbitals. $U_d$ indicates the strength of the on-site Coulomb repulsion between d holes and $U_p$ is that between p holes.

The values of band parameters were evaluated by several works [181,182,183,184,185]. We show an example: $U_d=10.5$, $U_p=4.0$ and $U_{dp}=1.2$ eV [182]. Here, $U_{dp}$ is the nearest-neighbor Coulomb interaction between holes on adjacent Cu and O orbitals and is small compared to $U_d$. $U_{dp}$ is neglected in this paper. We write ${\Delta }_{dp}={ϵ}_p-{ϵ}_d$. The number of sites is denoted as N, and the energy is measured in units of $t_{dp}$.

3.2. Optimization Variational Monte Carlo Method

3.2.1. Off-Diagonal Wave Function

The Gutzwiller wave function is

$${\psi }_G=P_G{\psi }_0,$$

(22)

where ${\psi }_0$ is a one-particle state. Our purpose is to improve the Gutzwiller function. We multiply the Gutzwiller function by an exponential-type operator. The wave function is given as as [50,73,186,187,188,189,190]

$${\psi }_{\lambda }=exp(-\lambda K){\psi }_G,$$

(23)

where K denotes the kinetic part of the Hamiltonian. $\lambda$ is a newly introduced real variational parameter [41,73,187,191]. There are other methods to improve the Gutzwiller function [43,192]. The following Jastrow operator is used [43],

$$P_{Jdh}=\prod _j\left(1-(1-\eta )\prod _{\tau }\left[d_j(1-e_{j+\tau })+e_j(1-d_{j+\tau })\right]\right),$$

(24)

where $d_j$ is the operator for the doubly-occupied site given as $d_j=n_{j\uparrow }{n}_{j\downarrow }$ and $e_j$ is that for the empty site given by $e_j=(1-n_{j\uparrow })(1-n_{j\downarrow })$. $\eta$ is the variational parameter in the range of $0\le \eta \le 1$. The wave function is

$${\psi }_{\eta }=P_{Jdh}{\psi }_G.$$

(25)

In this paper, we use the wave function of exponential type in Equation (23) because the energy is further lowered when we use this wave function [50]. The wave function for the d-p model is formulated similarly. An initial state ${\psi }_0$ contains many variational parameters (${\tilde{t}}_{dp}$, ${\tilde{t}}_{pp}$, ${\tilde{t}}_d^{\prime }$, and ${\widetilde{ϵ}}_p-{\widetilde{ϵ}}_d$):

$${\psi }_0={\psi }_0({\tilde{t}}_{dp},{\tilde{t}}_{pp},{\tilde{t}}_d^{\prime },{\widetilde{ϵ}}_p-{\widetilde{ϵ}}_d).$$

(26)

We use ${\tilde{t}}_{dp}=t_{dp}$ as the energy unit. We consider the following wave function that is improved from the Gutzwiller wave function [50,73,186,187,188,189,190]:

$${\psi }_{\lambda }=exp(-\lambda K){\psi }_G.$$

(27)

The expectation values are evaluated by using the auxiliary field method [73,191]. The kinetic part K also contains the band parameters $t_{pp}$, $t_d^{\prime }$ and ${ϵ}_p-{ϵ}_d$ as variational parameters:

$$K=K({\widehat{t}}_{pp},{\widehat{t}}_d^{\prime },{\widehat{ϵ}}_p-{\widehat{ϵ}}_d).$$

(28)

We take ${\widehat{t}}_{pp}={\tilde{t}}_{pp}$, ${\widehat{t}}_d^{\prime }={\tilde{t}}_d^{\prime }$ and ${\widehat{ϵ}}_p-{\widehat{ϵ}}_d={\widetilde{ϵ}}_p-{\widetilde{ϵ}}_d$, for simplicity. Thus, we have g, ${\tilde{t}}_{pp}$, ${\tilde{t}}_d^{\prime }$, ${\widehat{ϵ}}_p-{\widehat{ϵ}}_d={\widetilde{ϵ}}_p-{\widetilde{ϵ}}_d$, and $\lambda$ as variational parameters. The expectation values for this type of wave function are calculated on the basis of the variational Monte Carlo method. One can evaluate the expectation value correctly within statistical errors.

3.2.2. Antiferromagnetic Wave Function

The AF one-particle state ${\psi }_{AF}$ is formulated by the eigenfunction of the AF trial Hamiltonian:

$$H_{AF}=\sum _{ij\sigma }{t}_{ij}{c}_{i\sigma }^{†}{c}_{j\sigma }-{\Delta }_{AF}\sum _{i\sigma }{(-1)}^{x_i+y_i}\sigma n_{i\sigma },$$

(29)

where ${\Delta }_{AF}$ is the AF order parameter and $(x_i,y_i)$ represents the coordinates of the site i. With ${\psi }_{AF}$, the wave function is given as

$${\psi }_{\lambda ,AF}=exp(-\lambda K)P_G{\psi }_{AF}.$$

(30)

3.2.3. Superconducting Wave Function

We start from the BCS wave function

$${\psi }_{BCS}=\prod _k(u_k+v_k{c}_{k\uparrow }^{†}{c}_{-k\downarrow }^{†})|0\rangle ,$$

(31)

with coefficients $u_k$ and $v_k$ satisfying $u_k^2+{\left|v_k\right|}^2=1$. We choose $u_k/v_k={\Delta }_k/({\xi }_k+\sqrt{{\xi }_k^2+{\Delta }_k^2})$ for the gap function ${\Delta }_k$ and ${\xi }_k={ϵ}_k-\mu$. We assume ${\Delta }_k={\Delta }_{SC}(cos{k}_x-cos{k}_y)$. The Gutzwiller-projected BCS wave function is

$${\psi }_{G-BCS}=P_{N_e}{P}_G{\psi }_{BCS},$$

(32)

where $P_{N_e}$ indicates the operator to extract the state with $N_e$ electrons. The exponential-BCS wave function is given by

$${\psi }_{\lambda }=e^{-\lambda K}{P}_G{\psi }_{BCS}.$$

(33)

In this wave function, we perform the electron–hole transformation for down-spin electrons:

$$d_k=c_{-k\downarrow }^{†},d_k^{†}=c_{-k\downarrow };$$

(34)

and not for up-spin electrons: $c_k=c_{k\uparrow }$. The electron pair operator $c_{k\uparrow }^{†}{c}_{-k\downarrow }^{†}$ denotes the hybridization operator $c_k^{†}{d}_k$ in this formulation.

3.3. Correlated Superconductivity

We first discuss the superconducting (SC) state in the two-dimensional Hubbard model. In the optimization Monte Carlo method, the SC state becomes indeed stable when the Coulomb interaction U is large to be of the order of the bandwidth. We show the ground-state energy as a function of the superconducting order parameter $\Delta$ in Figure 2 (left). The simple Gutzwiller-projected BCS wave function predicted the possibility of superconductivity in the Hubbard model, and the improved wave function also shows a stability of the SC state.

We show the SC and antiferromagnetic (AF) order parameters as a function of U in Figure 3. The AF order parameter has a peak when $U/t\sim 10$, which is of the order of the bandwidth, and the SC one also has a peak at $U_c$ that is greater than the bandwidth. This indicates that there is the possibility of high-temperature superconductivity in the strongly correlated region.

The AF correlation is maximized at $U\sim U_c$ and decreases when U is larger than $U_c$. We show schematic pictures in Figure 4, where the SC condensation energy as a function of U is shown in the left panel, and the AF and SC gap functions are shown in the right panel. There is a crossover from weakly correlated region to the strongly correlated region. The superconducting state is most favorable when the AF correlation is gradually suppressed in the strongly correlated region. Thus, high temperature superconductivity is highly promising in the strongly correlated region where U is as large as the bandwidth D or larger than D.

3.4. Stability of Antiferromagnetic State

3.4.1. Hubbard Model

Let us examine the stability of AF state. There are two parameters U and $t^{\prime }$, and there is the AF region in the parameter space. High temperature superconductivity is expected near the boundary between the AF phase and the paramagnetic phase. We show the AF condensation energy $\Delta E_{AF}$ as a function of $1-n_e$ in Figure 5a for $t^{\prime }=0$ and Figure 5b for $t^{\prime }=-0.2t$. The AF region becomes larger as $|t^{\prime }|$ increases. When $t^{\prime }=-0.2t$, the AF region extends up to about 20% doping. From the competition between superconductivity and AF order, $t^{\prime }=0$ is most favorable for superconductivity.

3.4.2. Three-Band d-p Model

In general, in the three-band d-p model, the AF correlation is very strong and the AF state is more stable than in the single-band Hubbard model. This is because d electrons are localized and easily form magnetic order [13]. To investigate the possibility of high temperature superconductivity in the d-p model, it is necessary to reveal regions with weak AF order. There are many parameters in the d-p model to control the strength of the AF correlation. Among them, the Coulomb repulsion between d electrons $U_d$, the level difference ${\Delta }_{dp}={ϵ}_p-{ϵ}_d$, and the hole density x are important. The AF region is shown in Figure 6 where $U_d$ and ${ϵ}_p-{ϵ}_d$ are varied, and the hole density is fixed at 0.1875. The AF region increases when the hole density decreases. We expect that high temperature superconductivity will occur near the boundary between AFM and PM regions. This boundary exists in the region when ${\Delta }_{dp}$ is small. High temperature superconductivity is likely occur when ${\Delta }_{dp}$ is small. There is a “on-site attractive region” when ${\Delta }_{dp}$ is large where two d electrons prefer to occupy the same site. In this region, a charge-density wave or an s-wave superconducting state will be realized.

We proposed to introduce the transfer integral $t_d^{\prime }$ to control the strength of AF correlation [74]. We show the AF region at half-filling in the $t_{pp}-t_d^{\prime }$ plane in Figure 7. As $-t_d^{\prime }$ increases, there is a phase transition from the AF insulator to the paramagnetic insulator (PMI). We expect that $t_d^{\prime }$ and $t_{pp}$ will play an important role to suppress AF correlation when holes are doped in the d-p model.

3.5. Phase Diagram for the Hubbard Model

We discuss the phase diagram when carrier holes are doped in the CuO${}_2$ plane. We evaluate the energy lowering when we include the order parameter $\Delta$. We define

$$\Delta E=E(\Delta =0)-E({\Delta }_{min}),$$

(35)

where $E(\Delta )$ takes a minimum at $\Delta ={\Delta }_{min}$. We show $\Delta E$ as a function of the hole doping rate x in Figure 8 where we put $U/t=18$ and $t^{\prime }=0$. This phase diagram contains several interesting features. There are three phases: antiferromagnetic insulator (AFI), coexistent state (AFSC) and superconducting phase (SC). When the hole doping rate x is large, e.g., $x>0.09$, the pure d-wave stat is stable. There is the possibility of high (and room) temperature superconductivity in this phase. In the underdoped region, approximately $0.06<x<x_{dSC}$ with $0.08<x_{dSC}<0.09$, there is the coexistent state of antiferromagnetism and superconductivity. This is the mixed phase of AF and SC. $x_{dSC}$ could not be determined precisely. There is the possibility that both the AFSC and SC states are found for $x_{dSC}<x<0.09$, but the SC solution will have lower energy. There is the AFSC-SC transition at $x=x_{dSC}$. The AFI state exists near half-filling for about $x<0.06$, where doped holes form clusters and localize.

The existence of AFI phase is closely related to the phase separation [140,141] when the hole density is very small. In the phase-separated phase, the doped holes are localized and cannot be conductive. The existence of AFI phase is determined by the quantity

$${\delta }^{2}E(N_e)\equiv [E(N_e+\delta N_e)-2E(N_e)+E(N_e-\delta N_e)]/{(\delta N_e)}^2,$$

(36)

where $E(N_e)$ is the ground-state energy with $N_e$ electrons. ${\delta }^{2}E(N_e)$ is approximately the second derivative of the energy $E(N_e)$ and is proportional to the charge susceptibility. When ${\delta }^{2}E(N_e)$ is negative, the phase separation occurs. As shown in Figure 8, the phase separation occurs for $x<0.06$. Concerning the phase separation, the parameter $t^{\prime }$ is important because the phase separation region decreases as $-t^{\prime }$ increases. Thus, the AFI phase will decrease as $-t^{\prime }$ increases. The phase separation disappears for $t^{\prime }=-0.2t$.

4. Summary

We have discussed the possibility of high temperature superconductivity in many-electron systems. The critical temperature $T_c$ may increase as the characteristic energy of the interaction increases. Empirically, $T_c$ is proportional to the inverse of the effective mass of electrons. $T_c$ is low when the effective mass is very heavy. A candidate of high (room) temperature superconductivity may be in materials with strong electron correlation and with small effective mass enhancement. From this view point, the repulsive Coulomb interaction can be a candidate of the origin of high temperature superconductivity.

We have shown phase diagrams for the 2D Hubbard model and the three-band d-p model. The diagram in Figure 8 exhibits the characteristic property of cuprate superconductors. This supports that the origin of high temperature superconductivity is the strong correlation between electrons. That is, the mechanism of high-$T_c$ superconductivity is the electron-pair formation due to the strong on-site repulsive Coulomb interaction. The competition between antiferromagnetism and superconductivity is important in realizing high temperature superconductivity. High-$T_c$ superconductivity is expected in the region near the boundary between AF phase and paramagnetic phase. In the phase diagram for the Hubbard model, the SC phase exists near the AF phase, and AF order and superconductivity coexist where the doping rate is approximately 0.05∼0.06 $<x<x_{dSC}$ and $0.08<x_{dSC}<0.09$. We expect that this coexistence may be related to anomalous metallic behavior in the underdoped region. The AF phase near half-filling is insulating, which is approximately for $x<0.06$. There is the pure d-wave phase for $x>x_{dSC}$.

In the d-p model, the AF region exists in the multi-dimensional parameter space. The AF-PM boundary is a multi-dimensional region in this space. Since we expect that superconductivity occurs near the boundary, high temperature superconductivity is more likely to occur in the d-p model. There is the AF–PM boundary when the level difference ${\Delta }_{dp}$ is small. Thus, $T_c$ of high temperature cuprates will be high when ${\Delta }_{dp}$ is small. This tendency is consistent with experimental $T_c$ of cuprates.

We give a comment on the crossover between weakly correlated region and strongly correlated region. We expect that this crossover is universal in the sense that similar phenomena occur in nature. There may be a universal class. It will include the Kondo effect [194,195,196], QCD [197], BCS-BEC crossover [198], sine-Gordon model [199,200,201,202], and Gross–Neveu model [203].