Analytical Descriptions of High-T_c Cuprates by Introducing Rotating Holes and a New Model to Handle Many-Body Interactions

1. Introduction

First of all, note that, as the abstract mentioned, the present paper is written under the condition that our previously published article [1] was understood that describes the T_c formula analytically, although the present paper will provide the review sections.

Although several significant advancements have been presented, from the initial discovery of a superconductor, the most impressive discoveries are CuO₂-based superconductors (i.e., high-T_c cuprates) [2].

This is because, prior to this result, superconductors generally require significantly high refrigeration because of their lower critical temperature (~20 K). However, because they have higher T_c than LN₂, the high-T_c cuprates received considerable attention and interests from condensed matter physics researchers and researchers in technologies who demonstrated interest in the technical merits when applied to superconducting magnetic energy storage and energy transmission [3–5].

Thus, initial results demonstrated that high-T_c cuprates involved researchers from many condensed matter physics and related technologies.

However, condensed matter physics researchers investigated high-T_c cuprates for much deeper reasons, i.e., they are the first case at which the standard band model and the Bardeen–Cooper–Schieffer (BCS) theory are not applied, which indicates that novel physical phenomena occurred. (Recent H-based superconductors [6] with extremely high pressures have high potential to be applied to the BCS theory.) Moreover, many claimed that it is related to many-body interactions [7], which made many theoretical researchers’ approaches to the mechanism difficult. It is obvious that, similar to the BCS theory, the use of quantum field theory is not adequate because quantum field theory is extremely abstract and does not reflect the fact that a phenomenon in condensed matter physics involves many actual physical constants.

Although many studies about the experiments have been reported [8–13], (in particular, STM and STS [15] experimental methods to date revealed many aspects in high-T_c cuprates,), no theory describes all of the experimental data. These theories are divided to two methods: either Fermi-liquid model or resonating valence bond (RVB) model [16–19].

However, these theories have undetermined parameters, which inevitably leads to numerical or fitting methods. We must mention that they are insufficient because many related and actual physical parameters (i.e. physical constants) are involved when the properties of high-T_c cuprates are considered. For example, several researchers claim that, because of the existence of magnetic-field interactions, the natural force to combine a Cooper pair must be spin interactions. However, as mentioned in this study and our previous study [1], magnetic-field interactions are not generally only the spin interactions. For example, the spin-fluctuation [19] model is a numerical one; in this sense, this model is similar to the Hubbard-like model [20]. These models have multiple parameters to determine or to fit; thus, they do not reflect actual physical picture the high-T_c cuprates originally have.

Furthermore, if the interaction was defined as spin interactions, they could not explain why other multiple physical parameters such as phonons are related [21].

Although multiple theories exist discussing the nature of force to combine a Cooper pair and the origin of pseudo-gap using RVB model or Hubbard-like model, few theoretical articles analytically address and explain experiments data, in addition to the anomaly metal phase and the transition temperature T₀ at which the anomaly metal phase appears.

Briefly, the understanding of high-T_c cuprates requires

• Analytical calculations of many-body interactions. Most theories use a numerical or fitting method; however, these approaches cannot clarify the physical picture in high-T_c cuprates.
• To understand the nature of force to combine a Cooper pair over long distance.

However, research-related challenges have prevented a complete investigation of the abovementioned issues. If the calculations can be analytically solved, condensed matter physics will make considerable progress in developing then methods for fabricating compounds with higher critical temperatures could be developed through condensed matter fields. Thus, uncovering the physical mechanics of high-T_c cuprates is urgently required, and has motivated the present study. We thus provide new answers to the above questions.

Combining with our previous study [1], we will propose a concept of macroscopic Boson. This paper will describe the mechanism of high-Tc cuprates, using only this concept, with the consideration of many-body interactions.

As the contents of this paper, first we introduce a concept of macroscopic Boson, in which its mass and spin are described. Then using the partition function, the pseudo-gap energy is explained. Moreover, the method to handle many-body interaction will be introduced. Next, as the review, we obtain the formula of Tc [1]. The calculations of obtaining formulas of T^* and T₀ are positioned, analyzing anomaly metal phases. Section 3 is Method in which more concrete calculation methods are indicated. Then Result and Discussion sections are followed. Finally, the paper concludes the entire contents in section 6.

2. Theory

2.1. Introduction of new particle and pseudo-gap relating to new particle

2.1.1. Introduction of a macroscopic Boson

When considering a CuO₂ surface as the most important point [22] and when the refrigeration is sufficient such that a hole’s wavelength becomes larger than that of width of the surface, it is assumed that 2D is completely formed. This indicates that, on the surface, an angular momentum must be conserved; thus, each hole takes a circle by self-rotating. At this time, because this rotating circle has magnetic field energy, we consider that a new particle has been created. Note that the creations of the particles implies a phase transition, which will be described later in this paper. This is related to the electron nematic phase [27]. Going forward, we refer to this new particle as a “macroscopic Boson”; the schematic is shown in Figure 1.

For a literature support of the assumption of a macroscopic Boson, please refer to [23]. Moreover, this fact corresponds to the fact that, in a CuO₂ surface, a local persistent current exists [27].

2.1.2. Calculating the mass of a macroscopic Boson

First, let us calculate the mass of a macroscopic Boson. Using a magnetic flux, magnetic field energy is represented as follows:

$$2\mathrm{U}=\frac{1}{2}I{\Phi }_0$$

(1)

where I and Φ₀ denote a current surrounding a macroscopic Boson and a magnetic flux of a macroscopic Boson having a unique value. In this equation, the magnetic flux is assumed to be quantized because each angular momentum is conserved as mentioned.

$${\Phi }_0=\frac{h}{e}$$

(2)

where h and e denote the Planck constant and the charge of a hole, respectively. Note that this study used both the constant h and ħ as Planck constants. Note that this quantization implies that each flux behaves as a particle, as discussed.

In this current, the cyclotron angular frequency is introduced.

$$\mathrm{I}=\frac{e}{T}=2\pi e{\omega }_c=2\pi e\frac{e{B}_0}{m}=2\pi e^2\frac{{\mu }_0{H}_0}{m}$$

(3)

where ω_c, B₀, and u₀ denote the cyclotron angular frequency, the constant and unique value of magnetic field in a macroscopic Boson, and the magnetic permeability in the vacuum, respectively. Thus, eq. (1) becomes

$$\mathrm{U}=\frac{1}{2}2\pi e^2\frac{{\mu }_0{H}_0}{m}\frac{h}{e},$$

(4)

where considering a flux of eq. (2)

$$H_0=\frac{h}{e}\frac{1}{{\mu }_0\pi {\eta }^2},$$

(5)

where η is the approximated radius of a macroscopic Boson.

Because the magnetic field B₀ is expressed as eq. (5), the rest energy, i.e. the mass of a macroscopic Boson is formed as follows:

$$2\mathbf{U}=\mathsf{\pi }{\mathit{e}}^2\frac{{\mathit{\mu }}_0}{\mathit{m}}\frac{\mathit{h}}{\mathit{e}}\frac{1}{{\mathit{\mu }}_0\mathit{\pi }{\mathit{\eta }}^2}\frac{\mathit{h}}{\mathit{e}}=\frac{{\mathit{h}}^2}{\mathit{m}{\mathit{\eta }}^2}.$$

(6)

2.1.3. Spin of a macroscopic Boson

Let us now consider the spin of a macroscopic Boson to obtain partition function, which creates all the anomaly metal properties. As mentioned, when a macroscopic Boson is created, a complete 2D motion can be considered. That is, among the x-y-z axes, we cannot consider z-components; therefore, using the Pauli matrix, this case considers only the x- and y-components.

$$s_x=\frac{ħ}{2}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$$

(7-1)

$$s_y=\frac{ħ}{2}(\begin{array}{cc}0& -i\\ i& 0\end{array})$$

(7-2)

where i denotes the imaginary unit.

In this study, a spin angular momentum is defined as the determinant from the Pauli matrix.

Thus, each determinant is as follows:

$$\mathrm{d}\mathrm{e}\mathrm{t}{s}_x=-\frac{1}{2}ħ,$$

(8-1)

$$\mathrm{d}\mathrm{e}\mathrm{t}{s}_y=\frac{1}{2}ħ.$$

(8-2)

Therefore, a net spin angular momentum of a macroscopic Boson is calculated as follows:

$$\mathbf{s}\equiv \mathbf{d}\mathbf{e}\mathbf{t}{\mathit{s}}_{\mathit{x}}+\mathit{d}\mathit{e}\mathit{t}{\mathit{s}}_{\mathit{y}}=0·ħ.$$

(9)

The above result indicates that, although a single hole behaves as a Fermion, this macroscopic Boson on 2 D behaves similar to a Boson. Thus, the name of this particle is derived from this fact.

Let us consider this by another view:

It is necessary to consider a magnetic-flux property in terms of the electromagnetics. As described, a macroscopic Boson is simply a magnetic flux ${\Phi }_i$in a 2D sheet. Moreover, electromagnetic energy is expressed by the following Hamiltonian [33];

$$\mathrm{H}=(\mathrm{n}+\frac{1}{2})ħ\omega$$

While time-dependent energy is described by the first term in this equation, a static magnetic field energy, i.e., the energy of a macroscopic Boson, is equal to the second term, i.e., the zero-point energy [33]. On the other hand, the liquid He (i.e., quantum liquid) has also the zero-point energy [33]. Because the liquid He is consisted of Bose particles, a macroscopic Boson can be considered to be a boson.

As described, as long as considering 2D, the statistic property would alter.

2.1.4. Obtain the partition function

Because a macroscopic Boson on 2D follows the Bose’s partition function, we simply must consider the following:

$$f_r=\frac{1}{\exp \left(\frac{E_i-E_F}{k_{B}T}\right)-1},$$

(10)

where E_i, $\left|E_F\right|$, $k_B$,and T denote energy, a chemical potential (i.e. $E_F<0$), the Boltzmann constant, and temperature, respectively. An important point is that the exponential function is approximated as a Maclaurin series,

$${\mathit{f}}_{\mathit{r}}\approx \frac{1}{\frac{{\mathit{E}}_{\mathit{i}}-{\mathit{E}}_{\mathit{F}}}{{\mathit{k}}_{\mathit{B}}\mathit{T}}+1-1}=\frac{{\mathit{k}}_{\mathit{B}}\mathit{T}}{{\mathit{E}}_{\mathit{i}}-{\mathit{E}}_{\mathit{F}}}.$$

(11)

This abovementioned partition function is very important because all properties in the anomaly metal phase in CuO₂-based superconductors are described using this partition function. We will see how this equation describes properties of the anomaly metal phase later.

Moreover, this equation has another expression. Because we are now considering the chemical potential from Bosons and the general boson partition function, the following equation generally holds:

$$E_F=E+k_{B}Tln\left(\frac{N_A}{n_i}\right),$$

(12)

where the absolute value of the second term must be dominate over the value of the first term because the chemical potential is negative, and where $N_A$ denotes accepter concentration. Moreover, $\frac{N_A}{{2n}_i}$ indicates a doping parameter in this study. The number 2 is attached because of the presence of spin. Therefore, because n_i indicates the concentration of lattices, $\frac{N_A}{n_i}$ of ln is less than the value of the number 1 as long as we consider the image in which holes are doped in a Mott insulator.

Using the equation above, the partition function, eq. (11), is translated as follows:

$${\mathit{f}}_{\mathit{r}}=-{\left[\mathbf{ln}\left(\frac{{\mathit{N}}_{\mathit{A}}}{{\mathit{n}}_{\mathit{i}}}\right)\right]}^{-1}.$$

(11-2)

2.1.5. Calculate the pseudo-gap energy

Let us calculate the pseudo-gap energy, which is directly related to the mass of a macroscopic Boson. First, we define the carrier concentration of macroscopic Bosons considering a 2D energy state density.

$$D_{2 }\left(E\right)=\frac{m}{\pi {ħ}^2}\equiv p_0,$$

(13)

$$\mathrm{n}=\frac{1}{d}\int D_2(E)f_{r}dE,$$

(14)

where $D_2(E)$, n, and d denote energy state density in 2D, particle concentration, and width of the 2D sheet, respectively. An important point to note is that the parameter d [m] is consistently substituted by the number 1; however, the reason of the appearances in certain equations clarify the meaning of these equations. The integral for concentration (14) is simply as follows because the energy state density in 2D is constant as indicated in eq. (13) and because partition function f_r is represented by eq. (11-2). In the process of this calculation of eq. (14), an energy E₀ appears as follows:

$$E_0=-\frac{d}{p_0}{n}_0\times \ln (\frac{N_A}{n_i}),$$

(15)

This energy E₀ is assumed to be essentially equal to the pseudo-gap energy. Combined with the mass of a macroscopic Boson, this pseudo-gap energy is represented as follows:

$$E_0=-U\times ln\left(\frac{N_A}{n_i}\right)=-\frac{1}{2}\frac{h^2}{m{\eta }^2}\mathrm{l}\mathrm{n}(\frac{N_A}{n_i}).$$

(16-1)

The abovementioned pseudo-gap energy equation has a coefficient for doping function ln. This factor is identical to the zero-point energy:

$$\frac{1}{2}ħ\omega =\frac{1}{2}\frac{h^2}{m{\eta }^2}.$$

(16-2)

The above equation implies that this derived energy E₀ merely indicates a potential. In general, however, an energy gap appears or disappears involving a photon’s emission or absorption, which has a momentum. This fact indicates that, for a potential to become a general energy gap, the potential is given the product of the fine-structure constant α, which includes characteristic impedance Z₀ for electromagnetic waves. Typically, the fine-structure constant α is determined as follows:

$$\mathsf{\alpha }=\frac{Z_0{e}^2}{4\pi ħ}=\frac{1}{137.0}.$$

(17)

In eq. (17), the impedance Z₀ works as the specific impedance to electromagnetic waves. Thus, the net pseudo-gap energy ${\left|\Delta \right|}_0$ is derived as follows, which will give the temperature of pseudo-gap T^* as discussed later.

$${\left|\mathit{\Delta }\right|}_0=-\frac{1}{2}\frac{{\mathit{h}}^2}{\mathit{m}{\mathit{\eta }}^2}\mathit{\alpha }\times \mathbf{ln}(\frac{{\mathit{N}}_{\mathit{A}}}{{\mathit{n}}_{\mathit{i}}}).$$

(18)

2.2. Superconductivity with consideration of many-body interactions

It is necessary to describe why macroscopic Bosons undertake Bose–Einstein (BE) condensation by forming a pair from two macroscopic Bosons, although they have been already general Bosons such as Cooper pairs. In the previously published paper [1], we reported a new attractive force to combine particles from local current in a CuO₂ cell [27]. This local current is equal to both rotational and self-current, which creates the mass of macroscopic Bosons; hence, the result of the previous paper agrees with the descriptions in the present paper. Therefore, in this section, based on the understanding that two macroscopic Bosons form a pair, we describe why BE condensation occurs considering many-body interactions between Bosons.

2.2.1. Description of the model and the principle to many-body interaction

There are many-body interactions among the carriers in various materials. In particular, this fact is essential to high-T_c cuprates because the general band theory cannot be applied. The many-body interactions of carriers indicate there are many local temperatures T_i in the materials, where i is index for a location. In other words, only in a temperature T_i, thermal equilibrium can be assumed. Figure 2 shows our model for handling many-body interactions. In this figure, a radius a_i forms a sphere shell, which has differential number dN and local temperature T_i. Moreover, in the center, a macroscopic Boson is presented. The immediately outer particles out of dN yield a pressure to this sphere shell, which is equal to the kinetic energies of particles in dN (i.e., it is represented by a temperature T_i). However, the central macroscopic Boson provides force of expansion, which indicates electrostatic energy, i.e., Coulomb interactions. Moreover, this case adds magnetic interactions between macroscopic Bosons as an expansion force.

Considering that these forces of expansion should be balanced to a force of compression in a sphere shell, the following relation holds:

$$\mathbf{(}\mathbf{Coulomb} \mathbf{interaction} \mathbf{energy} \mathbf{and} \mathbf{magnetic} \mathbf{field} \mathbf{interaction} \mathbf{energy}\mathbf{)}=\frac{3}{2}{\mathit{k}}_{\mathit{B}}{\mathit{T}}_{\mathit{i}}\times \mathit{d}\mathit{N}$$

2.2.2. Calculate the principal equation of our model and the internal quantum state

Calculate the abovementioned principle equation. First, dN is represented as follows:

$$\mathrm{d}\mathrm{N}=\mathrm{g}\mathrm{f}\mathrm{d}\overrightarrow{k}=gf\left(\frac{1}{dv}\right),$$

(19)

where k, v, g, and f denote wave number, volume, state number, and partition function for the Boson, respectively. In the equation of dN, as mentioned, state number g and partition function f are given as follows:

$${f\equiv f}_r=-{[\ln (\frac{N_A}{n_i})]}^{-1}.$$

(11-2)

$$\mathrm{g}=\frac{1}{d}\int D_2\left(E\right)dE=p_0{E}_0,$$

(14)

$$D_2\left(E\right)=\frac{m}{\pi {ħ}^2}\equiv p_0,$$

(13)

$$E_0={\left|\Delta \right|}_0=-\frac{1}{2}\frac{h^2}{m{\eta }^2}\alpha \times \ln (\frac{N_A}{n_i}).$$

(18)

Thus, fg is given as follows:

$$\mathrm{f}\mathrm{g}=p_0\frac{1}{2}\frac{h^2}{m{\eta }^2}\alpha .$$

(20)

To calculate the left-hand side of the abovementioned balanced equation in principle, the electrostatic energy U_E is calculated as follows:

$$U_E=\frac{1}{2}{\epsilon }_0{(\frac{e}{4\pi {\epsilon }_0{a}_i^2})}^{2}dv,$$

(21)

where ${\epsilon }_0$ and${ a}_i$ denote the permittivity for the vacuum and the radius which dN is taking in the model.

At this time, a volume element of the integral is expressed as follows:

$$\mathrm{d}\mathrm{v}=\frac{1}{\mathrm{d}\overrightarrow{k}}=\frac{4\pi }{3}{a}_i^3.$$

(22)

Moreover, the magnetic interaction V_p from macroscopic Bosons is given as follows:

$${V_p=U}_{B}dN.$$

(23)

Consequently, the resultant equation is provided by

$$a_i^2=\frac{9{\epsilon }_0}{e^2}\left(3k_B{T}_i-2U_B\right)fg.$$

(24)

As shown in Figure 3, the central macroscopic Boson behaves under the model of the infinite well-potential. Thus, as every elementary quantum mechanics text [24] describes, the eigenvalue and wave function of it are presented as follows:

$${\mathit{\psi }}_{\mathit{i}}\left(\mathit{r}\right)=\sqrt{\frac{2}{2{\mathit{a}}_{\mathit{i}}}}\mathbf{s}\mathbf{i}\mathbf{n}\left(\frac{\mathit{i}\mathit{\pi }\mathit{r}}{2{\mathit{a}}_{\mathit{i}}}\right),$$

(25)

$${\mathit{E}}_{\mathit{i}}=\frac{1}{2\mathit{M}}{(\frac{ħ\mathit{i}\mathit{\pi }}{2{\mathit{a}}_{\mathit{i}}})}^2,$$

(26)

where M, i, and r denote the mass of a macroscopic Boson, index, and microscopic variable of sphere-coordinates, respectively. These equations indicate that a particle under the many-body interactions forms a stationary wave and that the wave function of the stationary wave and the eigenvalue (i.e., kinetic energy) are determined by a radius a_i..

2.2.3. Describe BE condensation and the superconducting transition

Using the abovementioned concept, we consider how BE condensation occurs. In addition to a sphere shell having temperature T_i,, another sphere shell having temperature T_j is considered. When we accept a combination of two macroscopic Bosons by a force F, these two Bosons must have the identical kinetic energy because, in general and as mentioned in our previous paper [1], a relative and attractive force appears only when their relative velocities become the same. In particular, this fact is applied when an attractive Lorentz force is generated between moving and charged particles whose velocities are identical. Thus, when forming a pair from two macroscopic Bosons, the eigenvalues, eq. (26), indexed by i and j becomes equal. That is,

$$\left|E_i-E_j\right|=0.$$

(27)

This indicates that an index i and j becomes equal, resulting in that all the radius a_i and eigenvalue E_i take the identical radius a₀ and E_B because of the arbitrary property of index i and j. Hence, if a pair forms, every energy of macroscopic Bosons undergoes the identical energy E_B, which indicates all the rest Bosons take pairs and BE condensation.

Moreover, as shown in Figure 4, considering index i to be equal j indicates that temperatures T_i and T_j must be equal. Even at this moment, positions r of wave functions, eq. (25), are common and thus the two sphere shells take the superposition, i.e. the relative distance ξ_G between the two sphere shells should be 0. Thus, the net coherence of two holes becomes on a cell order, 1 nm, as reported by many literatures.

Employing the abovementioned equation (24), an equation of the relative distance between sphere shells ξ_G for temperature T is derived as follows:

$$\frac{1}{4}{\mathit{\xi }}_{\mathit{G}}^2=\frac{9{\mathit{\epsilon }}_0}{{\mathit{e}}^2}\left(3{\mathit{k}}_{\mathit{B}}〈\mathit{T}〉-2{\mathit{U}}_{\mathit{B}}\right)\mathit{g}\mathit{f},$$

(28)

where U_B is substituted with pseudo-gap ${\left|\Delta \right|}_0$ in eq. (18). Note that, in this equation, considering BE condensation and single-particle picture, p₀ of gf in eq. (20) is redefined as the value 1.

As will be discussed in the Results section, temperatures at which ${\xi }_G^2\le 0$ indicates a superconductivity state (i.e., the net coherence of two holes is about 1 nm, which equals CuO₂ cell order) and the transition temperature T_c at which ${\xi }_G=0$ indicates a critical temperature.

2.3. Review to obtain the formula for T_c

Herein, we would like to note the reason why there are Fermi energy and chemical potential E_F [25]. Considering each pair and because these pairs have superposition, a single macroscopic wave with a converged phase is produced. However, at this stage, although we cannot consider each particle motion as a pair of two macroscopic Bosons, there is a non-zero temperature (i.e., $\mathrm{T}\le T_c$). This indicates that the internal particles of a macroscopic Boson (i.e., holes) collide with each other. Thus, only in the case of $\mathrm{T}\le T_c$, we consider the Fermi energy, i.e., $E_F\ge 0$.

2.3.1. Derivation of a general energy gap (review)

Let us review our previous study [1], which describes a force F to combine two particles and a critical temperature T_c on doping. Note that because this is a review to understand the stream of outlined derivations of a critical temperature T_c, certain equations in the calculation and derivation processes are left out. In case that our readers are interested in the detail, the paper can be downloaded as an Open Access paper.

First, we assume that a general energy gap $\left|\Delta \right|$ is proportional to both Fermi energy and Critical temperature as follows [1]:

$${\left|\Delta \right|}^2=k_B{T}_c{E}_F.$$

(29)

In this equation, the Fermi energy in a p-type material [40] is employed as follows:

$$E_F=E_i-k_{B}Tln(\frac{N_A}{n_i}).$$

(30)

Note that we are considering the carrier is a hole.

In this equation, a superconducting energy gap is introduced.

$$2E_i=k_B{T}_c$$

(31)

Substituting these energies and employing the state equation with the universal gas constant R, the following equations are obtained.

$${\left|\Delta \right|}^2=\frac{1}{2}{(k_B{T}_c)}^2\left\{1-2\frac{T}{T_c}\mathrm{l}\mathrm{n}(\frac{N_A}{n_i})\right\}.$$

(32)

and

$${\left|\mathit{\Delta }\right|}^2=\frac{1}{2}{({\mathit{k}}_{\mathit{B}}{\mathit{T}}_{\mathit{c}})}^2\left\{1-2\frac{1}{{\mathit{T}}_{\mathit{c}}}\frac{\left|{\mathit{\Omega }}_{\mathit{B}}\right|}{\mathit{R}}\frac{1}{{\mathit{\rho }}_{\mathit{s}}}\mathbf{l}\mathbf{n}(\frac{{\mathit{N}}_{\mathit{A}}}{{\mathit{n}}_{\mathit{i}}})\right\}.$$

(33)

where

$$\left|{\Omega }_B\right|=pV.$$

(34)

where ${\Omega }_B$ denotes a thermodynamic potential, and ρ_s is the concentration of Cooper pairs.

In this manner, a general expression of energy gap for temperatures is derived.

2.3.2. Generation of an attractive force that combines two carriers (review)

To consider the superconducting energy gap, it is necessary to mention a force F, which results in a combination of a Cooper pair. As previously mentioned, two charged particles generally experience an attractive force with each other when they are moving with the same velocity, i.e., when the relative energy or momentum is 0. As shown in Figure 5a–5d, two parallel conductors along which the same direction and same amount of a current are presented. From the electromagnetism, these current leads experience an attractive force with each other, which is attributed to the Lorentz force. When we shorten these leads to a wavelength of a carrier, this attractive force still exists. This indicates that two charged particles with identical wave numbers are attracted to each other. This attractive force stems from the Lorentz force.

2.3.3. Derivation of T_c (review)

Considering the principle of generating an attractive force and assuming that the wave function of a hole is a plane wave and that the magnetic field generated by the moving holes is derived from a linear current, the Lorentz force F is given as follows:

$$\mathrm{F}=q^2\frac{ħ{\mu }_0}{m^2}\frac{4{\pi }^2}{k^2}\beta {\left|\psi \right|}^2{k}^2\frac{1}{2r}sin\theta cos\phi =\frac{2q^2{\pi }^2{\mu }_0ħ}{m^2}\beta {\left|\psi \right|}^2\frac{1}{r}sin\theta cos\phi ,$$

(35)

where ψ, r, θ, ϕ, q, β, k m, and μ₀ denote wave function of a hole, relative distance of two holes, angle associated with the Lorentz force, angle related with two wave number of holes, the electric charge of a hole, constant, common wave number, the mass of an electron, and magnetic permeability of the vacuum, respectively.

Note that this equation employs the probability density flux as current density.

The energy u (i.e., superconducting energy gap) from the line integral of the above force F is represented as follows:

$$\mathrm{u}=-\frac{2q^2{\pi }^2{\mu }_0ħ}{m^2}\beta {\left|\psi \right|}^2\ln \left(r\right)\times sin\theta cos\phi +u_0 u_0\le 0,$$

(36)

where$u_0$ denotes an integral constant.

Furthermore, the derived superconducting energy gap u produces T_c with the combination of a general gap energy derived in eq. (33).

$$T_c=-4{\alpha }^{\text{'}2}\frac{\left|{\Omega }_B\right|}{R{\rho }_s}\ln \left(\frac{N_A}{n_i}\right)-{\theta }_D,$$

(37-1)

where

$$\mathsf{\alpha }=-\frac{2q^2{\pi }^2{\mu }_0ħ}{m^2}\ln \left(\xi \right)\times sin\theta cos\phi ,$$

(37-2)

$${\alpha }^{\text{'}}=\frac{1}{k_B{\theta }_D}\alpha =-\frac{1}{k_B{\theta }_D}\frac{2q^2{\pi }^2{\mu }_0ħ}{m^2}\ln \left(\xi \right)\times sin\theta cos\phi .$$

(37-3)

In this process, we added a Debye temperature θ_D and a net coherence ξ to the equation. Note that, as an integral constant in eq. (36), the BCS formula under a particular condition was employed. That is, in the formula T_c of the BCS theory, because the Boson combination energy in high-T_c cuprates is generally sufficiently large attributed to the short coherence (note that, the shorter the coherence is, the larger the magnetic field associated with the Lorentz force becomes), the large value of NV in the BCS formula of T_c makes the exponential function be almost the value 1. Thus, only the Debye temperature in the BCS formula is left. Concerning the thermodynamic potential, the following equation is applied under the condition of BE condensation.

$$\left|{\Omega }_B\right|=pV=\frac{2}{5}{E}_{F0},$$

(38-1)

$$2E_{F0}=E_{G0}.$$

(38-2)

where E_F0 and E_G0 denote the Fermi energy and band gap at zero temperature, respectively. Moreover, here the volume V is assumed to be the unit, i.e. the number 1. Thus, the critical temperature becomes

$$T_c=-4{\left(\frac{1}{k_B{\theta }_D}\right)}^2({\frac{2q^2{\pi }^2{\mu }_0ħ}{m^2}\mathrm{l}\mathrm{n}(\xi )\times sin\theta cos\phi )}^2\frac{E_{G0}}{5R{\rho }_s}\ln \left(\frac{N_A}{n_i}\right)-{\theta }_D.$$

(39)

Moreover, we derive a 2D critical temperature equation from the above. Thus, to conclude, the critical temperature equation is derived as follows:

$${〈{\mathit{T}}_{\mathit{c}}〉}_2=-4{\left(\frac{1}{{\mathit{k}}_{\mathit{B}}{\mathit{\theta }}_{\mathit{D}2}}\right)}^2({\frac{2{\mathit{q}}^2{\mathit{\pi }}^2{\mathit{\mu }}_0\mathit{ħ}}{{\mathit{m}}^2}\mathbf{l}\mathbf{n}(\mathit{\xi })\times \mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathit{c}\mathit{o}\mathit{s}\mathit{\phi })}^2\frac{{\mathit{E}}_{\mathit{G}0}{\mathit{n}}_{\mathit{q}}}{5\mathit{R}{\mathit{\sigma }}_{\mathit{s}}}\mathbf{ln}\left(\frac{\mathit{\sigma }}{{\mathit{n}}_{\mathit{i}2}}\right)-{\mathit{\theta }}_{\mathit{D}2}$$

(40)

where σ, σ_s, θ_D2, and n_q denote the surface density of carriers, the surface density of pairs, Debye temperature in 2D, and the number of layers. Note that all constants in the consequent equation have actual physical meaning and unit. This indicates that no numerical calculations or fitting methods are required. This fact is consistent everywhere in the present study.

Note that, in eq. (36) for the superconducting energy gap, the probability density function is interpreted as follows:

$${\left|\psi \right|}^{2}sin\theta cos\phi ={\left|\psi \right|}^{2}sin\theta \sin \left(\phi +\frac{\pi }{2}\right)=\frac{1}{2}{\left|\psi \right|}^2\left\{-\cos \left(\theta +\phi +\frac{\pi }{2}\right)+\mathrm{c}\mathrm{o}\mathrm{s}(\theta -\phi +\frac{\pi }{2})\right\}$$

(41)

which shows that the gap is anisotropic [26]

In Figure 6, a result of this review section is shown where used physical parameters are listed in Table 1. Note that for additional details, please see the Method section at which the full list of employed physical constants are presented. As shown, our derived critical temperature equation sufficiently agrees with a typical high-T_c copulate. Note that the reason why the band gap is relatively large is related to the property of the Mott insulator. For more details, please refer to [1].

2.4. Calculations for obtaining formulas for T^* and T₀

2.4.1. Derive the pseudo-gap temperature T ^*.

Now, we consider the relation between a general energy gap and temperature, as shown in eq. (32)

$$\mathrm{T}=-\frac{{\left|\Delta \right|}^2}{k_B^2}\frac{1}{T_c}\frac{1}{\ln \left(\frac{N_A}{n_i}\right)},$$

(32)

When the previously derived energy gap from a macroscopic Boson is substituted for an energy gap in the abovementioned equation, then variable temperature T must become a constant of pseudo-gap temperature T^*. Therefore, the temperatures T_c and T^* have a dependent relationship. Thus, as a formula of pseudo-gap temperature T^*, the following equation holds:

$${\mathit{T}}^{\mathit{*}}=-\frac{1}{{\mathit{k}}_{\mathit{B}}^2}\frac{1}{4}{\left(3.4\times {10}^{-21}\right)}^2{[\mathbf{ln}(\frac{{\mathit{N}}_{\mathit{A}}}{{\mathit{n}}_{\mathit{i}}})\}}^2\frac{1}{{\mathit{T}}_{\mathit{c}}}\frac{1}{\mathbf{ln}(\frac{{\mathit{N}}_{\mathit{A}}}{{\mathit{n}}_{\mathit{i}}})}=-\frac{1}{{\mathit{k}}_{\mathit{B}}^2}\frac{1}{4}{\left(3.4\times {10}^{-21}\right)}^2{[\mathbf{ln}(\frac{{\mathit{N}}_{\mathit{A}}}{{\mathit{n}}_{\mathit{i}}})\}}^{ }\frac{1}{{\mathit{T}}_{\mathit{c}}},$$

(42)

where to the equation of ${\left|\Delta \right|}_0$ of eq. (18) in creating eq. (42), each physical parameter was substituted. That is, the physical parameters m, h, and α in eq. (18) were given actual values. Note that radius η is approximated as 1 nm.

2.4.2. Derive the transition temperature T₀

In this study, we consider the anomaly metal phase properties in CuO₂–based superconductors. These properties are primarily determined by the transition temperature T₀, which is directly related to appearances of the Hall-effect coefficient RH. To obtain an equation for the temperature T₀, we consider derivations of the Hall-effect coefficient RH. The Hall-effect coefficient RH depends on absolute of energy, –uB_e,, where u and B_e denote self-magnetic moment of a macroscopic Boson and applied magnetic field, respectively. The absolute of energy, uB_e, involves Boltzmann statistics and thus it is related to concentration (i.e., the number) of macroscopic Bosons.

In the previously appeared concentration eq. (14), the calculation for energy integral, in turn, is actually conducted because we attempted to obtain temperature T dependence for RH

$$\mathrm{n}=k_{B}T\frac{1}{d}{p}_0{\int }_a^b\frac{dE}{E-E_F}=k_{B}T\frac{1}{d}{p}_0\times \ln (\frac{T_0}{T_c}),$$

(43)

where

$$\mathrm{a}=k_B{T_c b=k_B{T}_0}_{ }.$$

(43-2)

Note that the second form of fr in eq. (11-2) is not employed here. Eq. (43) is very important because the concentration n is proportional to the temperature T , which describes a property of the anomaly state. That is, the resistivity anomaly. Now we begin to calculate RH. As mentioned, considering an energy –uB_e, the Boltzmann statics is represented as follows:

$$\mathrm{n}=n_0\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{\mu B_e}{k_{B}T}),$$

(44)

where n₀ is concentration without an applied magnetic field. In this equation, the exponential function is approximated by the Maclaurin series.

$$\mathrm{n}\approx n_0(1-\frac{\mu B_e}{k_{B}T}).$$

(45)

In the above equation, the previously calculated concentration n, eq. (43), is applied.

$$k_{B}T\frac{p_0}{d}\times \ln (\frac{T_0}{T_c})=n_0(1-\frac{\mu B_e}{k_{B}T}).$$

(46)

Solving this equation for n₀ and using the general definition of RH, we reach an important equation.

$$R_H=\frac{\frac{\mu B_e}{k_{B}T}-1}{e{k}_{B}T\frac{p_0}{d}\times \ln (\frac{T_0}{T_c})}.$$

(47)

Composition of this equation presents a new temperature T₀, which indicates the appearance of RH.

$${\mathit{T}}_0\equiv \frac{\mathit{\mu }{\mathit{B}}_{\mathit{e}}}{{\mathit{k}}_{\mathit{B}}}.$$

(48)

2.4.3. Implement the formulation of T₀

To implement the formula T₀, it is necessary to obtain u and B_e in eq. (48). First, a magnetic moment u is generally defined as follows:

$$\mathsf{\mu }=\mathrm{I}\mathrm{S},$$

(49)

where I and S ($\approx {\eta }^2$ ) denote the self-current and the area in which a magnetic flux is presented. Seeing the schematic of Figure 1 of a macroscopic Boson (which assumes the motion of a hole to be a circle) and because a magnetic flux of it should be quantized as h/e, the magnetic flux of a macroscopic Boson is as follows:

$${\Phi }_0=B_0\pi {\eta }^2\equiv \frac{h}{e}.$$

(50)

That is,

$$B_0=\frac{h}{e}\frac{1}{\pi {\eta }^2},$$

(50-2)

(50-2)

where radius η is approximated on a cell of the CuO₂ surface. That is,

$$\mathsf{\eta }\approx 1\mathrm{n}\mathrm{m}$$

(51)

Moreover, assuming that a magnetic field among a macroscopic Boson is equal to the central magnetic field generated by a moving hole, a persistent current I in a magnetic moment is calculated as follows:

$$\mathrm{I}=\frac{1}{{\mu }_0}2\eta B_0.$$

(52)

Consequently, a magnetic moment u is derived as follows:

$$\mathsf{\mu }\approx \frac{2}{{\mathit{\mu }}_0}\mathit{\eta }\frac{\mathit{h}}{\mathit{e}}.$$

(53)

While an applied magnetic field B_e in the definition of T₀ is variable, the magnetic field B₀ is a constant derived by the physical constants. This fact allows us to introduce a variable quantum number N between B_e and B₀

$$B_0\frac{1}{N}=B_e.$$

(54)

Moreover, this variable integer N is undergone by the partition function fr.

$$\mathrm{N}=N_0{f}_r,$$

(55)

where eq. (11-2) is applied as fr.

Note that the magnetic field B₀ was calculated from eq. (50-2). The employment of partition function fr indicates that an application of B_e makes every direction of certain magnetic moments of macroscopic Bosons have the same orientation. In other words, prior to the application of magnetic field B_e, the directions of self-magnetic moments of each macroscopic Boson are random (i.e., up- or down-direction), although the conservations of angular momentum produces macroscopic Bosons. However, the application of magnetic field B_e presents all the directions of certain magnetic moments of macroscopic Bosons with the same orientation. Because the interaction between macroscopic Bosons with the same directed magnetic moment is repulsive, these Bosons now obtain the existences as single and independent particles. However, without an external applied magnetic field, why do our high-T_c cuprates become superconductive by forming many independent macroscopic Bosons? This can be understood by considering an analogy that every magnetic moment in a ferromagnetic material spontaneously acquires the same orientation under Curie temperatures. Thus, high-T_c cuprates have a property that is similar to a ferromagnetic material. We claim that this fact is related to the electronic nematic phase [27].

In this case, because macroscopic Bosons are formed in 2D CuO₂, weak interactions between the magnetic moments of macroscopic Bosons can justify the abovementioned calculation. The actual calculations of Curie temperatures with complete consideration of many-body interactions are presented in the Appendix of this study.

Assembling these facts, the conclusive equation of the transition temperature T₀ is derived, which depends on carrier doping.

$${\mathit{T}}_0\approx -\frac{1}{{\mathit{k}}_{\mathit{B}}}\left(\frac{2}{{\mathit{\mu }}_0}\mathit{\eta }\frac{\mathit{h}}{\mathit{e}}\right)\left(\frac{\mathit{h}}{\mathit{e}\mathit{\pi }{\mathit{\eta }}^2}\right)\frac{1}{{\mathit{N}}_0}\mathbf{ln}(\frac{{\mathit{N}}_{\mathit{A}}}{{\mathit{n}}_{\mathit{i}}}).$$

(56)

As described later, this equation of T₀ and the formula of critical temperature T_c [1] will be crucial factors when calculating properties of the anomaly metal phase.

2.5. Analyze anomaly metal phase

2.5.1. More comprehensive calculation of RH

Next, we derive dependences on temperature of RH. Up to the previous section, the general equation of RH was derived, which resulted in a definition of transition temperature T_0. In this equation, we introduce the following approximation to the general equation of RH.

$$\frac{\mu B_e}{k_{B}T}\gg 1.$$

(57)

According to this approximation, the general equation of RH becomes as follows:

$$R_H\approx \frac{\mu B_e}{e{\left(k_{B}T\right)}^2\frac{p_0}{d}\times \ln (\frac{T_0}{T_c})}.$$

(58)

Thus, the approximated equation of RH is determined by the applied magnetic fields B_e. That is, this RH equation depends on both quantum number N and the universal magnetic field B_0.

$$R_H\approx \frac{\mu B_0}{e{(k_{B}T)}^2\frac{p_0}{d}\times \ln (\frac{T_0}{T_c})}\frac{1}{N}.$$

(59)

Note that the universal magnetic field B₀ is one in a macroscopic Boson. Thus, in view of magnetic field energy, an application of magnetic field, which dominates over the universal magnetic field B₀ results in the destructions of macroscopic Bosons and makes the anomaly metal phase disappear. Moreover, the employment of quantum number N indicates that the RH equation is determined by doping. That is, variable integer N is expressed by the partition function fr, which indicates doping.

$$\frac{1}{N}=\frac{1}{N_0{f}_r}=-\frac{1}{N_0}\times \ln (\frac{N_A}{n_i}).$$

(60)

Considering this, the approximated RH equation becomes

$${\mathit{R}}_{\mathit{H}}\approx -\frac{\mathit{\mu }{\mathit{B}}_0}{\mathit{e}{\left({\mathit{k}}_{\mathit{B}}\mathit{T}\right)}^2\frac{{\mathit{p}}_0}{\mathit{d}}\times \mathbf{ln}(\frac{{\mathit{T}}_0}{{\mathit{T}}_{\mathit{c}}})}\frac{1}{{\mathit{N}}_0}\times \mathbf{ln}(\frac{{\mathit{N}}_{\mathit{A}}}{{\mathit{n}}_{\mathit{i}}}).$$

(61)

As reported in many studies [28], this derived equation of RH is proportional to ${(\frac{1}{T})}^2$.

In the Results section, we will depict this RH equation in terms of both doping parameters and temperatures T.

2.5.2. Calculate the electron specific heat coefficient in the anomaly metal phase

In turn, let us consider electron specific heat coefficient in the anomaly metal phase. Because electron specific heat coefficient is essentially equal to the average energy U_E, it is simply necessary to calculate the average energy using the partition function fr. Thus, average energy using partition function fr (eq. (11)) for energy integrals is determined as follows:

$$U_E=\frac{\int E{f}_{r}dE}{\int f_{r}dE}.$$

(62)

Note that the lower limitation a and the upper limitation b of these integrals are given as follows:

$$\mathrm{a}=k_B{T}_c b=k_B{T}_0.$$

(63)

Assuming the chemical energy for macroscopic Bosons (i.e., not Fermi energy for single holes) is sufficiently small, the calculation results in

$$U_E=\frac{k_B{T}_0-k_B{T}_c+E_F\times \ln (\frac{k_B{T}_0-E_F}{k_B{T}_c-E_F})}{\ln (\frac{k_B{T}_0-E_F}{k_B{T}_c-E_F})}\approx \frac{k_B(T_0-T_C)}{\ln (\frac{T_0}{T_c})}.$$

(64)

In general, electron specific heat coefficient is derived by differential in terms of temperature to the average energy. In this study, however, ΔT is employed rather than the differential for temperature. Moreover, this ΔT is assumed to be (T₀-T_c) in this study. Therefore, using the average energy U_E and ΔT, electron specific heat coefficient is expressed as a calculation process.

$${\gamma }_0=\frac{U_E}{{(\Delta T)}^2}=\frac{k_B}{T_0-T_c}\frac{1}{\ln (\frac{T_0}{T_c})}.$$

(65)

Furthermore, to obtain electron specific heat coefficient with the unit [J/mol K²], the Avogadro constant $N_0^A$ is considered because previously calculated average energy U_E indicates one for a macroscopic Boson. Consequently, the electron specific heat coefficient is derived as follows:

$$\mathsf{\gamma }=\frac{{\mathit{N}}_0^{\mathit{A}}{\mathit{k}}_{\mathit{B}}}{{\mathit{T}}_0-{\mathit{T}}_{\mathit{c}}}\frac{1}{\mathbf{ln}(\frac{{\mathit{T}}_0}{{\mathit{T}}_{\mathit{c}}})}.$$

(66)

2.5. Summary of the logical flow

(1)

First, assuming a macroscopic Boson, which is based on angular momentum conservation on a CuO₂ surface, its energy was calculated; the implementation of the integral of the concentration resulted in a pseudo-gap energy. During this process, the two types of partition equations fr were derived.

(2)

To handle many-body interactions, a sphere shell with a local temperature T_i and differential particle number dN is introduced. From the forces that are balanced for both inside and outside the shell, a basic statistic equation, inner wave function and eigenvalue in a shell were derived.

(3)

The generation principle of attractive force: “The Lorentz force is applied between two charged particles when their relative velocity is 0.” Considering this principle, the abovementioned statistic equation, inner wave function and inner eigenvalue realize the combination of a Cooper pair, and then BE condensation occurs.

(4)

Therefore, the superconducting energy gap and T_c were calculated. During this process, a general energy gap is derived.

(5)

Combining the general energy gap and the mass of a macroscopic Boson, the pseudo-gap temperature, T^*, formula was obtained.

(6)

The transition temperature T₀ at which anomaly metal phase appears was defined by the appearance of the Hall coefficient RH. Thus, to calculate RH, combining the Boltzmann statistics, particle concentration was implemented using the partition equation fr. Then, the general definition of RH and the concentration produced the equation of RH. Considering the form of this equation, the transition temperature T₀ was derived.

(7)

Because the resulted T₀ has the magnetic moment of a macroscopic Boson u and magnetic field B_e, these two factors were formulated. Thus, the T₀ formula was implemented.

(8)

The abovementioned derived RH equation was approximated, and electron specific heat coefficient γ was calculated. Of note, during this process, the average energy using partition equations fr was obtained.

3. Methods

Herein, we describe the detailed method for the Results section.

3.1. Calculation tool

We employed the MS Excel software.

3.2. Physical constants for calculations

Table 2 shows the primary physical constant in this study. Of note, although Debye temperatures for 3D and 2D are different, we employed 2D one

3.3. Resulted equations

3.3.1. Critical temperature

$${〈{\mathit{T}}_{\mathit{c}}〉}_2=-4{\left(\frac{1}{{\mathit{k}}_{\mathit{B}}{\mathit{\theta }}_{\mathit{D}2}}\right)}^2({\frac{2{\mathit{q}}^2{\mathit{\pi }}^2{\mathit{\mu }}_0\mathit{ħ}}{{\mathit{m}}^2}\mathbf{l}\mathbf{n}(\mathit{\xi })\times \mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathit{c}\mathit{o}\mathit{s}\mathit{\phi })}^2\frac{{\mathit{E}}_{\mathit{G}0}{\mathit{n}}_{\mathit{q}}}{5\mathit{R}{\mathit{\sigma }}_{\mathit{s}}}\mathbf{ln}\left(\frac{\mathit{\sigma }}{{\mathit{n}}_{\mathit{i}2}}\right)-{\mathit{\theta }}_{\mathit{D}2}.$$

(40)

The critical temperature is shown above again. Concerning anisotropic properties, sine and cosine are given the maximum values of 1. Table 2 lists the physical constant used except for concentrations.

3.3.2. How to determine n_i and ρ_s

In eq. (40), $\frac{\sigma }{n_{i2}}$ is identical for $\frac{N_A}{n_i}$, because the length along the c-axis, d, is consistently given the value of 1 by considering the 2D surface. Moreover, it is necessary to determine the values of 1/σ_s, i.e., 1/ρ_s when given the doping variable $\frac{{\mathrm{N}}_{\mathrm{A}}}{2{\mathrm{n}}_{\mathrm{i}}}$ as follows:

(How to determine n_i)

In this study, the concentration n_i indicates lattice concentration. Because the unit cell of the CuO₂ surface is of the 1-nm order, the following assumption is introduced

$${2n}_i=\frac{1}{d}\frac{1}{{({10}^{-9})}^2}=\frac{1}{d}\times {10}^{18},[1/{\mathrm{m}}^3]$$

(67)

Note that d has the unit of [m] and the consistent value of 1 because we are considering two dimensions.

Because the critical temperature equation (40) uses the universal gas constant,

$$\mathrm{R}=8.31\left[\mathrm{J}·{mol}^{-1}·K^{-1}\right],$$

(68)

the concentration n_i must be transformed into one with the unit [mol/L].

Thus, consider the following:

1)

Avogadro constant $N_A^0$

2)

1[L] = ${10}^{-3} [m^3]$

Therefore the concentration n_i is typically

$$n_i=\frac{1}{d}8.3\times {10}^{-10}[\mathrm{m}\mathrm{o}\mathrm{l}/\mathrm{L}].$$

(69)

(How to determine 1/ρ_s)

First, in the MS Excel sheet, the variable-doping ratio $\frac{N_A}{2n_i}$ is in the range of 0.005–0.5. Note that the number 2 appears due to spins. Then, $\frac{N_A}{n_i}$ is calculated based on the abovementioned variable doping ratio.

1/ρ_s should be determined by the constant concentration, eq. (69)

$$\frac{1}{{\rho }_s}=x\frac{1}{n_i},$$

(70-1)

where x denotes dimensionless variable. To give eq. (70-1) the meaning, variable x is provided as

$$\mathrm{x}=\frac{N_A}{n_i}.$$

(70-2)

3.3.3. Pseudo-gap temperature and transition temperature at which an anomaly metal phase occurs

We list the results of each transition temperatures, which will be shown in the Results section.

$${\mathit{T}}^{\mathit{*}}=-\frac{1}{{\mathit{k}}_{\mathit{B}}^2}\frac{1}{4}{\left(3.4\times {10}^{-21}\right)}^2{[\mathbf{ln}(\frac{{\mathit{N}}_{\mathit{A}}}{{\mathit{n}}_{\mathit{i}}})\}}^{ }\frac{1}{{\mathit{T}}_{\mathit{c}}}.$$

(42)

$${\mathit{T}}_0\approx -\frac{1}{{\mathit{k}}_{\mathit{B}}}(\frac{2}{{\mathit{\mu }}_0}\mathit{\eta }\frac{\mathit{h}}{\mathit{e}})(\frac{\mathit{h}}{\mathit{e}\mathit{\pi }{\mathit{\eta }}^2})\frac{1}{{\mathit{N}}_0}\mathbf{ln}(\frac{{\mathit{N}}_{\mathit{A}}}{{\mathit{n}}_{\mathit{i}}}).$$

(56)

Of note, $N_0\approx 1.0\times {10}^6$.

3.3.4. Physical results of the anomalous metal phase

(Hall effect coefficient)

$${\mathit{R}}_{\mathit{H}}\approx -\frac{\mathit{\mu }{\mathit{B}}_0}{\mathit{e}{\left({\mathit{k}}_{\mathit{B}}\mathit{T}\right)}^2\frac{{\mathit{p}}_0}{\mathit{d}}\times \mathbf{ln}(\frac{{\mathit{T}}_0}{{\mathit{T}}_{\mathit{c}}})}\frac{1}{{\mathit{N}}_0}\times \mathbf{ln}(\frac{{\mathit{N}}_{\mathit{A}}}{{\mathit{n}}_{\mathit{i}}}).$$

(61)

Of note, $N_0=1.1\times {10}^2$. Because B₀ is constant, the variation of integer N₀ indicates variation in the applied magnetic field B_e.

Moreover, in the abovementioned resulting equation, the following constants were employed.

$$\mathsf{\mu }\approx \frac{2}{{\mu }_0}\eta \frac{h}{e},$$

(53)

$$B_0=\frac{h}{e}\frac{1}{\pi {\eta }^2},$$

(50-2)

$$\frac{m}{\pi {ħ}^2}\equiv p_0,$$

(13)

(Electron specific heat coefficient)

$$\mathsf{\gamma }=\frac{{\mathit{N}}_0^{\mathit{A}}{\mathit{k}}_{\mathit{B}}}{{\mathit{T}}_0-{\mathit{T}}_{\mathit{c}}}\frac{1}{\mathbf{ln}(\frac{{\mathit{T}}_0}{{\mathit{T}}_{\mathit{c}}})}.$$

(66)

3.3.5. Results of the many-body interaction model

$$\frac{1}{4}{\mathit{\xi }}_{\mathit{G}}^2=\frac{9{\mathit{\epsilon }}_0}{{\mathit{e}}^2}\left(3{\mathit{k}}_{\mathit{B}}〈\mathit{T}〉-2{\mathit{U}}_{\mathit{B}}\right)\mathit{g}\mathit{f},$$

(28)

where

$$\mathrm{g}\mathrm{f}\equiv p_0\frac{1}{2}\frac{h^2}{m{\eta }^2}\alpha .$$

(20)

Of note, in eq. (20), considering the BE condensation and single-particle picture, $p_0\equiv 1$

$${U_B=E}_0={\left|\Delta \right|}_0=-\frac{1}{2}\frac{h^2}{m{\eta }^2}\alpha \times \ln (\frac{N_A}{n_i}).$$

(18)

The doping variable is fixed as a constant only in the abovementioned equation.

$$\frac{N_A}{{2n}_i}\equiv 0.16 (optimal)$$

Note that, precisely, when ${\xi }_G=0$, it should be considered that the doping is the cross point of Tc-dome and T^*.

4. Results

First, Figure 7 shows the entire depictions of T_c, T^*, and T_o on doping because of analytical calculations. Generally, the agreements with the experiments are good. Moreover, in Figure 8, the result of theoretical calculations of the Hall coefficient RH. As shown, the lower doping, the higher RH, and the RH behave as non-linear on temperatures.

Because of the statistic equation for the many-body interactions, Figure 10 shows superconductivity state up to a critical temperature ~140 K. In this figure, the state that relative distance ${\xi }_G$ between two spherical shells (i.e. two macroscopic Bosons) considering the many-body interactions is under 0 indicates the superconductivity state. From the further temperatures higher than the critical temperature, the relative distance ${\xi }_G$ becomes much larger as a change of non-continuity. Obviously, a transition occurs at ~140 K. This result accurately agrees with the experiments such as [28]. Furthermore, Figure 9 shows a result of theoretical calculation for the electron specific heat coefficient. According to the experiments [29,30], the calculation values are valid; moreover, it takes a maximum at a higher doping.

Importantly, in Figure 10, the magnetic field interaction U_B in our statistic equation is substituted by the pseudo-gap energy at optimum doping. Thus, as many studies claim, the many-body interactions in terms of macroscopic Bosons are some of the reasons why high-T_c cuprates exhibit a considerably higher critical temperate. Moreover, for the determination of T_c, the pseudo-gap energy that is derived from the mass of a macroscopic Boson is crucial.

5. Discussion

5.1. Macroscopic Boson and high-T_c cuprates

We propose a particle describing high-T_c cuprates is not a normal hole but a macroscopic Boson, which is formed by the conservation of angular momentum in 2D and by rotational motion of a hole itself. The concept of a macroscopic Boson, as mentioned, provided a unique partition function; this partition function can explain every property in the anomaly metal phase. Moreover, the presence of this Boson gives substantial reason why high-T_c cuprates have significantly high critical temperature when considered with many-body interactions.

5.2. Anomaly metal phase and transition temperature T₀

Thus far, to understand the mechanism of a high-T_c cuprate, it was important to study the source of pseudo-gap energy. Although this is true, another important factor that should be understood is the source of the transition temperature T₀, which defines the anomaly metal phase appearance. As mentioned, all equations that describe the anomaly metal phase have the parameter T₀ and T_c. Therefore, the excessive focus on the origin of pseudo-gap energy made most researchers less careful of the source of the transition temperature T₀, and this attitude confused researchers when considering the mechanism.

5.3. Highlights of the process for the materials to undergo superconductivity

Let us review the process, which describes the mechanism from forming a macroscopic Bosons to undergoing BE condensation. First, high-T_c cuprate reaches the transition temperature T₀ with a lower or no refrigeration_. At this stage, because the wavelength of a hole along c-axis becomes longer than the width of a 2D CuO₂ surface, the net 3D disappears and the conservation of angular momentum forms a macroscopic Boson, which indicates the rotation of a hole producing a magnetic field energy. Thus, this magnetic field energy gives a mass of macroscopic Boson.

By further refrigeration, our established statistic equation results in the following:

1. Many-body interactions, including the magnetic field energy of macroscopic Bosons and Coulomb interactions, result in very short relative distance of two holes (i.e., the net coherence of ~1 nm) as a result of all the sphere shells being superposed. Note that, at this stage, the paring of two macroscopic Bosons indicates the pairing of two holes.

2. Simultaneously, two holes gain a strong combination of the Lorentz force because the relative kinetic energy among two holes becomes 0; note that all Cooper pairs take the identical energy and thus BE condensation is produced, which is the source of the Meissner effect.

Although the derivation of a macroscopic wave function inevitably results in the London equation using the GL equation [31]; herein, let us review the reason why the Meissner effect is derived by another approach, thus stressing the converged and constant phase ${\theta }_0$.

Under an applied magnetic field B (i.e., vector potential A), we can derive the Aharonov–Bohm (AB) effect [32] from the initial macroscopic wave function.

$${\psi }_A=\left|\psi \right|exp[({\theta }_0+\frac{2q}{ħ}\int Ads)j\},$$

(71)

where q, j, and ${\theta }_0$ denote the hole charge, imaginary unit and converged phase of the macroscopic wave function, respectively.

From eq. (71), it is derived that

$$({\theta }_0+\frac{2q}{ħ}\int Ads)=2n\pi ,$$

(72)

where n is the integer. Assuming n = 0, we have

$${\theta }_0=-\frac{2q}{ħ}\int Ads,$$

(73)

and considering center-of-mass motion,

$${\theta }_0=2k_{0}x,$$

(74)

Substituting eq. (74) in eq. (73) and differentiating both sides of eq. (73), we obtain

$${2k}_0=-\frac{2q}{ħ}A.$$

(75)

The probability density flow is then defined as follows:

$$j_s=q{\left|\psi \right|}^2\frac{ħk_0}{m},$$

(76-1)

$$\int {\left|\psi \right|}^{2}dv=1.$$

(76-2)

Substituting eq. (75) in eq. (76-1), we derive the following London equation:

$$j_s=-q^2{\left|\psi \right|}^2\frac{1}{m}A.$$

(77)

This is the identical result from approaches by the GL equation [36].

5.4. The reason why high-T_c cuprates have significantly high critical temperature

As mentioned, an attractive force is the Lorentz force when two charged particles have no relative kinetic energy. However, as shown in Figure 11, this concept can be satisfied in s-wave pair and d-wave pair. Considering this schematic, the pair symmetry of high-T_c cuprates is not very important. Rather, it is crucial to focus on an irregular many-body interactions in high-T_c cuprates with an explanation of the significantly high critical temperature.

As per the model employed to handle many-body interactions in terms of charged particles, it is normally impossible for two particles to take their relative distance shorter than ~10^-7 m. In this case, however, our employed equation in many-body interactions has magnetic field interaction $U_B$ in eq. (28) because of the presence of macroscopic Bosons (i.e. pseudo-gap energy) ad Coulomb interactions. Therefore, this fact renders relative distance between two macroscopic Bosons to be almost 0 up to a high temperature, which makes the net coherence of two holes become the order on the cell of a CuO₂ surface (i.e. ~1 nm). This fact indicates that the combination energy becomes very strong.

This is demonstrated as shown in Figure 10, which results in a critical temperature of ~140 K. Considering $U_B$ in eq. (28) in our model equation to handle many-body interactions is pseudo-gap energy, eq. (18), which is essentially equal to the mass of a macroscopic Boson, the parameter η [m] (i.e., radius of a Boson and order on a CuO₂ cell) determines the critical temperature. This parameter determines both a Debye temperature and a band gap. Thus, this fact does not contradict the critical current equation (40) in this review section or our previous study [1]. Furthermore, as per our derived statistic equation, the larger U_B is, the higher a critical temperature T_c, and actual high-T_c indicates that U_B is sufficiently large, which is caused by the fact that the parameter η [m] is sufficiently small, in addition to optimum doping.

In eq. (28), given the value of 0 for${ \xi }_G$, immediately the doping variable becomes fixed and the maximum critical temperature T_{c, max} is derived;

$$k_B{T}_{c,max}=\frac{2}{3}{U}_{B,0},$$

(78)

where $U_{B,0}$ indicates the pseudo-gap of eq. (18) for maximum doping.

The calculation of quantities by eq. (78) is shown in Figure 12. In this figure, the horizontal axis implies the η of the radius of a macroscopic Boson. This parameter indicates the unit cell order of the CuO₂ surface. An important point is that, considering the parameter η is proportional to the lattice constant and although every high-T_c cuprate has macroscopic Bosons, differences in lattice constants render their critical temperature to be variable. Thus, if the type of material among the high-T_c cuprates differs, then the critical temperature is different.

To conclude, the existence of a macroscopic Boson indicates that:

1)

It causes the anomaly metal phase in high-T_c cuprates.

2)

Irregular many-body interactions are caused by it, which results in a high critical temperature higher than LN₂.

Note that, if we consider electron-doping in a Mott insulator, carrier concentration dominates over the lattice concentration n_i considering local electrons at each lattice in the Mott insulator; thus, the sign of the function ln in eq. (18) of pseudo-gap energy (i.e., U_B in eq. (28)) is altered. Hence, the sign of U_B in eq. (28) becomes the opposite, which makes electron-doping unable to have a high critical temperature because, on the contrary, U_B would prevent the enhancement of critical temperatures T_c.

5.5. Image of Cooper paring of two holes when $T\le T_C$

Figure 13 is an image that a hole on 2D of CuO₂ cell takes a circle, which in turn becomes a macroscopic Boson. When two macroscopic Bosons are close to each other and when the relative velocity between the two holes is zero, these two holes take the identical and rotational velocity and take the identical angular frequency as shown in Figure 14. Therefore, when the attractive force principle is satisfied, in which the fact that relative velocity is zero is the source of an attractive force between them, the two holes take rotations, keeping the constant relative distance. This fact is represented in Figure 15. That is, these holes take parallel motions. This corresponds to the d-wave pairing.

5.6. Consideration of significances in this paper

We believe that this study is significant because:

1)

It clarified why high-T_c cuprates have actual high critical temperature higher than LN_2.

2)

It demonstrated that all puzzles, including the properties of anomaly metal phase reported in previous articles, have been attributed to the presence of a macroscopic Boson.

To date, multiple theoretical investigations were reported to explain the mechanism of high-T_c cuprates but most of them used numerical computing or fitting methods; however, a general understanding of how the mechanism worked was largely unclear. Therefore, we proposed a detailed explanation of the mechanism that has been proposed for a comprehensive understanding of high-T_c cuprates.

Anticipated results and spillover effects:

1)

The analytical and physical understanding of high-T_c cuprates described in this study will promote the search for and synthesis of new materials exhibiting higher critical temperature near room temperature than standard materials at any given pressure.

2)

>2) All fields of condensed matter physics rely on statistical methods. Therefore, pure analytical (not numerical) approaches can be applied to many-body interactions. Our model that handles many-body interactions will provide new results to unsolved problems in condensed matter physics. For example, the analysis of many-body interactions of magnetic quanta would solve the primary problems of physics and superconducting technologies such as analytical formulation of critical current density.

6. Conclusion

This study described theoretically high-T_c cuprate properties such as the transition temperatures on doping, Hall effect or electron specific heat coefficient on doping. Moreover, it established a novel model to handle general many-body interactions, which explained why the high-T_c cuprates exhibit a significantly high critical temperature.

In general, the derived resultant equations predicted values accurately agree with data from experimental studies with no numerical calculations and fitting methods.

As discussed in the Discussion, consider the summary of significances in the present study.

1)

It has uncovered the source of mysteries in high-T_c cuprates, i.e., the presence of a macroscopic Boson.

2)

It has succeeded in describing the anomaly metal phase with a pure theory, which has no fitting or numerical calculations and which agrees with experiments.

3)

It has established a new model to handle general many-body interactions; using this model, this study has clarified why high-T_c cuprates have considerably high critical temperatures.

The resistivity on lower doping in the anomaly metal phase is not discussed in this study. However, an equation for conductivity, which takes linearly temperature dependence (i.e., non-linearly resistivity), was obtained in the theoretical section of this study because the carrier concentration in eq. (43), which lineally depends on temperatures, indicates the conductivity. However, the non-lineally resistivity in the anomaly metal phase, which appears only on low doping and mobility from the experiments is unclear because it is directly related to superconductivity (i.e., resistivity = 0). Therefore, because it does involve macroscopic Bosons, magnetic flux quanta and I-V characteristic, the subject is complex. Thus, we expect additional investigations on the subject involving magnetic flux quanta and critical current density in future.