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

2. Theory

Let us consider a theoretical structure briefly. This theory is based on the emergence of a Macroscopic Boson under the transition temperature T₀. Moreover, the mass of this Boson is related to the pseudo-gap temperature T^*. Then for calculations of the critical temperature T_c, there are two ways in the derivation: the one comes from the combination of the macroscopic Bosons (i.e., Cooper pairs) and the other is originated from handling many-body interaction of the macroscopic Bosons. However, this paper will discuss that these two ways are essentially same. Afterwards, this paper will consider the anomaly metal phases based on the partition function of the Bosons. The common point is that every property in the cuprates is represented only by the assumption of the macroscopic Bosons.

Moreover, as a material, the present theory assumes that

i)

it has a CuO₂ 2-dimentional surface, and

ii)

its carries are holes (i.e., electron carries are not handled in this paper).

2.1. Introduction of New Particle and Pseudo-Gap Relating to the New Particle

2.1.1. Introduction of a Macroscopic Boson

When considering a CuO₂ surface as the most important point [25] 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 imply a phase transition T₀, which will be described later in this paper. This is related to the electron nematic phase [26]. 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 [27]. Moreover, this fact corresponds to the fact that, in a CuO₂ surface, a local persistent current exists [26].

2.1.2. Calculating the Mass of a Macroscopic Boson

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

$$2\mathrm{U}={\displaystyle \frac{1}{2}}I{\Phi }_0-\overrightarrow{{\mu }_s}·\overrightarrow{H}={\displaystyle \frac{1}{2}}I{\Phi }_0-{\mu }_{s,z}{H}_z$$

(1)

where I, Φ₀, $\overrightarrow{{\mu }_s}$ and $\overrightarrow{H}$ denote a current surrounding a macroscopic Boson, the magnetic flux of a macroscopic Boson having a unique value, the spin magnetic moment of a hole, and the generated magnetic field in the macroscopic Boson. Note that it is assumed that the magnetic field direction is along the z-axis. Moreover, if we do not substitute the more concrete equation of ${\mu }_{s,z}$, the second term is formed generally in terms of the electromagnetism. If we assumed that the particle is a fermion, then ${\mu }_{s,z}$ would have both the positive and the negative numbers. However, as will be mentioned later, at a temperature T₀, the identical and uniform particles are emerged. This implies that the uniformed energy U requires that ${\mu }_{s,z}$ be zero. Therefore, the particle must be a boson.

Considering the same reason, the flux must be quantized as

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

(2)

where h and e denote the Planck constant and the charge of a hole, respectively. Note that this study used both the constants h and ħ as Planck constants.

In the current of Equation (1), the cyclotron angular frequency is introduced.

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

(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, Equation (1) becomes

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

(4)

where considering the flux of Equation (2)

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

(5)

where η is the approximated radius of a macroscopic Boson.

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

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

(6)

2.1.3. Spin of a Macroscopic Boson

Let us now consider the spin of a macroscopic Boson. As mentioned, Equation (1) claims that the second term of this equation must be zero. This implies that the direction of spins of a hole and that of the magnetic field is perpendicular. Thus, if the direction of the magnetic field is assumed to be z-axis, then the directions of hole spins must be x-or y-axis.

Therefore, using the Pauli matrix, this case considers only the x- and y-components.

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

(7-1)

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

(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:

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

(8-1)

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

(8-2)

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

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

(9)

Next, let us consider the orbital angular momentum l:

$$l_z={\eta }^2{m}_h{\omega }_c,$$

(10)

where

$${\omega }_c\equiv {\displaystyle \frac{eB}{m_h}},\mathrm{and}$$

(11)

m_h is the mass of a hole, and η is the radius of a macroscopic Boson.

Thus,

$$l_z={\eta }^{2}eB.$$

(12)

The magnetic flux is provided as

$${\Phi }_B={\displaystyle \frac{h}{e}}=\pi {\eta }^{2}B.$$

(13)

Accordingly,

$$l_z=2ħ.$$

(14)

To conclude, the net spin j is

$$\mathit{j}=0·\mathit{ħ}+2\mathit{ħ}=2\mathit{ħ},$$

(15)

which implies the macroscopic Boson is a boson.

The above result indicates that, although a single hole behaves as a Fermion, this macroscopic Boson on 2 D behaves as a boson. Thus, the name of this particle is derived from this fact. As described, as long as considering 2D, the statistic property would alter.

2.1.4. Obtain the Partition Function of Macroscopic Bosons

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

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

(16)

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,

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

(17)

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:

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

(18)

where the absolute value of the second term must dominate the value of the first term because the chemical potential is negative, and where $N_A$ denotes accepter concentration. Moreover, ${\displaystyle \frac{N_A}{{2n}_i}}$ indicates a doping parameter in this study. The number 2 is attached because of the presence of spins. Therefore, because n_i indicates the concentration of lattices, ${\displaystyle \frac{N_A}{n_i}}$ of the function 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 (That is, if the carrier is an electron, ${\displaystyle \frac{N_A}{n_i}}$ of the function ln is more than the value of the number 1 in the Mott insulator). Using the equation above, the partition function, Equation (17), is translated as follows:

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

(17-2)

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.

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

(19)

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

(20)

where $D_2\left(E\right)$, 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 (20) is simply because the energy state density in 2D is constant as indicated in Equation (19) and because partition function f_r is represented by Equation (17-2). In the process of this calculation of Equation (20), an energy E₀ appears as follows:

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

(21)

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:

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

(22-1)

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

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

(22-2)

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:

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

(23)

In Equation (23), 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.

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

(24)

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 [26]. 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. Note that it is assumed that the number N is sufficiently large and thus we can consider the differential dN.

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

$$(\mathrm{Coulomb} \mathrm{interaction} \mathrm{energy} \mathrm{and} \mathrm{magnetic} \mathrm{field} \mathrm{interaction} \mathrm{energy}) = {\displaystyle \frac{3}{2}}{k}_B{T}_i\times dN$$

2.2.2. Calculate the Principal Equation of Our Model and the Internal Quantum State

Calculate the abovementioned principle-equation in the previous subsection. First, dN is represented as follows:

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

(25)

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:

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

(17-2)

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

(19-2)

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

(19)

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

(24)

Thus, fg is given as follows:

$$fg=p_0{\displaystyle \frac{1}{2}}{\displaystyle \frac{h^2}{m{\eta }^2}}\alpha .$$

(26)

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

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

(27)

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:

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

(28)

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

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

(29)

Consequently, the resultant equation is provided by

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

(30)

As shown in Figure 3, the central macroscopic Boson behaves under the model of the infinite well-potential. Thus, as every elementary quantum mechanics textbook [e.g.,[28]] describes, the eigenvalue and wave function of it are presented as follows:

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

(31)

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

(32)

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, Equation (32), indexed by i and j becomes equal. That is,

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

(33)

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, Equation (31), 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. This physical picture is described as that one-unit cell has two-combined macroscopic Bosons and that, over the entire 2D surface, these unit cells having two macroscopic Bosons become uniform. Thus, by BE condensation, it can be assumed that the particle number is one and that $dN\to 0$. Therefore, the established shells vanish and thus the radius of a₀ →η, which implies the rest interactions will be only for the neighboring particles, not many-body interactions.

As mentioned, the occurrence of BE condensation and forming two-combined Bosons are equivalent. Thus, a critical temperature can be obtained both by the way of the combination of the particles and by the approach of the many-body interactions.

Employing the abovementioned equation (30), an equation of the relative distance between sphere shells ξ_G for temperature T is derived as follows:

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

(34)

where U_B is substituted with pseudo-gap ${\left|\Delta \right|}_0$ in Equation (24). Note that, in this equation, considering BE condensation and single-particle picture, p₀ of gf in Equation (26) 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 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

Note that this section is the review section for Ref. [1]

Herein, we would like to note the reason why there are Fermi energy and chemical potential E_F [29] in this paper. At this stage that two-macroscopic Bosons take a combination at a unit cell, although we cannot consider many-body interactions, there is a non-zero temperature, which is originated from the interaction for the neighboring Cooper pairs (i.e., $\mathrm{T}\le T_c$). This is equivalent to the fact that the neighboring holes in the Cooper pairs between unit cells collide with each other, which implies that each Cooper pair is repeatedly formed and destroyed [1]. 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]:

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

(35)

In this equation, the Fermi energy in a p-type material [30] is employed as follows:

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

(36)

Note that we are considering the carrier is a hole.

In this equation, a superconducting energy gap is introduced.

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

(37)

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

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

(38)

and

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

(39)

where

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

(40)

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 was derived. Note that interactions between Cooper pairs and Cooper pairs is assumed to be relatively small (i.e., that is merely local and neighboring collisions), which justifies the employment of the state equation.

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–d, 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:

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

(41)

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:

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

(42)

where $u_0$ denotes an integral constant.

$$\alpha \equiv -{\displaystyle \frac{2q^2{\pi }^2{\mu }_0ħ}{m^2}}\mathrm{l}\mathrm{n}(\xi )\times sin\theta cos\phi ,$$

(43)

where ξ denotes the coherence of a Cooper pair.

$$u=\alpha \beta {\left|\Psi \right|}^2+u_0.$$

(44)

It is assumed that $\left|\Delta \right|$ is proportional to T_c. That is, $\left|\Delta \right|$ is proportional to the energy u.

$${\alpha }^{\prime }\left|\Delta \right|=u={\displaystyle \frac{1}{2}}{k}_B{T}_c,$$

(45)

$${\alpha }^{\prime }\left|\Delta \right|=\alpha \beta {\left|\Psi \right|}^2+u_0.$$

(46)

Therefore, the relationship between α and ${\alpha }^{\prime }$ is a linear dependence

That is,

$${\alpha }^{\prime }\equiv \rho \alpha .$$

(47)

According to the analysis of the dimension, the constant ρ has the unit [1/J].

Thus, we can give the equation that

$$\rho \equiv {\displaystyle \frac{1}{k_B{\theta }_D}},$$

(48)

where $k_B$ and ${\theta }_D$ denote the Boltzmann constant and the Debye temperature, respectively.

Equation (45) is squared:

$${\alpha }^{\mathrm{\text{'}}2}{\left|\Delta \right|}^2={\displaystyle \frac{1}{4}}{k}_B^2{T}_c^2.$$

(49)

That is,

$${\left|\Delta \right|}^2={\displaystyle \frac{k_B^2{T}_c^2}{4{\alpha }^{\prime 2}}}.$$

(50)

From Equation (39),

$${\displaystyle \frac{k_B^2{T}_c^2}{4{\alpha }^{\prime 2}}}={\left(k_B{T}_c\right)}^2\{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{T_c}}{\displaystyle \frac{\left|{\Omega }_B\right|}{R}}{\displaystyle \frac{1}{{\rho }_s}}\ln \left({\displaystyle \frac{N_A}{n_i}}\right)\}.$$

(51)

By the organization of the above equation and the substitution of ${\alpha }^{\prime }$,

$$T_c=-4{\left[{\displaystyle \frac{1}{k_B{\theta }_D}}\right(-{\displaystyle \frac{2q^2{\pi }^2{\mu }_0ħ}{m^2}}\mathrm{l}\mathrm{n}\left(\xi \right)\times sin\theta cos\phi \left)\right]}^2{\displaystyle \frac{\left|{\Omega }_B\right|}{R}}{\displaystyle \frac{1}{{\rho }_s}}\mathrm{l}\mathrm{n}\left({\displaystyle \frac{N_A}{n_i}}\right).$$

(52)

In this equation, the thermodynamics potential is replaced by that of zero temperature, ${\left|{\Omega }_B\right|}_0$ , and the integral constant $T_{c,m}$ is instead introduced. This implies that when $\left|{\Omega }_B\right|\to {\left|{\Omega }_B\right|}_0$, the difference is renormalized to $T_{c,m}$:

$$T_c=-4{\left[{\displaystyle \frac{1}{k_B{\theta }_D}}\right(-{\displaystyle \frac{2q^2{\pi }^2{\mu }_0ħ}{m^2}}\mathrm{l}\mathrm{n}\left(\xi \right)\times sin\theta cos\phi \left)\right]}^2{\displaystyle \frac{{\left|{\Omega }_B\right|}_0}{R}}{\displaystyle \frac{1}{{\rho }_s}}\ln \left({\displaystyle \frac{N_A}{n_i}}\right)-T_{c,m}.$$

(53)

where

$$T_{c,m}\equiv {\theta }_D\left[1\right]$$

(54)

and

$${\left|{\Omega }_B\right|}_0=p_{0}V,$$

(55)

where p₀ and V denote the pressure and volume, respectively.

Again,

$$T_c=-4{\alpha }^{\prime 2}{\displaystyle \frac{{\left|{\Omega }_B\right|}_0}{R{\rho }_s}}\ln \left({\displaystyle \frac{N_A}{n_i}}\right)-{\theta }_D,$$

(56-1)

where

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

(56-2)

$${\alpha }^{\prime }={\displaystyle \frac{1}{k_B{\theta }_D}}\alpha =-{\displaystyle \frac{1}{k_B{\theta }_D}}{\displaystyle \frac{2q^2{\pi }^2{\mu }_0ħ}{m^2}}\mathrm{l}\mathrm{n}(\xi )\times sin\theta cos\phi .$$

(56-3)

In this process, we added a Debye temperature θ_D and a net coherence ξ to the equations. Note that, to an integral constant T_c,m in Equation (53), 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, which 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.

$${\left|{\Omega }_B\right|}_0=p_{0}V={\displaystyle \frac{2}{5}}{E}_{F0},$$

(57)

where E_F0 denotes the Fermi energy at zero temperature. Moreover, here the volume V is assumed to be the unit, i.e., the number 1. Thus, the critical temperature becomes

$$T_c=-4{\left({\displaystyle \frac{1}{k_B{\theta }_D}}\right)}^2({{\displaystyle \frac{2q^2{\pi }^2{\mu }_0ħ}{m^2}}\mathrm{l}\mathrm{n}\left(\xi \right)\times sin\theta cos\phi )}^2{\displaystyle \frac{2F_{F0}}{5R{\rho }_s}}\ln \left({\displaystyle \frac{N_A}{n_i}}\right)-{\theta }_D.$$

(58)

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

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

(59)

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 Equation (42) for the superconducting energy gap, the probability density function is interpreted as follows:

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

(60)

which shows that the gap is anisotropic [31].

Note that we assume that ${\theta }_{D2}$ and E_F0 are not varied very much over kinds of the cuprates. 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 [32]. Note that the reason why the Fermi energy 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 again consider the relation between a general energy gap and temperature, as shown in Equation (38)

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

(38)

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:

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

(61)

where to the equation of ${\left|\Delta \right|}_0$ of Equation (24) in creating Equation (61), each physical parameter was substituted. That is, the physical parameters m, h, and α in Equation (24) 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 macroscopic Boson as particles. To obtain an equation for the temperature T₀, we consider first the concentration dependent on the magnetic fields. Next, derivations of the Hall-effect coefficient RH are discussed. The concentration 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. That is, 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 Equation (20), the calculation for energy integral, in turn, is actually conducted because we attempted to obtain the concentration dependent on temperature T.

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

(62)

where

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

(62-2)

Note that the second form of fr in Equation (17-2) is not employed here. Equation (62) is very important because the concentration n is proportional to the temperature T, which describes a property of the anomaly metal state. As mentioned, considering an energy –uB_e, the Boltzmann statics is represented as follows:

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

(63)

where n₀ is the concentration of holes and n implies that the concentration of the macroscopic Bosons whose orbital momenta are the same directions as that of the magnetic field B_e.

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

(64)

Using the above equation,

$$n=n_0\mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{T_0}{T}})\approx n_0[1-{\displaystyle \frac{T_0}{T}}]$$

(63-2)

Note that, in the above equation, ${\displaystyle \frac{T_0}{T}}\le 1$ is formed as the condition, because the concentrations n and n₀ must be positive. When $T\to \infty$, the number of holes is equal to that of the macroscopic Bosons, which implies that the net number of the macroscopic Bosons takes zero. This is equivalent to zero magnetic fields when considering Equation (63). However, after the refrigeration, in the boundary condition, ${\displaystyle \frac{T_0}{T}}\equiv 1$, obviously a transition temperature is defined, which implies the emergence of a macroscopic Boson.

Now we begin to calculate RH.

In Equation (63), the exponential function is approximated by the Maclaurin series.

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

(65)

In the above equation, the previously calculated concentration n, Equation (62), is applied.

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

(66)

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

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

(67)

Considering the above equation (67), we also derive the transition temperature T₀, Equation (64). In short, the transition temperature T₀ implies appearances of both a macroscopic Boson and anomaly metal phase

2.4.3. Implement the Formulation of T₀

To implement the formula T₀, it is necessary to obtain u and B_e in Equation (64). First, a magnetic moment u is generally defined as follows:

$$\mu =IS,$$

(68)

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 the magnetic flux of it should be quantized as h/e, the magnetic flux of a macroscopic Boson is as follows:

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

(69)

That is,

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

(69-2)

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

$$\eta \approx 1\mathrm{nm}$$

(70)

This implies that the magnetic field B₀ is universally constant.

Moreover, assuming that a magnetic field in 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:

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

(71)

Consequently, a magnetic moment u is derived as follows:

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

(72)

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

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

(73)

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

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

(74)

where Equation (17-2) is applied as fr.

However, without an external applied magnetic field, high-T_c cuprates generally achieve superconductivity. 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 [26].

In this case, because macroscopic Bosons are formed in 2D CuO₂ surface, the weak interactions between macroscopic Bosons can justify the abovementioned calculation. The appearance of temperature T in Equation (63) guarantees the existence of these weak interactions. (The actual calculations of Curie temperatures with complete consideration of many-body interactions are presented in the Appendix A of this study.)

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

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

(75)

As described later, this equation of T₀ and the formula of critical temperature T_c 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. In this equation, we introduce the following approximation to the general equation of RH.

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

(76)

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

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

(77)

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.

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

(78)

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.

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

(79)

Considering this, the approximated RH equation becomes

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

(80)

As reported in many studies [33], this derived equation of RH is proportional to ${\left({\displaystyle \frac{1}{T}}\right)}^2$.

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 (Equation (17)) for energy integrals is determined as follows:

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

(81)

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

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

(82)

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

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

(83)

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.

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

(84)

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:

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

(85)

2.6. Summary of the Logical Flow

(1)

First, assuming a macroscopic Boson, which is based on angular momentum conservation on a CuO₂ surface, its energy, the mass, and the net spin were 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 that the combination of a Cooper pair results in BE condensation.

(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 and the macroscopic Bosons appear was defined by the Boltzmann statistics: Combining the Boltzmann statistics, particle concentration was implemented. 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 derived RH equation was approximated, and electron specific heat coefficient γ was calculated.

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 constants 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

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

(59)

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 constants used except for concentrations.

3.3.2. How to Determine n_i and ρ_s

In Equation (59), ${\displaystyle \frac{\sigma }{n_{i2}}}$ is identical for ${\displaystyle \frac{N_A}{n_i}}$, 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 ${\displaystyle \frac{{\mathrm{N}}_{\mathrm{A}}}{2{\mathrm{n}}_{\mathrm{i}}}}$ 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

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

(86)

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 (59) uses the universal gas constant,

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

(87)

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] = ${10}^{-3} \left[m^3\right]$

Therefore, the concentration n_i is typically

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

(88)

(How to determine 1/ρ_s)

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

1/ρ_s should be determined by the constant concentration, Equation (88)

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

(89-1)

where x denotes dimensionless variable. To give Equation (89-1) the meaning, variable x is provided as

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

(89-2)

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.

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

(61)

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

(75)

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

3.3.4. Physical Results of the Anomalous Metal Phase

(Hall effect coefficient)

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

(80)

Of note, $N_0=1.1\times {10}^2$, for example. 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.

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

(72)

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

(69-2)

$${\displaystyle \frac{m}{\pi {ħ}^2}}\equiv p_0,$$

(19)

(Electron specific heat coefficient)

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

(85)

3.3.5. Results of the Many-Body Interaction Model

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

(34)

where

$$gf\equiv p_0{\displaystyle \frac{1}{2}}{\displaystyle \frac{h^2}{m{\eta }^2}}\alpha .$$

(26)

Of note, in Equation (26), considering the BE condensation and single-particle picture, $p_0\equiv 1$

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

(24)

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

$${\displaystyle \frac{N_A}{{2n}_i}}\equiv 0.16 \left(optimal\right).$$

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.

Let us consider how Figure 7 relations each transition to the existence of a macroscopic Boson: In the equation of T_c, considering that the coherence ξ is essentially equal to the radius η of a macroscopic Boson does relation the critical temperature T_c to the macroscopic Boson. Furthermore, to the Equation (38) ($T=-{\displaystyle \frac{{\left|\Delta \right|}^2}{k_B^2}}{\displaystyle \frac{1}{T_c}}{\displaystyle \frac{1}{\mathrm{l}\mathrm{n}\left(\frac{N_A}{n_i}\right)}}$), substituting the energy of a macroscopic Boson results in the temperature T^*. Additionally, considering Equation (63) ($n=n_0\exp \left(-{\displaystyle \frac{\mu B_e}{k_{B}T}}\right)$), the value of e^-1 provides the transition temperature T₀ (i.e., Equation (75)), which implies a creation of macroscopic Boson at that transition temperature.

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.

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 two the sphere shells being superposed. Note that, at this stage, the paring of two macroscopic Bosons indicates the pairing of two holes.

2. Simultaneously, considering the short relative distance, two holes gain a strong combination of the Lorentz force when 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 [37]; 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 [38] from the initial macroscopic wave function.

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

(89)

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

From Equation (89), it is derived that

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

(90)

where n is the integer, and considering center-of-mass motion,

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

(91)

Substituting Equation (91) in Equation (90) and differentiating both sides of Equation (90), we obtain

$${2k}_0=-{\displaystyle \frac{2q}{ħ}}A.$$

(92)

The probability density flow is then defined as follows:

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

(93-1)

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

(93-2)

Substituting Equation (92) in Equation (93-1), we derive the following London equation:

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

(94)

This is the identical result from approaches by the GL equation.

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 for 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 Equation (34) 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 Equation (34) in our model equation to handle many-body interactions is pseudo-gap energy, Equation (24), 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. 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 Equation (34), given the value of 0 for${ \xi }_G$, immediately the doping variable becomes fixed and the representative critical temperature T_{c, rep} is derived;

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

(95)

where $U_{B,0}$ indicates the pseudo-gap of Equation (24) for maximum doping. Thus, T_c implies the representative critical temperature.

The calculation of quantities by Equation (95) is shown in Figure 12. In this figure, the horizontal axis implies 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 Equation (24) of pseudo-gap energy (i.e., U_B in Equation (34)) is altered. Hence, the sign of U_B in Equation (24) 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 implies 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 almost 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)

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’ 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 Equation (62), which lineally depends on temperatures, indicates the conductivity. However, this equation employs the critical temperature T_c and thus, over the lower doping, in which the resistivity appears, this equation is not available because, over the lower doping, critical temperatures are not defined. Thus, this will be a natural follow-up.

Moreover, the theory this paper presented seems not to describe electron doping cuprates. Furthermore, very recently, 2-dimentional CDW phases dependent on doping were measured. Thus, it is necessary to research these facts as follow-ups. However, in Appendix B of this paper, a preliminary calculation of the 2-dimensional CDW will be presented.

Appendix A.1. Introduction

The purpose of this appendix A is to confirm the proposed new model to handle many-body interactions described in the main text by applying another physical phenomenon. For example, we now introduce transitions of ferromagnetic material, i.e., Curie temperature.

Before conducting an actual calculation, we will briefly discuss certain background information to understand the significance of this appendix. Concerning transition phenomena, many studies have been reported. In particular, the Ising model is the most famous and basic. According to our literature review, however, few studies exist, which accurately predicted that the transition temperatures agreed with the experimental data. Moreover, many statistic physics textbooks claim that the Ising model in 2D provides an equation of transition temperature but there is no known model in 3D. If we follow the existing theory, a calculation of transition temperature indicates the evaluation of exchange interaction. However, this interaction is quite abstract and thus it difficult to evaluate in every ferromagnetic material. A general formula to determine a transition temperature has not been obtained because the partition function considering many-body interactions cannot be mathematically calculated.

In this Appendix A, using our established model for many-body interactions, we predict the actual values of transition temperatures, which sufficiently agree with experimental values. These calculations do not involve any numerical calculation or fitting method. Here, we provide a new model for statistical physics considering many-body interactions.

Appendix A.2. Predictions of Curie Temperature Using Our Employed Model to Handle Many-Body Interactions

As shown in Figure A1, a magnetic moment $\stackrel{⃑}{\mu }$ is located in the center of a sphere shell dN at which the temperature is T_i. Similar to that of the main text, the following balanced relation holds:

$$(\mathrm{magnetic} \mathrm{field} \mathrm{interaction} \mathrm{from} \mathrm{magnetic} \mathrm{moments}) = {\displaystyle \frac{ 3}{2}}{k}_B{T}_{i}dN$$

(A-1)

In this equation, the left-hand side is given as follows:

$-\overrightarrow{\mu }·\overrightarrow{B}$As every basic text describes, a magnetic field B is represented as follows:

$$\overrightarrow{B}=-{\displaystyle \frac{{\mu }_0}{4\pi }}[{\displaystyle \frac{\overrightarrow{\mu }}{r^3}}-{\displaystyle \frac{3(\overrightarrow{\mu ·}\overrightarrow{r)}\overrightarrow{r}}{r^5}}],$$

(A-2)

where r is radius of the sphere shell dN.

Figure A1. A schematic of our model to apply a ferromagnetic material. Basically, the concept to handle many-body interactions is the same as the case presented in the main text. That is, force of expansion from the central magnetic moment $\overrightarrow{\mathit{\mu }}$ is balanced to force of compression from the immediately outer locations, which are equal to kinetic energies in the differential number dN. Note that this case does not include the magnetic field interaction using macroscopic Bosons. Calculating the balanced equation results in a statistic equation that involves many-body interactions.

Figure A1. A schematic of our model to apply a ferromagnetic material. Basically, the concept to handle many-body interactions is the same as the case presented in the main text. That is, force of expansion from the central magnetic moment $\overrightarrow{\mathit{\mu }}$ is balanced to force of compression from the immediately outer locations, which are equal to kinetic energies in the differential number dN. Note that this case does not include the magnetic field interaction using macroscopic Bosons. Calculating the balanced equation results in a statistic equation that involves many-body interactions.

In this equation, the first term indicates ferromagnetism, while the second term indicates antiferromagnetism. Because the present case is to handle a ferromagnetic material, we employ the first term. Moreover, the directions of two magnetic moments $\stackrel{⃑}{\mu }$ are assumed to be parallel, i.e., the scalar product between two $\stackrel{⃑}{\mu }$ is positive. Accordingly, the above equation becomes

$$-\overrightarrow{\mu }·\left[-{\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{\overrightarrow{\mu }}{r^3}}\right]={\displaystyle \frac{3}{2}}{k}_B{T}_{i}dN.$$

(A-3)

Moreover, as mentioned, dN is expressed as follows considering the volume element of the integral:

$${\displaystyle \frac{{\mu }_0}{4\pi }}{\left|\overrightarrow{\mu }\right|}^2{\displaystyle \frac{1}{r^3}}={\displaystyle \frac{3}{2}}{k}_B{T}_{i}dN={\displaystyle \frac{3}{2}}{k}_B{T}_i\times gfd\overrightarrow{k},$$

(A-4-1)

$$\mathrm{d}\overrightarrow{k}={\displaystyle \frac{1}{dv}}={\displaystyle \frac{1}{\frac{4\pi }{3}{r}^3}}.$$

(A-4-2)

Thus, an important equation is derived as follows:

$${\mu }_0{\left|\overrightarrow{\mu }\right|}^2={\displaystyle \frac{9}{2}}{k}_B{T}_{i}gf={\displaystyle \frac{9}{2}}{k}_B{T}_{i}g{\displaystyle \frac{1}{\exp \left(-\frac{E_i-E_F}{k_B{T}_i}\right)-1}}.$$

(A-5)

In this Bose statistic equation, E_i denotes the zero-point energy of phonon, i.e., the Debye temperature and $E_F$ is a chemical potential, which is equal to the Gibbs free energy, but especially in this case implies only an internal energy. Therefore, this chemical potential is derived from electron specific heat coefficient γ as follows:

$${\mu }_0{\left|\overrightarrow{\mu }\right|}^2={\displaystyle \frac{9}{2}}{k}_{B}g{\displaystyle \frac{T}{\exp \left[-\frac{1}{k_{B}T}\left(\frac{3}{2}{k}_B{\theta }_D+\gamma T^2\right)\right]-1}}.$$

(A-6)

In this case, a transition temperature of T_c is assumed to be obtained by considering the extremum from this equation. Hence, to calculate differentials, T_i is considered to be a variable continuous temperature T because there are now no dependent parameters on the index i except T_i. Therefore, the following equation is calculated.

$${\displaystyle \frac{d}{dT}}{\mu }_0{\left|\overrightarrow{\mu }\right|}^2=0.$$

(A-7)

Consequently, the following equation is obtained:

$${\displaystyle \frac{\gamma T^2-\frac{3}{2}{k}_B{\theta }_D}{k_{B}T}}=-1,$$

(A-8-1)

$$\mathrm{T}\equiv T_c=-{\displaystyle \frac{k_B}{2\gamma }}+\sqrt{{\displaystyle \frac{3k_{B{\theta }_D}}{2\gamma }}}\approx \sqrt{{\displaystyle \frac{3k_{B{\theta }_D}}{2\gamma }}}.$$

(A-8-2)

Table A1 lists the physical constants of a ferromagnetic metal Fe.

Table A1. Fe physical constants.

Debye temperature ${\theta }_D$	470 K
Electron specific heat coefficient γ	$8.4\times {10}^{-27}$J/K2

Table A1. Fe physical constants.

Debye temperature ${\theta }_D$	470 K
Electron specific heat coefficient γ	$8.4\times {10}^{-27}$J/K2

Employing these physical constants, the transition temperature T_c for the metal Fe is calculated as follows:

$$T_c\approx 1.08\times {10}^3 \mathrm{K}.$$

(A-9)

Because measurements of the transition report 1043 K, the agreement is sufficient.

Then, we consider the transition temperature of the ferromagnetic Ni. The material Ni has much less thermal conductivity, unlike the metal Fe. This indicates that a chemical energy, i.e., the internal thermal energy is allowed to be ignored. Thus, from Equation (A-8-1), the T_c-equation is simply expressed as follows:

$${\displaystyle \frac{3}{2}}{k}_B{\theta }_D\approx k_B{T}_c.$$

(A-10)

Because the Debye temperature of Ni is reported as 450 K, T_c is calculated as follows:

$$T_c\approx 675 \mathrm{K}.$$

(A-11)

Compared with a measured transition value 627 K, the agreement can be considered to be sufficient.

Appendix B.1. Introduction

Very recently, in the phase diagram in the cuprates, a 2-dimentional Charge Density wave (CDW) transition was found [39]. Although some are mentioning that the recent phase diagram has been more complex, considering the existence of the macroscopic Boson will result relatively easily in the CDW transition. For example, this CDW appears for a parallel magnetic field for the c-axis [40,41], which indirectly implies the existence of a macroscopic Boson. Because of the existence of the macroscopic Boson, now we consider only 2-dimentional CDW. This appendix B would like to derive this CDW transition.

Appendix B.2. Calculations

The CDW is, in short, a stational wave, and thus we employ Figure 3 and its equations in the main body.

The eigenvalue of the stational wave in Figure 3 is again

$$E_i={\displaystyle \frac{1}{2M}}{\left({\displaystyle \frac{ħ\pi i}{2a_i}}\right)}^2.$$

(B-1)

Using equation, the following assumption is provided.

$$M{c}^2\equiv k_B{T}^{*}$$

(B-2)

where M is the mass of a macroscopic Boson and c is the seed of light.

In this appendix, the parameter a_i is assumed to be

$$a_i\equiv 1.6\times {10}^{-6}\mathrm{m}.$$

(B-3)

The above E_i is a merely energy level, different from two energy levels. Thus, as mentioned in the main body, the fine structure constant α is used to derive a gap of the CDW.

$$k_B{T}_{CDW}\equiv \alpha E_i=\alpha {\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{ħi\pi }{2a_i}}\right)}^2({{\displaystyle \frac{k_B{T}^{*}}{c^2}})}^{-1},$$

(B-4)

In the above equation,

$$i\equiv N_0,$$

(B-5)

where

$$N=N_0{f}_r.$$

Thus,

$$T_{CDW}=\alpha {\displaystyle \frac{1}{2k_B}}{\left\{{\displaystyle \frac{\pi ħ}{2a_i}}N[{-\mathrm{l}\mathrm{n}\left({\displaystyle \frac{N_A}{n_i}}\right)]}^{+1}\right\}}^2\times {\left({\displaystyle \frac{k_B{T}^{*}}{c^2}}\right)}^{-1}$$

(B-6)

In this paper, the phase of the CDW is defined as total macroscopic Bosons are condensed on the condition that Equation (B-2) is formed. Thus, N=1 is assumed.

In Figure A2, the result is shown, which well agrees with the experiments [39]. Note that the doping of the maximum is 0.12 (i.e., 1/8), which is also agreed [42,43]. Note that, for the method to create this figure, refer to Method section in the main body. Moreover, the result is not a numerical or fitting calculation.

Note that, importantly, in the superconductivity, the total macroscopic Bosons are condensed but the difference in the CDW is the condensation having the condition Equation (B2). This implies that the macroscopic Bosons are condensed in the CDW phase but they do not take a Cooper pair.