RETRACTED: Fractional Model of Electron–Phonon Interaction

Abstract

Based on the derivation of the equation of state for systems with a fractional power spectrum, the relationship between the van der Waals constant and the fractional derivative order has been established. The fractional model of electron–phonon interaction has received additional consideration, which may be pertinent when interpreting the experimental results. This model is valuable for describing superconductivity at high temperatures because it predicts relatively large values for the electron–phonon interaction constant.

1. Introduction

The concept of so-called fractional calculus, which is founded on the mathematical technique of fractional order integro-differentiation, has permeated physics, which has a growing interest in systems with a non-quadratic dependence of energy on momentum [1,2,3,4,5]. Systems described by fractional order differential equations exhibit fractional-power nonlocality and deviate from the quadratic dependence of energy on momentum. Specifically, such a deviation results in fractional-power probability distributions. These distributions exhibit a remarkable degree of stability [6,7].

Researchers’ interest in systems with a Hamiltonian containing a non-quadratic momentum power has increased recently. Due to the non-ideal nature of real-world systems, quadratic dependences of energy on momentum are rarely observed. As a result of the non-quadratic dependence of energy on momentum, some fairly intriguing results are observed. As previously established by Kadanoff, in the case of a non-quadratic dependence of electron energy on momentum [8], the time of electron revolution in a magnetic field and, consequently, the resonant frequency depends on the applied electric field. This effect, as noted by Kadanoff, can be used to investigate the structure of iso-energetic surfaces in semiconductors. When taking into account high-frequency electromagnetic phenomena in semiconductors with a non-quadratic dispersion law for current carriers, intriguing results were also obtained [9]. In addition, the electronic unpredictability of a one-dimensional superlattice was taken into account [10]. Due to the non-quadratic dispersion law, unpredictability existed. In conclusion, it was shown that the divergence from quadratic dependence causes the waves to decelerate down [11]. This was accomplished by analysing nonlinear electromagnetic waves in a stochastic electron plasma with a non-quadratic energy-to-momentum dependence.

Dyson’s works on phase variations in ferromagnets included consideration of nonlocal interactions of the power type [12,13]. In addition, fractional kinetic equations for these nonlocal interactions in a crystal lattice have been discovered [14]. As non-integer power of momentum Hamiltonian systems was examined, the characteristics of fractional quantum discovered were taken into account [15].

Notably, graphene [16], a recently discovered two-dimensional modification of carbon atoms, is a system with a Hamiltonian having a non-square momentum power. Indeed, the graphene electron dispersion law is linear [17].

In-depth examination of a potential method for introducing and physically interpreting fractional integro-differentiation in quantum statistical physics rests on the concept of a fractional power dependence of energy on momentum [15]. This concept is also developed in the current endeavour. The two main achievements of the present study are the relationship between the van der Waals constant and the fractional derivative order and predicting rather large values for the electron–phonon interaction constant, which may be useful for describing superconductivity at high temperatures.

2. Equation of State for Systems Having a Fractional-Power Spectrum

The starting point for this study is the fractional equation for the Green’s function $\mathfrak{G}$ in the following form [18]:

$$\left[i\frac{\partial }{\partial t}-{\mathit{\nabla }}_x^{\nu }\right]\mathfrak{G}(x,x^{*})=\delta (x-x^{*})$$

(1)

where $x$ means $s\tau$ and $x^{*}$ means $-s\tau$. Here and below, it is assumed that $\hslash =2m=1$, and the dimensionless time variables $\tau =t/t_0$ and coordinates $s=r/r_0$ are used, while $t_0$ and $r_0$ are some characteristic time- and length-scales of the system, respectively. ${\mathit{\nabla }}_x^{\nu }$ is the fractional Riesz derivative defined as [1].

$${\mathit{\nabla }}_x^{\nu }f(x)=\frac{1}{\mathsf{\Gamma }(2-\nu )\mathrm{c}\mathrm{o}\mathrm{s}\left[\frac{\pi }{2}(2-\nu )\right]}\frac{{\partial }^2}{\partial x^2}{\int }_{-\infty }^{+\infty }\frac{f\left(\xi \right)d\xi }{|x-\xi {|}^{\nu -1}}$$

(2)

where $1<\nu \le 2$. The conditions under which Equation (1) holds have been similarly formulated in previous studies [15]. The system described by Equation (1) has a fractional power spectrum.

$$\omega =|p{|}^{\nu }$$

(3)

where p is a dimensionless variable related to the dimensional one by relation $p=q/q_0$ and $q_0$ is some characteristic momentum of the system. In what follows, the system of units in which $\frac{q_0^2}{2m}=q_0=1$ is used.

The equation of state for a system with spectrum given by Equation (3) has been obtained in the form [15].

$$P=\frac{1}{\alpha }n+2\frac{1}{\alpha }{e}^{\alpha \mu }\int \left[1-e^{-\alpha (p^2-|p{|}^{\nu })}\right]e^{-\alpha |p{|}^{\nu }}\frac{d^{3}p}{(2\pi {)}^3}$$

(4)

where P is the pressure, p is the momentum, $\alpha =1/kT$, μ is the chemical potential, and n is the ideal gas concentration, i.e., gas with spectrum given by Equation (3) at $\nu =2$.

It can be observed from Equation (4) that taking into consideration the momentum power function in the dispersion law results in the appearance of additional pressure in the equation of state, i.e., it accounts for the interaction between particles. This additional pressure is determined by a distribution function that is essentially a Weibull distribution (stretched exponent), which is intimately tied to the fractal features of the system [6,7]. This additional pressure is zero at $\nu =2$, and the equation of state for an ideal gas is obtained.

Consider Equation (4) in more detail to make some general remarks about the nature of this interaction. Equation (4) is written as:

$$P-P_{\nu }=\frac{N}{V}kT$$

(5)

where $P_{\nu }=2\frac{1}{\alpha }{e}^{\alpha \mu }\int \left[1-e^{-\alpha (p^2-|p{|}^{\nu })}\right]e^{-\alpha |p{|}^{\nu }}\frac{d^{3}p}{(2\pi {)}^3}$, is extremely similar to the van der Waals equation, which does not, however, account for the finiteness of particle sizes. This is because the considered approximation does not account for collisions or the fact that particles cannot pass through one another. If one disregards the impacts of particles with limited sizes, the van der Waals equation can be expressed as [19]:

$$P-\frac{N^{2}a}{V^2}=\frac{N}{V}kT$$

(6)

Comparing Equation (6) with Equation (5) yields:

$$n^{2}a=2\frac{1}{\alpha }{e}^{\alpha \mu }\int \left[1-e^{-\alpha (p^2-|p{|}^{\nu })}\right]e^{-\alpha |p{|}^{\nu }}\frac{d^{3}p}{(2\pi {)}^3}$$

(7)

In order to obtain an expression for the chemical potential, one can resort to an approximation, namely, use the expression for the chemical potential of a Boltzmann ideal gas. This approximation is justified in many cases [19]. The chemical potential of the Boltzmann gas, up to insignificant constant values, is the value equal to [19].

$$\mu =\frac{1}{\alpha }\mathrm{l}\mathrm{n}\left[\frac{N}{V}{\alpha }^{3/2}\right]$$

(8)

From Equation (8), $n={\alpha }^{-3/2}{e}^{\mu \alpha }$. Then, upon dividing Equation (7) by n:

$$na=2\sqrt{\alpha }\int \left[1-e^{-\alpha (p^2-|p{|}^{\nu })}\right]e^{-\alpha |p{|}^{\nu }}\frac{d^{3}p}{(2\pi {)}^3}$$

(9)

In order to perform precise calculations, it is required to utilise dimensional quantities. In the system of units, in which $q_0^2/2m\ne 1,q_0\ne 1,\hslash \ne 1$, $P_{\nu }$ is written as:

$$P_{\nu }=2\frac{q_0^3}{\alpha }{e}^{\alpha \mu }\int \left[1-e^{-\alpha \frac{q_0^2}{2m}(p^2-|p{|}^{\nu })}\right]e^{-\alpha \frac{q_0^2}{2m}|p{|}^{\nu }}\frac{d^{3}p}{(2\pi \hslash {)}^3}$$

(10)

Additionally, in dimensional variables:

$$P_{\nu }=2\frac{1}{\alpha }{e}^{\alpha \mu }\int \left\{1-e^{-\alpha \frac{q_0^2}{2m}[(q/q_0{)}^2-(|q|/q_0{)}^{\nu }]}\right\}{e}^{-\alpha \frac{q_0^{2-\nu }}{2m}|q{|}^{\nu }}\frac{d^{3}q}{(2\pi \hslash {)}^3}$$

(11)

Then, for the van der Waals constant, follows the expression relating it to the parameter $\nu$:

$$n^{2}a=2\frac{1}{\alpha }{e}^{\alpha \mu }\int \left\{1-e^{-\alpha \frac{q_0^2}{2m}[(q/q_0{)}^2-(|q|/q_0{)}^{\nu }]}\right\}{e}^{-\alpha \frac{q_0^{2-\nu }}{2m}|q{|}^{\nu }}\frac{d^{3}q}{(2\pi \hslash {)}^3}$$

(12)

The chemical potential of the Boltzmann gas in dimensional variables has the form [19]:

$$\mu =\frac{1}{\alpha }\mathrm{l}\mathrm{n}\left[\frac{n}{2}{\left(\frac{2\pi {\hslash }^2}{m}\alpha \right)}^{3/2}\right]$$

(13)

From Equation (13), $n=2{\left(\frac{2\pi {\hslash }^2}{m}\alpha \right)}^{-3/2}{e}^{\mu \alpha }$. Then, upon dividing Equation (12) by the latter expression for n and keeping the second n in n² intact:

$$\begin{array}{cc}a& =\frac{\sqrt{\alpha }}{n}{\left(\frac{2\pi {\hslash }^2}{m}\right)}^{3/2}\int \left\{1-e^{-\alpha \frac{q_0^2}{2m}[(q/q_0{)}^2-(|q|/q_0{)}^{\nu }]}\right\}{e}^{-\alpha \frac{q_0^{2-\nu }}{2m}|q{|}^{\nu }}\frac{d^{3}q}{(2\pi \hslash {)}^3}\\ & =\frac{1}{n\alpha \nu \sqrt{{\pi }^3}}\left[{\left(\frac{q_0^2\alpha }{2m}\right)}^{\frac{6-3\nu }{2\nu }}\mathsf{\Gamma }\left(\frac{3}{\nu }\right)-\mathsf{\Gamma }\left(\frac{3}{2}\right)\right].\end{array}$$

(14)

Equation (14) connects the van der Waals constant, which determines the interaction between particles, to the parameter $\nu$. Considering the possibility of characterising real systems using the approach of fractional integro-differentiation, which is carried out by applying Equation (1) to the spectrum given by Equation (3), this relationship is crucial.

One can go as follows to gather information about the interaction potential of gas particles. It was proven that the equation of state describes the system of interacting fermions in the Hartree–Fock approximation [20,21,22].

$$P-\frac{1}{2}{n}^{2}v=\frac{N}{V}kT$$

(15)

where $v=\int U\left(r\right)dr$ is the self-consistent field of all particles and $U\left(r\right)$ is the potential energy of particle interaction. Comparing Equation (15) with Equation (4), and using Equation (14), yields the following relations in dimensional variables:

$$\frac{1}{2}{n}^{2}v=\frac{n}{\nu \sqrt{{\pi }^3}}\left[{\left(\frac{q_0^2\alpha }{2m}\right)}^{\frac{6-3\nu }{2\nu }}\mathsf{\Gamma }\left(\frac{3}{\nu }\right)-\mathsf{\Gamma }\left(\frac{3}{2}\right)\right]$$

(16)

or

$$v=\frac{2}{n\nu \sqrt{{\pi }^3}}\left[{\left(\frac{q_0^2\alpha }{2m}\right)}^{\frac{6-3\nu }{2\nu }}\mathsf{\Gamma }\left(\frac{3}{\nu }\right)-\mathsf{\Gamma }\left(\frac{3}{2}\right)\right]$$

(17)

or

$$\int U(r)dr=\frac{2}{n\nu \sqrt{{\pi }^3}}\left[{\left(\frac{q_0^2\alpha }{2m}\right)}^{\frac{6-3\nu }{2\nu }}\mathsf{\Gamma }\left(\frac{3}{\nu }\right)-\mathsf{\Gamma }\left(\frac{3}{2}\right)\right]$$

(18)

This final equation establishes a relationship between the self-consistent field and the parameter $\nu$.

In conclusion, it is important to emphasise one fascinating point. As shown by Equations (16)–(18), the sign of the interaction energy is governed by the expression in brackets and, by extension, by the temperature $1/\left(k\alpha \right)$. The van der Waals constant is a positive quantity, indicating that the interaction between particles is repulsive, as shown by the formulas. As it turns out, this circumstance is not always the case. In fact, for this scenario to continue, the thermal energy $1/\alpha$ must be less than the characteristic energy $q_0^2/\left(2m\right)$ or $q_0^2\alpha /\left(2m\right)\ge 1$. This condition establishes the temperatures at which the preceding curves are valid, namely, $T\le \frac{q_0^2}{2mk}$.

If this criterion is not met, however, the system undergoes major change. This alteration consists mostly of a shift in the nature of particle interactions. It turns negative, indicating that particles are attracted. Physically, it appears that this behaviour can be explained as follows: express the typical momentum $q_0$ in terms of the characteristic length using the formula $q_0=\hslash /l_0$. Interaction between particles is proportional to the distance between them. The distance at which the sign of the interaction energy flips from positive to negative is denoted by $l_0$. At $r=l_0$, the interaction energy is plainly equal to zero. Such a circumstance occurs in real atomic and molecular gases [19,20,21]. The thermal energy determines which side of $l_0$ will have the greatest average distance between particles. At low temperatures, T, particles do not scatter across great distances, and the average distance between particles remains below the characteristic length, $l_0$; hence, the interaction energy is repulsive. At temperatures ${\hslash }^2/\left(2m{l}_0^{2}k\right)<T$, the opposite is observed.

3. Relationship between Energy, Pressure and Volume

Deriving the relationship between energy, pressure, and volume for a system with a fractional-power spectrum is of interest. Only the qualitative dependences of the quantities are determined in the section that follows. Therefore, minor constant factors have been omitted from the following formulas. Whenever it becomes necessary to keep permanent members, this shall be expressed explicitly. It was demonstrated that the density of states for a system with a fractional-power spectrum had the form [15].

$$\rho (\omega )=\frac{1}{\nu }{\omega }^{\frac{D}{\nu }-1}$$

(19)

where D is the dimension ($D=\mathrm{1,2},3$) of the phase space. The $\nu$ parameter is essential, so it will be preserved in all expressions. Consider a three-dimensional structure. Then:

$$\rho (\omega )=\frac{1}{\nu }{\omega }^{\frac{3}{\nu }-1}$$

(20)

The number of particles with energies in the range from $\omega$ to $\omega +d\omega$ is then the quantity.

$$dN=\frac{V{\omega }^{\frac{3}{\nu }-1}d\omega }{\nu exp\left[\alpha \right(\omega -\mu \left)\right]+1}$$

(21)

The quantities $\alpha$ and $\mu$ determine the system’s thermodynamic state; hence, they cannot be neglected. The spin constant is absent from the final formulation. Note that all the formulas presented in this section have identical forms for Fermi and Bose statistics, with the exception of the sign in the denominator of Equation (21). Therefore, the following expression is true for the Bose gas:

$$dN=\frac{V{\omega }^{\frac{3}{\nu }-1}d\omega }{\nu \mathrm{e}\mathrm{x}\mathrm{p}\left[\alpha \right(\omega -\mu \left)\right]-1}$$

(22)

Integration of Equation (21) renders the total number of particles in the gas with respect to energy:

$$N=\frac{V}{\nu }{\int }_0^{\infty }\frac{{\omega }^{\frac{3}{\nu }-1}d\omega }{\mathrm{e}\mathrm{x}\mathrm{p}\left[\alpha \right(\omega -\mu \left)\right]\pm 1}$$

(23)

Introducing the integration variable $\zeta =\alpha \omega$ yields:

$$n=\frac{N}{V}=\frac{1}{\nu {\alpha }^{3/\nu }}{\int }_0^{\infty }\frac{{\zeta }^{\frac{3}{\nu }-1}d\zeta }{\mathrm{e}\mathrm{x}\mathrm{p}(\zeta -\alpha \mu )\pm 1}$$

(24)

Equation (24) defines the chemical potential μ as a function of temperature $1/\alpha$ and density $N/V$. For a large Landau thermodynamic potential (grand potential), from the well-known expression [19], $\partial \Omega =\mu \partial N$, it follows:

$$\Omega =-\frac{V}{\nu \alpha }{\int }_0^{\infty }{\omega }^{\frac{3}{\nu }-1}\mathrm{l}\mathrm{n}\{1\pm \mathrm{e}\mathrm{x}\mathrm{p}[\alpha (\mu -\omega )]\}d\omega .$$

(25)

Integrating by parts:

$$\Omega =-\frac{\nu }{3}\frac{V}{\nu }{\int }_0^{\infty }\frac{{\omega }^{\frac{3}{\nu }}d\omega }{\mathrm{e}\mathrm{x}\mathrm{p}\left[\alpha \right(\omega -\mu \left)\right]\pm 1}.$$

(26)

Using expression (21), the total energy of the system is:

$$E=\int \omega dN=-\frac{V}{\nu }{\int }_0^{\infty }\frac{{\omega }^{\frac{3}{\nu }}d\omega }{\mathrm{e}\mathrm{x}\mathrm{p}\left[\alpha \right(\omega -\mu \left)\right]\pm 1}.$$

(27)

Expression (26) coincides with expression (27) up to the factor $-\nu /3$. Keeping in mind also that $\Omega =-PV$ follows:

$$PV=\frac{\nu }{3}E.$$

(28)

It is interesting to note that under the dispersion law, the value of the thermal energy coefficient is dictated by the degree of energy dependence on momentum. For $\nu =2$ follows the relation for the state of an ideal gas.

4. Fractional Model of Electron–Phonon Interaction

Using the same methodology, investigate the topic of electron–phonon interaction. The model that accounts for the electron–phonon interaction is based on the quadratic dependency of the energy on momentum and offers the dependence of the interaction amplitude on momentum in the form [23]:

$$A\left(q\right)\sim q^{1/2}.$$

(29)

However, this model does not take into account all the details and various mechanisms of the electron–phonon interaction [24], and, in order to improve the agreement with experimental data, this function is modified, namely, the degree of momentum dependence is changed already at a purely phenomenological level [15,24]. It is of interest to consider the fractional model of the electron–phonon interaction. In doing so, an elementary approach to this problem is used [23].

Consider a system of interacting ions and electrons. The ions are located at the sites of the crystal lattice, and the electrons are assumed to be free with a certain degree of accuracy. The starting point for consideration is the fractional equation for the Green’s function [see Equation (1)]. As was said, systems described by such equations have a spectrum given by Equation (3). Taking this into account, the total energy for such systems can be written as:

$$E=|p{|}^{\nu }+U\left(r\right),$$

(30)

where $U\left(r\right)$ is the potential energy of ions, or in dimensional variables.

$$E=\frac{q_0^{2-\nu }}{2m}|q{|}^{\nu }+U\left(r\right).$$

(31)

Typically, ions are portrayed as harmonic oscillators. When the momentum dependency becomes a fractional-power law, it is more probable that deviations from the harmonic rule will occur. Consider this problem in its general form, in accordance with a previous study in which the class of Hamiltonians in the below form has been considered [15].

$$E=\left|p{|}^{\nu }\right|r{|}^{{\nu }^{\prime }}+\left|r{|}^{\kappa }\right|p{|}^{{\kappa }^{\prime }}$$

(32)

Consider the case when ${\nu }^{\prime }={\kappa }^{\prime }=0$. Then, it follows the Hamiltonian:

$$E=|p{|}^{\nu }+|r{|}^{\kappa }$$

(33)

Taking this into account, the expression for the total energy can be rewritten in the form:

$$E=\frac{q_0^{2-\nu }}{2m}\left|q{|}^{\nu }+\frac{\epsilon }{2}\right|r{|}^{\kappa },$$

(34)

where $\epsilon$ is the elasticity constant. Taking into account Equation (34), the total energy of the wave of deviations of the lattice nodes from the equilibrium position can be represented as:

$$E=\int \frac{q_0^{2-\nu }}{2m}n{\left|m\frac{\partial {\xi }_q}{\partial t}\right|}^{\nu }dr+\int \frac{\epsilon }{2}{\left|\Delta {\xi }_q\right|}^{\kappa }dr.$$

(35)

The wave of deviations of nodes $\Delta {\xi }_q$ from the equilibrium position is related to the gradient by the relation.

$$\Delta {\xi }_q={\xi }_q\left(r+r_0\right)-{\xi }_q\left(r\right)=r_0\frac{\partial {\xi }_q}{\partial r}.$$

(36)

In the latter equation, the higher order terms are truncated.

With this in mind, it finally follows:

$$E=\frac{q_0^{2-\nu }}{2}n{m}^{\nu -1}\int {\left|\frac{\partial {\xi }_q}{\partial t}\right|}^{\nu }dr+\int \frac{\epsilon }{2}{\left|\frac{\partial {\xi }_q}{\partial t}\right|}^{\kappa }dr.$$

(37)

If the wave of deviations is represented in the form of a standard exponent, ${\xi }_q=A{\varphi }_q^{+}{e}^{-i\omega t+\frac{i}{\hslash }qr}$, where A is the amplitude of deviations, ${\varphi }_q^{+}$ is the phonon production operator, then:

$$E=\frac{q_0^{2-\nu }}{2}n{m}^{\nu -1}{A}^{\nu }{\omega }^{\nu }V+\frac{\epsilon }{2}{\left(\frac{q}{\hslash }\right)}^{\kappa }{A}^{\kappa }V.$$

(38)

Since the energy of the phonon is $\hslash \omega$, and $\hslash \omega =cq$ and $c^2=\epsilon /\left(nm\right)$, an expression that determines the dependence of the amplitude on the momentum follows:

$$cq=\frac{q_0^{2-\nu }}{2}n{m}^{\nu -1}{A}^{\nu }(cq{)}^{\nu }V+\frac{c^{2}nm}{2}{\left(\frac{q}{\hslash }\right)}^{\kappa }{A}^{\kappa }V.$$

(39)

In the case of a quadratic Hamiltonian, i.e., when $\nu =\kappa =2$, it follows the well-known result [24]:

$$A\left(q\right)=\frac{\hslash }{\sqrt{nmcqV}}.$$

(40)

Consider $\nu =\kappa$. In this case, from Equation (39), it follows a more general dependence of the amplitude.

$$A\left(q\right)=\frac{\hslash }{{\left[\frac{1}{2}\left(q_0^{2-\nu }{m}^{\nu -2}+c^{2-\nu }\right)\right]}^{1/\nu }}{\left(\frac{q^{1-\nu }}{mnV{c}^{\nu -1}}\right)}^{1/\nu }.$$

(41)

The amplitude of the electron–phonon interaction is determined by the amplitude of the wave of deviation ${\xi }_q$ of the perturbed density of lattice sites $n_q(r,t)$ from the equilibrium value n, $n_q(r,t)=n(r,t)-n\sim n$. The latter is proportional to the gradient $\partial {\xi }_q/\partial r$. Hence, (see Equation (8) in Ref. [25]):

$$n_q\left(r,t\right)=\frac{\mathrm{const}}{(mnV{c}^{\nu -1}{)}^{1/\nu }}{q}^{1/\nu }\left[{\varphi }_q^{+}\mathrm{e}\mathrm{x}\mathrm{p}\left(-i\omega t+\frac{i}{\hslash }qr\right)+\epsilon c\right].$$

(42)

Taking this expression into account, for the Hamiltonian of the interaction of the electron gas with the resulting lattice vibration, the following expression is obtained:

$$H_{e-ph}=\sum _{pq}\mathrm{const}\cdot q^{1/\nu }\left({ϵ}_{p-q}^{+}{ϵ}_p{\varphi }_q^{+}+{ϵ}_{p+q}^{+}{ϵ}_p{\varphi }_q^{+}\right),$$

(43)

which, supplemented by the Hamiltonian of free particles, is the well-known expression for the Fröhlich Hamiltonian [26,27], but with the amplitude of the electron–phonon interaction:

$$A\left(q\right)\sim q^{1/\nu }.$$

(44)

With the help of merely changing the parameter $\nu$ in Equation (41), it is possible to understand the outcomes of actual experiments.

It is worth noting here that a detailed derivation of the Hamiltonian for electron–phonon coupling given by Equation (43) with $\nu =2$ can be found in the work by Heid [25]. In the said derivation procedure, to obtain Equation (43), $n_q~q^{1/\nu }$ given by Equation (42), is to be used instead of $n_q~q^{1/2}$.

The theory of superconductivity is particularly interested in various forms of electron–phonon interaction. In fact, one crucial factor that controls the critical temperature of the transition to the superconducting state is the amplitude of the electron–phonon interaction. It is well known that the traditional Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity, which is an idealised model, is unable to accurately describe the properties of all traditional superconductors [28]. For instance, the BCS theory made serious mistakes when calculating the parameters of magnesium d-boride, an unusual compound with a very high temperature of transition. The BCS does not cover the situation of a strong electron–phonon contact, which is an issue. Some enhancements of BCS make it possible to find formulas for calculating the critical temperature that are more precise [28]. There have also been some unsuccessful attempts to use the spectrum provided by Equation (3) to characterise high-temperature superconductivity [29].

In light of the foregoing, it is concluded that it would be interesting to apply a fractional model of superconductivity to graphene and other substances, where the conditions of high-temperature superconductivity are realised. The parameter of this model would be a more general expression [see Equation (44)], and it would follow that the amplitude of the electron–phonon interaction at $1<\nu <2$ would be greater than in the case of $\nu =2$.

5. Conclusions

It is established that the fractional-differential approach has become a useful tool in contemporary physics for describing complex systems with a variety of characteristics. This is supported by numerous works that employ this methodology. In the near future, this technique will undoubtedly become one of the most practical ways to describe various classical and quantum systems.

In general, fractional operators permit taking into consideration the non-locality of the power-type interaction, which can be completed particularly easily using fractional Equation (3) and the available parameter $\nu$.

The main results of this study can be summarised as follows:

• A study was conducted on the equation of state for systems having a fractional power spectrum. The relationship between the van der Waals constant and the fractional derivative order is determined.
• Consideration is given to the fractional model of electron–phonon interaction, which may be relevant when interpreting the results of actual investigations. This model predicts rather large values for the electron–phonon interaction constant and, therefore, is useful for describing superconductivity at high temperatures.