Can Anyon statistics explain high temperature superconductivity?

TL;DR

This paper proposes that Anyon statistics can provide a plausible explanation for high temperature superconductivity phenomena, potentially offering new insights into the underlying physics.

Contribution

It introduces a novel approach by applying Anyon statistics to explain high temperature superconductivity, which is a departure from traditional theories.

Findings

Anyon statistics can model high temperature superconductivity phenomena

The approach offers a new perspective on the mechanisms behind superconductivity at high temperatures

Potential implications for designing new superconducting materials

Abstract

In this paper, we find a reasonable explanation of high temperature superconductivity phenomena using Anyon statistics.

Equations

$$\left|\Delta \right|=\left|V\right|\sum _k\frac{\left|\Delta \right|}{2\sqrt{(ϵ(k)-\mu {)}^2+{\Delta }^2}}\left[f(-E_K)-f(E_K)\right]\text{,}$$

$${\left(\frac{1}{(1-p)f(E_k)}+1\right)}^{1-p}{\left(\frac{1}{pf(E_k)}-1\right)}^p=e^{\frac{\sqrt{(ϵ(k)-\mu {)}^2+{\Delta }^2}}{k_{B}T}}\text{.}$$

$$f(E_K)=\frac{1}{1+e^{\frac{\sqrt{(ϵ-\mu {)}^2+{\Delta }^2}}{k_{B}T}}}\text{,}$$

$$1=\left|V\right|\sum _k\int \frac{1}{2\sqrt{(ϵ(k)-\mu {)}^2+{\Delta }^2}}\left[\frac{1}{1+e^{\frac{-\sqrt{(ϵ(k)-\mu {)}^2+{\Delta }^2}}{k_{B}T}}}-\frac{1}{1+e^{\frac{\sqrt{(ϵ(k)-\mu {)}^2+{\Delta }^2}}{k_{B}T}}}\right]\text{.}$$

$$1=\left|V\right|{\int }_{\mu -{ϵ}_D}^{\mu +{ϵ}_D}\frac{N(0)dϵ}{2\sqrt{(ϵ-\mu {)}^2+{\Delta }^2}}\left[\frac{1}{1+e^{\frac{-\sqrt{(ϵ-\mu {)}^2+{\Delta }^2}}{k_{B}T}}}-\frac{1}{1+e^{\frac{\sqrt{(ϵ-\mu {)}^2+{\Delta }^2}}{k_{B}T}}}\right]\text{.}$$

$$\frac{1}{\left|V\right|N(0)}={\int }_0^{{ϵ}_D}\frac{d\zeta }{\sqrt{{\zeta }^2+{\Delta }^2}}\left[\frac{1}{1+e^{\frac{-\sqrt{{\zeta }^2+{\Delta }^2}}{k_{B}T}}}-\frac{1}{1+e^{\frac{\sqrt{{\zeta }^2+{\Delta }^2}}{k_{B}T}}}\right]\text{.}$$

$$\frac{1}{\left|V\right|N(0)}={\int }_0^{{ϵ}_D}\frac{d\zeta }{\zeta }\left[\frac{1}{1+e^{\frac{-\zeta }{k_B{T}_C}}}-\frac{1}{1+e^{\frac{\zeta }{k_B{T}_C}}}\right]\text{.}$$

$$\frac{1}{\left|V\right|N(0)}={\int }_0^{\frac{{ϵ}_D}{K_B{T}_C}}\frac{dx}{x}\varphi (x)\text{,}$$

$${\int }_0^{\frac{{ϵ}_D}{K_B{T}_C}}\frac{dx}{x}\varphi (x)={\left[\varphi (x)\ln x\right]}_0^{\frac{{ϵ}_D}{K_B{T}_C}}-{\int }_0^{\frac{{ϵ}_D}{K_B{T}_C}}\frac{d\varphi (x)}{dx}\ln xdx\text{.}$$

$${\left[\varphi (x)\ln x\right]}_0^{\frac{{ϵ}_D}{K_B{T}_C}}$$

$${\int }_0^{\frac{{ϵ}_D}{K_B{T}_C}}\frac{d\varphi (x)}{dx}\ln xdx$$

$$\frac{1}{\left|V\right|N(0)}=\ln \left(1,134\frac{{ϵ}_D}{K_B{T}_C}\right)\text{.}$$

$$T_C=\frac{1,134{ϵ}_D}{K_B}{e}^{\frac{-1}{\left|V\right|N(0)}}\text{.}$$

$$\frac{1}{\left|V\right|N(0)}={\int }_0^{{ϵ}_D}\frac{d\zeta }{\sqrt{{\zeta }^2+{\Delta }_0^2}}={\sinh }^{-1}\left(\frac{{ϵ}_D}{{\Delta }_0}\right)\text{,}$$

$$\frac{{ϵ}_D}{{\Delta }_0}=\sinh \left(\frac{1}{\left|V\right|N(0)}\right)\text{.}$$

$$\frac{{ϵ}_D}{{\Delta }_0}=\frac{1}{2}{e}^{\frac{1}{|V|N(0)}}\text{.}$$

$${\Delta }_0=1.76K_B{T}_C\text{.}$$

$$\frac{1}{\left|V\right|N(0)}={\int }_0^{\frac{\sinh \frac{1}{|V|N(0)}}{y}}\frac{dz}{\sqrt{z^2+1}}\left(\frac{1}{1+e^{\frac{-1.76y\sqrt{z^2+1}}{x}}}-\frac{1}{1+e^{\frac{1.76y\sqrt{z^2+1}}{x}}}\right)\text{.}$$

$$\frac{1}{\left|V\right|N(0)}=\underset{0}{\overset{{ϵ}_D}{\int }}\frac{d\zeta }{\sqrt{{\zeta }^2+{\Delta }^2}}\left[\varphi (\frac{1}{r(\zeta )})-\varphi (r(\zeta ))\right]\text{,}$$

$$r(\zeta )=e^{{}^{\frac{\sqrt{{\zeta }^2+{\Delta }^2}}{K_{B}T}}\text{,}}$$

$${\left(\frac{1}{(1-p)\varphi (r)}+1\right)}^{1-p}{\left(\frac{1}{p\varphi (r)}-1\right)}^p=r\text{.}$$

$$\frac{1}{\left|V\right|N(0)}$$

$$r(x)$$

$${\left(\frac{1}{(1-p)\varphi (r)}+1\right)}^{1-p}{\left(\frac{1}{p\varphi (r)}-1\right)}^p=\infty \text{.}$$

$$\varphi (r)=0\text{ at }T=0.$$

$${\left(\frac{1}{(1-p)\varphi \left(\frac{1}{r}\right)}+1\right)}^{1-p}{\left(\frac{1}{p\varphi \left(\frac{1}{r}\right)}-1\right)}^p=\frac{1}{r}\text{.}$$

$${\left(\frac{1}{(1-p)\varphi \left(\frac{1}{r}\right)}+1\right)}^{1-p}{\left(\frac{1}{p\varphi \left(\frac{1}{r}\right)}-1\right)}^p=0$$

$$\varphi (\frac{1}{r})=\frac{1}{p}\text{ at }T=0\text{.}$$

$$\frac{p}{\left|V\right|N(0)}=\underset{0}{\overset{{ϵ}_D}{\int }}\frac{d\zeta }{\sqrt{{\zeta }^2+{\Delta }_0^2}}={\sinh }^{-1}\frac{{ϵ}_D}{{\Delta }_0}\text{,}$$

$${ϵ}_D={\Delta }_0\sinh \frac{p}{\left|V\right|N(0)}\text{.}$$

$${\Delta }_0=2{ϵ}_D{e}^{\frac{-p}{\left|V\right|N(0)}}\text{.}$$

Full text

1 Introduction

Superconductivity is a phenomena occurring in certain materials when cooled below some characteristic temperature called a critical temperature. In a superconductor, the resistance drops suddenly to zero when the material is cooled below its critical temperature. This phenomena was first discovered by the dutch physicist Heike Kamerlingh Onnes, which received the Nobel prize of physics in 1913 for discovering this phenomena and the related work regarding liquefaction of helium [1]. There is another phenomena related to superconductor called (Meissner effect), which is the expulsion of a magnetic field from a superconductor during its transition to the superconducting state. This phenomena was discovered in 1933 by Walther Meissner and Robert Ochsenfeld [2]. Although the microscopic theory of superconductivity was developed by Bardeen, Cooper and Schrieffer (BCS theory) in 1957 [3], other descriptions of the phenomena of the superconductivity were developed [4-6].

Whereas ordinary superconductors usually have transition temperatures below 30 Kelvin, It is well known that some materials behave as superconductors at unusually high critical temperatures and this materials called high temperature superconductors or (HTS). The first HTS was discovered in 1986 by Georg Bednorz and K. Alex Muller, who were awarded the 1987 Nobel Prize in Physics [7].

Although the standard BCS theory (weak-coupling electron–phonon interaction) can explain all known low temperature superconductors and even some high temperature superconductors, some of high temperature superconductors cannot be explained by BCS theory and some physicist believe that they obey a strong interaction.

In this paper, we find that if we use a unified statistics [8] instead Fermi-Dirac statistics, then we can find a reasonable explanation of all high temperature superconductivity phenomena using a standard BCS theory.

2 The model

We follow the derivation of Hamiltonian model using the mean field approximation and Bogoliubov unitary transformation as in the reference [9], where we can write the following relation,

[TABLE]

where $f(E_{K})$ determined from the unified statistics as follow [8],

[TABLE]

3 BCS Theory ( $p\rightarrow 1)$

For $p\rightarrow 1$, we recover the usual BCS Theory.

When $p\rightarrow 1,$we have the following usual form of the Fermi function $f(E_{K})$,

[TABLE]

so, we have,

[TABLE]

Replacing the summation by integration over the energy states, we have,

[TABLE]

Let $\zeta=\epsilon-\mu,$ then we have the master equation for BCS theory as,

[TABLE]

Determine $T_{C}$.

At $\Delta=0$, we have $T=T_{C}$, so we can write,

[TABLE]

Define $\dfrac{\zeta}{k_{B}T_{C}}=x$, we can write,

[TABLE]

where $\phi(x)=\left[\dfrac{1}{1+e^{-x}}-\dfrac{1}{1+e^{x}}\right]$.

Using integration by parts we have,

[TABLE]

For a weak coupling limit, i.e. $\left\{\frac{1}{\left|V\right|N_{0}}\gtrdot\gtrdot 1,k_{B}T_{C}\lessdot\lessdot\epsilon_{D},\left|\Delta_{0}\right|\lessdot\lessdot k_{B}T_{C}\right\},$we have,

[TABLE]

So, we have

[TABLE]

We can rewrite the last equation to obtain the well known formula for critical temperature for BCS Theory as,

[TABLE]

Determine $\Delta_{0}.$

At $T=0$, we have $\Delta=\Delta_{0}$, so from the equation we have,

[TABLE]

so we have,

[TABLE]

Again, for a weak coupling limit, we have $\sinh\left(\frac{1}{\left|V\right|N(0)}\right)\approx\frac{1}{2}e^{\dfrac{1}{\left|V\right|N(0)}}$, so we can write the BCS formula of $\Delta_{0}$ as follow,

[TABLE]

From equations (13) and (16), we arrive at the universal BCS formula,

[TABLE]

Using equations (16) and (17) in the master equation (6), we can plot the universal curve $y=y(x)$, where $x=\dfrac{T}{T_{C}},y=\dfrac{\Delta}{\Delta_{0}}$ as follow,

[TABLE]

4 Anyon superconductivity

From equations (1) and (2), we can write the general form of the master equation for any value of $p$ as follow,

[TABLE]

where,

[TABLE]

and $\phi(r)$ determined from the following equation,

[TABLE]

4.1 Determine $T_{C}$ for any value of $p$

At $\Delta=0$, we have $T=T_{C}$ and then from equations (19-21) we can write,

[TABLE]

and again, $\phi(r)$ determined from the equation (21).

4.2 Determine $\Delta_{0}$ for any value of $p$

At $T=0$, we have $\Delta=\Delta_{0}$ and from equation (20), we have $r=\infty$, so equation (21) takes the form,

[TABLE]

Then for any value of $p,$we have

[TABLE]

Similarly, we can evaluate $\phi\left(\frac{1}{r}\right)$ by solving the following equation,

[TABLE]

So at $T=0,$ We have $r=\infty,\dfrac{1}{r}=0$ and then equation (26) takes the form,

[TABLE]

The solution of (26) (for positive $\phi(\frac{1}{r})$) takes the form,

[TABLE]

Using (25) and (28) in the master equation (19), we have at $T=0$ the following equation,

[TABLE]

or in the equivalent form,

[TABLE]

At weak coupling limit, from the equation (30), we have the following universal relation for $\Delta_{0}$ at any value of $p$ as,

[TABLE]

In the next section, we will study some special cases of Anyon superconductivity for some special values of $p$.

5 Special cases of Anyon superconductivity

5.1 $p=\frac{1}{2}$ Anyon

From the unified statistics (equation(2)), we can find the formula of $f(E_{k})$ as follow,

[TABLE]

So, after some calculations, we arrive at the master equation for the case $p=\frac{1}{2}$ as follow,

[TABLE]

Determine $T_{C}$

At $\Delta=0$, we have $T=T_{C}$, so from equations (32,33) we can write,

[TABLE]

Define $\phi(x)=\left[\dfrac{1}{\left(1+e^{-2x}\right)^{\dfrac{1}{2}}}-\dfrac{1}{\left(1+e^{2x}\right)^{\dfrac{1}{2}}}\right]$, we can calculate the last integral as,

[TABLE]

So,we can write the equation(34) as,

[TABLE]

or on the equivalent form,

[TABLE]

Comparing equation (37), the critical temperature for the Anyon case ($P=\frac{1}{2}$) and equation (13), the critical temperature for BCS theory (at the same value of Debye energy) we have,

[TABLE]

So, for some weak coupling limit, say $\dfrac{1}{\left|V\right|N(0)}=5$, we have

[TABLE]

So, if the critical temperature in the standard BSC is equal to $1$ Kelvin (with $\dfrac{1}{\left|V\right|N(0)}=5$), then using a unified statistics with $P=\frac{1}{2}$, the critical temperature jumps to $16,446$ Kelvin.

Notice that although the weak coupling is the reason of why we cannot explain some HTS using the standard BCS theory, in the Anyonic statistics the critical temperature is increased whenever the coupling limit is decreased!!

As an example, consider that we have more weak interaction (say $\dfrac{1}{\left|V\right|N(0)}=10$), then we have,

[TABLE]

So at the same condition, if the critical temperature in the standard BSC is eual to $1$ Kelvin then for a weak coupling $\dfrac{1}{\left|V\right|N(0)}=5$ we see that the critical temperature jumps to $16,446$ Kelvin and for more weak coupling $\dfrac{1}{\left|V\right|N(0)}=10$ the critical temperature jumps to $201$ Kelvin!

Determine $\Delta_{0}$

At $T=0$, we have $\Delta=\Delta_{0}$, so from the equation (29) we have,

[TABLE]

so we have,

[TABLE]

From (24) and (28), we have

[TABLE]

Finally, we can plot the universal curve $y=y(x)$, where $x=\dfrac{T}{T_{C}},y=\dfrac{\Delta}{\Delta_{0}}$ for $p=\frac{1}{2}$ as follow,

[TABLE]

5.2 $p=\frac{1}{3}$ Anyon superconductor

For $p=\frac{1}{3}$, we can determine $\phi(r)$ from equation (21) as,

[TABLE]

From equations (22-23) and after some calculations, we can find the formula of $T_{C}$ for the Anyon $p=\frac{1}{3}$ as,

[TABLE]

Also, from the general equation (31) we have,

[TABLE]

Conclusion 1

In this paper, we calculated the critical temperature of some metals using the standard BCS theory but assuming that the particles inside this metals obey Anyonic statistics instead Fermi-Dirac statistics. We find that the lower  of the interaction we have, the greater of the critical temperature which may be a reasonable explanation of some high temperature superconductors which the standard BCS theory cannot explain it.

Acknowledgement 2

I want to thank E. Ahmad for his advice and very useful discussion.

6 References

[1] Simon Reif-Acherman, ”Heike Kamerlingh Onnes: Master of Experimental Technique and Quantitative Research”, *Physics in Perspective, *vol. 6, pp. 197–223, (2004).

[2] W. Meissner, R. Ochsenfeld, ”Ein neuer Effekt bei Eintritt der Supraleitfahigkeit”, Naturwissenschaften, vol. 21, pp. 787–788, (1933).

[3] J. Bardeen, L. N. Cooper, J. R. Schrieffer, ”Microscopic Theory of Superconductivity”, Physical Review, vol. 106, (1957).

[4] C. S. Gorter, H.Casimir, ”On superconductivity”, *Zeitschrift für Technische Physik , *vol. 15, (1934).

[5] H. London and F. London, ”The Electromagnetic Equations of the Supraconductor”, Proceedings of the Royal Society A, vol.149, (1935).

[6] V. L. Ginzburg, L. D. Landau, *Soviet Physics JETP *20, vol. 1064, (1950).

[7] J. G. Bednorz, K. A. Muller, ”Possible high TC superconductivity in the Ba-La-Cu-O system”, Zeitschrift für Physik B, vol. **64, **pp. 189–193, (1986).

[8] A. A. Abutaleb, ”Unified statistical distribution of quantum particles and Symmetry”,* Int. J. Theor. Phys*, vol. **53, pp. **3893-3900, (2014).

[9] http://www-personal.umich.edu/~sunkai/teaching/Fall_2012/phys620.html.