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Can Anyon Statistics Explain High Temperature Superconductivity?

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Can Anyon Statistics Explain High Temperature Superconductivity? Can Anyon statistics explain high temperature superconductivity? Ahmad Adel Abutaleb Department of Mathematics, Faculty of science, University of Mansoura, Elmansoura 35516, Egypt August 10, 2018 Abstract In this paper, we find a reasonable explanation of high temperature superconductivity phenomena using Anyon statistics. 1 Introduction Superconductivity is a phenomena occurring in certain materials when cooled below some characteristic temperature called a critical temperature. In a su- perconductor, 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 liquefac- tion of helium [1]. There is another phenomena related to superconductor called (Meissner effect), which is the expulsion of a magnetic field from a supercon- ductor 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 be- low 30 Kelvin, It is well known that some materials behave as superconductors at unusually high critical temperatures and this materials called high tempera- arXiv:1702.00692v1 [physics.gen-ph] 27 Jan 2017 ture 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 inter- action) 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. 1 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 approxi- mation and Bogoliubov unitary transformation as in the reference [9], where we can write the following relation, ∆ ∆ = V | | [f( EK ) f(EK )] , (1) | | | | 2 (ǫ(k) µ)2 + ∆2 − − Xk − p where f(EK ) determined from the unified statistics as follow [8], (ǫ(k) µ)2 + ∆2 1 1−p 1 p − +1 1 = ep kB T . (2) (1 p)f(Ek) pf(Ek) − − 3 BCS Theory ( p 1) → For p 1, we recover the usual BCS Theory. → When p 1,we have the following usual form of the Fermi function f(EK ), → 1 f(EK )= , (3) (ǫ µ)2 + ∆2 − 1+ ep kBT so, we have, 1 1 1 1= V . | | Z 2 (ǫ(k) µ)2 + ∆2 (ǫ(k) µ)2 + ∆2 − (ǫ(k) µ)2 + ∆2 Xk − − − − p p k T p k T 1+ e B 1+ e B (4) Replacing the summation by integration over the energy states, we have, µ+ǫD N(0)dǫ 1 1 1= V . 2 2 2 2 2 2 | | Zµ−ǫD 2 (ǫ µ) + ∆ (ǫ µ) + ∆ − (ǫ µ) + ∆ − − − − p p k T p k T 1+ e B 1+ e B (5) 2 Let ζ = ǫ µ, then we have the master equation for BCS theory as, − 1 ǫD dζ 1 1 = . (6) 2 2 2 V N(0) Z0 ζ + ∆2 ζ + ∆2 − ζ + ∆2 | | − p pk T p k T 1+ e B 1+ e B Determine TC . At ∆ = 0, we have T = TC, so we can write, ǫ 1 D dζ 1 1 = . (7) V N(0) Z ζ ζ − ζ | | 0 − k T k T 1+ e B C 1+ e B C ζ Define = x, we can write, kB TC ǫD 1 dx = KBTC φ(x), (8) V N(0) Z x | | 0 1 1 where φ(x)= . 1+ e−x − 1+ ex Using integration by parts we have, ǫD ǫD ǫD KBTC dx KBTC KBTC dφ(x) φ(x) = [φ(x) ln x]0 ln xdx. (9) Z0 x − Z0 dx 1 For a weak coupling limit, i.e. ⋗⋗1, kBTC ⋖⋖ǫD, ∆0 ⋖⋖kBTC ,we |V |N0 | | have, n o ǫD KBTC ǫD [φ(x) ln x]0 = ln , (10) KBTC ǫD dφ(x) ∞ dφ(x) KBTC ln xdx ln xdx 0, 12563. (11) Z0 dx ≈ Z0 dx ≈− So, we have 1 ǫ = ln 1, 134 D . (12) V N(0) KBTC | | We can rewrite the last equation to obtain the well known formula for critical temperature for BCS Theory as, 3 1 − 1, 134ǫD V N(0) TC = e| | . (13) KB Determine ∆0. At T = 0, we have ∆ = ∆0, so from the equation we have, 1 ǫD dζ ǫ = = sinh−1 D , (14) V N(0) Z0 2 2 ∆0 | | ζ + ∆0 q so we have, ǫ 1 D = sinh . (15) ∆ V N(0) 0 | | 1 1 1 V N(0) Again, for a weak coupling limit, we have sinh |V |N(0) 2 e| | , so ≈ we can write the BCS formula of ∆0 as follow, ǫD 1 1 = e |V |N(0) . (16) ∆0 2 From equations (13) and (16), we arrive at the universal BCS formula, ∆0 =1.76KBTC . (17) Using equations (16) and (17) in the master equation (6), we can plot the T ∆ universal curve y = y(x), where x = ,y = as follow, TC ∆0 1 sinh |V |N(0) 1 dz 1 1 = y . V N(0) Z √ 2 √ 2 − √ 2 0 z +1 1.76y z +1 1.76y z +1 | | − 1+ e x 1+ e x (18) 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, ǫD 1 dζ 1 = φ( ) φ(r(ζ)) , (19) V N(0) Z ζ2 + ∆2 r(ζ) − | | 0 p 4 where, ζ2 + ∆2 pK T r(ζ)= e B , (20) and φ(r) determined from the following equation, 1 1−p 1 p +1 1 = r. (21) (1 p)φ(r) pφ(r) − − 4.1 Determine TC for any value of p At ∆ = 0, we have T = TC and then from equations (19-21) we can write, ǫD KBTC 1 dx 1 = φ( ) φ(r(x)) , (22) V N(0) Z x r(x) − | | 0 r(x) = ex, (23) and again, φ(r) determined from the equation (21). 4.2 Determine ∆0 for any value of p At T = 0, we have ∆ = ∆0 and from equation (20), we have r = , so equation (21) takes the form, ∞ 1 1−p 1 p +1 1 = . (24) (1 p)φ(r) pφ(r) − ∞ − Then for any value of p,we have φ(r)=0at T =0. (25) 1 Similarly, we can evaluate φ r by solving the following equation, 1−p p 1 1 1 +1 1 = . (26) 1 1 − r (1 p)φ pφ r r − 1 So at T =0, We have r = , = 0 and then equation (26) takes the form, ∞ r 1−p p 1 1 +1 1 = 0 (27) 1 1 − (1 p)φ pφ r r − 5 1 The solution of (26) (for positive φ( r )) takes the form, 1 1 φ( )= at T =0. (28) r p Using (25) and (28) in the master equation (19), we have at T = 0 the following equation, ǫD p dζ ǫ = = sinh−1 D , (29) V N(0) Z 2 2 ∆0 | | 0 ζ + ∆0 q or in the equivalent form, p ǫ = ∆ sinh . (30) D 0 V N(0) | | At weak coupling limit, from the equation (30), we have the following uni- versal relation for ∆0 at any value of p as, p − V N(0) ∆0 =2ǫD e| | . (31) In the next section, we will study some special cases of Anyon superconduc- tivity for some special values of p. 5 Special cases of Anyon superconductivity 1 5.1 p = 2 Anyon From the unified statistics (equation(2)), we can find the formula of f(Ek) as follow, 2 f(E )= . (32) k 1 2 (ǫ µ)2 + ∆2 2 − 1+ e p kBT So, after some calculations, we arrive at the master equation for the case 1 p = 2 as follow, 6 ǫD 1 dζ 1 1 = . 2 2 V N(0) Z0 ζ 2 1 − 1 | | + ∆ 2 2 2 2 2 2 p 2 ζ + ∆ 2 ζ + ∆ − p p 1+ e kBT 1+ e kB T (33) Determine TC At ∆ = 0, we have T = TC, so from equations (32,33) we can write, ǫD 1 dx 1 1 = KBTC . (34) 2 V N(0) Z x 1 − 1 | | 0 e−2x 2 e2x 2 (1 + ) (1 + ) 1 1 Define φ(x)= , we can calculate the last inte- 1 − 1 e−2x 2 e2x 2 (1 + ) (1 + ) gral as, ǫD ǫD ∞ KBTC dx KBTC dφ(x) ǫD 1.535ǫD φ(x) [φ(x) ln x]0 dx ln x = ln +0.4228 = ln . Z0 x ≈ −Z0 x KBTC KBTC (35) So,we can write the equation(34) as, 1 1.535ǫ = ln D , (36) 2 V N(0) KBTC | | or on the equivalent form, 1 − 1.535ǫD 2 V N(0) TC = e | | . (37) KB Comparing equation (37), the critical temperature for the Anyon case (P = 1 2 ) and equation (13), the critical temperature for BCS theory (at the same value of Debye energy) we have, 7 1 TC p= 1 2 2 V N(0) =1.35e | | . (38) TCBCS 1 So, for some weak coupling limit, say = 5, we have V N(0) | | TC 16, 446TC .
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