Carsten Timm: Theory of superconductivity

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We will now discuss the physics encoded by the RG flow equations. First, note that the quantity is invariant under the RG flow: dC dl = 4π 2 y dy dl − 2 dK πK 2 − 1 dK dl K dl C := 2π 2 y 2 − 2 − ln K (7.72) πK = 4π 2 y 2 (2 − πK) − 8π 2 y 2 + 4π 3 y 2 K = 0. (7.73) Thus C is a first integral of the flow equations. We can calculate C from the initial values y 0 , K 0 and obtain 2π 2 y 2 = 2π 2 y0 2 + 2 ( 1 π K − 1 ) + ln K (7.74) K 0 K 0 ⇒ y = √y 0 2 + 1 ( 1 π 3 K − 1 ) + 1 K 0 2π 2 ln K . (7.75) K 0 The RG flow is along curves decribed by this expression, where the curves are specified by y 0 , K 0 . These parameters change with temperature as given in Eqs. (7.70) and (7.71). These initial conditions are sketched as a dashed line in the figure. We see that there are two distinct cases: y T separatrix T = T c 0 2/ π K • For T < T c , K flows to some finite value K(l → ∞) > 2/π. This means that even infinitely large pairs feel a logarithmic attraction, i.e., are bound. Moreover, the fugacity y flows to zero, y(l → ∞) = 0. Thus large pairs are very rare, which is consistent with their (logarithmically) diverging energy. • For T > T c , K flows to K(l → ∞) = 0. Thus the interaction between a vortex and an antivortex that are far apart is completely screened. Large pairs become unbound. Also, y diverges on large length scales, which means that these unbound vortices proliferate. This divergence is an artifact of keeping only the leading order in y in the derivation. It is cut off at finite y if we count vortex-antivortex pairs consistently. But the limit K → 0 remains valid. At T = T c we thus find a phase transition at which vortex-antivortex pairs unbind, forming free vortices. It is called the Berezinskii-Kosterlitz-Thouless (BKT) transition. In two-dimensional films, vortex interactions thus suppress the temperature where free vortices appear and quasi-long-range order is lost from the point T = Tsingle c vortex where η = − 1 k B T β 4π γα = 1 2π 1 K 0 ! = 1 4 ⇒ K 0 = 2 π to the one where K(l → ∞) = 2/π and y 0 , K 0 lie on the “separatrix” between the two phases, C = 2π 2 y 2 0 − 2 πK 0 − ln K 0 ! = 0 − 1 − ln 2 π ⇒ (7.76) 2 πK 0 + ln K 0 = 1 + ln 2 π + 2π2 y 2 0. (7.77) Clearly the two criteria agree if y 0 = 0. This makes sense since for y 0 = 0 there are no vortex pairs to screen the interaction. In addition, a third temperature scale is given by the mean-field transition temperature T MF , where α ! = 0. 62
quasi−long−range order free vortices no condensate 0 T T K( l c T c single vortex oo ) = 2/π K 0 = 2/π In the low-temperature phase, the largest pairs determine the decay of the correlation function ⟨ψ ∗ (r)ψ(0)⟩ for large r. Thus we find T MF α= 0 ⟨ψ ∗ (r)ψ(0)⟩ ∼ = ψ 2 0 ( r r 0 ) −η (7.78) with η = 1 1 2π K(l → ∞) . (7.79) We note that the exponent η changes with temperature (one could say that the whole low-temperature phase is critical) but assumes a universal value at T c : There, lim K = 2 ⇒ η(T c ) = 1 l→∞ π 4 . (7.80) Due to the screening of the vortex interaction, the condensate is less stiff than it would be in the absence of vortices. This is described by the renormalization of K from K 0 to K(l → ∞). It is customary but somewhat misleading to express this as a renormalization of the superfluid density n s , which is after all proportional to K 0 . Assuming that the temperature dependence of α and thus of the bare superfluid density is negligible close to T c , the renormalized superfluid density n 0 s ∝ ψ 2 0 = − α β ∝ K 0 (7.81) n s (T ) := K K 0 n 0 s (7.82) obtains its temperature dependence exclusively from K/K 0 . (The argument l → ∞ is implied here and in the following.) For T < T c but close to the BKT transition we have The invariant C = 2π 2 y 2 0 − 2 πK 0 Thus we can write, close to T c , K = 2 + ∆K. (7.83) π − ln K 0 is an analytic function of temperature and its value at T c is [ C c = 2π 2 y 2 − 2 ] πK − ln K with some constant b := dC/dT | T =Tc . On the other hand, [ C = 2π 2 y 2 − 2 πK ]T − ln K ( ) 2 2 = − π ( 2 π + ∆K) − ln π + ∆K 1 = − 1 + π 2 ∆K − ln 2 ( π − ln 1 + π ) 2 ∆K T =T c = −1 − ln 2 π . (7.84) C ∼ = −1 − ln 2 π + b (T − T c) (7.85) π ∼= −1 + ✚ ✚ π2 ∆K − ✚ 2 4 ∆K2 − ln 2 π − π ✚2 ∆K ✚✚ + 1 2 π 2 4 ∆K2 = −1 − ln 2 π − π2 8 ∆K2 . (7.86) 63
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Theory of Superconductivity Carsten
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Contents 1 Introduction 5 1.1 Scope
Page 5 and 6:
1 Introduction 1.1 Scope Supercondu
Page 7 and 8:
2 Basic experiments In this chapter
Page 9 and 10:
Superconductivity with rather high
Page 11 and 12: • In 1979, Frank Steglich et al.
Page 13 and 14: 3 Bose-Einstein condensation In thi
Page 15 and 16: We note the identity g n (y) = ∞
Page 17 and 18: We find a phase transition at T c ,
Page 19 and 20: gives a reasonable approximation fo
Page 21 and 22: We now consider the force F = −eE
Page 23 and 24: 5 Electrodynamics of superconductor
Page 25 and 26: Thus the supercurrent flows in the
Page 27 and 28: where φ j is the polar angle of el
Page 29 and 30: with the penetration depth √ √
Page 31 and 32: N s is the total number of particle
Page 33 and 34: 6.2 Ginzburg-Landau theory for neut
Page 35 and 36: scope of this course, though. The i
Page 37 and 38: As above, we keep only terms up to
Page 39 and 40: Within the mean-field theories we h
Page 41 and 42: Including the superconducting contr
Page 43 and 44: The Gibbs free energy of the domain
Page 45 and 46: The magnetic flux Φ through this l
Page 47 and 48: Another useful quantity is the free
Page 49 and 50: We obtain e −ik yy (− 2 2m ∗
Page 51 and 52: has a simple interpretation: It is
Page 53 and 54: This is a Gaussian average since F
Page 55 and 56: Note that in two dimensions the cur
Page 57 and 58: Thus the energy is ∫ E 2 = 2E cor
Page 59 and 60: The energy of an isolated vortex-an
Page 61: Next, we have to express Z ′ in t
Page 65 and 66: which is valid within the film and
Page 67 and 68: attraction of the magnetic monopole
Page 69 and 70: Finally, the voltage measured for a
Page 71 and 72: In this case it is useful to expand
Page 73 and 74: It will be useful to write the elec
Page 75 and 76: (the minus sign is conventional) an
Page 77 and 78: Note that summing up the more and m
Page 79 and 80: 8.4 Effective interaction between e
Page 81 and 82: Re RPA, R V eff V c ( ) RPA q 0 Re
Page 83 and 84: diagrams describing this situation.
Page 85 and 86: where |0⟩ is the vacuum state wit
Page 87 and 88: This is called the BCS gap equation
Page 89 and 90: 10 BCS theory The variational ansat
Page 91 and 92: we obtain ( ) 2ξ k |u k | |v k | e
Page 93 and 94: 1 0 2 u k v 2 k k F k We also find
Page 95 and 96: 10.2 Isotope effect How can one che
Page 97 and 98: and expanding for small ∆T and sm
Page 99 and 100: For tunneling between two normal me
Page 101 and 102: In the limit k B T → 0 this becom
Page 103 and 104: tant. The first one is (u k ′u k
Page 105 and 106: for quasiparticle creation and anni
Page 107 and 108: 11 Josephson effects Brian Josephso
Page 109 and 110: Ginzburg-Landau theory also gives u
Page 111 and 112: Equation (11.25) can be used to stu
Page 113 and 114:
The averaged current is just Ī = I
Page 115 and 116:
where = 1. In the superconductor,
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with ˜k 1 := (−k 1x , k 1y , k 1
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S N S 2∆ 0 2eV eV In particular,
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But now the right-hand side is alwa
Page 123 and 124:
The fact that the gap closes at som
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where τ is the imaginary time, T
Page 127 and 128:
At lower temperatures, details beco
Page 129 and 130:
Also note that spin fluctuations st
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k y Γ X’ M hole−like Q 2 elect
Page 133 and 134:
Thus ∆ τσ (−k) = − 1 N or,
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It is plausible that in a crystal t
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Carsten Timm: Theory of superconductivity

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