Functional Integral Method for Superconducting Critical Phenomena

  • Hajime Takayama


By an unusual diagrammatic analysis Gaudin1 investigated a microscopic expression of the partition function of BCS interacting electrons. Later Langer2 developed his method, and derived the generalized Ginzburg—Landau (GL) free energy functional near the transition temperature T c 0 as
$$\rho \left( \left\{ {{\Psi }_{q}} \right\} \right)=\exp \left\{ -\Omega \left[ \underset{q}{\mathop{\sum L_{q}^{o}{{\left| {{\Psi }_{q}} \right|}^{2}}+\frac{1}{4}\delta \underset{{{q}_{1}}+{{q}_{2}}={{q}_{3}}+{{q}_{4}}}{\mathop{\sum }}\,\Psi _{{{q}_{1}}}^{*}\Psi _{{{q}_{2}}}^{*}{{\Psi }_{{{q}_{3}}}}{{\Psi }_{{{q}_{4}}}}}}\, \right] \right\}$$
$$L_{q}^{o}=\left( \pi /8{{T}_{co}} \right)\left( D{{q}^{2}}+\left| {{\omega }_{m}} \right| \right)+\eta $$
η=ln(T/T c 0)≃(T–T c 0)/T c 0;δ=0.212/N(0)T c 0;q=(q,ω m), with q the momentum and ω m the Matsubara frequency of electron pairs represented by the order parameter Ψq; Ω is the volume of the system; D = lv/3; and N(0) is the electronic density of states at the Fermi energy. The same diagrammatic analysis is easily applied to investigations of dynamic properties of superconductors, in particular, to calculations of a T τ product 〈T τ (J A (x) J B (y))〉, where J A (x) or J B (y) is a one-body operator of the type J A (x) = j A (x, x′) ψ†(x′) ψ(x)| x′→x′ ψ†(x) and ψ(x) being electron field operators and j A (x, x′) an appropriate operator corresponding to the physical quantity J A . Its Fourier transform is given by3
$${{\langle {{T}_{\tau }}\left( {{J}_{A}}{{J}_{B}} \right)\rangle }_{k}}=\left( T/\Omega Z \right)\underset{q}{\mathop{\prod }}\,\left[ \int J{{\Psi }_{q}} \right]\rho \left( \left\{ {{\Psi }_{q}} \right\} \right)\times \left\{ \underset{q}{\mathop{\sum }}\,{{\left| {{\Psi }_{q}} \right|}^{2}}{{\left| {{\Psi }_{q+k}} \right|}^{2}}{{K}_{A}}\left( k;q \right){{K}_{B}}\left( k;q \right)+\underset{q}{\mathop{\sum }}\,{{\left| {{\Psi }_{q}} \right|}^{2}}{{K}_{AB}}\left( k;q \right) \right\}$$


Partition Function Critical Phenomenon Diagrammatic Analysis Excess Conductivity Film Superconductor 
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Copyright information

© Springer Science+Business Media New York 1974

Authors and Affiliations

  • Hajime Takayama
    • 1
  1. 1.Department of PhysicsTohoku UniversitySendaiJapan

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