A Diagrammer's Note on Superconducting Fluctuation Transport for Beginners: I. Conductivity and Thermopower

TL;DR

This note provides a pedagogical, diagrammatic treatment of superconducting fluctuation transport in zero magnetic field using thermal Green functions and the Kubo framework. It connects microscopic fluctuation physics to Ginzburg-Landau theory via ladder resummations, Jonson-Mahan relations, and Ward identities, clarifying the correct heat-current vertex for Cooper pairs. In 2D, it yields explicit results for Gaussian GL fluctuations and the Aslamazov-Larkin process, including σ_{xx} = e^2/(16d)·T/(T−T_c) and α_{xx} ≈ (|e|/(2πd))·(τ_2/τ_1)·ln(T_Λ/(T−T_c)), illustrating logarithmic enhancements near Tc. The work emphasizes the consistency of current vertices and response functions across quasi-particle and pair channels, providing a foundational, instructional bridge between microscopics and GL phenomenology with potential extensions to magnetic-field cases in subsequent notes.

Abstract

A diagrammatic approach based on thermal Green function to superconducting fluctuation transport is reviewed keeping consistency with Ginzburg-Landau theory. The correct expression of the heat current vertex for Cooper pairs is clarified via Jonson-Mahan formula and Ward identities.

Figures (6)

• Figure 1: Effective interaction for Cooper pairs: The wavy and broken lines represent $-L$ and $-(-g)$. The solid line with arrow represents $G_0$. All the Feynman diagrams in this note are drawn by JaxoDraw.
• Figure 2: Current vertices for quasi-particles: The left black one is the charge current vertex. The right gray one is the heat current vertex. The broken line with arrow represents the coupling to the external field.
• Figure 3: Current vertices for Cooper pairs.
• Figure 4: Quasi-particle transport within relaxation-time approximation: The left Feynman diagram describes the charge response to the external electric field. The right describes the heat response to the electric field.
• Figure 5: Contours of the integral: These three panels are three different characterization of the same complex $z$-plane. The solid line with an arrow represents the contour of the integral. The horizontal and vertical gray lines represent the real and imaginary axes of the complex $z$-plane. The dots on the imaginary axis represent the fermionic thermal frequencies. The broken lines $C_0$ and $C_\lambda$ represent the cuts along ${\rm Im}\, z = 0$ and ${\rm Im}\, z = - \omega_\lambda$.