A Diagrammer's Note on Superconducting Fluctuation Transport for Beginners: II. Hall and Nernst Effects with Perturbational Treatment of Magnetic Field

TL;DR

The paper develops a diagrammatic, thermal Green's function framework to study superconducting fluctuation transport focusing on Hall and Nernst effects under a weak magnetic field perturbation. It combines Boltzmann, Kubo, GL, and AL formalisms to obtain electron- and Cooper-pair-fluctuation contributions, showing consistency with Boltzmann results in appropriate limits and highlighting the fluctuation-dominated regime near the superconducting transition. The Appendix reviews DC Hall conductivity for Dirac fermions in 2+1D, including the Chern-Simons action and Ishikawa–Matsuyama formula, emphasizing topological aspects and conditions under which interaction effects do not renormalize the Hall response. Together, the work links topological field-theory structures to measurable transport and sets the stage for non-perturbative magnetic-field treatment in Part III.

Abstract

A diagrammatic approach based on thermal Green function to superconducting fluctuation transport is reviewed focusing on Hall and Nernst effects. The treatment of weak magnetic field is carefully discussed within the linear order perturbation. In the Appendix the linear response theory for the DC Hall conductivity of the Dirac fermion in 2+1 space-time dimensions is reviewed. One focus is the Chern-Simons effective action for the gauge field. Another is the exact formula by Ishikawa and Matsuyama.

Figures (4)

• Figure 1: Diagrams with ${\bf A}$-linear contributions: (a) The solid line in upward direction represents the electron propagator $-G({\bf p+k/2}, i\varepsilon_n+i\omega_\lambda)$ and the broken line in downward direction represents $-G({\bf p+k/2}\leftarrow{\bf p-k/2}, i\varepsilon_n)$. Throughout the series of three Notes the Zeeman splitting is neglected so that the propagators for up-spin electrons and for down-spin electrons are degenerate. The upper black circle represents a component of $j^e_\mu({\bf k})$ where the momentum of the electron changes from ${\bf p+k/2}$ to ${\bf p-k/2}$ and the lower black circle represents a component of $j^e_\nu(0)$ where the momentum does not change. (b) The broken line in upward direction is $-G({\bf p+k/2}\leftarrow{\bf p-k/2}, i\varepsilon_n+i\omega_\lambda)$ and the solid line in downward direction is $-G({\bf p-k/2}, i\varepsilon_n)$. The upper black circle is $j^e_\mu({\bf k})$ and the lower black circle is $j^e_\nu(0)$. (c) The solid line in upward direction is $-G({\bf p+k/2}, i\varepsilon_n+i\omega_\lambda)$ and the solid line in downward direction is $-G({\bf p-k/2}, i\varepsilon_n)$. The upper black circle is $j^e_\mu({\bf k})$ and the lower gray circle is $- (e / m) \rho(-{\bf k}) A_\nu$ where the momentum changes from ${\bf p-k/2}$ to ${\bf p+k/2}$. (d) The solid line in upward direction is $-G({\bf p}, i\varepsilon_n+i\omega_\lambda)$ and the solid line in downward direction is $-G({\bf p}, i\varepsilon_n)$. The upper gray circle is $- (e / m) \rho(0) A_\mu$, where the momentum does not change, and the lower gray circle is $j^e_\nu(0)$. The integral of this diagram in terms of ${\bf p}$ is odd under the variable change $p_\nu \rightarrow - p_\nu$ and vanishes so that (d) does not contribute to the conductivity.
• Figure 2: Diagrams for a fixed gauge ${\bf A}=(A_x,0,0)$: The left diagram corresponds to the one in Fig. \ref{['fig:4-Loop']}-(a) and the right to Fig. \ref{['fig:4-Loop']}-(b). The broken line represents the coupling to the magnetic field (\ref{['jA-int']}). (left) The solid line in upward direction is $-G({\bf p}, i\varepsilon_n+i\omega_\lambda)$. The downward process is the product $[-G({\bf p}, i\varepsilon_n)]\cdot[-(-J_x A_x)] \cdot[-G({\bf p}, i\varepsilon_n)]$. The upper black circle is $J_x$ and the lower black circle is $(\partial J_y/\partial p_y)k_y/2$. (right) The solid line in downward direction is $-G({\bf p}, i\varepsilon_n)$. The upward process is the product $[-G({\bf p}, i\varepsilon_n+i\omega_\lambda)]\cdot[-(-J_x A_x)] \cdot[-G({\bf p}, i\varepsilon_n+i\omega_\lambda)]$. The upper black circle is $J_x$ and the lower black circle is $-(\partial J_y/\partial p_y)k_y/2$. Here $J_x \equiv (e/m)p_x$ and $\partial J_y/\partial p_y \equiv e/m$.
• Figure 3: AL process for a fixed gauge ${\bf A}=(A_x,0,0)$: These diagrams correspond to those in Fig. \ref{['fig:QP']}. The broken line represents the coupling to the magnetic field (\ref{['jA-int']}). (left) The wavy line in right-side is $-L({\bf q}, i\omega_m+i\omega_\lambda)$. The left-side process is the product $[-L({\bf q}, i\omega_m)]\cdot[-(-{\tilde{\Delta}}^e_x A_x)] \cdot[-L({\bf q}, i\omega_m)]$. The upper black circle is ${\tilde{\Delta}}^e_x$ and the lower black circle is $(\partial {\tilde{\Delta}}^e_y/\partial q_y)k_y/2$. (right) The wavy line in left-side is $-L({\bf q}, i\omega_m)$. The right-side process is the product $[-L({\bf q}, i\omega_m+i\omega_\lambda)]\cdot[-(-{\tilde{\Delta}}^e_x A_x)] \cdot[-L({\bf q}, i\omega_m+i\omega_\lambda)]$. The upper black circle is ${\tilde{\Delta}}^e_x$ and the lower black circle is $-(\partial {\tilde{\Delta}}^e_y/\partial q_y)k_y/2$. Here ${\tilde{\Delta}}^e_x \equiv 4 e N(0) \xi_0^2 q_x$ and $\partial {\tilde{\Delta}}^e_y/\partial q_y \equiv 4 e N(0) \xi_0^2$.
• Figure 4: (Left): Insulator and (Right): Metal.