Optical response of Higgs mode in superconductors at clean limit: formulation through Eilenberger equation and Ginzburg-Landau Lagrangian

TL;DR

This paper clarifies the microscopic origin of Higgs-mode generation in superconductors' optical response at the clean limit, resolving previous controversies by emphasizing the importance of treating optical frequency as a continuous variable in theoretical frameworks.

Contribution

The authors derive the effective Ginzburg-Landau Lagrangian from a path-integral approach and demonstrate the necessity of using continuous optical frequency to correctly predict Higgs-mode generation.

Findings

Finite Higgs-mode generation at clean limit confirmed

Effective action derived near $T_c$ using path-integral approach

Eilenberger equation in Keldysh formalism supports continuous frequency treatment

Abstract

Both macroscopic Ginzburg-Landau Lagrangian and microscopic gauge-invariant kinetic equation suggest a finite Higgs-mode generation in the second-order optical response of superconductors at clean limit, whereas the previous derivations through the path-integral approach and Eilenberger equation within the Matsubara formalism failed to give such generation. The crucial treatment leading to this controversy lies at an artificial scheme that whether the external optical frequency is taken as continuous variable or bosonic Matsubara frequency to handle the gap dynamics within the Matsubara formalism. To resolve this issue, we derive the effective action of the superconducting gap near $T_c$ in the presence of the vector potential through the path-integral approach, to fill the long missing blank of the microscopic derivation of the Ginzburg-Landau Lagrangian in superconductors. It is shown that only by taking optical frequency as continuous variable within the Matsubara formalism, can one achieve the fundamental Ginzburg-Landau Lagrangian, and in particular, the finite Ginzburg-Landau kinetic term leads to a finite Higgs-mode generation at clean limit. To further eliminate the confusion of the Matsubara frequency through a separate framework, we apply the Eilenberger equation within the Keldysh formalism, which is totally irrelevant to the Matsubara space. By calculating the gap dynamics in the second-order response, it is analytically proved that the involved optical frequency is a continuous variable rather than bosonic Matsubara frequency, causing a finite Higgs-mode generation at clean limit.