On the Puzzle of Odd-Frequency Superconductivity

TL;DR

This paper examines the theoretical foundations of odd-frequency superconductivity, emphasizing the importance of selecting the correct saddle-point in the path-integral approach to ensure stability and physical consistency.

Contribution

It clarifies the role of saddle-point solutions in the path-integral formulation, resolving previous issues with stability and Meissner effect in odd-frequency superconductivity.

Findings

Proper saddle-point selection ensures thermodynamic stability.

Path-integral framework accommodates general pairing types.

Analytic continuation extends the theory to real frequencies.

Abstract

Since the first theoretical proposal by Berezinskii, an odd-frequency superconductivity has encountered the fundamental problems on its thermodynamic stability and rigidity of a homogenous state accompanied by unphysical Meissner effect. Recently, Solenov {\it et al}. [Phys. Rev. B {\bf 79} (2009) 132502.] have asserted that the path-integral formulation gets rid of the difficulties leading to a stable homogenous phase with an ordinary Meissner effect. Here, we show that it is crucial to choose the appropriate saddle-point solution that minimizes the effective free energy, which was assumed {\it implicitly} in the work by Solenov and co-workers. We exhibit the path-integral framework for the odd-frequency superconductivity with general type of pairings, including an argument on the retarded functions via the analytic continuation to the real axis.