We study the pairing of fermions by an interaction consisting of a Hubbard repulsion, mimicking a screened Coulomb potential, and a dynamical phonon-mediated attraction. For such interaction, the gap equation allows even- and odd-frequency solutions ${\mathrm{\Delta }}_e$ and ${\mathrm{\Delta }}_o$. We show that odd-frequency pairing does not develop within the Eliashberg approximation due to over-critical pair breaking from the self-energy. When vertex corrections are included, the pairing interaction gets stronger, and ${\mathrm{\Delta }}_o$ can develop. We argue that even in this case keeping the self-energy is still a must as it cancels out the thermal piece in the gap equation. We further argue that ${\mathrm{\Delta }}_o$ is not affected by Hubbard repulsion and for strong repulsion is comparable to a reduced ${\mathrm{\Delta }}_e$. The resulting superconducting state is a superposition ${\mathrm{\Delta }}_e\pm i{\mathrm{\Delta }}_o$, which spontaneously breaks the time-reversal symmetry, despite that the pairing symmetry is an ordinary $s$ wave.

中文翻译：

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排斥相互作用的奇数频率配对和时间反演对称性破缺

我们通过由哈伯德斥力、模拟筛选库仑势和动态声子介导的吸引力组成的相互作用来研究费米子的配对。对于这种相互作用，间隙方程允许偶数和奇数频率解${\mathrm{\Delta }}_e$和${\mathrm{\Delta }}_{○}$. 我们表明，由于自能的过临界对破坏，奇数频率配对不会在 Eliashberg 近似内发展。当包括顶点校正时，配对相互作用变得更强，并且${\mathrm{\Delta }}_{○}$可以发展。我们认为，即使在这种情况下，保持自能仍然是必须的，因为它抵消了间隙方程中的热部分。我们进一步认为${\mathrm{\Delta }}_{○}$不受哈伯德斥力的影响，强斥力相当于减少${\mathrm{\Delta }}_e$. 产生的超导状态是叠加态${\mathrm{\Delta }}_e\pm \mathrm{一世}{\mathrm{\Delta }}_{○}$，它自发地破坏了时间反转对称性，尽管配对对称性是普通的$s$海浪。