In this paper we continue our analysis of the interplay between the pairing and the non-Fermi liquid behavior in a metal for a set of quantum-critical models with an effective dynamical electron-electron interaction $V\left({\mathrm{\Omega }}_m\right)\propto 1/{\left|{\mathrm{\Omega }}_m\right|}^{\gamma }$ (the $\gamma$ model). We analyze both the original model and its extension, in which we introduce an extra parameter $N$ to account for nonequal interactions in the particle-hole and particle-particle channel. In two previous papers [A. Abanov and A. V. Chubukov, Phys. Rev. B 102, 024524 (2020) and Y. Wu et al. Phys. Rev. B 102, 024525 (2020)] we considered the case $0<\gamma <1$ and argued that (i) at $T=0$, there exists an infinite discrete set of topologically different gap functions ${\mathrm{\Delta }}_n\left({\omega }_m\right)$, all with the same spatial symmetry, and (ii) each ${\mathrm{\Delta }}_n$ evolves with temperature and terminates at a particular $T_{p,n}$. In this paper we analyze how the system behavior changes between $\gamma <1$ and $\gamma >1$, both at $T=0$ and a finite $T$. The limit $\gamma \to 1$ is singular due to infrared divergence of $\int d{\omega }_{m}V\left({\mathrm{\Omega }}_m\right)$, and the system behavior is highly sensitive to how this limit is taken. We show that for $N=1$, the divergencies in the gap equation cancel out, and ${\mathrm{\Delta }}_n\left({\omega }_m\right)$ gradually evolve through $\gamma =1$ both at $T=0$ and a finite $T$. For $N\ne 1$, divergent terms do not cancel, and a qualitatively new behavior emerges for $\gamma >1$. Namely, the form of ${\mathrm{\Delta }}_n\left({\omega }_m\right)$ changes qualitatively, and the spectrum of condensation energies $E_{c,n}$ becomes continuous at $T=0$. We introduce different extension of the model, which is free from singularities for $\gamma >1$.

中文翻译：

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金属中量子临界点的超导电性和非费米液体行为之间的相互作用。三，γ= 1上的γ模型及其相图

在本文中，我们继续针对具有有效动态电子电子相互作用的一组量子临界模型分析金属中的配对和非费米液体行为之间的相互作用。 $V（{\mathrm{\Omega }}_{米}）\propto \mathrm{1个}/{\left|{\mathrm{\Omega }}_{米}\right|}^{\gamma }$ （ $\gamma$模型）。我们分析了原始模型及其扩展，在其中引入了一个额外的参数$ñ$解释了粒子-孔和粒子-粒子通道中的不平等相互作用。在前两篇论文中[A. Abanov和AV Chubukov，物理学。版本B 102，024524（2020）和Y.吴等人。 物理 版本B 102，024525（2020）]，我们考虑的情况$0<\gamma <\mathrm{1个}$ 并认为（i）在 $Ť=0$，存在拓扑上不同的间隙函数的无限离散集 ${\mathrm{\Delta }}_{ñ}（{\omega }_{米}）$，都具有相同的空间对称性，并且（ii）每个 ${\mathrm{\Delta }}_{ñ}$ 随着温度变化并终止于特定温度 ${Ť}_{p，ñ}$。在本文中，我们分析了系统行为如何在$\gamma <\mathrm{1个}$ 和 $\gamma >\mathrm{1个}$，两者都位于 $Ť=0$ 和一个有限的 $Ť$。极限$\gamma \to \mathrm{1个}$ 由于红外散度是奇异的 $\int d{\omega }_{米}V（{\mathrm{\Omega }}_{米}）$，并且系统行为对采取此限制的方式非常敏感。我们证明了$ñ=\mathrm{1个}$，则间隙方程中的差异被抵消，并且 ${\mathrm{\Delta }}_{ñ}（{\omega }_{米}）$ 通过逐步发展 $\gamma =\mathrm{1个}$ 都在 $Ť=0$ 和一个有限的 $Ť$。对于$ñ\ne \mathrm{1个}$，分歧项不会取消，并且出现了定性的新行为 $\gamma >\mathrm{1个}$。即形式${\mathrm{\Delta }}_{ñ}（{\omega }_{米}）$ 发生质的变化，凝聚能谱 ${Ë}_{C，ñ}$ 在变得连续 $Ť=0$。我们介绍了该模型的不同扩展，其中没有奇异之处$\gamma >\mathrm{1个}$。