Near a quantum-critical point, a metal reveals two competing tendencies: destruction of fermionic coherence and attraction in one or more pairing channels. We analyze the competition within Eliashberg theory for a class of quantum-critical models with an effective dynamical electron-electron interaction $V(\Omega_m) \propto 1/|\Omega_m|^\gamma$ (the $\gamma$-model) for $0 < \gamma <1$. We argue that the two tendencies are comparable in strength, yet the one towards pairing is stronger, and the ground state is a superconductor. We show, however, that there exist two distinct regimes of system behavior below the onset temperature of the pairing $T_p$. In the range $T_{cr} < T < T_p$ fermions remain incoherent and the density of states $N(\omega)$ displays "gap filling" behavior in which the position of the maximum in $N(\omega)$ is set by temperature rather than the pairing gap. At lower $T < T_{cr}$, fermions acquire coherence, and $N(\omega)$ displays a conventional "gap closing" behavior. We argue that the existence of the two regimes comes about because of special behavior of fermions with frequencies $\omega = \pm \pi T$ along the Matsubara axis. Specifically, for these fermions, the component of the self-energy, which competes with the pairing, vanishes in the normal state. We further argue that the crossover at $T \sim T_{cr}$ comes about because Eliashberg equations allow an infinite number of topologically distinct solutions for the onset temperature of the pairing within the same gap symmetry. Only one solution, with the highest $T_p$, actually emerges, but other solutions are generated and modify the form of the gap function at $T \leq T_{cr}$. Finally, we argue that the actual $T_c$ is comparable to $T_{cr}$, while at $T_{cr} < T < T_{p}$ phase fluctuations destroy superconducting long-range order, and the system displays pseudogap behavior.

中文翻译：

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金属中量子临界点处超导性与非费米液体之间的相互作用

在量子临界点附近，金属显示出两种相互竞争的趋势：费米子相干性的破坏和一个或多个配对通道中的吸引力。我们分析了 Eliashberg 理论中一类具有有效动态电子-电子相互作用 $V(\Omega_m) \propto 1/|\Omega_m|^\gamma$（$\gamma$-模型）的量子临界模型的竞争对于 $0 < \gamma <1$。我们认为这两种倾向在强度上是可比的，但配对的倾向更强，基态是超导体。然而，我们表明，在配对 $T_p$ 的起始温度以下存在两种不同的系统行为制度。在 $T_{cr} < T < T_p$ 范围内，费米子保持不相干，状态密度 $N(\omega)$ 显示“间隙填充” $N(\omega)$ 中最大值的位置由温度而不是配对间隙设置的行为。在较低的 $T < T_{cr}$ 时，费米子获得相干性，并且 $N(\omega)$ 显示出传统的“间隙关闭”行为。我们认为这两种机制的存在是由于频率为 $\omega = \pm \pi T$ 的费米子沿着 Matsubara 轴的特殊行为。具体来说，对于这些费米子，与配对竞争的自能成分在正常状态下消失。我们进一步认为 $T \sim T_{cr}$ 处的交叉是因为 Eliashberg 方程允许在相同间隙对称性内对配对的起始温度有无限数量的拓扑不同解。实际上只有一个具有最高 $T_p$ 的解决方案出现，但其他解决方案会生成并修改 $T \leq T_{cr}$ 处的间隙函数的形式。最后，我们认为实际的 $T_c$ 与 $T_{cr}$ 相当，而在 $T_{cr} < T < T_{p}$ 时相位波动会破坏超导长程有序，并且系统显示伪间隙行为.