Non-Fermi-liquid superconductivity: Eliashberg approach versus the renormalization group

We address the problem of superconductivity for non-Fermi liquids using two commonly adopted, yet apparently distinct, methods: (1) the renormalization group (RG) and (2) Eliashberg theory. The extent to which both methods yield consistent solutions for the low-energy behavior of quantum metals has remained unclear. We show that the perturbative RG beta function for the 4-Fermi coupling can be explicitly derived from the linearized Eliashberg equations, under the assumption that quantum corrections are approximately local across energy scales. We apply our analysis to the test case of phonon-mediated superconductivity and show the consistency of both the Eliashberg and RG treatments. We next study superconductivity near a class of quantum critical points and find a transition between superconductivity and a “naked” metallic quantum critical point with finite, critical BCS couplings. We speculate on the applications of our theory to the phenomenology of unconventional metals.

Figure 1

Frequency-dependent kernel $u\left(q_0\right)$ generated by boson exchange.

Figure 2

Diagrammatic representation of the Eliashberg equations.

Figure 3

One-loop corrections that contribute to (2.26): (dressed) tree-level boson exchange, anomalous dimension dressing, and BCS diagram. The straight line represents a fermion, while the wiggly line corresponds to a virtual boson.

Figure 4

Left: plot of $1/\lambda \left(x\right)$ as a function of $x$ for ${\alpha }_0/{\omega }_E=0.1$ showing a zero (equivalent to a divergence of $\lambda$ at a critical value $x_c$). Right: the critical value $x_c$ is obtained for various values of ${\alpha }_0/{\omega }_E$ (black curve). A linear relationship exists between $1/x_c$ and ${\alpha }_0/{\omega }_E$ when the latter ratio is small, consistent with the BCS prediction (blue line). At larger couplings, quantum effects introduce deviations from the BCS behavior.

Figure 5

${\tilde{\mathrm{\Delta }}}_{\text{diff}}\left(x\right)$ against $F\circ {\tilde{\mathrm{\Delta }}}_{\text{diff}}\left(x\right)$, for which we picked $x_{\mathrm{\Lambda }}=-12$, and $g_1=0.2$. The gap is solved to be at $x_{\text{gap}}=1.7006$.

Figure 6

Comparison between ${\tilde{\mathrm{\Delta }}}_{\text{diff}}\left(x\right)$ (red, dashed line) and $F\circ {\tilde{\mathrm{\Delta }}}_{\text{diff}}\left(x\right)$ (blue line), normalized by their values at $x_{*}$, with $x_{\mathrm{\Lambda }}=-10,g_1=0.27,\epsilon =0.1$, for which $x_{*}\approx 19.49$. In these variables, the NFL scale is at $x=0$ and the UV cutoff at $x_{\mathrm{\Lambda }}=-10$.