Renormalization group analysis on a neck-narrowing Lifshitz transition in the presence of weak short-range interactions in two dimensions

We study a system of weakly interacting electrons described by the energy dispersion $\xi(\mathbf{k}) = k_x^2 - k_y^2 - \mu$ in two dimensions within a renormalization group approach. This energy dispersion exhibits a neck-narrowing Lifshitz transition at the critical chemical potential $\mu_c=0$ where a van Hove singularity develops. Implementing a systematic renormalization group analysis of this system has long been hampered by the appearance of nonlocal terms in the Wilsonian effective action. We demonstrate that non-locality at the critical point is intrinsic, and the locality of the effective action can be maintained only away from the critical point. We also point out that it is crucial to introduce a large momentum cutoff to keep locality even away from the critical point. Based on a local renormalization group scheme employed near the critical point, we show that, as the energy scale $E$ is lowered, an attractive four-fermion interaction grows as $\log^2 E$ for $E > \mu$, whereas it retains the usual BCS growth, $-\log E$, for $E < \mu$. Starting away from the critical point, this fast growth of the pairing interaction suggests that the system becomes unstable toward a superconducting state well before the critical point is reached.