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We study a system of weakly interacting electrons described by the energydispersion $\xi(\mathbf{k}) = k_x^2 - k_y^2 - \mu$ in two dimensions within arenormalization group approach. This energy dispersion exhibits aneck-narrowing Lifshitz transition at the critical chemical potential $\mu_c=0$where a van Hove singularity develops. Implementing a systematicrenormalization group analysis of this system has long been hampered by theappearance of nonlocal terms in the Wilsonian effective action. We demonstratethat non-locality at the critical point is intrinsic, and the locality of theeffective action can be maintained only away from the critical point. We alsopoint out that it is crucial to introduce a large momentum cutoff to keeplocality even away from the critical point. Based on a local renormalizationgroup scheme employed near the critical point, we show that, as the energyscale $E$ is lowered, an attractive four-fermion interaction grows as $\log^2E$ 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 pairinginteraction suggests that the system becomes unstable toward a superconductingstate well before the critical point is reached.

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