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