RG Analysis on a Neck-Narrowing Lifshitz Transition in the Presence of Weak Short-Range Interactions in Two Dimensions

We present a pertrbative renormalization group (RG) analysis for a neck-narrowing Lifshitz transition in the presence of weak short-range interactions in two dimensions. The model we examine is described by the dispersion $\varepsilon(\mathbf{k}) = k_x^2-k_y^2$, which would be at the critical point of the neck-narrowing transition at zero chemical potential, $\mu=0$. At the critical point ($\mu=0$), we find that one-loop quantum corrections to the interaction vertex are non-analytic. This makes capturing the evolution of the low-energy effective theory, as the energy cutoff is progressively lowered, in terms of $\beta$-functions for local operators impossible. Thus we conjecture that any consistent RG description at the critical point of this neck-narrowing transition will involve non-local operators. Slightly away from the critical point ($\mu > 0$), where the Fermi surface has a narrow neck, we find that the quantum corrections are analytic only over a finite momentum range, which shrinks to zero as the chemical potential approaches zero. More importantly, within this analytic range, where a local RG description is possible, we show that the narrower the width of the neck, the larger the couplings of irrelevant interactions become, leading to the breakdown of perturbative RG.