Lifshitz transitions in two 2D systems with a single quadratic band touching point as the chemical potential is varied have been studied here. The effects of interactions have been studied using the renormalization group (RG) and it is found that at the transition a repulsive interaction is marginally relevant and an attractive interaction is marginally irrelevant. We corroborate the results obtained from the RG calculation by studying a microscopic model whose ground state and Greens functions can be obtained exactly. We find that away from the transition, the system displays an instability towards forming and excitonic condensate.
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