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Influence of short-range four-fermion interactions on quadratic and cubicnodal line fermion systems is studied by renormalization group theory. It isfound that arbitrarily weak four-fermion interaction could drive quadratic orcubic nodal line fermion system to a new phase. According to the initialconditions and value of fermion flavor, the system may appear three kinds ofinstabilities. First, quadratic or cubic nodal line is split into conventionalnodal lines, thus the system becomes nodal line semimetal. Second, finiteexcitonic gap is generated, and the system becomes an excitonic insulator.Third, the system is driven into superconducting phase. Thus, quadratic andcubic nodal line fermion systems are rare strong correlated fermion systems inthree dimension under the influence of four-fermion interactions. Thesetheoretical results may be verified in the candidates for quadratic and cubicnodal line fermion systems.

Possible instabilities in quadratic and cubic nodal line fermion systems with correlated interactions