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