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We consider Lifshitz criticality (LC) with the dynamical critical exponent$z=2$ in one-dimensional interacting fermions with a filled Dirac Sea. Wereport that interactions have crucial effects on Lifshitz criticality. Singleparticle excitations are destabilized by interaction and decay into theparticle-hole continuum, which is reflected in the logarithmic divergence inthe imaginary part of one-loop self-energy. We show that the system issensitive to the sign of interaction. Random-phase approximation (RPA) showsthat the collective particle-hole excitations emerge only when the interactionis repulsive. The dispersion of collective modes is gapless and linear.  If the interaction is attractive, the one-loop renormalization group (RG)shows that there may exist a stable RG fixed point described by two couplingconstants. We also show that the on-site interaction (without any otherperturbations at the UV scale) would always turn on the relevant velocityperturbation to the quadratic Lagrangian in the RG flow, driving the systemflow to the conformal-invariant criticality. In the numerical simulations ofthe lattice model at the half-filling, we find that, for either on-sitepositive or negative interactions, the dynamical critical exponent becomes$z=1$ in the infrared (IR) limit and the entanglement entropy is a logarithmicfunction of the system size $L$. The work paves the way to studyone-dimensional interacting LCs.

Quantum $z=2$ Lifshitz criticality in one-dimensional interacting fermions