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

## TL;DR

This paper investigates the effects of interactions on one-dimensional Lifshitz criticality with dynamical exponent z=2, revealing destabilization of single-particle excitations, emergence of collective modes, and a flow to z=1 in the IR limit.

## Contribution

It provides a detailed analysis of how interactions influence Lifshitz criticality in 1D fermions, including RG fixed points and numerical confirmation of z=1 IR behavior.

## Key findings

- Interactions destabilize single-particle excitations.
- Collective particle-hole modes emerge with repulsive interactions.
- Numerical simulations show z=1 in the IR limit and logarithmic entanglement entropy.

## Abstract

We consider Lifshitz criticality (LC) with the dynamical critical exponent $z=2$ in one-dimensional interacting fermions with a filled Dirac Sea. We report that interactions have crucial effects on Lifshitz criticality. Single particle excitations are destabilized by interaction and decay into the particle-hole continuum, which is reflected in the logarithmic divergence in the imaginary part of one-loop self-energy. We show that the system is sensitive to the sign of interaction. Random-phase approximation (RPA) shows that the collective particle-hole excitations emerge only when the interaction is 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 coupling constants. We also show that the on-site interaction (without any other perturbations at the UV scale) would always turn on the relevant velocity perturbation to the quadratic Lagrangian in the RG flow, driving the system flow to the conformal-invariant criticality. In the numerical simulations of the lattice model at the half-filling, we find that, for either on-site positive or negative interactions, the dynamical critical exponent becomes $z=1$ in the infrared (IR) limit and the entanglement entropy is a logarithmic function of the system size $L$. The work paves the way to study one-dimensional interacting LCs.