# Renormalization group flows in a Lifshitz-like four fermi model

## TL;DR

This paper analyzes the renormalization group flows in a Lifshitz-like four-fermion model, revealing fixed points and phase transitions, with implications for Lorentz invariance violations and potential supersymmetric extensions.

## Contribution

It provides a detailed study of RG flows and fixed points in a Lifshitz-like four-fermion model, including a new fixed point with $z=1$ scaling and Lorentz violation.

## Key findings

- Identification of a nontrivial RG flow in one marginal and one relevant coupling.
- Discovery of a new fixed point with $z=1$ scaling and Lorentz violation.
- Flow from Lifshitz fixed point to the new fixed point in the broken symmetry phase.

## Abstract

We study renormalization group flows in the Lifshitz-like $N$-flavour four fermi model discussed in 0905.2928. In the large-$N$ limit, a nontrivial flow occurs in only one of all possible marginal couplings and one relevant coupling, which provides the scale for Lorentz invariance violations. We discuss in detail the phase diagram and RG flows in the space of couplings, which includes the Lifshitz fixed point, the free field fixed point and a new fixed point characterized by $z=1$ scaling and a violation of Lorentz invariance, which cannot be tuned away by adjusting a parameter. In the broken symmetry phase, the model flows from the $z=3$ Lifshitz-like fixed point in the ultraviolet to this new fixed point in the infrared. However, in a modified version of the present model, which has an effective ultraviolet cut-off much smaller than the Lorentz invariance violating scale, the infrared behaviour is governed by an approximately Lorentz invariant theory, similar to the low energy limit of the usual relativistic Nambu$-$Jona-Lasinio model. Such a modified model could be realized by a supersymmetric version of the present model.