# Instability caused by the fermion-fermion interactions combined with rotational and particle-hole asymmetries in three-dimensional materials with quadratic band touching

## TL;DR

This paper uses renormalization group analysis to study how four-fermion interactions, rotational, and particle-hole asymmetries influence the low-energy phases of three-dimensional quadratic band touching systems, revealing instabilities and phase diagrams.

## Contribution

It provides a comprehensive, unbiased analysis of the combined effects of interactions and asymmetries on phase stability in 3D quadratic band touching materials.

## Key findings

- Four-fermion interactions destabilize Gaussian fixed points at low energies.
- Asymmetries and interactions induce superconductivity instability under certain conditions.
- Phase diagrams illustrate the states influenced by interactions and asymmetries.

## Abstract

We investigate the role of four-fermion interactions, rotational and particle-hole asymmetries, and their interplay in three-dimensional systems with a quadratic band touching point by virtue of the renormalization group approach, which allows to treat all these facets unbiasedly. The coupled flow evolutions of interaction parameters are derived by taking into account one-loop corrections in order to explore the behaviors of low-energy states. We find four-fermion interaction can drive Gaussian fixed points to be unstable in the low-energy regime. In addition, the rotational and particle-hole asymmetries, together with the fermion-fermion interactions conspire to split the trajectories of distinct types of fermionic couplings and induce superconductivity instability with appropriate starting conditions. Furthermore, we present the schematic phase diagrams in the parameter space, showing the overall behaviors of states in the low-energy regime caused by both fermionic interactions and asymmetries.