# Generic "Unnecessary" Quantum Critical Points with Minimal Degrees of Freedom

## TL;DR

This paper investigates minimal-degree, generic 'unnecessary' quantum critical points that can be avoided by symmetry-allowed deformations, providing new examples in three-dimensional topological phases with fewer degrees of freedom.

## Contribution

It identifies new minimal-degree examples of generic unnecessary quantum critical points in 3D topological insulators and superconductors, expanding understanding of their minimal configurations.

## Key findings

- Two examples of minimal-degree unnecessary quantum critical points in 3D systems.
- One example involves two (3+1)d massless Dirac fermions in a topological insulator.
- Another involves a single (3+1)d massless Dirac fermion in a topological superconductor.

## Abstract

We explore generic "unnecessary" quantum critical points with minimal degrees of freedom. These quantum critical points can be avoided with strong enough symmetry-allowed deformations of the Hamiltonian, but these deformations are irrelevant perturbations below certain threshold at the quantum critical point. These quantum critical points are hence unnecessary, but also unfine-tuned (generic). The previously known examples of such generic unnecessary quantum critical points involve at least eight Dirac fermions in both two and three spatial dimensions. In this work we seek for examples of generic unnecessary quantum critical points with minimal degrees of freedom. In particular, in three dimensional space, we identify two examples of such generic unnecessary quantum critical points. The first example occurs in a $3d$ interacting topological insulator, and it is described by $two$ $(3+1)d$ massless Dirac fermions in the infrared limit; the second example occurs in a $3d$ topological superconductor, and it is formally described only $one$ $(3+1)d$ massless Dirac fermion.