# The "non-regularizability" of gapless free fermion Hamiltonian protected by on-site symmetries

## TL;DR

This paper extends the Nielsen-Ninomiya theorem to show that certain gapless free fermion theories protected by symmetries cannot be regularized on a lattice, highlighting fundamental limitations in modeling these systems.

## Contribution

It generalizes the Nielsen-Ninomiya theorem to all minimal nodal free fermion theories with specific symmetries, establishing their non-regularizability.

## Key findings

- Proves all such fermion theories cannot be regularized on a lattice.
- Identifies symmetry protections that enforce non-regularizability.
- Links non-regularizability to quantum anomalies.

## Abstract

The non-regularizability of free fermion field theories, which is the root of various quantum anomalies, plays a central role in particle physics and modern condensed matter physics. In this paper, we generalize the Nielsen-Ninomiya theorem to all minimal nodal free fermion field theories protected by the time reversal, charge conservation, and charge conjugation symmetries. We prove that these massless field theories cannot be regularized on a lattice.