Minimal Doubling Fermion and Hermiticity

TL;DR

This paper investigates the properties of lattice fermion kinetic terms, focusing on symmetries like PT, R-hermiticity, and $ ext{γ}_5$-hermiticity, and their implications for fermion doubling and quantum corrections.

Contribution

It demonstrates that PT symmetry prevents fermion doubling in lattice kinetic terms and analyzes how minimal doubling fermions break PT symmetry and R-hermiticity, leading to complex couplings.

Findings

PT symmetry prevents fermion doubling

Minimal doubling fermions break PT symmetry and R-hermiticity

Quantum corrections induce complex couplings

Abstract

We analyze the lattice fermion kinetic term using PT symmetry, R-hermiticity, and $\gamma_{5}$-hermiticity. R-hermiticity is a condition for Hermite action and it is related to $\gamma_{5}$-hermiticity and PT symmetry. Assuming that a translation-invariant kinetic term with continuum and periodic function does not have PT symmetry, it can have R-hermiticity or $\gamma_{5}$-hermiticity. We prove that a kinetic term with continuum and periodic function that is PT symmetric does not reduce doublers. As a simple example, we analyze the two-dimensional two-flavor Gross-Neveu model with minimal doubling fermions. The minimal doubling fermions break PT symmetry and R-hermiticity, hence complex or non-Hermite coupling constants are caused by quantum correction.