Naive Lattice Fermion without Doublers

TL;DR

This paper proposes a method to avoid fermion doubling on the lattice by using forward finite differences, which breaks hermiticity but preserves chiral symmetry, and demonstrates its effectiveness through analytical and numerical analysis.

Contribution

It introduces a lattice fermion formulation using forward differences that avoids doublers while maintaining chiral symmetry, challenging the traditional Nielsen-Ninomiya theorem constraints.

Findings

No doubling issue with first-order accuracy propagator.

Second-order accuracy also avoids fermion doubling.

Numerical implementation of Hybrid Monte Carlo confirms viability.

Abstract

We discuss the naive lattice fermion without the issue of doublers. A local lattice massless fermion action with chiral symmetry and hermiticity cannot avoid the doubling problem from the Nielsen-Ninomiya theorem. Here we adopt the forward finite-difference deforming the $\gamma_5$-hermiticity but preserving the continuum chiral-symmetry. The lattice momentum is not hermitian without the continuum limit now. We demonstrate that there is no doubling issue from an exact solution. The propagator only has one pole in the first-order accuracy. Therefore, it is hard to know the avoiding due to the non-hermiticity. For the second-order, the lattice propagator has two poles as before. This case also does not suffer from the doubling problem. Hence separating the forward derivative from the backward one evades the doublers under the field theory limit. Simultaneously, it is equivalent to breaking the hermiticity. In the end, we discuss the topological charge and also demonstrate the numerical implementation of the Hybrid Monte Carlo.