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 ${\ensuremath{\gamma}}_{5}$-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.
Highlights
It is hard to have an analytical solution in strongly coupled systems
After we only adopt one finite-difference scheme, it is equivalent to decoupling the nonphysical poles from the physical one
The result is consistent with the continuum physics avoiding the fermion doubling problem
Summary
It is hard to have an analytical solution in strongly coupled systems. For studying physics, people adopted lattice regularization for putting the systems on a lattice. [5,6], one used a one-sided lattice difference with the first-order accuracy to show the naive lattice fermion with a chiral symmetry In this explicit example, the lattice momentum operator is not Hermitian, except for the continuum limit [5]. We can explicitly study the doubling problem of the first-order accuracy and the second-order accuracy for the forward finite-difference Both cases lose Hermiticity on a lattice. We show that the second-order accuracy evades the fermion doubling problem without breaking the chiral symmetry. After we only adopt one finite-difference scheme, it is equivalent to decoupling the nonphysical poles from the physical one (under the field theory limit) This approach directly brings a broken of the Hermiticity. The result is consistent with the continuum physics avoiding the fermion doubling problem
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