Minimal Doubling Fermion and Hermiticity

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.

12 pages, 7 figures

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• Four-dimensional graphene and chiral fermions
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• Renormalization of minimally doubled fermions
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• Lattice fermions as spectral graphs
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• New conjecture on exact Dirac zero-modes of lattice fermions
• Minimal-doubling and single-Weyl Hamiltonians
• Non-${\gamma }_5$hermiticity fermions in two dimensions
• Lattice fermion formulation via Physics-Informed Neural Networks: Ginsparg-Wilson relation and Overlap fermions